Paper Title: Infinite dimensional metapopulation SIS model with generalized incidence rate.





Jean-François Delmas,  CERMICS, École des Ponts, France
jean-francois.delmas@enpc.fr


Kacem Lefki, Univ. Gustave Eiffel, Univ.    Paris Est Creteil, CNRS, F-77454 Marne-la-Vallée, France
  kacem.lefki@univ-eiffel.fr


Pierre-André Zitt, Univ. Gustave Eiffel, Univ.    Paris Est Creteil, CNRS, F-77454 Marne-la-Vallée, France
  Pierre-Andre.Zitt@univ-eiffel.fr

 


This work is partially supported by Labex Bézout reference ANR-10-LABX-58

[2020]47B65, 
47B38, 

34D05, 
35R15, 
37D35, 
92D30 


, 

§ ABSTRACT 
  We  consider an  infinite-dimension  SIS model  introduced by  Delmas,
  Dronnier and  Zitt, with a more  general incidence rate, and study its
  equilibria. Unsurprisingly, there exists at
  least one  endemic equilibrium if  and only if the  basic reproduction
  number is larger than 1.  When the pathogen transmission exhibits one
  way propagation, it is possible  to observe different possible endemic
  equilibria.  We characterize  in a general setting  all the equilibria,
  using a decomposition of the space into  atoms, given by the transmission  operator.
  We also
  prove that  the proportion of  infected individuals  converges
  to  an equilibrium,  which is  uniquely determined  by the
  support of the initial condition.

  We  extend those results to 
  infinite-dimensional SIS models with reservoir or  with immigration. 


Infinite dimensional metapopulation SIS model with generalized incidence rate
    Pierre-André Zitt
    August 5, 2024
=============================================================================



theorem
theoTheorem[section]
theoITheorem
propI[theoI]Proposition
prop[theo]Proposition
defi[theo]Definition
lemme[theo]Lemma
lem[theo]Lemma
coro[theo]Corollary
cor[theo]Corollary
nota[theo]Notation
rap[theo]Reminder
conj[theo]Conjecture
ques[theo]Question
ans[theo]Answer
defth[theo]Theorem-Definition
assumAssumption


remark 
ex[theo]Example
rqe[theo]Remark
rem[theo]Remark
cex[theo]Counter-example
Infinite dimensional metapopulation SIS model with generalized incidence rate
    Pierre-André Zitt
    August 5, 2024
=============================================================================





§ INTRODUCTION




 §.§ Model and relations with existing models


We consider  an inhomogeneous SIS epidemic model, where
individuals are either susceptible or infected. The homogeneous
model was introduced  by Kermack and McKendrick <cit.>, we
refer    to    the    monograph   of    Brauer,    Castillo-Chavez    et
Feng <cit.>  for an  analysis of this homogeneous  SIS model  and some  of its
variants. Let us  recast  the  model from <cit.>   in the  constant
population case: let
I(t) and   S(t) denote respectively the  number of the infected  and  susceptible)
individuals  at  time t ≥  0, in a population  of constant size  N=S(t)+I(t) >
0. The evolution of the number of infected is given by:

    I' = k  SI/N - γ I,

where  k ≥ 0 is   the infection  rate and  γ  > 0  the
recovery rate.



The assumption of homogeneity of  the population is not always satisfied
in practice, see for example: Trauer et al. <cit.> for a review
on  tuberculosis, <cit.>  on  the impact  of health  condition,
<cit.>  on  the  number  of  sexual  partners  in  a  sexually
transmissible  infection,  or  the review <cit.>  for  more
possible sources  of heterogeneity.  The inhomogeneous  SIS model from
Lajmanovich     and    Yorke <cit.>     generalizes    the
Kermack-McKendrick  model to  a  population divided  in n  sub-groups;
the same equation appears also when studying network of communities
linked by dispersal, see Mouquet and Loreau <cit.> and more
generally <cit.>.
Later,     Thieme <cit.>    and     Delmas,    Dronnier     and
Zitt <cit.>  introduced   a  variant  allowing  an infinite
number (possibly uncountable)   of
sub-groups  or  features.



We  follow  the model  given  by <cit.>  where the  transmission
operator    can   be    non-irreducible,   see    the   discussion    in
Section <ref> below and  allowing furthermore a more
general  incidence   rate,  see   Section <ref>.   The
heterogeneity   of    the   population    is   described    as   follow:
(Ω, , μ) is a measured space with a non-zero σ-finite
measure  μ: an  element x∈Ω corresponds  to a  particular
feature  (or  trait)  of   individuals.   We  assume  that
individuals with  the same feature behave  in the same way with  respect to
the  epidemic, and that  features stay 
constant during the whole infection process.   We also assume that for a
given feature x ∈Ω, the  size of the population μ( x) of feature x
remains constant over time. 



Let u(t,x) denote the proportion of individuals with feature x∈Ω that are
infected at time t≥ 0 among the population  of individuals with
feature  x.  Let Δ be the set of measurable functions defined
on Ω taking values in [0, 1]. The heterogeneous SIS dynamics
is given, for an initial condition h ∈Δ, by the evolution
equation on  the  Banach  space
 L^∞  of measurable  bounded  real-valued  functions defined  on
 Ω by:

    {[  u' = F(u),; u(0) = h∈Δ, ].

with

    F(u)= φ(u)  Tu - γ u,

 where F depends on: a bounded linear transmission operator T
 on  L^∞,  a bounded  real-valued positive  recovery rate
 function  γ  defined on  Ω,  and  a real-valued  function
 φ   defined  on      encoding  the   non-bilinearity  of   the
 incidence    rate.    The    hypotheses    on   the    parameter
 (T,   γ,   φ)   are  given   in   Assumptions <ref>
 and <ref>. Let us stress that the usual law of mass
   action  φ = 1-𝕀, with 𝕀 the identity map on ,
 satisfies the corresponding hypothesis from  Assumption <ref>
 summarized in Condition (<ref>). 

 
   
 Let k : Ω^2
→_+ be a kernel, that is a nonnegative measurable
function. The associated kernel operator T_k is defined as follow. For
h∈ L^∞ and  x ∈Ω,  we define:

    T_k (h) (x) = ∫_Ω k(x,y) h(x)  μ(  y).

The quantity k(x,y) represents  the transmission rate from individuals
with feature  y∈Ω  to those with  feature x∈Ω.  The
heterogeneous    SIS   model    from <cit.>   is    then   given
by (<ref>)  and (<ref>)  with  T=T_k  (under
some integral hypothesis on the kernel) and the usual law of mass action
φ =  1-𝕀.  This  in particular  encompasses the  Lajmanovich and
Yorke model.
 



 In     epidemiology,     equilibria      are     constant     solutions
 of (<ref>), that is, functions g∈Δ such that:

    F(g) = 0.

They play a  significant role in the long-time behavior  of the dynamics
of an  outbreak, see Theorem <ref> below.   Obviously, the
disease-free equilibrium  (DFE) g  = 0  is an  equilibrium. Any
other  equilibrium  is  called  endemic  equilibrium  (EE).   The
basic reproduction number denoted R_0 is defined by Heesterbeek
and  Dietz <cit.> as  “the expected  number of  secondary
cases  produced  by a  typical  infected  individual during  its  entire
infectious period,  in a  population consisting of  susceptibles only”.
Following <cit.>  (see also  the method  of the  next-generation
operator in Diekmann, Heesterbeek and Metz <cit.>), the basic
reproduction  number R_0  for  the SIS  model (<ref>)
with the usual incidence rate associated to φ=1- 𝕀 is
defined as the  spectral radius of the operator  T M_1/γ, where
the operator M_1/γ is the multiplication by 1/γ.
There usually is a threshold behavior  for the existence of EE according
to  the value  of R_0:  for R_0  ≤ 1  only the  DFE exists  as an
equilibrium, and  for R_0 > 1  there exists an EE.
This is not universal: for example, models with imperfect  vaccines
or exogenous  re-infections might
lead to backward  bifurcation and produce multiple  EE even  in the regime
R_0 < 1, see <cit.>  and more specifically <cit.> for
a SIS model. Nevertheless, we check that threshold behavior holds
    for    the     SIS    model (<ref>),    see
Theorem <ref>   below.    A  discussion   of   the
uniqueness of EE is given in Section <ref>.





 §.§ A taste of the  main results in the finite setting



Except in the trivial case where the population may be split in subpopulations
that do not interact at all, the existence of multiple equilibria is fundamentally
linked to asymmetry in the transmission dynamics. In this section,
we first explain this
phenomenon, and a related crucial decomposition of the space, in the simple
case where Ω is finite, to give a taste of the general results stated below. 


We consider a finite set   Ω, let    be the set of  subsets of Ω,
and μ a  finite measure
with support Ω.   The transmission  operator T is
identified  with a  matrix  K=(  K_x,y)_x,y∈Ω
where K_x,y is the infection  rate from individuals with feature y
to  those with  feature x;  in particular  it takes  into account  the
relative size of the sub-populations.   When Ω is a singleton and
φ = 1-𝕀, we recover Equation (<ref>) (with u = I/N
and K=k).   When Ω  is finite, we  recover the  Lajmanovich and
Yorke <cit.> model, and the same  framework can be used to
describe households models <cit.> and multi-host and vector-borne
diseases <cit.>.



In this finite case, the  study of the non-uniqueness for equilibria
relies   on  the   properties   of  the   oriented  transmission   graph
G_K=(Ω,      E_K)      with       the      set      of      edges
E_K={(y,x)∈Ω^2   K_x,y>0} given by the support of
the  transmission matrix  K.   An  edge from  y  to  x models  the
possibility of infection from the sub-population with feature y to the
sub-population  with  feature x;  in  particular  the graph  may  have
self-loops.    For   transmission   graph   models   see   for   example
<cit.>.

Strongly connected components of G_K will be called atoms —
the notion will be generalized in the infinite case. 
An atom is non-zero  unless it is a singleton with no self
loop.   Notice the  transmission matrix/operator  is irreducible  if and
only if the  graph G_K is strongly connected (that  is, Ω is an
atom), and  is said monatomic  if there is  a unique non-zero  atom. For
further  result on  monatomic  operators in the general case see  <cit.> and  references
therein;   we  refer   also   to   Corollary <ref>  for   a
characterization of  monatomic transmission  matrix using the  number of
EE.




In many examples  the transmission graph G_K is  symmetric (even though
the  transmission  K  might  not  be symmetric).  In  this  case,  all
(strongly) connected  components behave independently and  one can study
each connected components separately.
Cases where G_K  is not strongly connected occur
less frequently in  the literature; it has been mentioned  for example in a
multi-type SIR model by <cit.>.


Let us mention two examples of non symmetric transmission graphs.



  *  The  West Nile Virus,  presented in <cit.>,
  infects three species,  birds (B), humans (H) and  mosquitoes (M).  It
  is a vector-borne disease where  birds and mosquitoes serve as vectors
  for a transmission to humans.  In this model, mosquitoes infects birds
  and  humans while  biting them  and mosquitoes  get infected  by birds
  while biting them, and we assume  there is no infection from humans to
  mosquitoes, nor  between birds and  humans.  The graph G_K  given in
  Fig. <ref> has  only one non  zero atom {B,M}  and a
  zero atom {H}. In particular K is monatomic. 

  *   In  the   zoonosis  model  from <cit.>,  a
  pathogen  exists  in wild  animals  (W),  is transmitted  to  domestic
  animals (D) that  transmit themselves the pathogen to  humans (H). The
  graph G_K given in Fig. <ref> has three non-zero atoms:
  {W}, {D} and {H}.


In such cases where  G_K is not symmetric,  the picture is
richer: many endemic equilibria may exist, they may be entirely characterized
by the atoms contained in their support, and their basins of attraction
may be described explicitly.



Let us give a few additional definitions to state these
results more precisely, before giving the general statements below in
Theorem <ref> and <ref>.  Define
the future of a set A⊂Ω as the set (A) of
all the vertices in G_K reachable from A by a (possibly empty)
path using edges in E_K.  For two atoms A and B of G_K, we
write B ≼ A if B⊂(A); the relation
≼ is a partial order.  An antichain of atoms is a
set of atoms which are pairwise unordered.  The future of an antichain
is the future of the union of its elements.

In   the  West   Nile  Virus
model <ref>, the antichains of non-zero atoms are ∅ and
{B,M}; in the zoonosis model <ref>, the antichains of non-zero
atoms are: ∅, {H}, {D}  and {W}.

Finally, an atom is
supercritical if the  basic reproduction of the  SIS-model restricted to
the  atom is  larger  than  1; in  particular  a  supercritical atom  is
non-zero   and    an   atom   A   is    trivially   supercritical   if
K_x,x/γ(x)>1 for all x∈ A, where γ is the recovery rate
function,  and  the function   φ  satisfies the  regularity
Condition (<ref>)     below.



Our first main result, Theorem <ref>,
states (in the general possibly infinite setting) that
each equilibrium  is characterized
by a (different) antichain of supercritical atoms, and the DFE is associated to the
antichain ∅.   For example,  assuming for simplicity  that all
non-zero atoms are supercritical, we deduce  that in the West Nile Virus
model <ref>  there  is only  one  EE  and  that in  the  zoonosis
model <ref> there are  three EE.



Let us mention that a similar result  on the existence of multiple EE is
obtained  in Waters  et al. <cit.>  for a  waterborne parasites
that infect both humans and animals, such as Giardia infection in
rural Australia.  In  this model the animals and the  humans can be seen
as  non-zero   atoms  for  the   transmission,  and  the  water   as  an
environmental  reservoirs.    This  model  does  not   fit  exactly  the
metapopulation SIS  model (<ref>)-(<ref>), nor
the    model   with    external   disease    reservoir   presented    in
Section <ref> because the reservoir is between the animal
population and the human population.


Theorem <ref> also states  also that the support  of an
equilibrium is  given by  the future of  its corresponding  antichain of
supercritical atoms.   Furthermore, Corollary <ref> asserts
that for two equilibria g and g',  we have g≤ g' if and only if
(g)⊂(g').  This in  particular allows to  recover the
existence of a maximal equilibrium g^*, in the sense that if g is an
other equilibrium, then g≤ g^*.


For example,  assuming again for simplicity  that all
non-zero atoms are supercritical, we deduce  that  in  the  zoonosis
model <ref>, denoting by g_x the equilibrium characterized by
the antichain {x}, we have:

    (g_H)={H}⊂ (g_D)={D, H}⊂(g_W)={W,D,H}  and  
    0≠ g_H≨  g_D≨  g_W=g^*.




Our second main result is a full characterization of basins
of attraction of the various equilibria: we show in 
Theorem <ref> that,   starting   with  an   initial   condition
u(0)=h, the epidemics  converges in long time  towards the equilibrium
associated to the maximal antichain in ((h)) the future of the
support of the initial condition.

In the example  of  the West Nile  Virus model <ref>, assuming
that all non-zero atoms are  supercritical, we deduce that starting with
an initial  condition where only  the human population H  is infected,
the epidemic converges  to the DFE and thus dies  out, but starting with
an initial condition where the populations H and M (or simply M or
B) is infected,  the epidemic converges to the unique  EE g^*, whose
support is {B,M,H}. 

In the example  of the zoonosis model <ref>, assuming  again that all
the atoms are supercritical, we deduce from Theorem <ref>,
starting with an initial condition  u(0)=h, the epidemics converges in
long  time  towards the  equilibrium    whose support  is  the
support of h.










 §.§ Assumptions and main results


Recall the SIS model (<ref>)-(<ref>) with
parameter (T, γ,
φ).   We
shall 
consider  the following  assumptions  on the parameters.  For p∈ [1, +∞ ], let L^p denote the usual Lebesque space
of    measurable    function    defined   on    the    measured    space
(Ω, , μ) endowed with the  L^p norm ·_p, and
L^p_+ the subset of L^p of nonnegative functions. 


  The  measure  μ is  finite  and  non-zero;  the map T is  a bounded
  linear map  on L^∞ and there  exists p ∈ (1,+∞)  and a
  finite constant  C_p such  that for all f∈ L^∞:

    Tf_p≤ C_p   f_p;
  
the  function  γ
  belongs to  L^∞ and γ>0 a.e.; and  the  function φ : →
  is locally Lipschitz, nonnegative on [0, 1] and φ(1) = 0.




  Assumption <ref> holds; there exists  a finite constant C'_p
  such that for all
  f∈ L^∞:
  
    Tf_∞≤ C'_p   γ f_p;
  
 and the map φ is decreasing on [0,1] with φ(0) = 1.



The next two remarks are related to kernel operators. 
We also refer to Section <ref> for further  properties of the
operator T induced by those two assumptions. 



 
  
  Assume  Condition (<ref>) holds.   Since γ  is
  bounded, we  deduce that there  exists a finite constant  C”_p such
  that   Tf_∞≤   C”_p    f_p   for   all
  f∈  L^∞,  and  also  that  Condition (<ref>)
  holds.  According to <cit.>, we  deduce that T is
  a kernel operator (and the kernel is indeed nonnegative by Theorem 1.3
  therein).
 






  
  Recall the  definition of the kernel  operator T_k for k  a kernel
  given   in  Section <ref>.    We  check   that  the
  SIS model from <cit.>, see Assumption 1 therein, satisfies our
  Assumption <ref>.  In <cit.>,  the measure μ  is a
  probability measure on Ω, the function γ is positive and
  bounded,   and   the   mass-action   incidence  rate   is   associated
 to 
  φ = 1-𝕀. So the conditions on φ, γ and μ
  in Assumption <ref>  are clearly satisfied.  Therein,  we have
  T = T_k for k : Ω^2 →_+ a kernel such that:

    sup_x ∈Ω ∫_Ω(k(x,y)/γ(y)) ^q
     μ(  y) < +∞,

for some q∈ (1, +∞ ).  It  is then elementary to check that the
conditions on  the operator T=T_k from  Assumption <ref> are
satisfied  with p ∈ (1,+∞) given by 1/p + 1/q = 1.




We now give our main results.  Recall that
Δ={f∈ L_+^∞   1- f∈ L^∞ _+} is the
set of measurable functions taking their value a.e. in [0, 1].
Proposition <ref> below asserts that under
Assumption <ref>, for any initial condition h ∈Δ,
Equation (<ref>) has a unique global solution in
L^∞ given by the semi-flow
(ϕ(t, h))_t ∈_+ and that ϕ(t, h) belongs
to Δ for all t∈_+. The following result on long time convergence
appears below as Theorem <ref>
(see Section <ref> below  for a precise definition of the convergence involved).
[Longtime behavior]
  
  Let (T, γ, φ)  satisfy Assumption <ref>.
  The semiflow ϕ always converges to an equilibrium: 
  for any initial condition  h ∈Δ, there exists
  g∈Δ  such that F(g) = 0 and 

    _t → + ∞ϕ(t, h) = g
        in L^∞.


Under Assumption <ref>, we define  the basic reproduction number
R_0 as the spectral radius  ρ() of the power compact operator
   on  L^∞   given  by   f=  T(f/γ),   see
Lemma <ref>.  It  comes at no  surprise that if R_0≤  1, then
the zero function  is the  only equilibrium, so that all epidemic
disappear in the long  run, see Proposition <ref>. However,
if  R_0>1,  then  there  exists a  maximal  endemic  equilibrium,  say
g^*≠, see  Theorem <ref>.  If  furthermore
T is irreducible (which is equivalent to the existence and uniqueness,
up to  a scaling factor,  of v∈ L^∞ _+∖{} such
that Tv =R_0  v and that v is positive), the maximal equilibrium is
the only endemic equilibrium and  g in (<ref>) is equal to
g^*  as soon  as  the initial  condition h  is  non-zero, see  again
Proposition <ref>.   Those   results  appear   already  in
<cit.> in a slightly less general framework for 
R_0≤ 1 or T irreducible (or  quasi-irreducible).



The  main  result  of  the  paper is the description of  all the
endemic equilibria and their domain of
attraction: for any equilibrium g we give  all  the initial
conditions h∈Δ such that (<ref>) holds.
To do so, we shall rely on the decomposition of the state space in atoms
associated to the operator T given by Schwartz <cit.>, see
also   our    previous   work <cit.>,    which   is    recalled   in
Section <ref>,   and  Section <ref>
for  the  elementary case  where Ω is finite.
To  summarize,  a  measurable  set
A ∈  is invariant if the  support of the function  T_A is a
subset of A (up  to a set of zero measure); a set  is admissible if it
belongs to the σ-field generated by the invariant sets; the atoms
are the minimal admissible sets with positive measure (that is A is an
atom  if A  is  admissible  with μ(A)>0  and  if  B⊂ A  is
admissible  then  either  μ(B)=0  or μ(B)=μ(A)).   If  T  is
irreducible, then  Ω is an  atom. For  a measurable set  A, its
future (A)  is the smallest invariant  set containing A (up  to a
set of zero measure).  The set  of atoms (identifying atoms which differ
by  a  set of  zero  measure)  can be  endowed  with  an order  relation
≼: A  ≼ B when (A)  ⊂(B) (where
the   inclusion   holds  up   to   a   set   of  zero   measure).    See
Section <ref> for further details.  We say that an atom
A  is supercritical  if the  spectral  radius of  the operator  
restricted to  A, denoted R_0(A),  is strictly larger than  1. The
number of supercritical  atoms is finite; it is positive  if and only if
R_0>1, see <cit.>.   A supercritical antichain is  a finite
set of supercritical atoms which  are pairwise unordered with respect to
≼; we define its future as  the future of the union of its
atoms.  For example,  with two supercrtical atoms, say A  and B, the
supercritical antichains are ∅, {A} and {B}, with also
{A, B}  if A and  B are unordered.   Notice that R_0>1  if and
only if there exists a non empty supercritical antichain.



We give a complete characterization of equilibria, see
Theorem <ref> for a more complete statement. 


[Equilibria and supercritical antichains are in bijection]
  
  If (T,  γ, φ) satisfy Assumption <ref>,  then the
  set  of supercritical  antichains and  the  set of  equilibria are  in
  bijection. Furthermore the support of the equilibrium associated to a
  supercritical 
  antichain is given by its future.


The empty supercritical antichain corresponds  to the DFE g=. 
We  deduce  from  this  result  that if  g  and  g'  are  two
equilibria, then g≤ g' if  and only if (g) ⊂(g')
(up to a  set of zero measure) and (h)={h>0}  is the support of
the function h, see Corollary <ref>.

To complete this theorem we fully describe basins of attraction. To state
the result, we denote by T_A  the projection  of T on  a measurable set  A,
that is, the operator on  L^∞ defined  by T_Af=  A T(A  f) for
f∈ L^∞. Notice that if T satisfies Assumption <ref>,
so does  T_A. When this is  the case, we  say that g is  the maximal
equilibrium  of A when g is the maximal equilibrium of the SIS model
with T replaced by T_A.
Intuitively, 
from an  initial condition  h∈Δ, the  epidemic  converges  to an
 equilibrium  which  depends only  on the  support of  the initial
condition; it is the same as the one starting from the  “worst possible
case” where the whole population in  ((h))⊂Ω is infected. 
 [Basins of attraction of equilibria]
  The limiting  equilibrium g  of an  epidemic with  initial condition
  h∈Δ  from   Theorem <ref>  is   the  maximal
  equilibrium of ((h)).

This result  appears below as Theorem <ref>.  It  is a full
generalization of Theorem 4.13  in <cit.>, which only covers the
irreducible case T  where, if R_0>1, the endemic  equilibrium g is
unique and all epidemics with initial condition h≠ converge to
g in large time.   Proposition <ref> states that γ
times  the  epidemic converges  uniformly  to  γ g.   Thus,  when
γ > 0, the epidemic  converges uniformly to g, see also
Remark <ref>   when   furthermore    R_0<1.    This   uniform
convergence is no longer true a priori when γ = 0,
see Remark <ref> and Example <ref>.







 §.§ Model with an external disease reservoir



We consider an infinite-dimensional SIS model with an external disease
reservoir, called SISκ model  in <cit.>; it can be seen
as  an  extension  of   the  SIS  model (<ref>).   An
external  disease  reservoir is  a  particular  case of  environmentally
transmitted  diseases   where  the   population  of  pathogens   in  the
environment   is   assumed  to   be   constant   over  time,   see   for
example <cit.>  and references therein.  See  also the
example  of the  West  Nile Virus,  where birds  and  mosquitoes form  a
reservoir that is not infected  by humans, see <cit.>.  It also
encompasses some  SIS models with immigration  from <cit.>, see
Remark <ref> below.

Recall   I(t)  ≥ 0  and  S(t) ≥ 0  denote
respectively  the   number  of  infected  individuals   and  suscpetible
individuals      at      time       t      ≥      0.      According
to <cit.>,     the     corresponding     ordinary
differential  equations model,  including  infection  from the  external
disease reservoir, is given by:

    {
        S'   = μ_0 N - μ_0 S - k  SI/N- κ S  +γ I,
    
        I'   = k SI/N + κ S - (μ_0 +γ) I,
      .

  where N=I+S  is the total  population, μ_0 is the  healthy birth
  rate  and  the common  death  rate  of  the susceptible  and  infected
  populations, κ S  is the rate of disease  transmission from the
  reservoir, with κ>0.   Notice the total size  population N is
  constant in time. 

In an inhomogeneous setting,
with the measured space of types  (Ω, , μ) and  μ(Ω)∈ (0,
+∞ ), the proportion of infected individuals among the
individuals with feature x is given by 
u(t,x)=I(t,x)/N(x),  where  I(t,x) denotes  the number  of
infected individuals  with feature  x∈Ω  at time t∈_+ and N(x) the total size of the population with feature x ∈Ω, assumed constant over time. In the inhomogeneous SISκ model
inspired by (<ref>),  the function 
u=(u(t,x))_t∈_+, x∈Ω is 
solution   in L^∞ of  the ODE:

    {[ u' = F_κ(u),;    u(0) = h, ].

with initial condition h∈ L^∞ and:

    F_κ(u)=  φ(u) (Tu+κ) - γ u,

where    φ   is    a   continuous    function   on       and
κ∈    L^∞    _+. 
The particular case of SISκ model given by (<ref>)
corresponds to Ω={ω}, μ a Dirac mass at ω, T
the multiplication operator by k, γ and κ constant functions, and φ=1-𝕀.



This model can be related to SIS model with immigration, see the following remark.


  
  For   the   homogeneous   population,    we   link   the   SISκ
  model (<ref>)  with  the  SIS  model  with  immigration
  of <cit.>. Assume initially that the total population N(t) is not necessarily constant over time t. Let A ≥  0 be the  immigration rate,  p ∈  [0,1] the
  proportion of infected  individuals among the immigrants, and  d > 0
  the death rate among the population.  All those parameters are assumed
  to be  constant over  time.  We  assume that  the epidemic  induces no
  death (that is α =  0 in <cit.>) and that the
  incidence rate is the standard  mass-action.  Then, the SIS model with
  immigration given in <cit.> reduces to:

    {
       I'    = pA + k (N-I)I/N   - (d+γ)I, 
    
      N'    = A - d N.
        .

Since lim_t→∞ N(t)=A/d, and  since we
are interested  in the long  time equilibrium,  it is natural  to assume
that N start  at its equilibrium, that  is N(0)=A/d, so
that the population size is constant over time.
In this case, Equation (<ref>)  with u(t) = I(t)/N(0)
reduces to:

    u' = (1-u)(ku+pd) - (γ + (1-p)d)u.

The same arguments applied to an inhomogeneous population would lead to
a similar multi/infinite-dimensional ODE with u(t) replaced by a
function u(t,x), with x∈Ω the set of features, and ku
replaced by Tu with T the transmission operator, so that
(<ref>) becomes:

    u' = (1-u)(Tu+pd) - (γ + (1-p)d)u.

This            corresponds to the SISκ model (<ref>)-(<ref>)     with
φ= 1-𝕀,  κ = pd and γ replaced by
γ + (1-p)d. In conclusion the SIS model with immigration and the
SIS model with an external disease reservoir lead to the same ODE.




In  Proposition <ref> and
Corollary <ref>, we prove that the SISκ model
with reservoir of (<ref>) can  be analyzed
using the classical SIS model (<ref>) by adding a new
element $̊ to the set of featuresΩcorresponding to the
reservoir. In particular we provide a full description of the equilibria and
their domain of attraction for the  SISκmodel. 




 §.§ Discussion on the incidence rate


In this section, we discuss different models for the infection rate, and
more precisely for the functionφin (<ref>).

In an homogeneous population, Ross <cit.> considered the so called
law of mass actionβSI/N(which corresponds toφ= 1-𝕀in
the  SIS  model):  the  incidence  rate  is
proportional to the product of the proportion of susceptible individuals
and the  proportion of  infected individuals.   According to  Wilson and
Worcester  <cit.>,  it  corresponds   to  the  assumption  that
infected  individuals are  mixing  uniformly with  the susceptible  ones
throughout the population, see also Heesterbeek <cit.> for
an historical review.  Some epidemic models introduced in the literature
replace  the law  of  mass action  by various  incidence  rates, see  in
particular  the  survey  McCallum, Barlow  and  Homeo <cit.>.
Concerning    the    functionφ,    Assumptions <ref>
and <ref> below reduce to:

    φ is locally Lipschitz on ,
    decreasing on [0,1] with   φ(1) =
    0 and φ(0) = 1.

  In the 
examples below from the literature, the setΩis a singleton and the
transmission operatorTis thus a  constant, which is assumed to be
positive; so the conditionφ(0) = 1is a normalization convention on
the (constant) operatorTand could be replaced here by the more relevant conditionφ(0)>0. 


  *  London and Yorke <cit.> consider the incidence rate β
  SI(1 -cI), that is: 
  
    φ(u)  = (1-u)(1-au)  with    a>0

  for measles  epidemic (in  New York  City and  Baltimore from  1928 to
  1972)  in  order  to  eliminate the  systematic  differences  on  data
  observed between years  with many cases and years  with relatively few
  cases; however they do not provide a biologic or physical argument for
  such modification.  Notice that by  considering u/a instead  of u,
  one    can    assume   that    a    ≤    1,   in    which    case
  Condition (<ref>) holds.



  *  We recall that in the SIR model,
  once  infected, the  individuals  recover with  a permanent  immunity.
  Rose et al. <cit.> incorporate in  the SIR model (with constant
  population  N=S+I+R)   a  population-level  heterogeneity   for  the
  infection susceptibility given by  the gamma probability distribution;
  in <cit.> they   consider  the   incidence   rate
  β I S^α. Using data  from the 2009 H1N1 influenza outbreak,
  they observe that the higher-order models are more consistent with the
  data  than the  case  α=1.  In our  setting,  this model  would
  correspond  to   the  following  function  φ   with  satisfies
  Condition (<ref>):
  
    φ(u) = (1-u)^α  with  α≥ 1.

  *  Capasso and  Serio <cit.> study a SIR  model (with
  constant population N=S+I+R)  taking into
  account saturation  and “psychological” effects.  They  consider the
  incidence rate g(I)S. 
In our  setting,  this model  would
  correspond  to:

    φ(u) = (1-u) g(u)/u,

  where the conditions  on g translated into  our framework correspond
  to:  the function  g is  defined on  [0,1], nonnegative,  bounded,
  differentiable  with  g' bounded and such  that g(0) = 0,  g'(0)=1 and
  g(u) ≤ u  on _+.  Under those assumption,  the function 
  φ   is  Lipschitz   on   [0,1],   with  φ(1)=0   and
  φ(0)=1.  However  the monotonicity  condition on  φ on
  [0,1], which amounts  to g(u)≥ u(1-u)   g'(u) on  [0, 1], is
  not   satisfied   in   general.

  Notice   that
  Condition (<ref>) is indeed satisfied for the following
 functions g, where c>0:
 u/(1+cu) in <cit.>,
(1-exp(-cu))/c in <cit.> on SIR model for
Covid19     outbreak,  and log(1+cu)/c in Table 1 of the
survey <cit.> on pathogen transmission models. They
respectively correspond to:

    φ(u) = 1-u/1+cu, 
      φ(u) =  (1-u) 1-exp(-cu)/cu  and  φ(u)=(1-u)  log(1+cu)/cu·

  *  We recall that in the  SIRS model, once infected, the individuals
   recover with a temporary immunity. To exhibit qualitatively different
   dynamical behaviors,  Liu, Lewin and Iwasa <cit.>  introduced a
   SIRS model  (with constant population N=S+I+R)  where the incidence
   rate is  given by I H(I,S)  for some differentiable  function H
   such that H(I,0)=0 and ∂_S  H>0 for all I>0.  The latter
   condition reflects  the biologically  intuitive requirement  that the
   incidence  rate   be  an  increasing   function  of  the   number  of
   susceptibles. In our setting, this model would correspond to:
   
    φ(u)= H(u, 1-u),

   with φ differentiable and  φ(1)=0.
Notice that φ is decreasing on [0,1] if  ∂_S H>
∂_I H. 
The authors consider the  particular case
 φ(u) =  (1-u)^α  u^β -1 with α, β >
  0.  Condition (<ref>)  holds for β = 1 and
  α≥ 1, which is already considered in
  Point <ref>. 




§ NOTATIONS


 §.§ Ordered set


Let(E, ≤)be a (partially) ordered set.  Whenever it exists,
the supremum ofA⊂E, denoted bysup(A), is the
least upper bound ofA: for allx∈A,x ≤sup(A)and if
for somez∈Eone hasx≤zfor allx∈A, thensup(A) ≤z.  A collection(x_i)_i ∈ℐof
elements ofEis an antichain if for all distincti, j ∈ℐ, the elementsx_iandx_jare not
comparable for the order relation.



 §.§ Banach space and Banach lattice

Let(X, ·)be a complex Banach space not reduced
to{0}. An operatorTonXis a  bounded  linear (and thus
continuous) map fromXto itself. IfY ⊂Xis a subspace ofXsuch thatT(Y) ⊂Y, we denoteT|_Ythe restriction ofTto the subspaceY, that is an operator on the Banach space(Y, ·).
The  operator norm ofTis given by:

    T_X= sup{Tx   x ∈ X  s.t. x = 1},

its                              spectrum                             by(T) =  {λ∈     T - λ𝕀 has no
  bounded inverse}, where𝕀is  the identity operator onX.  Ifλ∈andx∈X0satisfyTx=λx, then the
elementxis an eigenvector  ofTandλ, which  belongs to(T) ,  is an eigenvalue  ofT.  The  spectral radius ofTis
defined by (see <cit.>):

    ρ(T) = sup{ |λ|   λ∈
        (T)} =lim_n→∞T^n_X^1/n.

By convention, we setT^0=𝕀.
The spectral radius
is commutative in the sense that ifTandSare two operators onX, we have:

    ρ(TS) = ρ(ST).

We define the spectral bound of the operatorTby:

    s(T) = sup{(λ)   λ∈(T)}.



LetX^⋆denote  the (continuous or topological)  dual Banach space
ofX, that is the set of  all the continuous linear forms onX.  Forx∈X,x^⋆∈X^⋆, let⟨x^⋆, x ⟩denote
the duality product and the norm ofx^⋆inX^⋆is defined byx^⋆=sup{⟨x^⋆,     x    ⟩      
x=1}.   For an  operatorT,  the dual  operatorT^⋆onX^⋆is                  defined                 by⟨T^⋆x^⋆, x ⟩=⟨x^⋆, Tx ⟩for allx∈X,x^⋆∈X^⋆.    It   is    well   known   thatT^⋆_X^⋆=T_Xand(T^⋆) = (T).




An ordered real Banach space(X, ·, ≤)is a real
Banach space(X, ·)with an order relation≤.  For
anyx ∈X, we define|x| = sup({x, -x})the supremum ofxand-xwhenever it exists.  Following <cit.>,
the ordered Banach space(X, ·, ≤)is a Banach
  lattice if:

  *  For  any x, y, z  ∈ X, λ≥  0 such that x  ≤ y, we
  have x + z ≤ y + z and λ x ≤λ y.

  *  For any x, y ∈ X, there exists  a supremum of x and y in X.

  *  For any x, y ∈ X such that |x| ≤ |y|, we have
  x≤y.


Let(X, ·,  ≤)be  a  real Banach  lattice.  We  denoteX_+  =  {x  ∈X      x  ≥0  }the  positive  cone  ofX.  Recall it is a closed
set.
We shall also consider the
dual coneX^⋆_+={x^⋆∈X^⋆    ⟨x^⋆, x
⟩≥0  for all  x∈X_+}. A  linear mapTonXis  positive ifT(X_+)  ⊂X_+.   According  to <cit.>
positive  linear maps  on  Banach  lattices are  bounded  (and thus  are
operators).

IfSandTare  two operators onX,
we writeT ≤Sif the operatorS - Tis positive.
If the operatorsT, SandS-Tare positive, then we have, see <cit.>:

    ρ(T) ≤ρ(S).




Any real Banach  latticeXand any  operatorTonXadmits  a natural
complex  extension.  The  spectrum  ofTwill  be  identified as  the
spectrum of its complex extension and denoted by(T), furthermore
by <cit.>, the spectral radius  of the complex
extension             is            also             given            bylim_n→∞ T^n_X^1/n, with·_Xstill defined by (<ref>). 
Moreover, by <cit.>, ifTis positive
(seen as an operator on the real Banach latticeX), thenTand its complex
extension have the same norm.




 §.§ Lebesgue spaces and essential limits


Let(Ω, ,  μ)be   a  measured  space  withμaσ-finite measure.       For      any𝒜  ⊂,  we denote  byσ(𝒜)theσ-field  generated by𝒜.   Iff, gare  two
real-valued  measurable   functions  defined   onΩ,   we  writef  ≤ga.e. (resp.f =  ga.e.)   whenμ({f  > g})  = 0(resp.μ({f ≠g  }) = 0), and denote(f  )= {f ≠0}the support off.    We  say that  a  real-valued
measurable functionfis nonnegative whenf ≥0a.e., we say
thatfis    positive,    denotedf    >    0a.e.,    whenμ({f ≤0}) = 0, and we say thatfis bounded if there existsM ≥0such that|f| ≤Ma.e.. 
IfA, B ⊂Ωare measurable sets,
we writeA ⊂Ba.e. (respA = Ba.e.) when_A ≤_Ba.e. (resp._A = _Ba.e.). LetL^0(Ω, , μ), simply
denotedL^0,   be the
set of[-∞, + ∞]-valued
measurable  functions defined  onΩ, where  functions which  are
a.e.  equal  are   identified. 
The elementsandofL^0denote the
functions which  are a.e. equal  respectively to 1  and to 0.   For the
sake of clarity, we will omit to write a.e. in the proofs.



Let(f_t)_t∈Tbe a family of measurable functions defined onΩtaking values in[-∞, +∞]. We recall thatf^*=_t∈T f_tis a measurable function such thatf_t≤f^*a.e. for allt∈Tand iffis measurable function
such thatf_t≤fa.e. for allt∈Tthenf ^*≤fa.e.(ifTis at most countable, then one can take_t∈T f_t=sup_t∈T f_t).  We now considerT=_+.
Let(f_t)_ t∈_+be a non-decreasing sequence, in the sense
that for allt≤swe havef_t≤f_sa.e., then if(t_n)_ n∈is a sequence converging to+∞, we have
that the sequence(f_t_n)_ n∈converges a.e. towardsf^*=_t∈_+ f_t, and thus we shall simply writef^*=_t→+∞ f_t. We leave to the reader the
definition ofand the limit of a non-increasing sequence of
measurable functions.  For the family(f_t)_ t∈_+, we
considerf^*_t=_s≥t f_sfor allt∈_+, and get
that the sequence(f^*_t)_t∈^+is non-increasing and write_t→∞ f_t= _t→∞
f^*_t.  We define in a similar way_t→∞ f_t.  Notice that ifg_t=f_ta.e. for allt∈_+, then the essential supremum/infimum limits
of(f_t)_t∈^+and(g_t)_t∈^+are a.e. equal.
Therefore, the essential supremum/infimum limits of sequences is well
defined on the spaceL^0.  We say the sequence of functions(f_t)_ t∈_+inL^0essentially converges if_t→∞ f_t=_t→∞ f_tinL^0, and write_t→+∞ f_tfor
this common limit (which is an element ofL^0).  When consideringT=instead ofT=_+, the analog of the essential convergence is
the a.e. convergence, that is the usual convergence inL^0.





For a measurable functionf, we writeμ(f)=∫f  μ=∫_Ωf(x)  μ(x)the integral
offwith respect toμwhen it is well defined. Whenfis
measurable and a.e. finite and nonnegative, we denotef μthe
measure on(Ω, )defined byf μ(A) = μ(_A f)for
any measurable setA.  Forp ∈[1, +∞], the Lebesgue spaceL^p(Ω, , μ)is the set of all real-valued measurable
functionsf∈L^0defined onΩwhoseL^p-norm,f_p = μ(|f|^p)^1/pifp<+∞andf_∞= (|f|)ifp=+∞, is finite.  When
there is no ambiguity we shall simply writeL^p(Ω),L^p(μ)orL^pforL^p(Ω, , μ).  The Banach spaceL^pendowed
with the usual orderf≤g, that isμ({f>g})=0, is a Banach
lattice.  The positive coneL^p_+is the subset ofL^pof
nonnegative functions; it is normal (as the norm·is
monotonic, that is,0≤f≤gimpliesf_p≤g_p, see <cit.>)
and reproducing (that is,L^p_+ -L^p_+=L^p).  Since the supports of
two functions which are a.e. equal are also a.e. equal, we get that
the support off∈L^pis well defined up to the a.e. equality; it
will still be denoted by(f).  Forp∈[1, +∞), the
dual ofL^pisL^qwhere1/p+1/q=1, with the duality product⟨g, f ⟩= ∫fg  μforf ∈L^pand g ∈L^q(forp=1, we use that the measureμisσ-finite).



For anyf ∈L^∞, we denote byM_fthe multiplication byf, which
can be seen as 
an operator onL^pforp ∈[1,+∞].  ForA ∈a
measurable set, we denote:

    M_A=M__A.

LetTbe an operator onL^p.  
The projection 
 ofTonA, denotedT_A, is the operator defined  by:

    T_A=M_A   T   M_A,

and, ifμ(A)>0, we denote byT|_Athe restriction of the operatorT_AtoL^p(A), where the setAis endowed with the trace ofonAand the measureμ|_A (·)=μ(A∩·).

We now assume thatμ(Ω)>0, so thatL^p(Ω)is not reduced
to  a singleton.  When there  is no  ambiguity on  the operatorT, we
simply writeρ(A)for the spectral radius ofT_A(and ofT|_Awhenμ(A)>0).  In  particular, we  haveρ(Ω)=ρ(T)andρ(A) = 0ifμ(A) = 0.  If the operatorTis positive, we also
have that:

    A ⊂ B   ⟹  ρ(A) ≤ρ(B).


 §.§ Decomposition of positive operators on L^p, with p∈ (1,
+∞ )

Recall the measureμisσ-finite and non-zero.
We recall the atomic decomposition from Schwartz <cit.> of a
positive operatorTonL^p, see also <cit.>.   A measurable setA ∈isT-invariant, or simply invariant when there is
no ambiguity,   ifM_A^c T  M_A = 0, which, see <cit.>, is equivalent
to:

    ⟨ g , Tf   ⟩=0.

for allf ∈L^p_+and  g ∈L^q_+such that(f) ⊂Aa.e.  and(g)  ⊂A^ca.e..    The  operatorTis
irreducible  if  its  only  invariant sets  are  a.e.  equal  toΩor∅.  A measurable  setAwith positive measure is
irreducible if  the operatorT|_AonL^p(A)is irreducible.
The future  of a setA  ⊂Ω, denoted(A),  is the
smallest invariant set  that containsA.  Ifis  an at most countable collection of subsets  ofΩ,  then  we denote  by()the
future of the union of the elements of:

    ()= (  ⋃_A∈  A) =  ⋃_A∈(A),

 where the last equality is <cit.>.

A set
is admissible  if it belongs  to theσ-field  generated by
the invariant  sets. An atom is  a minimal admissible set  with a
positive  measure  (that is,Ais  an  atom  ifAis  admissible,μ(A)>0and ifBis an  admissible set such thatB⊂Aa.e.then a.e.B=∅orB=A), and we identify two  atoms that are
a.e.  equal.  In  particular, if  the setAis  an atom  andBis
admissible then we have:

    μ(A ∩ B)>0  A⊂ B   a.e..

According to <cit.>, 
a measurable set  is admissible and irreducible with positive measure if
and only if it is an atom. 
Since the  atoms have  positive measure,  we get that  the set  of atoms  (up to
the a.e. equality),,   is at
most countable.
We shall also consider the (at most countable) set of non-zero atoms:

    ={ A ∈  ρ(A) >0}.






The relation≼on,
defined  byA  ≼Bif(A)  ⊂(B)a.e.  (or
equivalentlyA ⊂(B)a.e.), is  an order relation.  We end this
section by noticing that antichains  of atoms are characterized by their
future.

  Let  and ' be two antichains of atoms. Then, we have:
  
    ()=(')  ⟺  ='.

  Assume that ()=('). 
  Consider an atom  A∈. Since
  A⊂() = ('), 
  there exists A' ∈' such that we have
  μ(A ∩(A')) > 0, which implies A ≼ A' as A is
  an atom. Conversely there exists B∈ such that
  A'≼ B, and by transitivity A≼ B.  Since
   is an antichain, we obtain  A=B and thus A=A' is  an element
  of '.
  The reverse implication is trivial by (<ref>). 


 §.§ Power compact operators on L^p


A linear mapTon a Banach space is compact if the image of the
unit ball is relatively compact; it is then bounded. 
An  operatorTon a  Banach space  is power  compact if  there
existsk  ∈^*such  thatT^kis  compact.  We  recall  some
well-known properties of power compact  operators, see
<cit.> for  instance.


Let T be an operator on a Banach space. 

  * 
  The operator  T^⋆ is power compact if
  and only if the operator T is power compact.

  * 
  If T  is power compact,  then the
  set (T)  is at  most countable and  has no  accumulation points
  except possibly 0 (it is thus totally disconnected).




It  is well  known  that the  spectral radius  (and  more generally  the
spectra) is a  continuous function on the set of  compact operators with
respect         to         the        operator         norm,         see
<cit.>.   We  shall   however  use  a
weaker  result   from  Anselone <cit.>.   We  say a  familyof
operators onXis collectively compact if⋃_V∈ V(B)is relatively compact, whereBis the unit ball ofX.  The following
result  is  a direct  consequence  of  Proposition 4.1 and  Theorem 4.16
in <cit.>.
 

  Let I  be an interval  of   and (V_t)_t∈  I be a  family of
  collectively compact operators on a Banach  space X.  If t∈ I is
  such  that  lim_s→  t(V_s  -V_t)x=0  for  all
  x∈ X, then we have lim_s→ tρ(V_s)=ρ(V_t).






We give a result on compact operators in Lebesgue space. 
Recall thatμis a non-zeroσ-finite  measure    onΩ,  and thatL^pdenoteL^p(Ω, , μ).


 

  Let  p ∈  (1,+∞).
  A positive operator on L^p  which 
 is dominated by a compact operator  is  compact.

   Notice    that   L^p    has   an    order   continuous    norm   for
  p∈ [1, ∞ ), see <cit.>, that is,
  according to <cit.>, if (f_n)_n∈ is
  a     non-decreasing     sequence     of     L^p_+     such     that
  sup_n∈          f_n=f∈         L^p,          then
  lim_n→∞f_n-f_p=0.
  In particular, when p ∈ (1,+∞), the dual of L^p, that is isomorphic to L^q with 1/p + 1/q = 1, also has an order continuous norm.
 The lemma is then  a
  direct consequence of <cit.>.


We recall  in our framework some classical
results, see <cit.>.


Let T be a positive power compact operator on L^p with p ∈
(1,+∞). 

  * Krein-Rutman.
  If ρ(T)  is positive then ρ(T) is
  an  eigenvalue of
  T, and there exists  a corresponding nonnegative right eigenfunction
  denoted 
  v_T.
  * de Pagter. If T is irreducible then ρ(T) is positive unless T=0
    and (L^p)=1, that is, if A is measurable then either
    μ(A)=0 or μ(A^c)=0.
  * Schwartz.
      We have that for any admissible set A:
    
    ρ(A)=max_B∈,   B⊂ Aρ(B).





Following   <cit.>,  we   now   state   a 
technical result based on Collatz-Wielandt inequality and the
Krein-Rutman theorem,   giving a bound on  the spectral radius given  a strict
supersolution  of  the  eigenvalue  equation.

  Let S be a positive operator on L^p with p ∈ (1, + ∞).
  If there exists λ>0 and a non-null nonnegative function v ∈ L^p_+ such that
  Sv ≥λ v, then:
  
    ρ(S)≥ρ(S_A) ≥λ,

   where A=(v). Furthermore, if S is power compact, then we have:
  
  * 
    If Sv= λ v on A, then ρ(S_A) = λ.
  
  * 
    If  Sv - λ v is positive on A, then  ρ(S_A) >
    λ. 
  
  We       first      note       that      ρ(S)≥ρ(S_A)
  by (<ref>).     Multiplying    the    inequality
  Sv≥λ  v by A yields  that S_A v ≥λ v.
  The     fact     that     ρ(S_A)≥λ,     and     thus
  ρ(S)  ≥λ,   is  then  a  direct   consequence  of  the
  Collatz-Wielandt     inequality     <cit.>.

  We    now    assume    that    S   is    power    compact.     Since
  (S_A)^⋆  =   (S^⋆)_A,  we   shall  denote  them   simply  by
  S^⋆_    A.     Since    S_A≤     S,    we    deduce    from
  Lemma <ref>   that   the   operator  S_A,   and   thus
  S_A^⋆,  is power  compact.   We apply  the Krein-Rutman  theorem
  (Theorem <ref> <ref>)  to   the  power  compact
  operator   S_A^⋆,  which   has   a   positive  spectral   radius
  ρ(S_A^⋆)  = ρ(S_A)  >  0:  there exists  w  ∈ L^q_+  a
  non-null  eigenfunction  of  S_A^⋆  related  to  the  eigenvalue
  ρ(S_A).                                                     Since
  ρ(S_A) w  = S^⋆_ A  w= A S^⋆(A w),  we deduce
  that (w) ⊂ A  and thus ⟨ w,v ⟩>0.
  
We now consider the following duality product:

    w,S_Av - λ v = S_A^⋆ w,v - λw,v = (ρ(S_A) - λ)w,v.

Under the hypothesis that Sv = λ v on A=(v), the left
hand side of (<ref>) is zero, and thus
ρ(S_A)=λ. This gives Point <ref>.



To prove Point <ref>, since S_Av-λ v is by hypothesis
positive on A, and wA = w is nonnegative and not identically
equal to zero, we get w,S_Av - λ v>0. Then
use (<ref>) again to get  that
 ρ(S_A)>λ.


 §.§ Operators related to the SIS model

Let(Ω,    ,    μ)be   a    measured    space    with
finite non-zero  measureμ, that is,μ(Ω)∈(0,   +∞).  Recall we simply writeL^pforL^p(Ω, , μ). 
Observe              thatL^∞⊂L^pforp ∈(1,+∞)and  that  the
identity mapιfromL^∞toL^pis a  bounded injection.

Let  p ∈  (1,+∞).
Let  T be an  operator  from L^p  to
  L^∞. The  linear map  ι T is a  compact operator on
  L^p.

  Since L^p  is reflexive, see <cit.>, we
  get by <cit.> that  T and ι are
  weakly  compact.  By <cit.>, as  the space
  L^∞ is  an AM-space, it satisfies  the Dunford-Pettis property,
  thus, by <cit.>, the  operator ι T on
  L^p is then compact. 


Letp∈(1, +∞)andγ∈L^∞such thatγ>0a.e..    LetTbe  a   positive  operator   onL^∞such
that (<ref>)  holds   for  allf∈L^∞.   In
particular, asL^∞is dense  inL^p(for theL^p-norm), we
can extendTby  density into  a bounded linear  maponL^p.
Recall thatM_1/γdenotes the multiplication by1/γ.  Notice
that  under Assumption <ref>,  see (<ref>), the
linear mapTM_1/γ,  denoted by, can be seen  as a bounded
linear map  onL^∞, and it  can also be  extended by density
into   a    bounded   linear   maponL^p,    see   also
Fig. <ref>.




  Let (T, γ, φ) satisfy
Assumption <ref>. Then   is an operator  on
  L^p. If Assumption <ref> holds, then we have:

  *  is a  compact operator on 
  L^p;

  * T^2  is a  compact operator on 
  L^∞;

  * γ  is a  compact operator on 
  L^p;

  *  is an  operator
  on   L^∞ and  ^2 is compact;
 
  * (T)=()
and ()=(). 


  Suppose      Assumption <ref>      holds.       We      deduce
  from (<ref>) that there exists  a finite constant C
  such   that  Tf_∞≤   C   f_p  for   all
  f∈ L^∞. By density, we can extend T into an operator 
  from  L^p to  L^∞. This  gives that  =ι  and
  Point <ref> on   the
  compactness    property    of        is    a    consequence    of
  Lemma <ref>. As T=  ι and thus T^2=    ι, we also get that T^2 is compact, that is Point <ref>. 
  Using (<ref>), we can also
 extend TM_1/γ into an operator _1/γ
  from  L^p to  L^∞. Arguing as above gives
  Points <ref> and <ref>. 



We    now    prove Point <ref>.      Two   complex    Banach    spaces
(E,  ·)   and  (E',  ·')  are   compatible  if
(E”,  ·+ ·'),  with E”=E∩  E', is  a Banach
space, and E” is dense in E and in E'. Given two compatible
spaces E and E', two operators A  on E and A' on E' are said
to  be   consistent  if  A(E”)⊂  E”,
A'(E”)⊂ E” and Ax=A'x for all x∈ E”. If furthermore A
and A'  are compact,  then <cit.>  gives that
(A)=(A').  The proof therein relies on the spectrum to be at
most countable and  with no accumulation points except  possibly 0 and
that the spectral projections have   finite rank, see Theorems 5 and 6 p.579 in <cit.>. Since this also holds
for power compact operators, the results can be extended to A and A'
being power compact operators.

As μ  is a finite measure,  the
spaces L^p and  L^∞ are pairwise compatible.   Notice also the
operators  T   and  ,   as  well  as     and   ,  are
consistent. Then the equalities follow.


§ EQUILIBRIA AND RESTRICTION



We      consider     the      SIS
model (<ref>)-(<ref>)  onL^∞=L^∞(Ω, ,  μ)with parameter(T, γ, φ)such that
Assumption <ref> holds. In particular the measureμis
finite and non-zero. 
We consider the following subset ofL^∞:

    Δ={f ∈ L^∞  ≤ f ≤}.

Recall  thatgis  an  equilibrium ifgbelongs  toΔand
solves (<ref>), that is:

    F(g)=0.

 In particular,  the   functionis  an equilibrium. We  say thatg^*∈L^∞is the
maximal  equilibrium ifg^*is  an equilibrium  and all  other
equilibriumg∈L^∞are such thatg≤g^*.



The  existence   result  of  the  semi-flow  and  the   maximal  equilibrium
follows <cit.>  with slightly  more
general hypothesis  onTand is  obtained similarly,  see a proof in 
Section <ref> for  completeness. We shall refer to
this section for notations and definitions/properties of the semi-flow. 


  Let (T, γ,  φ) be parameters of the  SIS model satisfying
  Assumption <ref>.  The following properties hold. 

  * 
 Equation (<ref>)  in L^∞ with
  initial condition h∈Δ has a unique global
  solution given by the semi-flow
  ϕ(·, h)=(ϕ(t, h))_t∈_+. The semi-flow belongs to
  ^1(_+). 

  * 
  For all t ≥ 0 and h ∈Δ, we have
  ϕ(t, h) ∈Δ.

  * 
  The sequence
  (ϕ(t, ))_t∈_+ is non-increasing and converges
  essentially to a limit,  g^*, which is the maximal equilibrium:
  
    _t→+∞ϕ(t, )= g^*.

The    convergence     in
Point <ref> is not uniform  in general,
see Remarks <ref> and <ref> and  
Example <ref>.




The proof of the monotonicity of the maximal equilibrium in  the parameters is
given in Section <ref> for consistency of the
arguments. 


  For i=1,2, let (T_i, γ_i, φ_i) be parameters of the SIS
  model         satisfying        Assumption <ref> and denote
  g^*_i the corresponding maximal equilibrium.  If 
   T_1 ≥  T_2,
 φ_1  ≥φ_2 and γ_1   ≤γ_2, 
  then we have g^*_1 ≥ g^*_2.





LetAbe  a measurable  set.   Since(T,  γ, φ)satisfy
Assumption <ref>,  so  does(T_A,  γ,  φ),  whereT_A= T_A =  M_A T M_Ais the  projection ofTonA.   We shall now
focus on this restricted(T_A, γ, φ)-SIS model.  We set foru∈L^∞:

    F_A(u) = φ(u) T_A(u) - γ u,

and callg∈Δan equilibrium  ofAifF_A(g)=0. In this
case, notice that(g) ⊂Aa.e..  We also
denoteϕ_Athe   corresponding  semi-flow   andg^*_Athe
corresponding maximal equilibrium ofAgiven by Proposition <ref>.



Let (T, γ, φ)  satisfy Assumption <ref>.
  Let A⊂ B a.e. be measurable sets. We have  g^*_A≤ g^*_B. 

  Apply Lemma <ref> with  T_1 = T_B≥ T_A=T_2,
  φ_1 = φ_2 = φ and γ_1 = γ_2 = γ.



We now provide results on  equilibria and  semi-flows 
associated toTandT_A.





  Let (T,  γ, φ)  satisfy Assumption <ref>. Let
  A be a
  measurable set and g ∈Δ.

  * 
    If    g is  an
  equilibrium and (g)⊂ A  a.e., then g is an equilibrium
  of  A.

  * 
  If  A is invariant and g is an equilibrium
  of  A, then  g is an equilibrium (and  (g)⊂ A  a.e.).

  * 
  If A is invariant and  g is an equilibrium, then _A^c g
  is an equilibrium of A^c and of A^c ∩(g).

  If (g)⊂ A and  g is  an
  equilibrium, then we get that g is an equilibrium of A as:
  
    F_A(g)=φ(g) _A T (_A g) - γ g=
        _A ( φ(g) T ( g) - γ  g)=0.


Let A be an invariant set and h∈Δ. Since _A^c
T(_A h) = 0, we deduce that:

    F_A^c (_A^c h)= _A^c F(h).

  If furthermore (h)⊂ A, then we have:

    F_A (h)=  F(h) - φ(h) _A^c T (_A h)=  F(h).

  
  If g is an equilibrium of  A, then we get (g)⊂ A and,
  since A is  invariant, we deduce from (<ref>) that  g is an
  equilibrium.    If  g   is  an   equilibrium,  we   deduce
  from (<ref>)  that  _A^c  g  is  an  equilibrium  of
  A^c. Then use <ref> with T replaced by T_A^c and A
  by (g) to  deduce that _A^c g is also  an equilibrium of
  A^c ∩(g).

Let (T, γ, φ)  satisfy Assumption <ref>.
Let A be a measurable set and h ∈Δ.
The following properties hold:

  * _t→+∞ϕ_A(t, _A) = g^*_A.

  * 
  If  A  is invariant,  then we  have
  _A^c ϕ(·,h)  =  ϕ_A^c(·,  _A^c   h
  ).

  * 
  If    A   is    invariant    and
  (h)      ⊂      A       a.e.,      then      we      have
  ϕ(·,h) = ϕ_A(·, h ).

  As g^*_A is  an equilibrium of A, we get  (g^*_A) ⊂ A
  and thus we have g^*_A ≤_A ≤.  By the monotonicity of
  the semi-flow, see Lemma <ref> <ref>,
  we                                                                have
  g^*_A = ϕ_A(t,g^*_A) ≤ϕ_A(t,_A) ≤ϕ_A(t, ) for
  all             t             ∈_+.             Then
  Proposition <ref> <ref>     gives
  Point <ref>.


Let A be invariant. 
For simplicity, we  write ϕ instead of ϕ(t, h) and ϕ' for
its derivative.
By definition of the semi-flow ϕ, we deduce from (<ref>)
that:

    _A^cϕ' = _A^c F(ϕ)=  F_A^c
      (_A^cϕ)  .

The map t ↦_A^cϕ(t, h) is thus a solution of
Equation (<ref>), with  F replaced by F_A^c and initial
condition _A^c   h. By the uniqueness of
the semi-flow,  we get 
_A^cϕ(·,h) = ϕ_A^c(·, _A^c h ),
that is Point <ref>. 


We now assume that (h) ⊂ A.
By Point <ref>,  we have ( ϕ)⊂ A.
We deduce from (<ref>) that:

    ϕ'= F(ϕ)=F_A(ϕ).

Then, the same argument as above yields Point <ref>.


§ CHARACTERIZATION OF  EQUILIBRIA


In this  section, we assume that  Assumption <ref> holds for the
SIS model(T, γ, φ). In particular  the mapφrestricted to[0,1]is a decreasing bijection
  onto[0,1]. 
Recall the operators related toTdefined in
Section <ref> and their properties. 






 §.§ Equilibria, supports and spectral radius







For anyg ∈Δ, letL_gdenote the compact  operator onL^pdefined by:

    L_g = M_φ(g) γ.

This operator is associated to the linearizationM_φ(g) T - γof the dynamics neargin the same
way asγis associated to the linearizationT -
γof the dynamics  near0(asφ(0)=1).
Notice that when(T, γ, φ)satisfy
Assumption <ref>, then(M_φ(g) T, γ, φ)also does.
It is immediate to check that forg∈Δ:

    g is an equilibrium  ⟺  
    L_g   (γ g)=γ g.

Let (T, γ, φ)  satisfy Assumption <ref>.
Let g∈Δ be an equilibrium. We have the following properties.

  * φ(g) > 0 a.e.. 
  * 
          The operators , γ and L_g have the
          same invariant sets,  irreducible sets, atoms and non-zero
          atoms. 
	
  *  The set (g) is invariant.
          
        If  g≠,  then the  following  additional  properties
        hold. Let h∈Δ be an equilibrium. Set A= (g).
        
  * ρ(L_g)  ≥ρ((L_g)_A) = 1.
        
  * 
          If  h≤ g, then either
          ρ((L_h)_A) = 1 and h=g, or ρ((L_h)_A) >1
          and (h)≠(g) a.e..
        
  * h≤ g ⟺ (h) ⊂(g) a.e..


As a consequence of Point <ref>
we directly get the following corollary.

Let (T, γ, φ) that satisfy Assumption <ref>. Two equilibria with the same support are equal.

Since φ(g) T(g) = γ g, φ(g) does not vanish on
(g). On the complement set, g=0 so φ(g)=1 does not
vanish either.  Since φ is nonnegative, we get
Point <ref>.



Point <ref>   is   a   direct   consequence   of   the
characterization         of         invariant         sets         given
by (<ref>)  as  1/γ and  φ(g)  are
positive          by          Point <ref>          and
Thereom <ref> <ref>     on    non-zero     atoms.
Point <ref>      is      a     direct      consequence
of (<ref>), <cit.> (which state that the
support  of  the  eigenfunction   γ  g  is  L_g-invariant)  and
Point <ref>.


Point <ref> follows directly from Lemma <ref>
with S=L_g, v=γ g 
and λ = 1. 

The proof of Point <ref> follows similar
lines. Let h≤ g be two equilibria with h≠.
The two eigenvalue equations written for h and g yield
(L_g-L_h) (γ g) + L_h(γ(g-h)) = γ (g-h), 
which we rewrite as:

    L_h (γ(g-h)) = γ (g-h) +  (φ(h)-φ(g)) Tg.

On  the  right  hand  side,  φ(h)  -  φ(g)  and  Tg  are
nonnegative,  and by  strict  monotonicity of  φ  they are  both
positive on  B = (g-h) ⊂(g).  If B is  empty, then
g=h and  we are  back to  Point <ref>.  If  not, we
apply  Lemma <ref> <ref>  to S  =  L_h,
λ   =    1   and   v=γ (g-h)≠    which   is   non-negative:
ρ((L_h)_B)> 1.  If B was a  subset of (h) this would imply
ρ((L_h)_(h))       >1,       a      contradiction       with
Point <ref>. So B is not a subset of (h), or
in other words (h) ≠(g).

            Finally            let           us            prove
Point <ref>. Clearly  h≤ g  implies that
(h) ⊂(g). To prove  the reverse implication,  let us
assume that (h)⊂(g).  Since F( max(g,h)) ≥ 0, by
Lemmas <ref> <ref>   and <ref>,
the semi-flow starting from  max(g,h) is non-decreasing and converges
to    an   equilibrium    g̃,    which   therefore    satisfies
max(g,h)  ≤g̃.   Since  (max(g,h))  = (g)  is
invariant    by    Point <ref>,   we    deduce    from
Lemma <ref> <ref>                     that
(g̃)⊂(g)         and        thus
(g̃) = (g).  Since g≤g̃, by the previous
point,   the  functions   g  and   g̃  must   be  equal,   so
g = max(g,h), or in other words h≤ g.


 §.§ Maximum equilibria and critical vaccination

Recall the  notations of  Section <ref>.  LetAbe a
measurable set.  Notice the linear map(γ)_A = (T_A)_1/γis an operator
onL^p.  Following <cit.>,  we  then  define the  basic
  reproduction number  ofAas  the spectral radius of  this operatorR_0(A)   =  ρ((γ)_A),   and   simply   writeR_0forR_0(Ω).  Notice  that, by (<ref>),  the mapA ↦R_0(A)is non-decreasing, that is, for anyA, Bmeasurable
sets withA ⊂Ba.e., we haveR_0(A) ≤R_0(B).



The following result generalizes <cit.>,
and is proved similarly, 
see Section <ref> for details. 

Let (T, γ, φ)  satisfy Assumption <ref>.
Then we have the following properties. 

  * 
  If  R_0 ≤ 1, then we have g^* =, and for all h∈Δ:
  
    _t→ +∞ϕ(t, h)=.

  *  If R_0 >  1, then the maximal equilibrium g^*  is non-null,
 (that  is, μ((g^*)) >  0).

  * 
  If R_0 >  1 and  T is
  quasi-irreducible, that  is, T  = T_A with  A an  irreducible set,
  then we  have (g^*) = A  a.e. and g^* is  the unique non-null
  equilibrium.

  * 
  If R_0 >  1, T is
  quasi-irreducible, that  is, T  = T_A with  A an  irreducible set,
  and h∈Δ, then we
  have _t→ +∞ϕ(t, h)= if
  (h) ∩ A=∅ a.e. and:
  
    _t→ +∞ϕ(t, h)=g^*
          if  μ((h) ∩ A)>0.


In  the next  remarks  and  examples, we  explore  the  uniformity of  the
convergence                in               Point <ref>
and Proposition <ref> <ref>.



  Assume  (T,  γ,  φ)  satisfies  Assumption <ref>,
  R_0<1 and  γ >  0.  By  Lemma <ref> we
  also have  s(T - γ)  < 0.  Then, mimicking  the proof of
  <cit.> and  using that 1-φ≥ 0,  we get
  that   for  all   c  ∈   (0,   -s(T  -   γ)),  there   exists
  θ∈_+  such that, for all  h ∈Δ, t  ≥ 0, we
  have:
  
    ϕ(t,h)_∞≤θh_∞ -ct.

   We assume  that R_0≤ 1 and  γ = 0.   Consider the
   function  v  defined by  v(t)  =  exp(-tγ) ∈Δ  for
   t ≥ 0.  As v'(t) - F(v(t))=  - φ(v(t))   T v(t) ≤ 0,
   we        deduce        from        Lemma <ref>        and
   Proposition <ref> that  ϕ(t,) ≥ v(t)  for all
   t≥                 0.                  We                 obtain
   1≥ϕ(t,)_∞≥v(t)_∞  =1  as
   γ = 0.  Thus the semi-flow (ϕ(t,))_t∈_+
   does not converge to g^*= in L^∞.

   Notice  the  same  conclusion  holds (with  the  same  arguments)  if
   R_0≤   1   is   replaced   by   the   more   general   condition
   g^*<1.
 [A uniform convergence when γ=0]
  
  If g^*=1 and γ = 0  it is possible  for the
  semi-flow  (ϕ(t,))_t∈_+  to  converge  to   g^*  in
  L^∞.  Consider  the  particular case:  Ω=(0,  1]  with
  μ=ν  +  δ_1,  where  ν  is  the  Lebesgue  measure  and
  δ_1 the Dirac  mass at 1; Tf= f(1)  for all f∈ L^p;
  γ(x)=x/2  and φ(r)=1-r.  In this  case, {1}  is the
  only  atom,  and  R_0({1})=2.  We get  g^*(x)=1/(1+x)  and  thus
  g^*=1. (Notice that g^*(0+)= g^* and
  γ(0+)=γ.)
  Elementary     calculus     give      that,     for     t≥     0,
  ϕ(t,)(1) = 1/(2--t/2) and for x∈ (0, 1):
 
    ϕ(t,)(x) =  2+ (x-1) -(x+1)t/2/x+1 ϕ(t,)(1) ,

 so that  ϕ(t, ) -  g^*_∞≤-t/2.   So the
 semi-flow (ϕ(t,))_t∈_+ converges  to g^*  in L^∞.


 
 
We  now  focus on  critical  vaccination.   Letφ_0defined  byφ_0(r)=1   -rand_kthe   kernel    operator   form
Remark <ref>        withkandγsatisfying (<ref>)   so  that(_k,  γ,   φ_0)satisfies  Assumption <ref>. Letη∈Δseen as  a
perfect vaccination strategy: the SIS model(_k M_η, γ, φ_0)(which  indeed satisfies  Assumption <ref>)  corresponds to  the
initial SIS model,  where forx ∈Ω, a
proportion1-η(x)of the population is vaccinated and thus does not
spread the disease, see  <cit.> and references therein. In
this setting vaccinating  the population amounts to  replace the measureμbyημ.


Motivated by this example, we shall consider the effective
reproduction               number               defined
by:

    R_e(η)  
      =  ρ( γ  M_η) ,

forη∈Δ(notice that(TM_η, γ, φ)satisfies
Assumption <ref> andγ  M_η=(TM_n)_1/γ). 
Following  <cit.>,  we  shall be  interested  in  critical
vaccinationηfor whichR_e(η)  =  1.   It is  observed  in
<cit.>        that         for        the        SIS model(_k,     γ,    φ_0),     the    vaccination     strategyη=φ_0(g^*)is  critical.  We  generalize this result  (with a
shorter  proof  based  on  the  fact  thatR_e(φ(g))=ρ(L_g),
see (<ref>)) for more general  operatorsTand functionsφ.





[Equilibria and critical vaccination]
  Let (T, γ, φ) that satisfy Assumption <ref>. Let h ∈Δ be an equilibrium.  Then we
  have h = g^*  ⟺  R_e(φ(h)) ≤ 1.  If
  furthermore  R_0 > 1, then we have:
  
    h = g^*  ⟺  R_e(φ(h)) = 1.

  First, remark that ρ(L_g) = R_e(φ(g)), where L_g is
  defined by (<ref>).

If  R_0 ≤ 1, then by Proposition <ref>, we have
  g^* =. As R_e(φ(0)) = R_0 ≤ 1 by
  Assumption <ref>, we directly get  the equivalence
  h = g^*  ⟺   R_e(φ(h)) ≤ 1.

  If    R_0    >     1,    then    we    have     g^*≠    by
  Proposition <ref>.
According to Lemma <ref> <ref>, if
h≠,  we have
R_e(φ(h))=ρ(L_h)≥ 1, and by (<ref>) 
and <ref> that if h≠ g^* then ρ(L_h)
≥ρ((L_h)_A) >1 with A=(g^*). 



To complete the  proof, that is, R_e(φ(g^*))=1,  we shall assume
that ρ(L_h)>1 and  show that h≠ g^*.   Informally the idea
is to follow the unstable direction  at the equilibrium h to construct
a trajectory leading to another  equilibrium.  Since L_h is a positive
compact    operator,     thanks    to    the     Krein-Rutman    theorem
(Theorem <ref> <ref>), we can consider an eigenvector
u∈ L^p_+ of L_h related to ρ(L_h).  Since the set
A=(h)                is               invariant                by
Lemma <ref> <ref>, we have:

    ρ(L_h) u  = L_h u = (L_h)_A u + L_h( A^cu).

If uA^c was equal to , ρ(L_h) > 1 would be an eigenvalue of (L_h)_A,  
contradicting Lemma <ref> <ref>. As A
is invariant and uA^c≠, multiplying (<ref>)
by A^c gives ρ(L_h) uA^c  =  (L_h)_A^c (
  uA^c),  showing  that  uA^c is an eigenvector of (L_h)_A^c, so
ρ((L_h)_A^c)= ρ(L_h) >1. 

Since
(L_h)_A^c  =  M_φ(h)  (γ)_A^c =
(T_A^c)_1/γ   (as    h=0   on   A^c and φ(0)=1),    we   may   apply
Proposition <ref>   with   T=T_A^c:  there   exists   a
λ>0    and     w∈    L^∞_+     such    that
T_A^c w - γ w = λ  w.  Without loss of generality, we can
assume  that  w_∞  is  small  enough   to  ensure  that
w  ∈Δ  and, as φ  is continuous  with φ(0)=1,
that φ(  w_∞) ≥  1- δ with  δ>0 small
enough so  that δ(λ+  γ_∞ )  ≤λ.
Note that (w) ⊂ A^c.  Since h+ w = hA + wA^c,
and        A=(h)        is        invariant,        we        get
φ(h+  w) Th = φ(h)  Th.  Since h is  an equilibrium, we
obtain that:

    F(h +  w)
    = φ(h+ w) T(h+ w) - γ (h+ w) 
         =  φ(h+ w)  Tw - γ w 
       ≥   (1-δ) T_A^cw - γ w 
       ≥( λ - δ(λ + γ_∞
        ) )   
        w
       ≥ 0.

By Lemmas <ref> <ref>
and <ref>, this implies that the trajectory starting from h+ w
converges monotonously to an equilibrium g. Since h≤ h+
w ≤ g≤ g^* and w≠, we get that 
h≠ g^* as claimed. 


 §.§ Equilibria and antichains of atoms


We  now  focus  on  the   characterization  of  equilibria.   We  recall
from (<ref>) that the the future  of an antichainof atoms
(which      is     at      most     countable)      is     given      by () = ( ⋃_A ∈ A )= ⋃_A ∈ ( A ).
The  set of
supercritical atoms:

    ={A∈    R_0(A)>1}

is finite by <cit.>.  We  say an antichainof atoms
is supercritical if  all its elements are supercritical  atoms, that is,⊂.  We   denote  bythe   (finite)  set  of
supercritical antichains.   For a setA, let_Adenote the
(possibly empty)  supercritical antichain given by  the maximal elements
of{B ∈    B ⊂A  a.e.}.  Notice that
whenAis  admissible, we  get by (<ref>)  that_Ais
non-empty if and only ifR_0(A)>1.  Forh∈Δ, we simply write_hfor_(h).



The following theorem generalizes the uniqueness result of
Proposition <ref> when the operatorTis not
necessarily quasi-irreducible. Recall that, by Lemma <ref> <ref>,
equilibria are characterized by their support.

[Equilibria and supercritical antichains are in bijection]
  
  Let (T, γ, φ)  satisfy Assumption <ref>.  The set
  of  the equilibria  and the  set  of supercritical  antichains are  in
  bijection through the equivalent relations:

    (g)= ()
        ⟺  _g=  ,

where g∈Δ is an  equilibrium and ∈ a supercritical
antichain. Furthermore,  if g≠,  then the equilibrium  g is
the maximal equilibrium of (_g).


We divide the proof in two lemmas. 


  If g ∈Δ is an equilibrium, then  we have (g) =
  (_g) a.e.. 
  In particular if g and h are two equilibria, we get:
  
    _g=_h
        ⟺  g=h.

  As the set (g) is invariant by
  Lemma <ref> <ref> and as every
  element of _g is included in (g), we have
  (_g) ⊂(g).

By construction of _g, every atom B⊂(g) with R_0(B)>1
is included in (_g).
This implies by (<ref>) that, with A=(g) ∩(_g)^c an invariant (by Lemma <ref> <ref>) and thus admissible set:

    R_0(A) = max_ B⊂ A,   B∈ R_0(B) ≤ 1.

Then Lemma <ref> <ref>
gives that _(_g)^c g is an equilibrium of (_g)^c and
 of A. Then Proposition <ref> <ref> (with
 the SIS model
 (T_A, γ, φ)) implies
 that _(_g)^c   g=, that is (g) ⊂(_g). 
Thus, we get (g)=(_g). The second part of the lemma is
then a
direct consequence of Corollary <ref> and
Lemma <ref>.  

  For any supercritical antichain , there exists an
  equilibrium g∈Δ such that  = _g. If  is non
  empty, then  g≠ is the maximal equilibrium of ().

  If   is  empty, then  taking  the equilibrium  g=, we  get
  =_g.   We   assume  now   that     is  not   empty.   Let
  g be the maximal equilibrium on (). It is also an
  equilibrium   by    Lemma <ref> <ref>   and
  (g)⊂().  For  any A∈, we have  g^*_A ≤ g
  by Lemma <ref>, and g^*_A  is positive on A
  by Proposition <ref> <ref>  since A is
  a  supercritical atom  and thus  an irreducible  set with  R_0(A)>1.
  This implies that A⊂(g) and thus (A) ⊂(g)
  as  (g)   is  invariant.    Then  use (<ref>)   to  get
  ()  ⊂(g),  so   that  ()=(g),  and  thus
  ()=(_g)  by  Lemma <ref>.   The  two  antichains
    and  _g  have  the  same future  and  are  thus  equal  by
  Lemma <ref>. The proof is then complete.


Let⊂Δdenote the set of equilibria of the SIS model(T, γ, φ). 

The map g ↦_g from  to  is one-to-one by Lemma <ref> and onto by Lemma <ref>.
The equivalence given by (<ref>) is a direct consequence of
Lemmas <ref> and <ref>. Use the last
part of Lemma <ref> to get the last part of the theorem. 


 §.§ Monatomicity and order relation via equilibria


Let(T,  γ,   φ)that  satisfy  Assumption <ref>.
Consider the SIS model(T, λγ, φ)with recovery rate γmultiplied  by a real parameterλ>0.   The
reproduction  number  of  a  measurable   setAfor  this  model  isρ((λγ)_A)= ρ(( γ)_A  
M_1/λ )=R_0(A)/λ. We  deduce from Theorem <ref>  that the  number of
equilibria of the SIS model(T, λγ, φ)is decreasing
withλ.



We say that the operatorTonL^pforp ∈(1,
+∞)is monatomic if it has exactly one non-zero atom,
that is,() = 1. 
Monatomicity  is a natural extension of  (quasi-)irreducibility,
see <cit.> and references therein. We complete the
characterization of monatomic  operatorTgiven
in <cit.> using the number of equilibria of the SIS models(T,
λγ, φ). 

[Criterium of monatomicity]
  
Let (T, γ, φ) satisfy Assumption <ref>.
The operator T is monatomic if and only if the two following properties hold:

  * 
  For  λ > 0, the
SIS model (T, λγ, φ)  has at most one
  non-null equilibrium. 

  * 
  There   exists  λ   >   0   such  that   the   SIS model
  (T,     λγ,      φ)     has     a     non-null
  equilibrium. 

 By  Theorem <ref>, we  deduce  that if   is  a
finite    antichain     of    non-zero    atoms    then,     for    all
λ∈ (0, min   _A∈   R_0(A))  there   exists  an   equilibrium
g^λ  for the  model  (T, λγ,  φ) such  that
()=(g^λ).

Then, Point <ref>  means that the number  of finite
antichains    of    non-zero    atoms    is    at    most    two,    and
Point <ref> that it is at  least two: so the two points
are equivalent to the number of  antichains being exactly two (one being
empty), that is T is monatomic.


§ CONVERGENCE AND ATTRACTION DOMAINS



In this  section, we  are interested  in the  behavior of  the semi-flow(ϕ(t,  h))_t∈_+of  Equation (<ref>) whentgoes to infinity for  an initial conditionh ∈Δ.   IfTis a
quasi-irreducible   kernel   positive   operator,  then   according   to
Proposition <ref> <ref>,    see   also
<cit.> whenTis an irreducible kernel operator,
the  semi-flow(ϕ(t,  h))_t∈_+converges   essentially     tog^*ifμ((h) ∩(g^*)) > 0andotherwise.   We   generalize  this  result  to  general
operators, see Section <ref> for a proof.

[Convergence to an equilibrium]
  Let (T, γ, φ) satisfy Assumption <ref>.  The semi-flow
  (ϕ(t,  h))_t∈_+ with initial condition h∈Δ
  converges essentialy to a limit, say g∈Δ; and g is an equilibrium and
  more precisely the maximal
  equilibrium of the set ((h)):
  
    _t→∞ϕ(t, h) = g
      and  _g=_((h))
    .


We derive  directly the  next corollary,  where the  maximal equilibriumg^*is possibly equal to.



   Let  (T, γ,  φ)  satisfy Assumption <ref>.   The
   semi-flow with  initial condition h∈Δ  converges  essentially
   to      the     maximal      equilibrium     g^*,      that     is,
   _t→∞ϕ(t,  h)  =  g^*,  if and only
   if  ((h)) contains all the supercritical atoms.


In                 particular,                we                 recover
Proposition <ref> asR_0≤1means
there  is no  supercritical atom, andR_0>1andTquasi-irreducible
means there is only one supercritical atom.   In  the previous  corollary, it  may however happen that  none  of   the  supercritical  atoms  is included  in(h), see the next example.


  Let Ω  = {a,b}  with the counting  measure, and  consider the
  SIS model (T, γ,  φ) with T identified  with the matrix
  [ 1 0; ⋆ 2 ]  (with ⋆  >  0), γ  =, φ(r)  =
  1-r. Notice that  {a} and {b} are non-zero  atoms, the former
  being  critical   with  ({a})=Ω   and  the   latter  being
  supercritical and invariant.  Thus,  there exists only two equilibria:
  =(0,0)  and  g^*=(0,1/2).  If  h≠,  then we  have
  lim_t→∞ϕ(t,  h) =  g^*.  For  h=(0,1), we
  have   (h)=(g^*),   but   for    h   =   (1,0)   we   have
  (h) ∩(g^*) = ∅.

 Let  (T, γ,  φ)  satisfy 
  Assumption <ref>.  For
 h ∈Δ and g the maximal equilibrium of ((h)), we have   that:

    lim_t→∞(ϕ(t,h)-g) γ_∞ =0.


In particular, whenγ> 0, the convergence given by Theorem <ref> is uniform.
Notice we have a stronger result if furthermoreR_0<1(and thusg=), see 
Remark <ref>. 


  We start with a preliminary result.  Set
  M the  norm of the operator   from L^p  to L^∞, which
  coincide with T on L^∞, see the proof of Lemma <ref>;
  it  is  finite  by  Assumption <ref>.  Let m>0. Let  h_1≥  h_2  be
  elements     of    Δ,     and    for     t∈_+     set
  f(t) = ϕ(t, h_1) - ϕ(t, h_2). We claim that:

    lim sup_t→∞f(t){γ≥ m} _∞≤M/m lim sup_t→∞f(t)_p.

Indeed, by monotonicity of the semi-flow (see Lemma <ref>),
we have ϕ(t, h_1) ≥ϕ(t, h_2) and thus f(t) ≥. We
also have, as φ is decreasing  on [0,1] and T is positive
that for t∈_+:

    f'(t)
         =  φ(ϕ(t, h_1)) Tϕ(t, h_1) - φ(ϕ(t, h_2))
        Tϕ(t, h_2) - γ f(t)
       ≤φ(ϕ(t, h_1)) Tf(t) - γ f(t)
       ≤  Tf(t) - γ f(t).

On {γ≥  m}, we get for v(t)=mt f(t) that  for all t ≥ 0:

    v'(t) ≤ (m - γ)  v(t)+
    T v(t) ≤ ι  v(t)_∞≤ M v(t)_p .

By (<ref>)   and (<ref>)   on   the   Bochner
integral,                 we                 deduce                 that
v(t)  ≤ v(0)  +  M ∫_0^t  v(s)_p   s on {γ≥  m}.  Since f  is
nonnegative, we get:

    f(t){γ≥ m}_∞≤-mth_1 -h_2_∞
      + M ∫_0^t -m(t-s)f(s)_p  t
      ≤  2 -mt 
      + M ∫_0^t -m sf(t-s)_p  t.

This gives (<ref>). 







Let   h  ∈Δ  and   g   be  the   maximal  equilibrium   of
A=((h)).       By     Lemma <ref>,      we     have
(ϕ(1,h))   =  A.    Let   h_1  =   max(ϕ(1,h),  g)   and
h_2  =   min(ϕ(1,h),  g).    We  thus   have  h_1≤   h_2  and
(g)=(h_2)⊂(h_1)=A. By Theorem <ref>
(and using  that g  is the  maximal equilibrium  on the  invariant set
((h_i))      for     i=1,      2),      we     get      that
_t→∞ϕ(t,  h_i)=g for  i=1, 2.   With
f(t)  =  ϕ(t,  h_1)  -  ϕ(t,  h_2),  we  get  by  the  dominated
convergence theorem that  lim_t→∞f(t)_p=0,
and                       by (<ref>)                       that
lim_t→∞f(t){γ≥
    m}_∞   =0 for all m>0 and thus
lim_t→∞ f(t) γ_∞   =0 as
f_∞≤  1.   Use   the
monotonicity        of        the         semi-flow        to        get
ϕ(t, h_1) ≥ϕ(t+1, h) ≥ϕ(t, h_2) and deduce
that (<ref>) holds. 


§ SIS MODEL WITH AN EXTERNAL DISEASE RESERVOIR



In  this section,  we  consider  the infinite-dimensional  inhomogeneous
SIS  model  with an  external  disease  reservoir, called  SISκmodel,   presented   in  Section <ref>.    The   functionu=(u(t,x))_t∈_+, x∈Ω,  whereu(t,x)is the proportion
of  infected  population  among  the population  with  featurex,  is
solution inL^∞of the ODE:

    {[ u' = F_κ(u),;    u(0) = h, ].

with initial conditionh∈L^∞and:

    F_κ(u)=  φ(u) (Tu+κ) - γ u,

whereφis    a   continuous    function   onandκ∈L^∞_+. 
To   study    solutions   of (<ref>)    and   the
corresponding equilibria,  that is  functionsg  ∈Δsuch thatF_κ(g)    =    0,    we    shall    use    the    formalism    of
Section <ref> by adding  a sub-population corresponding to
the reservoir  with type$̊.  Notice the case  φ([0, 1])={0}
(which is  not possible under Assumption <ref>)  is trivial, and
thus   we  shall   assume  there   exists  a∈   (0,  1)   such  that
φ(a)>0.



We  set  Ω_=̊Ω⊔{}̊ (assuming  without  loss  of
generality that ̸̊∈Ω),  _=̊σ(, {}̊) and
μ_$̊ a measure  on(Ω_,̊ _)̊which coincides  withμonΩand with positive  finite weight  on$̊.  For  a function
f_$̊ defined onΩ_$̊, we simply write f  for its restriction
to Ω (and  similarly, for a function f defined  on Ω, we
write f_$̊ for a function  defined onΩ_$̊ which coincides with
f on Ω,  the value of f_$̊ on$̊  being given when needed).
We  simply write  L^p_$̊  forL^p(Ω_,̊ _,̊  μ_)̊,
wherep∈[1,  +∞].  We define the positive  operatorT_$̊ on
L^∞_$̊, as an  extension ofTonΩ_$̊,
by:

    T_f̊_(̊x)= Ω(x)   Tf(x) 
      + f_(̊)̊ α_(̊x),

with  α_∈̊(L^∞  _)̊ _+  such that  α=κ/a on
Ω,  with  a∈  (0,  1), and  α_(̊)̊=b>0. For p ∈ (1,+∞), we  define
similarly  the  operator  _$̊  onL^p_$̊ based on the operator  on L^p, see Lemma <ref>.   We  also  define  the
function  γ_$̊  (which coincides  withγonΩby
definition)   such  thatγ_(̊)̊=   bφ(a) is assumed
to be positive (that is,φ(a)>0).    LetΔ_=̊{f_∈̊(L^∞_)̊_+     1-f_∈̊(L^∞_)̊_+}be the analogue ofΔforΩ_$̊. It is elementary
to check the following result.

  
  Let  (T, γ,  φ) satisfy  Assumption <ref>. Assume
  furthermore there exists  a∈ (0, 1) such  that φ(a)>0, and
  let       κ∈       L_+^∞.        A       function
  (u(t,    x))_t∈_+,    x∈Ω    is    a    solution
  to (<ref>)  related  to the  SISκ  model
  with  initial condition  h∈Δ  if  and only  if the  function
  (u_(̊t, x))_t∈_+, x∈Ω_, where u_(̊t, )̊=a for
  all t∈_+, is  a solution to (<ref>) related to
  the  SIS  model  with  parameter  (T_,̊γ_,̊φ)  on
  Ω_$̊ and  with initial conditionh_∈̊Δ_$̊ such that
  h_(̊)̊=a.



 We shall consider the supercritical atoms out of the individuals
infected by the reservoir:

    _=̊{A ∈   A∩((κ))=∅ a.e.}.

Based on  Theorems <ref> and <ref> for
the (T_,̊γ_,̊φ) SIS  model we can give a representation
of the  equilibria of SISκ  model, that  is, of the  solutions to
F_κ(g)=0  in  Δ,  prove that the equilibria are
characterized by their support, and explicit  their  attraction  domain.   For  a
function  h∈  L^∞  _+,  we   shall  denote  _,̊  h  the
  antichain    given   by   the   maximal    elements   of
{B   ∈_    B   ⊂((h))
   a.e.}. Notice that Assumption <ref> implies that
φ is positive on [0, 1). 


  
  Let (T,  γ, φ)  satisfy Assumption <ref>
  and κ∈ L_+^∞.
  The set of equilibria and the set of antichains in _$̊ are in
  bijection through the equivalent relations:

    (g) = ()∪((κ))   a.e.  ⟺  _,̊g =,

where⊂_$̊ is an antichain and g∈Δ an
equilibrium, that is, F_κ(g)=0. 

Furthermore,   the  semi-flow   (ϕ(t,   h))_t∈_+   solution
of (<ref>) with  initial condition  h∈Δ
is well  defined and it converges  a.e. to a limit,  say g∈Δ;
and g is an equilibrium and more precisely:
  
    _t→∞ϕ(t, h) = g
      and  _,̊g=_,̊ h
    .




  
We deduce the following  properties under the hypothesis of
Corollary <ref>. 
  
   
  * Notice that   _$̊ is empty if and only if 
      there exists a unique equilibriumg∈Δfor the
      SISκmodel. In this case,
     we have(g) = ((κ))a.e.     and_t→∞ 
     ϕ(t, h) = gfor allh∈Δ.
   
  *  Ifκ>0, then_$̊ is empty.
     
  * If g is an equilibrium  for the
      SISκ model, then g is positive on ((κ))
      and thus on (κ).  
  



If A∈ is  invariant for  , then  A seen  as an  element of
_$̊  is  also  invariant  for_$̊.  Furthermore,  the reservoir
{}̊ is an atom of μ_$̊ and, asΩis_$̊-invariant, we
get that {}̊ is a  _$̊-atom.  We deduce that
a  setB∈_$̊  is  admissible  for  _$̊  if   and  only  ifB ∩Ωis  admissible for.
In particular a setAis an atom of_$̊ if and only if either
A={}̊ or A ⊂Ω and A is an atom of .
We  denote by  _(̊A) the
future of a set A⊂Ω_$̊ with respect to_$̊. Notice that
_(̊A)=(A) for any A⊂Ω and that the future of the
reservoir is:

    _(̊{}̊)={}̊∪((κ)).




Recall the  basic reproduction number R_0(A)
of a measurable set for the SIS model (T, γ, φ) 
given in Section <ref>. We simply write R_(̊A) when
considering the  basic reproduction number of A ∈_$̊ for the 
 SIS model(T_,̊ γ_,̊ φ). Since(T_)̊_A=T_AforA⊂Ωmeasurable, we deduce thatR_(̊A)=R_0(A)for any A⊂Ωmeasurable.
We also have:

    R_(̊{}̊)=α_(̊)̊/γ_(̊)̊=1/φ(a)·

Note that under  Assumption <ref>, we haveφ((0, 1))=(0,
1)and thus the atom{}̊is super-critical for the(T_,̊
γ_,̊ φ)SIS model.





We  deduce from  Proposition <ref>.  that  a functiong∈Δis an equilibrium for the SISκmodel if and only ifg_∈̊Δ_$̊,  such that g_(̊)̊>0  is an equilibrium  for the
(T_,̊γ_,̊φ)  SIS  model on  Ω_$̊.  Notice  that
necessarilyg_(̊)̊=a.
The equilibria of the(T_,̊
γ_,̊ φ)SIS model 
whose support contains the reservoir{}̊are according to
Theorems <ref> in bijection will all the
supercritical antichains containing the atom{}̊. Those
supercritical  antichains
are exactly the antichains of_$̊ with the atom {}̊ added
to them.
Then, use (<ref>) and Theorems <ref>
and <ref> to conclude.



abbrv





§ A SHORT REMINDER ON INTEGRATION, DERIVATION AND ODE IN  BANACH
  SPACE

 



This section is devoted to the  definition and properties of the Bochner
integral, the  differentiation and differential equations in
Banach spaces. We consider  (X, ·) a real Banach space. 




 §.§ Integration in Banach spaces


We give a  short summary on the Bochner integral,
and  refer to  <cit.> and  <cit.> for  a more
detailed presentation.
We  consider the  Borel  σ-field  on   and  write
ν(  t)=  t  for  the Lebesgue  measure. Let I  be an interval of  .  A function
f   I ↦ X   is simple  if f= ∑_k=1^n  a_k A_k
  where n∈,  the a_k's belong to X and  the A_k's are Borel
  subsets of  I with finite  Lebesgue measure.  We define  its Bochner
  integral as:

    ∫ _I f ν
      = ∑_k=1^n a_k  ν(A_k).

Notice  the  integral  belongs  to  X  and  does  not  depend  on  the
representation    of   the    simple    function    f.   A    function
f    I  ↦  X  is  Bochner  measurable  (simply  called
measurable  in <cit.>) if  there exists  a sequence  of
simple functions (f_n)_n∈, with f_n    I ↦ X,
such that ν-a.e. lim_n→∞ f_n =f  (that is,
lim_n  →∞f(t)  -f_n(t)=0 t-a.e. on
I); it  is  furthermore  Bochner integrable  if  one  can find  such
approximating      sequence      (f_n)_n∈      so      that
lim_n  →∞∫  _I f(t)  -  f_n(t) 
t=0. In this case the Bochner integral of f is defined as:

    ∫ _I f ν
      = lim_n→∞∫ _I f_n ν
    ,

 where  the limit  holds in  the  Banach space.  The Bochner  integrable
 ∫ _I  f ν does not depend  on the approximating sequence
 (f_n)_n∈; we shall also denote it by ∫_I f(t)  t. 
Thanks to <cit.> X-valued continuous
function are Bochner measurable and    a.e. limits of 
 Bochner measurable function  are Bochner measurable. 
 According to <cit.>, a function f   I ↦ X is Bochner integrable if and only
 if it is Bochner measurable and f   I ↦_+
 is integrable; in this case we have:

    ∫ _I f ν≤∫ _I f ν.
 
When I̅=[a,b] with -∞≤ a<b≤ +∞  and f is
Bochner integrable on I, we simply 
write the Bochner integral as:

    ∫ _I f  ν=∫_a^b f ν.

 The Bochner integral enjoy many properties as the usual Lebesgue
 integral, as such we shall use the dominated convergence, see
 <cit.>, which we recall. 


[Dominated convergence theorem]
  Let I be a non-empty interval of  .  Let (f_n)_n ∈ be a
  sequence  of   Bochner-integrable  functions  defined  on   I  which
  converges   ν-a.e. to f   I →  X. Assume
  there        exists        a       Lebesgue-integrable        function
  g    I →_+ such that f_n≤ g ν-a.e.  for all n ∈.  Then the function f is Bochner-integrable and:

    ∫_I f  ν = lim_n→∞∫_I f_n ν  in X.







 §.§ Differential equations in Banach spaces


We now consider the derivation of  functions on a Banach space.  Let I
be  an interval  of   with non-empty  interior.  We  say that  a
function 
f      I  ↦  X is  differentiable  at a  point
t   ∈   I    if   the   following   limit,    f'(t),   exists   in
(X, ·):

    f'(t)= lim_s → 0
    t+s∈ If(t+s) - f(t)/s·

Notice that  if f is differentiable  at t, then it  is continuous at
t.  We say  that f is differentiable on I  if it is differentiable
at  any point  of  I, and  that  f  belongs to  ^1(I)  if it  is
differentiable  on I  and  f' is  continuous on  I.
We   have   the  following   fundamental   theorem   of  calculus,   see
<cit.>     and    <cit.>.  Assume  I=[a,b] with  -∞ <a<b<+∞ and
that f   I → X belongs to 𝒞^1(I), then f'
is Bochner-integrable on I and we have:

    f(b) -f(a)= ∫_a^b f' ν.
        





We now recall  some results on differential equations  in Banach spaces.
Let  F   X → X be locally-Lipschitz, that is,  for
all x ∈ X, there exists η > 0 and C finite such that for all
y ∈ X, we have:

    x-y≤η  ⟹  F(x) - F(y)≤ C
      x-y.

The  Picard-Lindelöf  theorem, see <cit.>, 
 ensures  the   existence   of
 (u,τ), with τ∈ (0,  +∞ ] and u∈^1([0, τ)) taking values in X,
 that  is a solution to  the Cauchy
problem:

    {[ u' = F(u),;  u(0) = x, ].

where the first equality in (<ref>)  holds in [0, τ) and
x∈ X is the so-called initial condition, and furthermore the solution (u,τ)
is  unique and  maximal (that  is, if  (u',τ') is  another solution
to (<ref>),   then    τ'≤τ   and    u'=u   on
[0, τ')).
We say the solution is global 
if  τ=+∞. 






We    end   this    section    with   a    comparison   theorem.     Let
(X,   ·,   ≤)   be    a   real   Banach   lattice. Let
D_1,  D_2   ⊂  X.   A  map   F    X  ↦   X  is
cooperative    on     D_1    ×    D_2    if     for    any
(x,y) ∈ D_1 × D_2 with x ≤ y and any ν∈ X^⋆_+,
we have:

    ⟨ν, x-y ⟩ = 0   ⟹  ⟨ν,
    F(x) - F(y) ⟩≤ 0.

We recall <cit.>.


  Assume that X_+ has non-empty interior. 
Let F     X → X be  locally-Lipschitz, D_1, D_2 ⊂ X and τ∈ (0, + ∞].
Let u     [0, τ) → D_1 and v     [0, τ) → D_2 be two 𝒞^1 maps.
Suppose that F is cooperative on D_1 × X or on X × D_2,
that  u(0) ≤ v(0), and that  u'(t) - F(u(t)) ≤ v'(t) -
F(v(t)) for all t∈ [0, τ).
Then, we have u(t) ≤ v(t) for all t ∈ [0, τ).





 

§ EXISTENCE AND REGULARITY OF THE SEMI-FLOW FOR THE SIS MODEL





We prove Proposition <ref>  and
Lemma <ref> 
in this section assuming
that  (T,  γ,  φ)  satisfies  Assumption <ref>. 
The
results and proofs are very close to those in <cit.>. 



 §.§ Existence of the semi-flow


Recall that Assumption <ref> holds with the function F defined
in (<ref>)  by F(h) = φ(h) Th - γ h    for h
∈ L^∞. Recall also the set  Δ={f∈ L^∞  ≤ f≤}. 


  
  Let F be the function  from L^∞ to
  L^∞ defined by (<ref>). 
 
 
  * 
   The function F is  locally-Lipschitz on
     (L^∞, ·_∞).
 
  * 
There exists a finite constant C_p such that for g,h∈Δ, we
have:

    F(g)-F(h)_p≤  C_p g-h_p.


   
  *   Let  (h_n)_n∈  be  a  monotonous
     sequence   of    elements   of   Δ.   Then    the   sequence
     (F(h_n))_n∈ converges   a.e. to  F(h),
     where h is the  a.e. limit of (h_n)_n∈.
 





Since φ is locally Lipschitz, we denote by K_r the
corresponding (finite) Lipschitz constant of φ on [-r, r] and
M_r=sup_[-r, r] |φ|≤  |φ(0)| + rK_r.

We prove Point <ref>. 
Let r>0 and u, v ∈ L^∞ with u_∞≤ r and
v_∞≤ r.  We have:

    F(u) - F(v)_∞    = φ(u)Tu - φ(v) Tv - γ (u-v)_∞
       ≤φ(u)_∞T(u-v)_∞
        +φ(u) -φ(v)_∞  v_∞
        + γ_∞ u-v_∞
       ≤ (M_r T_L^∞ + K_r r _L^∞  +
       γ_∞ ) u-v_∞   .

This concludes the proof of Point <ref>. 



We prove Point <ref> in a similar way.
Let u, v ∈Δ. We have:

    F(u) - F(v)_p    = φ(u)Tu - φ(v) Tv - γ (u-v)_p
       ≤φ(u)_∞T(u-v)_p
        +φ(u) -φ(v)_p T v_∞
        + γ_∞ u-v_γ^p,p
       ≤(M_1 _L^p + K_1 T_L^∞  + 
        γ_∞) u-v_p   .

This concludes the proof of Point <ref>. 




For simplicity,  we assume  that (h_n)_n  ∈  is non-decreasing.
Thus it  converges a.e. (that  is, in L^0) to  a limit, say  h, and
this limit belongs to Δ.  Since T is positive, we also get that
the  sequence  (Th_n)_n ∈  is  non-decreasing and  bounded  by
T∈ L^∞, thus it converges  a.e. (that is, in  L^0) to a
limit,  say  w∈  L^∞.    On  the  other  hand,  by  dominated
convergence, we  also get that  (h_n)_n ∈ converges to  h in
L^p, and  thus, as  is  bounded on L^p,
we   get   that    ( h_n)_n   ∈   converges    to   h   in
L^p. We thus deduce that  w= h= Th. Then use that φ
is continuous,  to deduce that  (F(h_n))_n∈ converges  a.e. to
F(h).  This gives Point <ref>.



We now prove that F is cooperative, see (<ref>),  
using that  φ  is  non-negative  on  [0,  1].


  The map F is cooperative  on Δ× L^∞ and on
  L^∞×Δ.





We first prove  that F is cooperative on Δ× L^∞.
Let u∈Δ and v∈ L^∞ with  u ≤ v.
Let ν∈ L^∞, ⋆_+ such that ⟨ν, u-v ⟩ = 0. 
Since v-u ≥ 0, we deduce that
for any h∈ L^∞:

    ⟨ν,  (u-v)h ⟩ = 0
 (see <cit.> for a
proof in a very similar setting). 
Then, using (<ref>)
with h=γ, we get: 

    ⟨ν, F(u) - F(v) ⟩
      = ⟨ν, φ(u) Tu - φ(v) Tv - γ (u-v)
      ⟩
      = ⟨ν, φ(u) Tu - φ(v) Tv 
      ⟩
      .
 
For s, t∈_+, we set Φ(s,t)=(φ(s) -φ(t))/(s-t) if
s≠ t and Φ(s,s)=0. We have:

    φ(u) Tu - φ(v) Tv = φ(u) T(u-v) +   (u-v) h
        with  
      h=Φ(u,v)   Tv.

As φ is locally-Lipschitz by Assumption <ref>
and as u,v and Tv belongs to L^∞, we deduce that h∈
L^∞. We deduce from (<ref>) that:

    ⟨ν, F(u) - F(v) ⟩ = ⟨ν, φ(u) T(u-v)
      ⟩ .

As we have φ≥ 0 on [0, 1] by Assumption <ref> and
u ≤ v, we have φ(u) T(u-v) ≤ 0. Thus, as ν is a positive linear form on L^∞, we have ⟨ν, F(u) - F(v) ⟩≤ 0.
Therefore the map F is cooperative on Δ× L^∞.

If (u,v)∈ L^∞×Δ
satisfy u≤ v, then using similar computations with h=Φ(v,u) T
u, one get  instead of (<ref>) that
 ⟨ν, F(u) - F(v) ⟩ = ⟨ν, φ(v) T(u-v)
 ⟩. Similar arguments yields then that
  F
is also cooperative on L^∞×Δ.






Mimicking  the proof  of <cit.>  (which in
particular  uses  that 
φ(1)=0),  we get  that any  solution of (<ref>)
with an initial condition in Δ remains in Δ.



  
The domain Δ is forward invariant for the differential equation
u'=F(u) in L^∞. 




We then conclude on the existence of global  solutions in Δ. 


  Any  maximal  solution  of  u'=F(u)  in  L^∞  with  initial
  condition u(0)=h∈Δ is global.



  The                  bounded                 open                  set
  U  =  {f  ∈  L^∞  f_∞  <  2  }  of
  L^∞ contains Δ,  and the map F is  Lipschitz  on U by
  Lemma <ref> <ref>.   As  the  set  Δ  is
  forward     invariant     by    Lemma <ref>,     one     can
  apply <cit.> to  conclude that any maximal solution
  to u'=F(u) with initial condition in Δ is global. 



Under  Assumption <ref>,   using  Picard-Lindelöf   theorem <cit.>  and
Lemma <ref>, which ensure the existence and uniqueness of
maximal   solution    to   u'=F(u) in L^∞ with initial
condition in Δ,    we   can   define    the   semi-flow
ϕ  _+  ×Δ→Δ,  where  the
L^∞-valued  function ϕ(·, h)=( ϕ(t,  h))_t∈_+
is   the  global   solution  to (<ref>)   with  initial
condition u_0=h∈Δ.  Notice  that ϕ(·, h)  belongs to
^1(_+) and statisfies the semi-group property:

    ϕ(t+s, h)=ϕ(t, ϕ(s, h))
        for all  
    s,t ∈_+   and  
    h∈Δ.





 §.§ Properties of the semi-flow

We now establish the following properties of the semi-flow. 
We stress in the next lemmas that Assumption <ref> holds. 


Let (T, γ, φ) satisfy Assumption <ref>. 



  *   If  h_1 ≤  h_2  belong  to
  Δ,  then  we  have   ϕ(t,h_1)  ≤ϕ(t,h_2)  for  all
  t ∈_+.

  *   Let  h ∈Δ.   The  function
  t ↦ϕ(t,h)  from ^+ to L^∞ is non-decreasing
  (resp.  non-increasing) if and only if we have F(h) ≥ (resp.
  F(h) ≤) in L^∞.
  
  * Let t∈_+. The function h↦ϕ (t, h)
    defined on Δ is Lipschitz with respect to
    ·_∞.
  
  *     Let   t∈_+.    The   function
    h  ↦ϕ  (t, h)  defined  on Δ  is continuous  with
    respect to the a.e. convergence, and, more generally,
    if (h_r)_ r∈_+ is a sequence of elements  of Δ such
    that  h=  _r→ +∞  h_r  exists (and  thus
    belongs               to               Δ),               then
    _r→ +∞ϕ(t, h_r)=ϕ(t, h).




  For   all  the   Points   but <ref>,   the  proof   mimic
  respectively the  proofs of Propositions  2.8, 2.10 and  2.11 (ii) from
  <cit.>.   Following  the  proof of  Proposition  2.11 (iii)  in
  <cit.>,  we see  that to  get Point <ref>,  it is
  enough to check the following claim:
   if (h_n)_n∈ is a monotonous sequence of
  elements of Δ, which thus  converges a.e. to a
  limit, say  h∈Δ,  then (ϕ(t, h_n))_n∈ converges
  also a.e. to  ϕ(t,h) for all t≥ 0.

  For simplicity, we assume
  that  the   sequence  (h_n)_n∈  is   non-decreasing.   From
  Point <ref>,   we  get   that  the   sequence
  (ϕ(s, h_n))_n∈ is also non-decreasing  and thus converges a.e.  to  a  limit   say  f_s∈Δ  for  all
  s∈_+.  We deduce from Lemma <ref> <ref>
  that  (F(ϕ(s, h_n)))_n∈  converges a.e.  towards F(f_s). Since  F is bounded on Δ (as  it is locally
  Lipschitz on L^∞) we deduce that the convergence also holds in
  L^p.      Since      the     identity      map     from
  (L^∞,         ·_∞         )         to
  (L^p, ·_ p   ) is continuous, we
  deduce  that   a  solution  to (<ref>)  in
  L^∞   is  also  a  solution   to (<ref>)  in
  L^p.  By  the  fundamental Theorem  of calculus,
  see (<ref>),   we   get  that   for  all
  s≥ 0:

    ϕ(s,h_n) = h_n+ ∫_0^s F(ϕ(r, h_n))  r
       and  ϕ(s,h) = h+ ∫_0^s F(ϕ(r, h))  r
        hold in  L^p.

By the dominated  comvergence Theorem <ref> (with X=L^p and
g the constant function on _+ equal to 1), we
deduce that  (f_s)_s∈_+ is Bochner integrable on bounded
intervals of _+ and  for all s≥ 0:

    f_s = h+ ∫_0^s F(f_r)  r
        holds in  L^p.

We deduce from (<ref>) and
Lemma <ref> <ref> that for all s≥ 0:

    f_s - ϕ(s,h)_ p ≤  C_p ∫ _0 ^s   f_r - ϕ(r,h)_ p   r.

Since f_r and ϕ(r,h) belong to Δ, we get that f_r -
  ϕ(r,h)_ p ≤  2, so that r↦f_r -
  ϕ(r,h)_ p is locally ν-integrable. 
We deduce from the Grönwall's inequality that f_s -
  ϕ(s,h)_ p =0 for all s≥ 0. This gives that
f_s=ϕ(s,h) for all s≥ 0, which proves the claim.



Following <cit.>, we prove that  the limit of
the semi-flow is an equilibrium.


Let (T, γ, φ) satisfy Assumption <ref>. 
Let h ∈Δ. If _t → +∞ϕ(t, h)
exists, then it belongs to Δ and is an equilibrium. 




  Let h^*=_t → +∞ϕ(t, h). 
By Lemma <ref> <ref> and
by (<ref>), we have for all s ∈_+ that:

    ϕ(s, h^*)=ϕ(s, _t→ +∞ϕ(t, h))
      = _t→ +∞ϕ(s,  ϕ(t, h))=
      _t→ +∞ϕ(t, h)= h^*.

Then use Lemma <ref> <ref> to get that
F(h^*)=0. 





 §.§ Proof of Proposition <ref>  and
Lemma <ref>



The solution to Equation (<ref>)  in L^∞ with
initial condition in Δ is
given by the semi-flow ϕ, see Section <ref> and
Lemma <ref> therein. This gives
Point <ref>. Point <ref> is
Lemma <ref>. 





Since F() = φ(1) T() - γ = - γ≤ 0 by
Assumption <ref>, we get  by
Lemma <ref> <ref> that  the semi-flow t
↦ϕ(t, ) is non-increasing. This implies that
g^*=_t→ +∞ϕ(t, ) exists. By
Lemma <ref>, we get that g^* is an equilibrium. 
Let h ∈Δ be an equilibrium.
We have h ≤, thus by
Lemma <ref> <ref> we have h =
ϕ(t,h) ≤ϕ(t,) for all t ≥ 0. Taking the essential
limit, we get  h ≤ g^*. 
This  gives Point <ref>.





  For h ∈ L^∞ and i=1,2, let
  F_i(h) = φ_i(h) T_i(h) - γ_i h and let ϕ_i be the
  semi-flow of Equation (<ref>) with the parameters
  (T_i, γ_i, φ_i).  By assumption, we have
  F_1(g^*_2) ≥ F_2(g^*_2) =  0. Thus, by
  Lemma <ref> <ref>, the semi-flow
  t ↦ϕ_1(t, g^*_2) is non-decreasing.  By
  Lemma <ref>, since the  essential limit 
  g=_t→ +∞ϕ_1(t, g^*_2) exists, it belongs to
  Δ and is an equilibrium (for the parameters
  (T_1, γ_1, φ_1)).  As g^*_1
  is the maximal equilibrium, we have g^*_1 ≥ g, and thus 
  g^*_1 ≥ g^*_2 as the semi-flow t ↦ϕ_1(t, g^*_2) is
  non-decreasing.




§ PROOF OF PROPOSITION <REF>





 §.§ On the support of the semi-flow




The following lemma is a generalization of <cit.>
where  T is assumed to be  irreducible.


  Let (T, γ, φ)  that satisfies Assumption <ref>
  with φ(0)>0. Let
  h ∈Δ.  We have (ϕ(t,  h))=((h)) a.e. for
  all t > 0.




  Since φ(0)>0,  there exists
  a,η∈ (0,  1) such that  a-φ(r)<0 for all r∈  [0, η].
  Notice  the   operator  Q=aT  -γ+  γ_∞  on
  L^∞  is positive and that  the invariant sets for  Q and T
 are the same.
 Set f=η h/2  and A=(h)=(f).  There
  exists c > 0 small enough such that for all t ∈ [0,c]:

    exp(at T_L^∞ )< 2.

Set for  t ≥ 0:

    u(t) = t(a  T-γ) f= - γ_∞  t tQ f.

By <cit.>  applied to the operator Q,  we get that
(u(t))=(u(t)γ_∞      t)=(A)    for    t>0.
Differentiating (<ref>) leads to:

    u'(t) - F(u(t)) = (a - φ(u(t))) T u(t).

We deduce from (<ref>)
and (<ref>) that
u(t)_∞ <η for  t∈ [0, c], and thus, by definition
of a and  η, that u'(t) - F(u(t))≤ 0 for  t∈ [0, c]. Then, since
u(0)=f≤  h, 
Theorem <ref> implies that u(t)≤ϕ(t, h) 
for  t ∈ [0,c], and thus (A)⊂(ϕ(t, h)) for
t∈ (0, c]. Then, use the semi-flow equation (<ref>) to
propagate the result to all t>0.

We deduce from Lemma <ref> <ref> that
(ϕ(t,h)) =(ϕ_(A)(t,h)) ⊂(A) for all t≥
0. This gives that  
(ϕ(t,h)) = (A) for all t> 0. 






 §.§ Preliminary results on the spectral radius and bound




We refer to <cit.> for results on the spectral bound
on    Banach     lattices    defined     by (<ref>).   



  
  Let (T, γ, φ)  that satisfy Assumption <ref>. Let
  δ :  Ω→ be  a measurable  positive function
  with  δ≥γ  and  δ  >  0.   Then  the
  quantities s(T - δ), s( - δ),
  ρ()-1 and  ρ(δ)-1   have
  the same sign.




  Notice     that    (T,     δ,    φ)     also    satisfies
  Assumption <ref>.  By  Lemma <ref> <ref>  with
  γ      replaced     by      δ,     we      have     that
  ρ(δ)=ρ().   So  it  is   enough  to  prove  that
  s( -  δ) and ρ(δ)-1  have the same  sign, and
  that s(T  - δ) and  ρ()-1 have the same  sign.  This
  can be done by  mimicking the proof of <cit.>
  based on <cit.>, noticing that the cone L^p_+ is normal and
  reproducing for p∈ [1, +∞  ], and that, by Lemma <ref>,
  the operators δ and  
are respectively 
 compact 
on  L^p and   power compact  on L^∞, and that the  linear
maps
-  δ   and  T   -  δ  are operators
respectively on 
L^p and
L^∞. 


Adapting the proof of <cit.> on kernel operators, we provide
a weaker link between ρ ( γ) -1 and s( -
γ) without the condition γ > 0. 


Let (T, γ, φ) that satisfy Assumption <ref>.
Then the following assertions are equivalent:


  * 
  s( T - γ) > 0 or equivalently s( - γ) > 0. 
	
  * 
          ρ () > 1 or equivalently  ρ()>1.
	
  * There exists λ > 0 and w ∈ L^∞_+ such that we have Tw - γ w = λ
          w.  




  Recall Assumption <ref> holds and p∈ (1, +∞ ).  Since
  s(A-(γ+ε))=s(A     -γ)      -ε     for
  ε∈  and A  equal to  T or  , we  deduce from
  Lemma <ref>     that    the     two    conditions     in
  Point <ref>  are  equivalent.   We  also  deduce  from
  Lemma <ref> <ref>   that   the   two   conditions   in
  Point <ref> are equivalent. So, we shall only consider
  the second ones.  It is immediate that Point <ref>
  implies  Point <ref>  as  L^∞⊂  L^p  and
  T and  coincide on L^∞.



We       assume        Point <ref>       and       prove
Point <ref>.    For   any   a∈_+,   we   denote
ψ(a)  =  ρ(V_a)  with V_a=(γ+a).   Notice  that
V_a=  M_1/(γ+a) for a>0. By
Assumption <ref>,  the operator  V_a on L^p is positive
 and that V_a ≥ V_b for 0≤ a≤ b.  Thus the
map ψ is non-increasing on _+ by (<ref>).
By Point <ref>, there exists ε > 0 such that
s(   -  (γ   +  ε))   =s(   -  γ)
-ε>  0,   therefore  we  have  ψ(ε)   >  1  by
Lemma <ref> applied  to δ  = γ  + ε.
We   thus   get   ψ(0)   =  ρ(γ)   >   1,   that   is
Point <ref>.



We     assume     Point <ref>    and     prove     Point
<ref>.  By   Point <ref>,  we  have
ψ(0)    >    1.     As    for    all   a    >    0,    we    have
ψ(a) ≤V_a_L^p≤ a^-1 _L^p. 
  We  deduce  that 
lim_a→∞ψ(a) =0. 




We now prove that ψ is continuous on _+.
Let B denote the unit ball
of L^p. Notice that M_1/(γ+a)(B) ⊂ M_1/γ
(B) for a∈_+ and thus ⋃_a∈_+
V_a(B)=V_0(B)=γ (B) is relatively compact in L^p, and thus the family
(V_a)_a∈_+ is collectively compact.
Thanks to Lemma <ref>,  the continuity of ψ holds if 
lim_|a-b| → 0(V_a -V_b)f_p=0 
for any f∈
L^p. This is indeed the case as, for f∈ L^p, we have:

    (V_a -V_b)f_p=γ((b-a)γ/(γ+a)
        (γ+b) f)_p
    ≤γ_L^p(b-a)γ/(γ+a)
        (γ+b) f_p,

and the right members goes to 0 as |a-b| goes to 0 
using  |b-a|γ/(γ+a)
(γ+b) ≤  1 and  dominated convergence. In conclusion, the
function  ψ is continuous on _+. 

Since
ψ(0) > 1 and 
lim_a→∞ψ(a) =0, we deduce from  the continuity of ψ, 
that 
there exists  λ > 0 such that ψ(λ) = 1.  Thus by
the Krein-Rutman Theorem <ref> <ref>  applied to the positive compact
operator V_λ on L^p, there exists
v ∈ L^p_+ such that 
V_λ v = v. Thanks to (<ref>), we have 
v_∞ =V_λ v_∞≤T(v/γ)_∞≤ C'_p  v_p, 
we deduce that v
belongs also to L^∞. 
Setting 
w = v/(γ + λ)∈ L^∞ _+, we get that 
w - γ w = λ w.  As  and T coincide on L^∞, we get 
Point <ref>.







 §.§ Proof of Proposition <ref> <ref>

Let g be  a non-zero equilibrium. 
By
Assumption <ref>, φ(g) < 1 on (g) (as φ(r)
<1 for r∈ (0, 1]).
Recall
 the operator S=γ is compact on L^p,
 see Lemma <ref> <ref>. Since g∈Δ is an
 equilibrium, we obtain that:

    S (γ  g)  =  γ   g/φ(g)> γ g
        on  (g).

We deduce from 
Lemma <ref> <ref>, with λ=1, 
and (<ref>) that
R_0 =ρ(S)≥ρ(S_(g))
>1.   In other  words, R_0≤  1 implies
that  is the only equilibrium. The last part of Point <ref>
is a consequence of
Proposition <ref> <ref> and the
monotonicity of the semi-flow from
Lemma <ref> <ref>. 



 §.§ Proof of Proposition <ref> <ref>

We  assume  that  we  have   R_0  >  1.   Similarly  to <cit.>,  we  will prove  that  there  exists a  non-zero  initial
condition  w  ∈Δ  such  that  the  semi-flow ϕ(.,  w)  is
non-decreasing.  As R_0 > 1,  there exists a ∈ (0, 1) such
that  ρ( aγ) >  1.  Thus, by
Proposition <ref> (with  (a T, γ,  φ)), there
exists λ >  0 and w ∈ L^∞_+  such that
a Tw -  γ w = λ w.  By  Assumption <ref>, the map
φ  is  continuous  with  φ(0) =  1;  thus  there  exists
η∈  (0,1)  such  that  for   all  r  ∈  [0,η],  we  have
φ(r)  ≥  a.   Without  loss of  generality,  we  assume  that
w_∞≤η, and thus w ∈Δ.  We deduce that:

    F(w) = φ(w) Tw - γ  w ≥ aTw - γ w
      =          λ          w           ≥          0.

By   Lemma <ref> <ref>,    the   semi-flow
t  ↦ϕ(t,  w)  is  thus non-decreasing  on  _+  and  its
essential   limit,  say   g,   exists and belongs to Δ.  It   is   an  equilibrium   by
Lemma <ref>. Let g^* denote  the maximal equilibrium. As we
have g^*≥  g ≥ w  and μ((w)) >  0, we   deduce that
μ((g^*)) > 0. 




 §.§ Proof of Proposition <ref> <ref>


We now assume  that T = T_A  with A an irreducible  set.  Notice the
set A is invariant and thus admissible; and it has positive measure as
R_0>0.  It is  thus a  (non-zero) atom  by <cit.>.  Let
g  be  a non-zero  equilibrium.    We  have  (g)  ⊂  A
by (<ref>).    Since  (g)   is  invariant,   see
Lemma <ref> <ref>, and  A is  an atom,
we  deduce  that  (g)=A.  Then   use  that  the  support  of  g
characterizes g, see Corollary <ref> to deduce that g
is the only non-zero equilibrium.

 

 §.§ Proof of Proposition <ref> <ref>

 By Point <ref>, we have (g^*) = A.
 On   A^c,  we   have   ϕ(t,h)'=  -γϕ(t,h),  so   that
 _t→∞ϕ(t,  h) A^c=.  So it is
 enough to prove  the result when (h)⊂(g^*) = A. As (h) is a non-empty set included in the invariant and irreducible set A, its future is equal to A. Thus, by
 Lemma <ref> and considering ϕ(1, h) instead of h, one
 can assume without loss of generality that (h) = (g^*) = A.


 For    ε∈   (0,    1],   we    consider   the    operator
 U_ε=φ(ε)      M_{h≥ε}γ     on     L^p(μ),    and     set     U_0=M_{h>0}γ.    Let B be  the unit ball in
 L^p.       Since      γ     is      compact      and
 lim_ε→ 0φ(  ε) {h  ≥ε}   =    {h>   0}   a.e.,   we    deduce   that
 ⋃  _ε∈  [0,  1]  U_ε(B)  is  relatively
 compact.        Thus        the       family        of        operators
 (U_ε)_ε∈ [0,1] is collectively compact.
 By dominated convergence, we also get that
lim_ε→ 0(U_ε -U_0) f_p=0. We
deduce from Lemma <ref> that the map ε↦ρ(U_ε) is
continuous at 0. Thus, there exists ε∈ (0,1) such that  ρ (U_ε) > 1.
Notice that ( φ(ε) {h ≥ε}T, γ, φ) satisfies Assumption <ref>.
   By Proposition <ref> with T replaced by
   φ(ε)  {h ≥ε} T,  there exists λ > 0 and w ∈
   L^∞_+ such that:

    φ(ε){h ≥ε} T w - γ w= λ w.

Without loss of generality, we can assume that w_∞≤ε. We also have:

    (w) ⊂{h≥ε}⊂ A.

Using (<ref>)
we get that:

    F(w)=φ(w) Tw - γ w≥φ(ε)  Tw - γ w
    ≥λ w≥ 0.

 We deduce that the map t ↦ϕ(t,w) is non-decreasing by
 Lemma <ref> <ref> and, as g^*
 is the only non-zero equilibrium,  that _t→∞ϕ(t, w)=g^* by Lemma <ref>. 
 As the semi-flow is monotone by
 Lemma <ref> <ref>, we deduce that
 ϕ(t, w) ≤ϕ(t, h)≤ϕ(t,) for all t∈_+. Then use
Proposition <ref> <ref> to conclude
that  _t→∞ϕ(t, h)=g^*. 




§ PROOF OF THEOREM <REF>


Let (T, γ, φ) that satisfy Assumption <ref>.
We keep notations from Section <ref>; so 
_A is   the
supercritical antichain given by the maximal elements of the
supercritical atoms included in the  set A.


  Let A = ((h)) and g^*_A  be the maximal equilibrium on A
  (notice  that A=∅  if  h=). Since  A is  invariant,
  g^*_A        is         also        an         equilibrium        by
  Lemma <ref> <ref>.
  
We will, as in the proof of Proposition <ref> <ref>, prove the existence of w ∈Δ with w ≤ h such that the semi-flow (ϕ(t, w))_t∈_+ is non-decreasing and converges essentially to g^*_A.
By Lemma <ref>  and considering ϕ(1, h)  instead of h,
one can  assume without loss of  generality that (h) =A.
If _A is empty, we get that g^*_A= and 
by Lemma <ref> <ref>
and <ref>, we have _t→+∞ϕ(t, _A) =, and by monotonicity of the semi-flow that
_t→+∞ϕ(t, h) =, which proves Theorem <ref> in this
case.



We now assume that _A is not empty. 
Let B∈_A. By considering T_B instead of T, mimicking 
the proof of Proposition <ref> <ref>,
see (<ref>) and (<ref>), we deduce that 
there exists a function w_B∈Δ such that (w_B)⊂ B,
w_B≤  h, F(w_B)≥ 0 and _t→∞ϕ_B(t,w_B)= g^*_B. 
Since _A is an antichain of atoms, we deduce that ((w_B))∩(w_B')⊂(B) ∩ B'=∅ for all B', B∈_A such that B≠ B'. Set
w=∑_B∈_A w_B≤ h. We have:

    F(w)
      =∑_B∈_A F(w_B) + (B)( φ(w)- φ(w_B)) T w_B
      =∑_B∈_A F(w_B)≥ 0,

where for  the first  equality we  used that  (B) is  invariant and
(w_B)  ⊂   B⊂(B),   and  for  the   second  that
φ(w)-φ(w_B)=0 on (B) for B∈_A.  Arguing as in
the end of the proof  of Proposition <ref> <ref>,
we deduce that the semi-flow (ϕ(t,w))_t∈_+ is non-decreasing
and that  _t→∞ϕ(t, w)=g∈Δ with
g  an equilibrium.



We now prove the equality _A = _g = _g^*_A.
Since w≤  g,
we deduce that _A ⊂_g. 
Since (w) ⊂ A and A is invariant,  we have   (g)
⊂ A by Lemma <ref>. In particular, g is an
equilibrium of A by Lemma <ref> <ref>, thus
we get g ≤ g^*_A and deduce that  _g ⊂_g^*_A⊂_A as (g^*_A) ⊂ A.
This proves the claim and that  g=g^*_A by Corollary <ref>. 

By monotonicity of the semi-flow, we have that ϕ(t,w) ≤ϕ(t,h) ≤ϕ(t, _A) for all t ≥ 0.
The left term converges essentially to g=g^*_A, and the  right term to
g^*_A by Lemma <ref>. This gives
 _t→∞ϕ(t,h)=g^*_A. 
This ends the proof. 





 Paper Title: Long-time Asymptotics for the Ablowitz-Ladik system with present of solitons.



Long-time Focusing Ablowitz-Ladik system]Long-time Asymptotics for the Ablowitz-Ladik system with present of solitons




School of Mathematics and Statistics, Fujian Normal University, Fuzhou, PR China

chenms@fjnu.edu.cn




School of Mathematical Sciences,
	Fudan University, Shanghai, PR China
faneg@fudan.edu.cn



School of Mathematical Sciences,
	Fudan University, Shanghai, PR China
wang_zy@fudan.edu.cn




[2020]Primary 54C40, 14E20; Secondary 46E25, 20C20





§ ABSTRACT 
Abstract]In this article, we investigate some problems about the initial-value problem of the focusing Ablowitz-Ladik system, and the initial data belongs to a discrete weighted ℓ^2 space. 
	On the one hand, we have proved the global well-posedness for the initial-value problem.
	Utilizing the fact that the initial data belongs to the discrete weighted ℓ^2 space, we construct a Riemann-Hilbert problem for the initial-value problem.
	By Fredholm theory, the Riemann-Hilbert problem is uniquely solved with the jump contour consisting of three circles that all center around the origin. 
	On the other hand, based on the global well-posedness and the Riemann-Hilbert problem, we allow the solution to have the higher-order solitons, and then analyze the long-time asymptotics for the solution in different sectors on the upper-half plane.



KEYWORDS: Global wellposedness, Fredholm theory, long-time asymptotics, higher-order soliton


MSC 2020: 35P25, 35Q15, 35B40, 37K40






[
    Zhaoyu Wang
    August 5, 2024
==================







§ INTRODUCTION


The Ablowitz-Ladik systems read:

    / tq_n=q_n+1-2q_n+q_n-1+σ| q_n|^2(q_n+1+q_n-1),  σ=±1

which is the completely integrable discretization of nonlinear Schrödinger equations:

    u_t=u_xx+σu^2u,

and it was first introduced by Ablowtiz and Ladik in the 1970s <cit.>. 
The inverse scattering transform of (<ref>) has been well investegated <cit.>. 
By the connection between Ablowitz-Ladik systems and orthogonal polynomials on the unit circle, Nenciu <cit.> have constructed the Lax pair for the Ablowitz-Ladik systems.
Using methods of algebraic geometry, Miller et al <cit.> studied the finite genus solutions for Ablowtiz-Ladik systems.
The multi-Hamilton structure is constructed by Gekhtman and Nenciu for the Ablowitz-Ladik system in the defocusing cases<cit.>.
In this article, our research focus on the global well-posedness and the Fredholm theory of the initial-value problem of Ablowitz-Ladik system in the focusing case, and based on these, we study the long-time asymptotics. 
The focusing Ablowitz-Ladik system reads, that is, (<ref>) with σ=1:

    / tq_n=q_n+1-2q_n+q_n-1+| q_n|^2(q_n+1+q_n-1).

We constrain the initial potential in the weighted ℓ^2 space: ℓ^2,1.
In addition, we suppose no spectral singularity for the corresponding Riemann-Hilbert (RH) problem, and do not restrict the poles of the RH problem to be simple.
Under these assumption, by analyzing the Laurent's expansion of the solution for RH problem on the discrete spectrum, we equivalently transform it into RH problem <ref> with the jump contour consisting of three circles. 
It is interesting that RH problem <ref> is a pure RH problem without discrete spectrum.
Utilizing the Vanishing Lemma, we develop the Fredholm theory for RH problem <ref>, which ensure the solvability of this RH problem. 

It is well acknowledged that on the complex plane, a matrix-valued RH problem is not always solved<cit.>. 
In 1989, Zhou have developed a Vanishing Lemma for the n× n RH problem whose jump contour is symmetric about the real line ℝ by Fredholm theory<cit.>. 
In Zhou's work, the jump matrix admits skew-Hermitian outside ℝ: v(z)=v(z̅)^†, and he specified it to be positively definite along ℝ: v|_ℝ>0. The Vanishing Lemma ensures the Fredholm alternative to be zero and consequently the solvability of the matrix-valued RH problem.
For some discrete integrable systems, the discrete and continuous spectrum are symmetric about the unit circle, like Ablowitz-Ladik systems<cit.>, Toda system<cit.>, discrete sine-Gordon equation<cit.>. 
For the discrete integrable systems, if ones also develop the vanishing lemma, then it ensures the solvability of the corresponding RH problem, which is the Lemma <ref>'s results.

The long-time asymptotic behavior for the defocusing nonlinear Schrödinger equation, the equation (<ref>) with σ=-1, has been studied thoroughly<cit.>.
In 2018, Borghese et al  have investigated the long-time asymptotics for the focusing nonlinear Schrödinger equation<cit.>.
Utilizing the dbar steepest descent method, they indicate that the leading terms of the asymptotic formula are determined by the solitons and the RH problem at the stationary phase points, and the error is (t^-3/4).
The dbar steepest descent is first introduced when handling the asymptotic analysis for the orthogonal polynomials<cit.>.
Recently, Chen et al have constructed the inverse scattering theory on the discrete weighted l^2 space for the defocusing Ablowitz-Ladik system, i.e. (<ref>) with σ=-1<cit.>, and based on this, they investigate the long-time asymptotic behavior for the initial-value problem of the defocusing Ablowitz-Ladik system on the background of lower regularity<cit.>. 
The results show that according to the long-time asymptotic behavior, the upper half plane (n,t) is divided into three sectors, two of which are fast decaying regions with the rate (t^-1) while the other one is the oscillatory region where the long-time asymptotic behavior is agree with that for the defocusing nonlinear Schrödinger equation<cit.>. 
The results also consist with the work of Stewart in 2023 that shows the long time asymptotic behavior by linear estimates for some small-norm initial potential <cit.>.





This article is organized as follows. 
In Section <ref>, we present the global well-posedness of the focusing Ablowitz-Ladik system in the discrete weight ℓ^2 space: ℓ^2,1. 
In Section <ref>, we introduce the direct scattering and construct RH problem <ref> for the initial-problem.
When the initial potential belongs to ℓ^2,1, it is verified that the reflection coefficient belongs to H^1, and the discrete spectrum are invariant about the mapping: λ↦λ̅^-1. 
In Section <ref>, we investigate the solvability of RH problem <ref>. 
Transforming it into a new RH problem for M̃, we apply Fredholm theory on M̃, and then uniquely solve the RH problem <ref>. 
In Section <ref>, we study the long-time asymptotic in the region I: ξ∈(-1,1).
In this region, the leading term of the asymptotic formula is the solitonic part q_n^𝒵_ξ(t), and the second one with decaying rate (t^-1/2) is determined by the reflection coefficient around stationary phase points, and the remaining part is dominated by (t^-3/4).
The result is consisting with those for nonlinear Schrödinger equation <cit.>.
In Section <ref>, we study the region II: ξ<-1 and region III: ξ>1.
In these regions, the leading term is also the soliton, but the second term in the region I vanishes and the remaining part is dominated by (t^-1).

List some notation exploited in this paper below:

	
  * The right-shifting operator E: Ef(n)=f(n+1), the left-shifting operator E^-1: E^-1f(n)=f(n-1). 
	
  * ℓ^1: {{a_n}_n=-∞^∞:∑_-∞^∞| a_n|<∞}, where a_n_1=∑_-∞^∞| a_n|.
	
  * ℓ^k,1: {{a_n}_n=-∞^∞:∑_-∞^∞(1+n^2)| a_n|^k<∞}, k=1,2, where the norm reads: a_n_k,1=(∑_-∞^∞(1+n^2)| a_n|^k)^1/k.
	
  * H^1: {f(z)∈ L^2(Σ):∂_θ f(e^θ)∈ L^2([0,2π])}, where Σ={e^θ: θ∈[0.2π]} is the unit circle at the origin and f_H^1=f_L^2(Σ)+∂_θ f_L^2(Σ).
	
  * The Pauli matrices:
	
    σ_1=[
    			0   1
    1   0
    		],  σ_2[
    			0   -
       0
    		],  σ_3=[
    			1   0
    0   -1
    		].

	
  * The phase function on λ∈ℂ:
	
    ϕ(λ,n,t)=- t(λ+λ^-1-2)+nlnλ.

	
  * The discrete spectrum 𝒵∪𝒵^-1:
	
    𝒵={λ_1,…,λ_l}.

	And
	
    𝒵_ξ^+=𝒵∩{ϕ>0},  𝒵_ξ=𝒵∩{ϕ=0}.

	




 §.§ Main results


The initial-value problem of the focusing Ablowitz-Ladik system reads:

    / tq_n=q_n+1-2q_n+q_n-1+| q_n|^2(q_n+1+q_n-1),
        q_n(t=0)=q_n(0)∈ℓ^2,1,

where n∈ℤ is the spatial variable, t∈ℝ is the time variable. 
When we take the initial data shown in (<ref>), the solution is globally well-posed, and in RH problem <ref>, the reflection coefficient r(λ) belongs to H^1. 
r(λ)∈ H^1 means that it is 1/2-Hölder continuous on the unit circle. 
In the residue condition of RH problem <ref>, the order of the poles is set to be α_l' for any l'=1,…,l, and we remove the poles by some triangular rational functions, and equivalently transform RH problem <ref> into RH problem <ref> without poles. 
By the Fredholm theory and Vanishing Lemma, we unique solve RH problem <ref>.

Based on the solvability of RH problem <ref>, we construct the long-time asymptotic formulas, and according to it, we divide the upper-half plane (n,t) into three sectors: two soliton regions and one solitonic-oscillatory region.

	The initial-value problem (<ref>) is uniquely solved, and the solution satisfies the following asymptotic properties on the upper-half plane (n,t):
	
		
  * In region I: ξ∈(-1,1), 
		
    q_n(t)   =(∏_l'=1^lλ_l'^2α_l')e^∫_S_1^S_2ln(1+r(s)^2)/2π s s(q^𝒵_ξ_n(t)+t^-1/2q^Osc_n(t)+(t^-3/4)),

		where q^𝒵_ξ is the soliton, and q^Osc is the oscillatory part defined in (<ref>).
		
  * In regions II and III: ξ<-1 and ξ>1, 
		
    q_n(t)∼ q_n^𝒵_ξ(t)+(t^-1).

		
	
	




	In the above theorem, we have presented the long-time asymptotics for the initial-value problem in region I III, and the results for the transition regions is undergoing.


Our analysis exploits the Lax representation:

    EX   =AX,   X/ t=BX,
    
    	A   =z^σ_3+Q,   B=σ_3/((z-z^-1)^2/2-Q(E^-1Q)+z^σ_3Q-
    	(E^-1Q)z^σ_3),

where Q is a skew-symmetry matrix

    Q=Q(n)=[
    		0   q_n
    -q_n   0
    	].

That a smooth function solves the initial-value problem (<ref>) is equivalent to the compatibility condition:

    A/ t=(EB)A-AB,

which is proved to be the focusing Ablowitz-Ladik system. 
Introducing

    c_n(t)=∏_k=n^∞(1+q_k(t)^2),

it is readily seen that c_-∞=lim_n→-∞c_n(t) is a conserved quantity.
By direct computation,

    / t c_-∞   =/ t∏_n=-∞^∞(1+q_n(t)^2)=c_-∞∑_n=-∞^∞/ tq_n(t)q_n(t)+q_n(t)/ tq_n(t)/1+q_n(t)^2

and taking (<ref>) into (<ref>), we obtain that

    / t c_-∞≡ 0,

that is, c_-∞ is independent on t.




§ GLOBAL WELL-POSEDNESS

In this section, we consider the solution of the initial value problem (<ref>). 
We see that if the initial potential q(0) is in a given weighted ℓ^2 space and the conserved quantity c_-∞ is finite, then the solution is globally well-posed, which we shown in Proposition <ref>. 

	When q(0)∈ℓ^2,1 and c_-∞ is finite, the initial value problem (<ref>) admits a unique solution q(t)∈ℓ^2,1 for t∈ℝ, and the solution is Lipschitz continuous about t. 


	By the definition of c_-∞, notice the initial potential satisfies q(0)_∞<c_-∞.
	We discuss the solution of (<ref>) in the Banach space:
	
    B(q(0),c_-∞){a∈ℓ^∞:a-q(0)_∞<c_-∞}.

	Since the right-hand side of (<ref>) is Lipschitz continuous, by a standard argument about ODEs in a Banach space, there exist t_1>0 such that for t∈(-t_1,t_1), (<ref>) admits a unique solution, and t_1 is determined by the conserved quantity c_-∞.
	Moreover, noticing that q(-t_1/2)_∞,q(t_1/2)_∞≤ c_-∞, we can extended the solution to t∈(-t_1/2,3t_1/2)∪(-3t_1/2,t_1/2)=(-3t_1/2,3t_1/2) by investigating the solution of (<ref>) in the ball B(q(-t_1/2),c_-∞) and B(q(-t_1/2),c_-∞), respectively;
	repeating the procedure, we uniquely extended the solution q(t) to t∈ℝ, and q(t) belongs to B(0,c_-∞). 
	
	Since we have proved the solvability of (<ref>) and the uniqueness of solution in B(0,c_-∞), we claim that the solution q(t) does not blow up in ℓ^2,1 for t∈ℝ. 
	By (<ref>), 
	
    / tq_n(t)≲q_n(t)+q_n-1(t)+q_n+1(t),

	i.e.,
	
    / tq_n(t)^2≲q_n(t)^2+q_n-1(t)^2+q_n+1(t)^2.

	Summing up the both sides in the above inequality, we obtain that there exists positive constant C such that
	
    / tq(t)^2_1,1≤ Cq(t)_2,1^2.

	Integrate both sides of (<ref>) on the interval: [0,t] for t>0 (or [t,0] for t<0), it is readily seen that
	
    q(t)_2,1-q(0)_2,1≤ C∫_0^tq(τ)_2,1τ,

	therefore by Gronwell's inequality, q(t)_2,1≤q(0)_2,1e^Ct never blows up.
	Additionally, in the similar way, for any ϵ>0, we obtain that
	
    q(t±ϵ)-q(t)_2,1≲±∫_0^±ϵq(τ)_2,1τ,

	which means that the solution q(t) is Lipschitz continuous in ℓ^2,1 about t.
	This proves the results.





§ RIEMANN-HILBERT PROBLEM






 §.§ Jost solutions and modified Jost solutions


Denote  X^±(z,n,t) as the Jost solution for the Lax pair, and it satisfies:

    X^±(z,n,t)∼ z^nσ_3e^-/2(z-z^-1)^2tσ_3.

Notice that the Jost solution satisfies that

    X^+(z,n,t)≡ c_n(t)^-1,   X^-(z,n,t)≡ c_-∞c_n(t)^-1.

Recalling that c_-∞ is positive, by (<ref>), we see that X^±(z,n,t) are invertible. 
By uniqueness of the fundamental solution, there is a unique 2×2 matrix-valued function S(z) called scattering matrix such that

    X^-(z,n,t)=X^+(z,n,t)S(z).
 
We note that the matrix Q(n) is skew-Hermitian, therefore, by the Lax pair (<ref>), it follows that

    X^±(z,n,t)=σ_2X^±(z̅^-1,n,t)σ_2.

We naturally introduce the modified Jost solutions

    Y^±=Y^±(λ,n,t)=[
    		1   0
    0   z
    	]X^±(z,n,t)z^-nσ_3e^ t/2(z-z^-1)^2σ_3[
    		1   0
    0   z^-1],

λ=z^2, and it follows that Y^±∼ I as n→±∞.
By (<ref>) and (<ref>), we see that

	
    Y^±(λ,n+1,t)   =(I+Q(n,t))λ^σ_3/2Y^±(λ,n,t)λ^-σ_3/2,
    ∂_tY^±(λ,n,t)   =B̃(λ,n,t)Y^±(λ,n,t)+(λ+λ^-1-2)/2Y^±(λ,n,t)σ_3,


where

    B̃=σ_3/(λ+λ^-1-2/2I-Q(E^-1Q)+Q-(E^-1Q)λ^σ_3)

and it follows that Y^± are single-valued functions. 
By (<ref>), it is readily seen that

    λ^-n+1/2σ_3Y^±(λ,n+1,t)λ^n+1/2σ_3-λ^-n/2σ_3Y^±(λ,n,t)λ^n/2σ_3=λ^-(n/2+1)σ_3[
    			0   q_n(t)
    -q_n(t)   0
    		]Y^±(λ,n,t)λ^n/2σ_3,

and then taking summation on both sides of (<ref>), we rewrite Y^±(λ,n,t) in the form of summation 

	
    Y^+(λ,n,t)=I-∑_k=n^∞λ^n-k-2/2σ_3[
    			0   q_n(t)
    -q_n(t)   0
    		]Y^+(λ,k,t)λ^-n-k/2σ_3,
       Y^-(λ,n,t)=I+∑_k=-∞^n-1λ^n-k-2/2σ_3[
    			0   q_n(t)
    -q_n(t)   0
    		]Y^-(λ,k,t)λ^-n-k/2σ_3.


Recalling the assumption that q(0)∈ℓ^2,1∈ℓ^1, as shown in <cit.>, [Y^+_1,Y^-_2] is holomorphic on D_-={|λ|<1} and continuously extended to D_-∪Σ, Σ={|λ|=1}, while [Y^-_1,Y^+_2] is holomorphic on D_+={|λ|>1} and continuously extended to D_+∪Σ.

By (<ref>), taking the WKB expansion, it follows that the modified Jost solutions satisfy asymptotic properties at λ=0 and λ=∞:

	
    [Y^+_1,Y^-_2](λ,n,t)∼[
    			c_n^-1(t)+O(λ)   q_n-1(t)+O(λ)
    c_n^-1(t)q_n(t)λ+O(λ^2)   1+O(λ)
    		],  λ→0,
       [Y^-_1,Y^+_2](λ,n,t)∼[
    			1+O(1/λ)   -c_n^-1(t)q_n(t)1/λ+O(1/λ^2)
    -q_n-1(t)+O(1/λ)   c_n^-1(t)+O(1/λ)
    		],  λ→∞.





 §.§ Scattering coefficients


Introduce scattering coefficients:

    a(λ)=c_n(t)[Y^-_1,Y^+_2](λ,n,t),   b(λ)=e^ϕ(λ,n,t)c_n(t)[Y^+_1,Y^-_1](λ,n,t).

Factually,

    a(λ)=S_1,1(z),   b(λ)=zS_2,1(z),

which follows by (<ref>), (<ref>) and (<ref>).
Since Y^± is single-valued, we see that a(λ), b(λ) are also single-valued functions of λ. 
By the analyticity of Y^±, it follows that a(λ) is holomorphic on D_+ and continuously extended to D_+∪Σ;
in addition, b(λ) is continuous on Σ. 
By the symmetric property (<ref>) and the definition (<ref>), it is readily seen that

    ă(λ)a(λ̅^-1)=c_n(t)[Y^+_1,Y^-_2](λ,n,t),

and that ă(λ) is holomorphic on D_-. 
What is more, utilizing the asymptotic property (<ref>), we obtain that

    lim_λ→∞a(λ)=1,  ă(λ)=1.

The reflection coefficient is defined as

    r(λ)=b(λ)/a(λ),

and we claim that when q(0)∈ℓ^2,1, the corresponding reflection coefficient r belongs to H^1 as shown in Proposition <ref>. 

	Suppose that q(0)∈ℓ^2,1 and no spectral singularity, i.e., a admits no zero on Σ. 
	Let r be the corresponding reflection coefficient as defined in (<ref>).
	We assert that r∈ H^1. 

Before we prove Proposition <ref>, we introduce

    Ỹ=Ỹ^±(λ,n)=λ^-nσ_3/2Y^±(λ,n,0)λ^nσ_3/2,

and it is a little abuse of notation that we simply note q_n=q_n(0) in the proof.
Substituting (<ref>) into (<ref>), we obtain that

    a(λ)=c_n(0)[Ỹ^-_1,Ỹ^+_2](λ,n),   b(λ)=c_n(0)[Ỹ^+_1,Ỹ^-_1](λ,n).

and that Y^± satisfy the following Volterra summation equation:

    Ỹ^±=I+𝒯^±Ỹ^±,

where for any 2×2 matrix function Ỹ=Ỹ(n)

	
    𝒯^-Ỹ(λ,n)=∑_k=-∞^n-1[
    			0   q_k
    -q_k   0
    		]λ^(k+1)σ_3Ỹ(k),
       𝒯^+Ỹ(λ,n)=-∑_k=n^+∞[
    			0   q_k
    -q_k   0
    		]λ^(k+1)σ_3Ỹ(k).


Recalling that q∈ℓ^2,1, we obtain that the Volterra summation equation (<ref>) uniquely solved by

    Y^±=∑_k=0^+∞(𝒯^±)^kI.

In addition, by (<ref>), it is readily seen that for λ∈Σ, λ=λ̅^-1 and 𝒯^± admits the symmetry on the unite circle

    σ_2𝒯^±Ỹσ_2=𝒯^±(σ_2Ỹσ_2).


	Since a have no zeros on the jump contour Σ, if we prove that
	
    a,b∈ H^1,

	then, we obtain the results.
	Moreover, by (<ref>), to prove (<ref>), we only have to verify
	
    Y^±(·,0)∈ H^1.

	
	By direct computation, we obtain that for any m∈ℤ^+,
	
		
    (𝒯^-)^2m-1(λ,n)=(-1)^m∑_-∞<k_2l-1<⋯<k_1<n[
    				0
    ∏_Even jq_k_j∏_Odd jq_k_jλ^A_2m-1],
       (𝒯^-)^2m(λ,n)=(-1)^m∑_-∞<k_2m<⋯<k_1<n[∏_Odd jq_k_j∏_Even jq_k_jλ^1-A_2m
    0
    			],
       (𝒯^+)^2m-1(λ,n)=(-1)^m∑_n≤ k_1≤⋯≤ k_2m-1<+∞[
    				0
    ∏_Even jq_k_j∏_Odd jq_k_jλ^A_2m-1],
       (𝒯^+)^2m(λ,n)=(-1)^m∑_n≤ k_1≤⋯≤ k_2m<+∞[∏_Odd jq_k_j∏_Even jq_k_jλ^1-A_2m
    0
    			],

	
	where A_m=∑_j=1^m(-1)^j-1k_j+1.
	Denoting N(m,λ,n) as the non-zero entry of (𝒯^-)^k(λ,n), we obtain that for any g(λ)∈ L^(Σ)
	
    ∫_ΣN(m,λ,n)g(λ)λ/2π=∑_-∞<k_2m-1<⋯<k_1<n(∏_Even jq_k_j)(∏_Odd jq_k_j)ĝ(A_2m-1)
           ≤∑_-∞<k_2m-1<⋯<k_1<nq_k_1⋯ q_k_2m-1ĝ(A_2m-1)≤q_1^2m-2/(2m-2)!q_2ĝ_2≤(1+π^2/3)^m-1q^2m-1_2,1/(2m-2)!g_L^2(Σ),

	where ĝ=ĝ(n) are the Fourier coefficients of g, thus, 
	
    N(m,·,n)_L^2(Σ)≤(1+π^2/3)^m-1q^2m-1_2,1/(2m-2)!.

	We also have
	
    ∂_θ N=∑_j=1^2m-1(-1)^j-1N_j,
       N_j=N_j(m,λ,n)=∑_-∞<k_2m-1<⋯<k_1<n(∏_Even j'q_k_j')(∏_Odd j'q_k_j')k_jλ^A_2m-1.

	Using the same technique in (<ref>), we have
	
    ∫_ΣN_j(m,λ,n)g(λ)λ/2π   ≤∑_-∞<k_2m-1<⋯<k_1<n(∏_j' jq_k_j')(k_jq_k_j)ĝ(A_2m-1)
       ≤q_1^2m-1/(2m-2)!q_2,1ĝ_2≤(1+π^2/3)^m-1q^2m-1_2,1/(2m-2)!g_L^2(Σ).

	Considering (<ref>) and (<ref>), we obtain that
	
    ∂_θ N(m,·,n)_L^2(Σ)≤(2m-1)(1+π^2/3)^m-1q^2m-1_2,1/(2m-2)!.

	By (<ref>) and (<ref>), it follows that
	
    N(m,·,n)_H^1≤2m(1+π^2/3)^m-1q^2m-1_2,1/(2m-2)!,

	i.e.,
	
    (𝒯^-)^2m-1(·,n)_H^1≤2m(1+π^2/3)^m-1q^2m-1_2,1/(2m-2)!.

	Applying the technique for the estimate in (<ref>), we estimate the others in (<ref>) and obtain that
	
    (𝒯^±)^m(·,n)_H^1≤(m+1)(1+π^2/3)^m-1/2q^m_2,1/(m-1)!.

	Additionally, by (<ref>), it is readily seen that for any n∈ℤ^+,
	
    σ_2(𝒯^±)^mIσ_2=𝒯^±(σ_2(𝒯^±)^m-1Iσ_2)=⋯=σ_2(𝒯^±)^mIσ_2,

	thus, 
	
    (𝒯^±)^m(·,n)=[
    			0   1
    -1   0
    		](𝒯^±)^m(·,n)∈ H^1,

	and
	
    𝒯^±)^m(·,n)_H^1≤(m+1)(1+π^2/3)^m-1/2q^m_2,1/(m-1)!.

	
	By (<ref>), (<ref>) and (<ref>), applying the Dominated Convergence Theorem and taking n=0, we obtain that (<ref>). 
	This proves the results.





 §.§ Discrete spectrum


Under the assumption of no spectral singularity, the holomorphic function a(λ) admits at most  finite zeros the discrete spectrum 𝒵∈ D_+, and denote the zero's order at λ_l' as α_l'>0. 
By the symmetric relation (<ref>), it is clear that ă(λ) admits an α_l'-order zero at each λ=λ̅_l'^-1, l'=1,…,l. 

Since λ_l' is the α_l'-order pole of a(λ), by the first formula in (<ref>), it turns out that there exist constants: β_λ_l',0,β_λ_l',1,…,β_λ_l',α-1, such that for any α=0,…,α_l'-1,

    ∂_λ^αY^-_1(λ_l')=∑_α'=0^α(α
    α'
    	)∂_λ^α-α'Y^+_2(λ_l')(∑_j=0^α'(α'
    j
    	)β_λ_l',j∂_λ^α'-j(e^-ϕ(λ_l'))),

where  (α
α'
) denotes the binomial coefficient and β_λ_l',00. 
Similarly utilizing (<ref>), for λ̅_l'^-1∈ D_-, there exist constants β_λ̅_l'^-1,0,…,β_λ̅_l'^-1,α_l'-1, such that for any fixed α belonging to 0,…,α_l'-1, 

    ∂_λ^αY^-_2(λ̅_l'^-1)=∑_α'=0^α(α
    α'
    	)∂_λ^α-α'Y^+_1(λ̅_l'^-1)(∑_j=0^α'(α'
    α”)β_λ̅_l'^-1,j∂_λ^α'-j(e^ϕ(λ̅_l'^-1))),

especially, β_λ̅_l'^-1,0=-β_λ_l',00. 





 §.§ The reconstruction formula

In this part, we construct the Riemann-Hilbert problem and the reconstruction formula. 
Set the 2×2 matrix-valued function

    M=M(λ,n,t)=(P[Y^+_1,Y^-_2/ă])(λ,n,t),   λ∈ D_-,
    (P[Y^-_1/a,Y^+_2])(λ,n,t),   λ∈ D_+,

where P=P(n,t)=[
	1   0
c_n(t)q_n-1(t)   c_n(t)
] and ă=ă(λ)=a(λ̅^-1).
We claim that M is the unique solution of RH problem <ref>. 
By the analyticity of Y^± and (<ref>), we obtain the analyticity of M.
By the asymptotic behavior (<ref>) of Y^± at the infinity, we derive the normalization of M at λ→∞.
Now, we focus on the residue condition.
Recalling that λ=λ_l' is an α_l'-order zero of a(λ), it is naturally an α_l'-order pole of M(λ,n,t).
If denote λ=λ'_-αM as the coefficient of (λ-λ')^-α in the Laurent's expansion of M at λ=λ', then by the argument of complex analysis, we obtain the formula to compute the Laurent's coefficients:

    λ=λ_l'_-αM=1/(α_l'-α)!∂_λ^α_l'-α[M(λ)(λ-λ_l')^α_l']|_λ=λ_l'.

Considering (<ref>), (<ref>) and (<ref>), we obtain the residue condition (<ref>). In addition, the residue condition (<ref>) is verified similarly. 
The jump condition is a consequence of (<ref>), (<ref>) and (<ref>). 

	Find a 2×2 matrix-valued function M such that
	
		
  * Analyticity: M is holomorphic on ℂ∖Σ.
		
  * Normalization: As λ→∞, M∼ I+(λ^-1).
		
  * Residue condition: On the α_l'-order pole λ=λ_l'∈𝒵, we have
		
			
    λ=λ_l'_-αM=∑_j=0^α_l'-α∂_λ^α_l'-α-jM(λ_l')/(α_l'-α-j)![
    					0   0
    p_λ_l',je^ϕ(λ_l')   0
    				],
 
			where α=0,…,α_l'-1 and p_λ_l',j is a polynomial of (n,t) of order at most j
			
    p_λ_l',j
    				=∑_j'=0^j∑_j”=0^j'∂_λ^j”((·-λ_l')^α_l'/a)|_λ=λ_l'β_λ_l',j'-j”∂_λ^j-j'e^ϕ(λ_l')/(j-j')!(j'-j”)!j”!e^ϕ(λ_l').

			And λ=λ̅_l'^-1 is also a α_l'-order pole of M on D_-, and
			
    λ=λ̅_l'^-1_-αM=∑_j=0^α_l'-α∂_λ^α_l'-α-jM(λ̅_l'^-1)/(α_l'-α-j)![
    					0   p_λ̅_l'^-1,je^-ϕ(λ̅_l'^-1)
    0   0
    				],
 
			
		
		where
		
    p_λ̅_l'^-1,j=∑_j'=0^j∑_j”=0^j'∂_λ^j”((·-λ̅_l'^-1)^α_l'/ă)|_λ=λ̅_l'^-1β_λ̅_l'^-1,j'-j”∂_λ^j-j'e^-ϕ(λ̅_l'^-1)/(j-j')!(j'-j”)!j”!e^-ϕ(λ̅_l'^-1)

		
  * Jump condition: If we set the boundary value of M=M(λ,n,t) on λ∈Σ as M_±=M_±(λ,n,t), then
		
    M_+=M_-V,   V=V(λ,n,t)=[
    				1+| r(λ)|^2   r(λ)e^-ϕ(λ,n,t)
    r(λ)e^ϕ(λ,n,t)   1
    			].

	


By the asymptotic property (<ref>) of Y^± at λ→0 and the definition  (<ref>), we find that the solution of (<ref>) is recovered by the solution of RH problem <ref>, i.e. the reconstruction formula,

    q_n(t)=[M(0,n+1,t)]_1,2.








§ FREDHOLM ALTERNATIVE, PURE RH PROBLEM AND THE SOLVABILITY


In this section, utilizing some triangular matrix function [
	1   0
fe^ϕ   1
] and [
	1   f̆e^-ϕ
0   1
], we remove the poles in the original RH problem and transform it into a new RH problem without singularity.
In the new RH problem, we find that the jump matrix is self-adjoint, and as a consequence, using the Fredholm theorem, we prove that the RH problem is uniquely solved. 



 §.§ Remove the poles

In this section, we construct lower/upper triangular functions [
	1   0
fe^ϕ   1
], [
	1   f̆e^-ϕ
0   1
], such that f, f̆ are rational function and M[
	1   0
fe^ϕ   1
], M[
	1   f̆e^-ϕ
0   1
] are analytic on D̃_+, D̃_- respectively.
The regions are as shown in FIGURE <ref>.

Before proving the results for the region D̃_+ and D̃_- respectively, we consider the pole at each λ_l' or λ̅_l'^-1 belonging to the discrete spectrum as the following proposition.


	There exist polynomials f_λ_l'(λ), f_λ̅_l'^-1(λ) of (λ-λ_l)^-1, (λ-λ̅_l^-1)^-1 with the order not more than α_l', respectively, such that for l'=1,…,l and α=1,…,α_l',
	
    λ=λ_l'_-α(M[
    			1   0
    f_λ_l'e^ϕ   1
    		])≡0,   λ=λ̅_l'^-1_-α(M[
    			1   f_λ̅_l'^-1e^-ϕ
    0   1
    		])≡0.



	Set 
	
    f_λ_l'(λ)=∑_s=1^α_l'f_λ_l',s(λ-λ_l')^-s.

	Rewrite the first equality in (<ref>), 
	
    0≡λ=λ_l'_-α(M[
    			1   0
    f_λ_l'e^ϕ   1
    		])=λ=λ_l'_-αM+λ=λ_l'_-α(M[
    			0   0
    f_λ_l'e^ϕ   0
    		]).

	Recalling that the second column of M is holomorphic on λ∈ D_+, we rewrite the second term on the right-hand side of (<ref>):
	
    λ=λ_l'_-α(M[
    			0   0
    f_λ_l'e^ϕ   0
    		])=∑_j=0^α_l'-α∂_λ^α_l'-α-jM/(α_l'-α-j)![
    			0   0
    ∑_j'=0^jf_λ_l',α_l'-j'∂_λ^j-j'e^ϕ/(j-j')!   0
    		]|_λ=λ_l'.
 
	Substituting (<ref>) and (<ref>) into (<ref>), we obtain that
	
    f_λ_l',s=∑_s'=0^α_l'-s∂_λ^s'((λ-λ_l')^α_l'/a(λ))|_λ=λ_l'β_λ_l',α_l'-s-s'/(α_l'-s-s')!s'!

	solve (<ref>), i.e.
	
    f_λ_l'(λ)=∑_s=1^α_l'∑_s'=0^α_l'-s∂_λ^s'((λ-λ_l')^α_l'/a(λ))|_λ=λ_l'β_λ_l',α_l'-s-s'(λ-λ_l')^-s/(α_l'-s-s')!s'!.

	For the existence of f_λ̅_l'^-1, the proof is parallel.
	We complete the proof. 

As shown in Proposition <ref>, we have proved that the pole of M can be removed by the rational triangular function at each point of the discrete spectrum, which means in RH problem <ref>, the residue condition can be transformed into the jump.
Based on this, we want to do more, and find a rational triangular function on D_+ or D_- to remove the poles at the discrete spectrum uniformly, which is helpful when we prove the unique solvability of the newly obtained RH problem. 
By the theory of generalized Vandermonde matrix, we obtain the rational function on D_+ and D_- as the proposition below.


	There exists a rational function f(λ), f̆(λ) such that M[
		1   0
fe^ϕ   1
	], M[
		1   f̆e^-ϕ
0   1
	] are holomorphic on D_+, D_-∖{0}, respectively. 
	


	Notice that when we take l=1, it is Proposition <ref>; thus, Proposition <ref> is a generalized version of Proposition <ref>. To prove the results, we introduce the generalized Vandermonde matrix <cit.>
	
    V(λ_1,…,λ_l,α_1,…,α_l)=[
    			1   λ_1   ⋯   λ_1^α_1   ⋯   λ_1^τ
    
    			0   1   ⋯   α_1λ_1^α_1-1   ⋯   τλ_1^τ-1
    ⋯   ⋱   ⋱   ⋯   ⋯   ⋯
    
    			0   ⋯   0   1   ⋯   τ!/α_1!λ_1^τ-α_1
    ⋯   ⋯   ⋯   ⋯   ⋯   ⋯
    
    			1   λ_l   ⋯   λ_l^α_l   ⋯   λ_l^τ
    
    			0   1   ⋯   α_lλ_l^α_l-1   ⋯   τλ_l^τ-1
    ⋯   ⋱   ⋱   ⋯   ⋯   ⋯
    
    			0   ⋯   0   1   ⋯   τ!/α_l!λ_l^τ-α_l],

	where τ=∑_l'=1^lα_l'. The generalized Vandermonde matrix is invertible and its determinant is
	
    V(λ_1,…,λ_l,α_1,…,α_l)=∏_j> l(λ_j-λ_k)^α_jα_k.

	
	To construct f(λ), it is sufficient to construct a polynomial g(λ) of λ such that
	
    f(λ)=g(λ)∏_l'=1^lf_λ_l'(λ),   g(λ)=∑_s=0^τ-1g_sλ^s,

	and for any l'=1,…,l, α=1,…,α_l',
	
    λ=λ_l'_-αf(λ_l')=f_λ_l',α.

	Seeing (<ref>), since f is consisting of f_λ_l' and the holomorphic function g, we learn that f admits an α_l'-order pole at λ=λ_l'.
	Calculate the Laurent's coefficients of f: 
	
    λ=λ_l'_-αf(λ_l')   =1/(α_l'-α)!∂_λ^α_l'-α((·-λ_l')^-α_l'f_λ_l'g∏_l” l'f_λ_l”)(λ_l')
       =∑_s=0^α_l'-α1/s!f_λ_l',α+s∂_λ^s(g∏_l” l'f_λ_l”)(λ_l').

	By comparing (<ref>) and (<ref>), it is sufficient to solve the following linear system with τ variables g_0,…,g_τ-1:
	
    ∂_λ^s(g∏_l” l'f_λ_l”)(λ_l')=δ_0,s, 
    		  δ_0,s=
    			1   s=0
    
    			0   s0
    		,   s=0,…,α_l'-1.

	Equivalently, we rewrite (<ref>) in the form of matrix:
	
    blkdiag({J_l'(λ_l')}_l'=1^l) V(λ_1,…,λ_l,α_1,…,α_l)(g_0,…,g_τ-1)^t=(1,α_1-10,…,0,…,1,α_l-10,…,0)^t

	where blkdiag(·) denotes the block diagonal matrix, and J_l' is the α_l'×α_l' matrix function:
	
    J_l'=[∏_l” l'f_λ_l”   0   ⋯   0
    ∂_λ∏_l” l'f_λ_l”   ∏_l” l'f_λ_l”   ⋱   ⋮
    ⋮   ⋱   ⋱   0
    ∂_λ^α_l'∏_l” l'f_λ_l”/α_l'!   ⋯   ∂_λ∏_l” l'f_λ_l”   ∏_l” l'f_λ_l”].

	We default f_λ_l”(λ_l')0 for all l” l', or let f_λ_l” plus a constant to ensure its non-zero property at λ_l';
	therefore, the block diagonal matrix in (<ref>) is invertible;
	in addition, by the fact that the generalized Vandermonde matrix is invertible, it is readily seen that the linear system (<ref>) is uniquely solved.
	We have confirmed the existence of f. 
	
	For the existence of f̆, since M[
		1   0
fe^ϕ   1
	](λ) is holomorphic on λ∈ D_+, it follows that σ_2 M[
			1   0
fe^ϕ   1
		](λ̅^-1)σ_2 is holomorphic on λ∈ D_-.
	By (<ref>) and the symmetry shown in (<ref>), it is readily seen that
	
    σ_2 M[
    					1   0
    fe^ϕ   1
    				](λ̅^-1)σ_2=   σ_2Pσ_2^2e^-ϕσ_3/2σ_2^2[Y^-_1,Y^+_2]σ_2^2diag(a,1)σ_2^2e^ϕσ_3/2σ_2^2[
    					1   0
    fe^ϕ   1
    				]σ_2(λ̅^-1)
    
    			=   σ_2Pσ_2(e^-ϕ/2σ_3[Y^+_1,Y^-_2]diag(1,ă)e^ϕ/2σ_3)(λ)[
    				1   -f(λ̅^-1)e^-ϕ(λ)
    0   1
    			]
    
    			=   (σ_2P̅σ_2 P^-1)M(λ)[
    				1   -f(λ̅^-1)e^-ϕ(λ)
    0   1
    			];

	therefore, by the fact that the matrix function σ_2P̅σ_2 P^-1 is invertible, it follows that when we take f̆(λ)=-f(λ̅^-1), M[
		1   f̆e^-ϕ
0   1
	] is holomorphic on D_-. 
	We complete the proof.




 §.§ The solvability

We have constructed the rational triangular matrix functions in Proposition <ref> to remove poles of M, however since ϕ admits an essential singularity at λ→∞, the newly obtained matrix function is not normalized at the infinity, i.e., M[
	1   0
fe^ϕ   1
] diverges at λ→∞. 
Moreover, because the essential singularity of ϕ at λ=0, M[
	1   f̆e^-ϕ
0   1
] also admits an essential singularity at λ=0. 
To avoid these essential singularities, we choose a circle Σ̃_1 such that its radius is greater than 1 and the region D̃_+ between Σ̃_1 and Σ contains all the discrete spectra in D_+.
Writing Σ̃_2=(Σ̃_1)^-1 and D̃_-=(D̃_+)^-1, by the symmetry of the discrete spectrum, we claim that D̃_- contains all the discrete spectra belonging to D_-.
Taking the transformation:

    M̃=M
    		I    λ∈(D_+∪ D_-)∖(D̃_+∪D̃_-),
    [
    			1   0
    fe^ϕ   1
    		]   λ∈D̃_+,
    [
    			1   f̆e^-ϕ
    0   1
    		]   λ∈D̃_-,

we obtain a matrix-valued function M̃ that is the solution of RH problem <ref>.
In RH problem <ref>, it is easily to find that the jump matrix admits an upper-lower triangular factorization:

    V   =(I-w_-)^-1(I+w_+)
    
    	w_-   =0    on Σ̃_1,
    [
    			0   -f̆e^-ϕ
    0   0
    		]    on Σ̃_2,
    [
    			0   -(r+f)e^-ϕ
    0   0
    		]    on Σ,
    	,
    
    	w_+   =[
    			0   0
    fe^ϕ   0
    		]    on Σ̃_1,
    0    on Σ̃_2,
    [
    			0   0
    (r+f)e^ϕ   0
    		]    on Σ.



	Find a 2×2 matrix-valued function M̃ such that
	
		
  * M̃ is holomorphic on ℂ∖Σ̃, where Σ̃=Σ∪Σ̃_1∪Σ̃_2.
		
  * As λ→∞, M̃∼ I+(λ^-1).
		
  * On the jump contour Σ̃,
		
    M̃_+=M̃_-Ṽ,  Ṽ=[
    					1   0
    fe^ϕ   1
    				]   λ∈Σ̃_1,
    [
    					1   -f̆e^-ϕ
    0   1
    				]   λ∈Σ̃_2,
    [
    					1+f+r^2   (r+f)e^-ϕ
    (f+r)e^ϕ   1
    				]   λ∈Σ.

		
		
	


Introduce the Cauchy-type integral operator:

    C^Γ f(λ)=∫_Γf(ζ)/ζ-λζ/2π,

where Γ consists of finite oriented Jordan curves with finite intersection points.  


	

Denote C_±^Γf as the boundary values of C^Γf. 
It is well known that C_±^Γ are bounded operators on L^2(Γ).
Taking Γ=Σ̃=Σ∪Σ̃_1∪Σ̃_2 and a function f on the jump contour Σ̃,
we define the bounded operator:

    T^Σ̃f=C_-^Σ̃(C_+^Σ̃(fw)w_+)+C_+^Σ̃(C_-^Σ̃(fw)w_-),

where w=w_++w_-. By rationally approximating w_+, w_- in the 1st, 2nd term on the right-hand side of (<ref>), respectively, we obtain that both C_-^Σ̃(C_+^Σ̃(· w)w_+) and C_+^Σ̃(C_-^Σ̃(· w)w_-) are compact operators on L^2(Σ̃). 
Exactly, we simply rewrite (<ref>) as

    T^Σ̃=(C_w)^2,
 
where

    C_wf=C_+^Σ̃(fw_-)+C_-^Σ̃(fw_+),

and then by the compactness of T^Σ̃, -T^Σ̃=(+C_w)(-C_w) is Fredholm. 
In addition, referring to Corollary <ref> that the equation C_wμ=μ admits no non-trivial solution, the Fredholm alternative is zero, and therefore, we obtain that both -T^Σ̃ and -C_w are invertible with the relation

    (-C_w)^-1=(-T^Σ̃)^-1(+C_w).

Thus, RH problem <ref> is uniquely solvable and the solution of RH problem <ref> is

    M̃(λ)=I+∫_Σ̃[(-C_w)^-1Iw](ζ)/ζ-λζ/2π,

which is also well known as Beals-Coifman solution.





	Setting Γ⊂ℂ as a jump contour consisting of finite Jordan contours and the unit circle with at most finite intersections, we assume that the jump contour is invariant under the mapping λ↦λ̅^-1.  
	Suppose a 2×2 matrix-valued function V^Γ=V^Γ(λ) satisfies
	
		
  * V^Γ admits the symmetry on Γ∖Σ:
		
    V^Γ(λ̅^-1)=(V^Γ(λ))^†;

		
  * on λ∈Σ, 
		
    V^Γ(λ)>0;

		
  * V^Γ admits the factorization:
		
    V^Γ=(I-w^Γ_-)^-1(I+w^Γ_+),

		where w^Γ_± are some continuous 2× 2 matrix-valued functions on Γ.
	
	Define the Cauchy-type integral operator for f∈ L^2(Γ):
	
    C_w^Γf=C^Γ_+(fw^Γ_-)+C^Γ_-(fw^Γ_+),

	where
	
    w^Γ=w^Γ_++w^Γ_-,

	then, we claim that there is no non-trivial solution μ∈ L^2(Γ) for the equation:
	
    C_w^Γμ=μ.

	



	Suppose μ is a non-trivial solution. 
	It is readily seen that C^Γ(μ w^Γ)(λ) is analytic outside Γ and satisfies
	
		
  * on λ∈Γ, 
		
    C^Γ_+(μ w^Γ)=C^Γ_-(μ w^Γ)V^Γ,

		
  * as λ→∞, 
		
    C^Γ(μ w^Γ)(λ)∼(λ^-1).

	
	The holomorphism is the direct result of μ, w^Γ∈ L^2(Γ). 
	For (<ref>), it is well-known that when Γ consists of finite Jordan curves,  C^Γ_± are bounded on L^2(Γ) and 
	
    C_+-C_-=,

	therefore, 
	
		
    C^Γ_+(μ w^Γ)=C_w^Γμ+(C^Γ_+-C^Γ_-)(μ w^Γ_+)=μ(I+w^Γ_+),
       C^Γ_-(μ w^Γ)=C_w^Γμ+(C^Γ_–C^Γ_+)(μ w^Γ_-)=μ(I-w^Γ_-),

	
	i.e. we obtain (<ref>). 
	(<ref>) is the direct consequence of the compactness of Γ. 
	
	Now, we investigate the analytic function on ℂ∖Γ:
	
    Φ=C^Γ(μ w^Γ).

	By the above argument, it follows that Ψ=Ψ(λ)=Φ(λ̅^-1)^† is holomorhpic on ℂ∖Γ and satisfies that
	
		
  * on λ∈Γ, 
		
    Ψ_+(λ)=(V^Γ(λ̅^-1)^†)^-1Ψ_-(λ),

		
  * as λ→∞,
		
    Ψ(λ)∼(1),

	
	thus by (<ref>), (<ref>) and (<ref>), it follows that ΦΨ admits no jump over Γ∖Σ, that is, on λ∈Γ∖Σ,
	
    Φ_+(λ)Ψ_+(λ)=Φ_-(λ)(V(λ̅^-1)^†)^-1Ψ_+(λ)=Φ_-(λ)Ψ_-(λ).

	On Σ, Recalling the asymptotic property (<ref>) of Φ, we have that at λ=0,
	
    Ψ(λ)∼(λ),

	therefore, Φ(λ)Ψ(λ)λ^-1 is holomorphic on the unit disc {λ<1}, and by Cauchy's Integral Theorem, the integral
	
    0   =∫_ΣΦ_-(λ)Ψ_-(λ)λ^-1λ/2π=∫_Σ[Φ_-(V^Γ)^†Φ_-^†](λ)λ^-1λ/2π=∫_0^2π[Φ_-(V^Γ)^†Φ_-^†](e^θ)θ/2π.

	However, recalling that V^Γ|_Σ>0, we obtain that the right-hand side of (<ref>) vanishes if and only if Φ(λ) tends to zero almost everywhere on the boundary of {λ<1}, which by (<ref>) is equivalent to that μ is vanishing on the support of w^Γ.
	This controverts to the assumption.
	We have proved the results.



	Set the Cauchy-type integral operator C_w as defined in (<ref>).
	There is no non-trivial solution in L^2(Σ̃) for the equation below
	
    C_wμ=μ.

	



	(<ref>) shows that Ṽ admits a proper factorization.
	And since r∈ H^1 and f is a rational function, by (<ref>) and (<ref>), w_± are 1/2-Hölder continuous on Σ̃.
	By the choice of f and f̆ in Proposition <ref>, Ṽ satisfies that on Σ̃_1∪Σ̃_2,
	
    Ṽ(λ̅^-1)=Ṽ(λ)^†.

	Recalling that Ṽ is self-adjoint on Σ, it follows that
	
    Ṽ|_Σ>0.

	Thus, the results is the consequence of Lemma <ref>.
	We have proved the results.



	We have proved the solvability of RH problem <ref>.
	And by (<ref>) and (<ref>), the solution of the initial-value problem is written by M̃ in the form,
	
    q_n(t)=[M̃(0,n+1,t)]_1,2.

	Especially, under the reflectionless condition, i.e., r≡0, the RH problem is uniquely solved, which is equivalent to the solvability of RH problem <ref> with r≡0, and we denote the solution with the discrete spectral data (𝒵,) as 
	M^(𝒵,). 
	And the soliton solution is
	
    q_n(t|𝒵,)=[M^(𝒵,)(0,n+1,t)]_1,2.

	
	It is readily seen that M^(𝒵,) is unique. 
	In fact, we assume that there is another solution M^(𝒵,)' solves RH problem <ref> for r≡0, then we have that by Proposition <ref>, M^(𝒵,)(M^(𝒵,)')^-1 is entire, and by Liouville's Theorem, M^(𝒵,)(M^(𝒵,)')^-1≡ I on the whole plane, i.e., 
    M^(𝒵,)≡ M^(𝒵,)'.

	



	
	

	
	
	
		
			
			
			
		
	
	
		
		
	
	
		
		
	
	
		
		
	
	
		
		
	
	
		
		
	
	
	




§ LONG-TIME ASYMPTOTICS FOR Ξ<1

Previously, we have presented the direct scattering transform and constructed the RH problem for the initial-value problem that is unique solvable. 
When ξ=n/2t<1, there are two stationary phase points on the jump contour

    S_1=-ξ+√(1-ξ^2),   S_2=-ξ-√(1-ξ^2),

and the Taylor's expansions of ϕ at S_1 and S_2 are

    ϕ(λ,n,t)-ϕ(S_j,n,t)∼   (-1)^j t√(1-ξ^2)S_j^-2(λ-S_j)^2+(λ-ξ^3)
    	,  λ→ S_j.

In this section, we take a series of RH transformations for M. 
Under the condition ξ=n/2t<C_0 for some positive constant C_0<1, it asserts that the asymptotic behavior of q_n(t) is solitonic and oscillatory, that is, as t→+∞.
	
    q_n(t)   =(∏_l'=1^lλ_l'^2α_l')e^∫_S_1^S_2ln(1+r(s)^2)/2π s s(q^𝒵_ξ_n(t)+t^-1/2q^Osc_n(t)+(t^-3/4)),

where q^𝒵_ξ is the solitonic part

    q^𝒵_ξ_n(t)=[M^(𝒵_ξ)]_1,2(0,n+1,t),

and q^Osc is the oscillatory part

    q^Osc_n-1(t)=   -1/√(2)(1-ξ^2)^1/4(T_1^2M^(𝒵_ξ)_1,2(S_1,n,t)γ_2(r(S_1))
       +T_1^-2M^(𝒵_ξ)_1,1(S_1,n,t)γ_1(r(S_1))+T_2^2M^(𝒵_ξ)_1,2(S_2,n,t)γ_1(r(S_2))
    		+T_2^-2M^(𝒵_ξ)_1,1(S_2,n,t)γ_2(r(S_1)))

In addition, M^(𝒵_ξ) is defined as (<ref>), and for j=1,2

    (γ_j(r(S_1)))=(-1)^j((r(S_1))-π/4-(Γ(iν(r(S_1))))),
       (γ_j(r(S_2)))=(-1)^j-1((r(S_2))+π/4+(Γ(iν(r(S_2))))).




At the beginning, we introduce the scalar function that admits a jump on an upper-half arc S_1S_2 of the unit circle Σ,

    T(λ)=T(λ,ξ)=e^∫_S_1^S_2ln(1+r(ζ)^2)/ζ-λζ/2π∏_λ_l∈𝒵_ξ^+(λ-λ_l/λ-λ̅_l^-1)^α_l,  ξ∈(-1,1),

where 𝒵_ξ^+=𝒵∩{Reϕ>0} and the integral is on the arc from S_2 to S_1.
We assert that T(λ) admits properties shown in the following proposition. 

	The function defined by (<ref>) satisfies the following properties:
	
		
  * T(λ) is a meromorphic function on λ∈ℂ∖S_1S_2, and admits α_j-order pole at λ=λ_j.
		
  * For λ∈ℂ∖S_1S_2, T(λ) admits the symmetry
		
    T(λ̅^-1)=T(0)T^-1(λ)

		
  * For λ∈S_1S_2, the boundary value T_±(λ) of T(λ) satisfy
		
    T_+(λ)=T_-(λ)(1+r(λ)^2).

		
  * As λ→∞,
		
    T(λ)∼ 1+(λ^-1).

		
  * On the neighborhood of S_j, T(λ) admits the following asymptotic property,
		
    T(λ)∼∏_λ_l∈𝒵_ξ^+(S_j-λ_l/S_j-λ̅_l^-1)^α_l(λ-S_2/λ-S_1)^ν_je^α_j(S_j)+(λ-S_j^1/2),

		where 
		
    ν_j=ln(1+r(S_j)^2)/2π,  α(λ)=∫_S_2^S_1ln(1+r(s)^2)-ln(1+r(S_j)^2)/s-λ s/2π.

	


	(a) is directly verified by (<ref>). 
	For (b), we have
	
    T(λ)T(λ̅^-1)   =exp(∫_S_1^S_2ln(1+r(ζ)^2)/ζ-λζ/2π+∫_S_1^S_2ln(1+r(ζ)^2)/ζ-λ̅^-1ζ/2π)∏_λ∈𝒵_ξ^+(λ-λ_j/λ-λ̅_j^-1·λ^-1-λ_j^-1/λ^-1-λ̅_j)^α_j
       =exp(∫_S_1^S_2ln(1+r(ζ)^2)/ζ-λζ/2π+∫_S_1^S_2ln(1+r(ζ)^2)/(ζ̅^-1-λ̅^-1)ζ̅^2ζ̅/2π)∏_λ∈𝒵_ξ^+λ_j^-2α_j
       =exp(∫_S_1^S_2ln(1+r(ζ)^2)/ζ-λ(1-λ/ζ)ζ/2π)∏_λ∈𝒵_ξ^+λ_j^-2α_j
       =exp(∫_S_1^S_2ln(1+r(ζ)^2)ζ^-1ζ/2π)∏_λ∈𝒵_ξ^+λ_j^-2α_j=T(0).

	For (c), by Sokhotski-Plemelj formula, we have that for any λ∈S_1S_2,
	
    ∫_S_1^S_2ln(1+r(ζ)^2)/ζ-λ_+ζ/2π-∫_S_1^S_2ln(1+r(ζ)^2)/ζ-λ_-ζ/2π=ln(1+r(λ)^2),

	where λ_± means the ± limits of λ∈S_1S_2. 
	Since r∈ H^1, (d) is the direct result of the definition (<ref>).
	
	The remaining is (e). We refer to a conformal mapping (<ref>) that transforms the unit circle to the straight line, and then discuss it on the real line ℝ. We take
	
    δ(λ)=δ(λ,ξ)=T(λ)∏_λ_l∈𝒵_ξ^+(λ-λ_l/λ-λ̅_l^-1)^-α_l=e^∫_S_1^S_2ln(1+r(ζ)^2)/ζ-λζ/2π.

	Changing the variable on Σ: λ→λ', s→ s',
	
    λ=√(S_1S_2)√(S_1)+√(S_2)λ'/√(S_2)+√(S_1)λ',   s=√(S_1S_2)√(S_1)+√(S_2)s'/√(S_2)+√(S_1)s',

	we obtain that
	
    ∫_S_2^S_1f(s)-f(S_1)/(2π)(s-λ)ds=√(S_2)+√(S_1)λ'/2π∫_-∞^0f(s)-f(S_1) s'/(√(S_2)+√(S_1)s')(s'-λ'),

	where f(s)=ln(1+| r(s)|^2).
	Since r(s)∈ H^1⊂ L^∞(Σ), it naturally follows that
	
    F(s')=f(s)-f(S_1)/√(S_2)+√(S_1)s',   s'≤0,
    
    			0,   s'>0,

	belongs to H^1(ℝ) and F(s'=0)=0;
	therefore, we apply Lemma 23.3 in <cit.> and get that for any λ∈ℂ∖ℝ
	
    |∫_-∞^+∞F(s')/(2π)(s'-λ')ds'-∫_-∞^+∞F(s')/2π s'ds'|≲∥ F∥_H^1|λ'|^1/2,

	i.e.
	
    δ(λ)∼(λ-S_2/λ-S_1)^ν_je^α_j(S_j)+(λ-S_j^1/2),

	which is an equivalence of (<ref>).
	These complete the proof. 



 §.§ Transform to M^(1)

By the property of functions M and T, as usual, we take the RH transformation:

    M^(1)=MT^σ_3.

In the new RH problem below, the jump matrix admits a proper factorization and the exponential functions e^ϕ, e^-ϕ in the residue condition do not blow up.

We assert that (<ref>) solves the following RH problem.

	Find a 2×2 matrix-valued function M^(1) s.t.
	
		
  * M^(1)(λ) is meromorphic on ℂ∖Σ.
		
  * As λ→∞,
		
    M^(1)∼ I+(λ^-1)

		
  * On λ∈Σ, 
		
    M^(1)_+=M^(1)_-V^(1),

		where
		
    V^(1)=[
    					1+r^2   rT^-2e^-ϕ
    rT^2e^ϕ   1
    				]   λ∈Σ-S_1S_2,
    [
    					1   r̅/1+r^2T_+^-2e^-ϕ
    r/1+r^2T_-^2e^ϕ   1+r^2
    				]   λ∈S_1S_2.

		
  * Residue condition: On the poles 𝒵∪𝒵^-1, we have
		
			
    λ=λ_l'_-αM^(1)=∑_j=0^α_l'-α∂_λ^α_l'-α-jM^(1)(λ_l')/(α_l'-α-j)![
    						0   p^T_λ_l',je^-ϕ(λ_l')
    0   0
    					]    λ_j∈𝒵_ξ^+,
    ∑_j=0^α_l'-α∂_λ^α_l'-α-jM^(1)(λ_l')/(α_l'-α-j)![
    						0   0
    p^T_λ_l',je^ϕ(λ_l')   0
    					]    λ_j∈𝒵∖𝒵_ξ^+,
    				
       λ=λ̅_l'^-1_-αM^(1)=∑_j=0^α_l'-α∂_λ^α_l'-α-jM^(1)(λ̅_l'^-1)/(α_l'-α-j)![
    						0   0
    p^T_λ̅_l'^-1,je^ϕ(λ̅_l'^-1)   0
    					]   λ_j∈𝒵_ξ^+,
    ∑_j=0^α_l'-α∂_λ^α_l'-α-jM^(1)(λ̅_l'^-1)/(α_l'-α-j)![
    						0   p^T_λ̅_l'^-1,je^-ϕ(λ̅_l'^-1)
    0   0
    					]   λ_j∈𝒵∖𝒵_ξ^+,
 
			where p^T_λ_l',j and p^T_λ̅_l'^-1,j are polynomials of (n,t) of order at most j.
		
		
	

That M^(1)=MT^σ_3 satisfies the analyticity, normalization, and jump condition in RH problem <ref> is easily verified by RH problem <ref> and the first three items in Proposition <ref>.
The remaining to check is the Residue condition on 𝒵. 
On the one hand for λ_l'∈𝒵∖𝒵_ξ^+, 
since T is analytic at λ_l', we calculate the Laurent's coefficients of M^(1)_2,

    λ=λ_l'_-αM^(1)_1   =∂_λ^α_l'-α((·-λ_l')^α_l'M_1T)(λ_l')/(α_l'-α)!
       =∑_j=0^α_l'-α∂_λ^α_l'-α-j((·-λ_l')^α_l'M_1)(λ_l')/(α_l'-α-j)!∂_λ^jT(λ_l')/j!
       =∑_j=0^α_l'-α∂_λ^jT(λ_l')/j!λ=λ_l'_-α-jM_1.

By (<ref>) and Leibniz Rule, it follows that

    ∂_λ^α M_2(λ_l')/α!=∑_j=0^α∂_λ^α-j M_2^(1)(λ_l')/(α-j)!∂_λ^jT(λ_l')/j!.

Thus, considering (<ref>), (<ref>), and (<ref>), we obtain that there are polynomials p^T_λ_l',j of (n,t) of order at most j, j=0,…,α_l'-1, such that

    λ=λ_l'_-αM^(1)_1=∑_j=0^α_l'-α∂_λ^α_l'-α-jM^(1)_2(λ_l')/(α_l'-α-j)!p^T_λ_l',je^ϕ(λ_l'),  α=1,…,α_l'.

On the other hand for λ_l'∈𝒵_ξ^+, by the transform (<ref>), we see that at any λ_l'∈𝒵_ξ^+, the pole of M^(1) is on the second column while the first column is holomorphic. 
Calculate the Laurent's coefficients,

    λ=λ_l'_-αM^(1)_2=∑_j=0^α_l'-αλ=λ_l'_-α-jT^-1∂_λ^jM_2(λ_l')/j!.

By Leibniz Rule, We also have that for any natural number α<α_l',

    ∂_λ^α M^(1)_1(λ_l')/α!   =(α!)^-1∂_λ^α[(·-λ_l')^α_l'M_1T/(·-λ_l')^α_l'](λ_l')
       =∑_j=0^α∂_λ^α-j[(·-λ_l')^α_l'M_1](λ_l')/(α-j)!∂_λ^j[(·-λ_l')^-α_l'T]/j!
       =∑_j=0^αλ=λ_l'ℒ_α-j-α_l'M_1∂_λ^j[(·-λ_l')^-α_l'T]/j!.

By (<ref>), (<ref>), and (<ref>), it is readily seen that there are polynomials p^T_λ_l',j of (n,t) of order at most j such that for any λ_l'∈𝒵_ξ^+,

    λ=λ_l'_-αM^(1)_2=∑_j=0^α_l'-α∂_λ^α_l'-α-jM^(1)_1(λ̅_l'^-1)/(α_l'-α-j)!p^T_λ_l',je^-ϕ(λ_l'),  α=1,…,α_l'.

Combining (<ref>) and (<ref>), we obtain the residue condition (<ref>), and we also verify (<ref>) in the similar way. 
Thus, we complete the assertion. 





 §.§ Split the jump contour

In this part, applying the ∂̅-RH transformation, we split the jump contour Σ along the arcs S_1S_2 and Σ∖S_1S_2. 
In the follow, properly choose the jump contour Σ^(2)=Σ^(2)_1∪Σ^(2)_3∪Σ^(2)_4∪Σ^(2)_6as shown in FIGURE <ref> such that it consists of line segments S_j(1+e^±π/4(-ϵ_0,ϵ_0)) and arcs centering at the original.
Positive number ϵ_0 is so small enough that the ⋃_k=1,3,4,6Ω_k is away from the discrete spectrum.

Introduce the continuous functions on ℂ∖{0}

	
    R_1(λ)=r(λ/λ),   R_6(λ)=R_1(λ̅^-1),
       R_3(λ)=(r̅/1+r^2)(λ/λ),   R_4(λ)=R_3(λ̅^-1),


and the 2×2 matrix-valued function

    M^(2)=M^(1)ℛ,  ℛ=
    		I   λ∈Ω_2∪Ω_5,
    [
    			1   0
    -R_1T^2e^ϕ   1
    		]   λ∈Ω_1,
    [
    			1   -R_3T^-2e^-ϕ
    0   1
    		]   λ∈Ω_3,
    [
    			1   0
    R_4T^2e^ϕ   1
    		]   λ∈Ω_4,
    [
    			1   R_6T^-2e^-ϕ
    0   1
    		]   λ∈Ω_6.

Utilizing RH problem <ref> and the transformation (<ref>), ones easily check that M^(2) uniquely solves the following ∂̅-RH problem, where the jump contour Σ^(2) as shown in FIGURE <ref>.

	Find a function such that
	
		
  * M^(2) is continuous on ℂ∖{Σ^(2)∪𝒵∪𝒵^-1} and holomorphic on (Ω_2∪Ω_5)∖{𝒵∪𝒵^-1}.
		
  * As λ→∞, M∼ I+(λ^-1).
		
  * On Σ^(2),
		
    M^(2)_+=M^(2)_-V^(2),

		where
		
    V^(2)=[
    					1   0
    R_1T^2e^ϕ   1
    				]   λ∈Σ^(2)_1,
    [
    					1   R_3T^-2e^-ϕ
    0   1
    				]   λ∈Σ^(2)_3,
    [
    					1   0
    R_4T^2e^ϕ   1
    				]   λ∈Σ^(2)_4,
    [
    					1   R_6T^-2e^-ϕ
    0   1
    				]   λ∈Σ^(2)_6.

		
  * On the poles on 𝒵∪𝒵^-1, 
		
			
    λ=λ_l'_-αM^(2)=∑_j=0^α_l'-α∂_λ^α_l'-α-jM^(2)(λ_l')/(α_l'-α-j)![
    						0   p^T_λ_l',je^-ϕ(λ_l')
    0   0
    					]    λ_j∈𝒵_ξ^+,
    ∑_j=0^α_l'-α∂_λ^α_l'-α-jM^(2)(λ_l')/(α_l'-α-j)![
    						0   0
    p^T_λ_l',je^ϕ(λ_l')   0
    					]    λ_j∈𝒵∖𝒵_ξ^+,
    				
       λ=λ̅_l'^-1_-αM^(2)=∑_j=0^α_l'-α∂_λ^α_l'-α-jM^(2)(λ̅_l'^-1)/(α_l'-α-j)![
    						0   0
    p^T_λ̅_l'^-1,je^ϕ(λ̅_l'^-1)   0
    					]   λ_j∈𝒵_ξ^+,
    ∑_j=0^α_l'-α∂_λ^α_l'-α-jM^(2)(λ̅_l'^-1)/(α_l'-α-j)![
    						0   p^T_λ̅_l'^-1,je^-ϕ(λ̅_l'^-1)
    0   0
    					]   λ_j∈𝒵∖𝒵_ξ^+.
 
			
		
		
  * ∂̅ condition: On Ω_1∪Ω_3∪Ω_4∪Ω_6,
		
    ∂̅M^(2)=M^(2)∂̅ℛ.

	




Rewrite λ∈ℂ in the polar coordinates: λ=ρ e^θ.
We note that the function R_j are constant along the ray ℝ^+e^θ, thus utilizing the formula ∂̅=e^iθ(∂_ρ+ρ^-1∂_θ)/2, we calculate

	
    ∂̅R_1   =/2ρ^-1e^θ∂_θ(r(e^θ)),
    ∂̅R_3   =/2ρ^-1e^θ∂_θ(r(e^θ))-r(e^θ)^2∂_θ(r(e^θ))/(1+r(e^θ)^2)^2.


Through (<ref>), it is readily seen that the L^1-norm of ∂̅R_j on any circle {λ>0} is well dominated by the norm of r∈ H^1, which is applied to those estimates in Section <ref>.




 §.§ The factorization of the ∂̅-RH problem

In this part, we decompose the solution for ∂̅-RH problem into a product of solutions for an RH problem and a ∂̅ problem. i.e.,

    M^(2)=M^(2)_DM^(2)_RH.



	Find a 2×2 matrix-valued function M^(2)_RH satisfying the ∂̅-RH problem <ref> with ∂̅_λℛ≡ 0.


	Find a 2×2 matrix-valued function M^(2)_D such that
	
		
  * M^(2)_D is continuous on ℂ and analytic on Ω_2∪Ω_5.
		
  * ∂̅ condition: On Ω_1∪Ω_3∪Ω_4∪Ω_6, 
		
    ∂̅M^(2)_D=M^(2)_Dℛ̃,  ℛ̃=M^(2)_RH∂̅ℛ(M^(2)_RH)^-1.

	
	




  §.§.§ Reflection coefficient vanishes


In this part, we discuss the RH problem <ref> when the reflection coefficient vanishing and denote its solution as M^(𝒵)=M^(𝒵)(λ,n,t).

	Find a 2×2 matrix-valued function M^(𝒵) solves RH problem <ref> with r≡0.

For any λ_l'∈𝒵_ξ^+, by the method to prove Proposition <ref>, there are two rational functions f_λ_j^𝒵=f_λ_j^𝒵(λ), f_λ̅_j^-1^𝒵=f_λ̅_j^-1^𝒵(λ) such that M^(𝒵)[
	1   f_λ_l'^ξe^-ϕ
0   1
], M^(𝒵)[
	1   0
f_λ̅_l'^-1^ξe^ϕ   1
] are holomorphic at λ=λ_l', λ=λ̅_l'^-1, respectively.
Similarly, for λ_l'∈𝒵∖(𝒵_ξ^+∪𝒵_ξ), there are rational functions f_λ_j^𝒵=f_λ_j^𝒵(λ), f_λ̅_j^-1^𝒵=f_λ̅_j^-1^𝒵(λ) such that M^(𝒵)[
	1   0
f_λ_l'^ξe^ϕ   1
], M^(𝒵)[
	1   f_λ̅_l'^-1^ξe^-ϕ
0   1
] are holomorphic at λ=λ_l', λ=λ̅_l'^-1, respectively.
By these facts, choosing a small enough positive constant ϵ_1, we construct the matrix-valued function

    M^(3)=M^(𝒵)[
    			1   f_λ_l'^ξe^-ϕ
    0   1
    		],   λ-λ_l'<ϵ_1,  λ_l'∈𝒵_ξ^+,
    [
    			1   0
    f_λ̅_l'^-1^ξe^ϕ   1
    		],   λ-λ̅_l'^-1<ϵ_1,  λ_l'∈𝒵_ξ^+,
    [
    			1   0
    f_λ_l'^ξe^ϕ   1
    		],   λ-λ_l'<ϵ_1,  λ_l'∈𝒵∖(𝒵_ξ^+∪𝒵_ξ),
    [
    			1   f_λ̅_l'^-1^ξe^-ϕ
    0   1
    		],   λ-λ̅_l'^-1<ϵ_1,  λ_l'∈𝒵∖(𝒵_ξ^+∪𝒵_ξ),
    
    		I,   otherwise.

Clearly, by the choice of M^(3), it solves a RH problem without residue condition that the jump condition is over Σ^(3)=⋃_λ∈𝒵∖𝒵_ξ{λ-λ_l'=ϵ_1 or λ-λ̅_l'^-1=ϵ_1}, and seeing the sign table of ϕ in FIGURE <ref>, if ϵ_1 is small enough, there exists a positive constant C such that

    V^(3)-I_L^∞(Σ^(3))≤ e^-Ct.

Explicitly, the RH problem reads as follows.

	Find a 2×2 matrix-valued function M(3), such that
	
		
  * M^(3) is holomorphic on ℂ∖(Σ^(3)∪𝒵_ξ∪𝒵_ξ^-1).
		
  * As λ→∞, M^(3)∼ I+(λ^-1).
		
  * On Σ^(3),
		
    M^(3)_+=M^(3)_-V^(3),

		where
		
    V^(3)=[
    					1   f_λ_l'^ξe^-ϕ
    0   1
    				],   λ-λ_l'=ϵ_1,  λ_l'∈𝒵_ξ^+,
    [
    					1   0
    f_λ̅_l'^-1^ξe^ϕ   1
    				],   λ-λ̅_l'^-1=ϵ_1,  λ_l'∈𝒵_ξ^+,
    [
    					1   0
    f_λ_l'^ξe^ϕ   1
    				],   λ-λ_l'=ϵ_1,  λ_l'∈𝒵∖(𝒵_ξ^+∪𝒵_ξ),
    [
    					1   f_λ̅_l'^-1^ξe^-ϕ
    0   1
    				],   λ-λ̅_l'^-1=ϵ_1,  λ_l'∈𝒵∖(𝒵_ξ^+∪𝒵_ξ).

		
  * The residue condition is as that of RH problem <ref> on 𝒵_ξ if 𝒵_ξ∅.
	


Set 

    M^(𝒵_ξ)=
    		I   𝒵_ξ=∅,
    
    		M^(𝒵_ξ,^T_ξ)   𝒵_ξ∅,

where M^(𝒵_ξ,^T_ξ) is as defined in Remark <ref>, and the discrete spectral data (𝒵_ξ,^T_ξ) is the restriction of (𝒵,^T) on 𝒵_ξ, where (𝒵,^T) is the discrete spectral data of RH problem <ref>.
Recalling the discussion in Remark <ref>, M^(𝒵_ξ) uniquely exists. 
Referring to Proposition <ref>, for any λ_l'∈𝒵_ξ, there is a rational function f_λ_l'=f_λ_l'(λ) such that

    M^(3)[
    		1   0
    f_λ_l'   1
    	],   M^(𝒵_ξ)[
    		1   0
    f_λ_l'   1
    	]

are holomorphic at λ=λ_l', and then it follows that

    M̃^(3)=M^(3)(M^(𝒵_ξ))^-1

is analytic at λ_l'.
Similarly, it is also analytic at λ̅_l'^-1.
Based on the above arguments, it is clear that (<ref>) solves a small-norm RH problem without poles

	Find a 2×2 matrix-valued function M̃^(3) such that
	
		
  * M̃^(3) is holomorphic on Σ^(3).
		
  * As λ→∞, M̃^(3)∼ I+(λ^-1).
		
  * On Σ^(3), M̃^(3)_+=M̃^(3)_-V^(3,R), where V^(3,R)=M^(𝒵_ξ)V^(3)(M^(𝒵_ξ))^-1.
	

By (<ref>) and the boundedness of M^(𝒵_ξ) on Σ^(3), for t≫0, the jump matrix V^(3,R) decays to the identity exponentially,

    V^(3,R)-I_L^∞(Σ^(3))≲ e^-Ct.

Then by the small norm theory for the RH problems <cit.>, it follows that

    M^(3)(λ)=(I+∫_Σ^(3)μ^(3)(s)(V^(3)(s)-I)/s-λ s/2π)M^(𝒵_ξ)(λ),

where μ^(3) is the solution in L^2(Σ^(3)) of 

    (-C_V^(3,R)-I)μ^(3)=I,   C_V^(3,R)-I=C^Σ^(3)_-(·(V^(3,R)-I)).

It is well understood that the projection operator C^Σ^(3)_- is bounded on L^2(Σ^(3)), then it follows that the bound C_V^(3)-I_L^2(Σ^(3)) of the linear operator of C_V^(3)-I is dominated by

    C_V^(3,R)-I_L^2(Σ^(3))≤C^Σ^(3)_-_L^2(Σ^(3))V^(3,R)-I_L^∞(Σ^(3))≲ e^-Ct.

By (<ref>), for t≫0, the solution μ^(3)=(-C_V^(3,R)-I)^-1I for (<ref>) uniquely exists and as t→∞,

    μ^(3)_L^2(Σ^(3))≤(1).

By (<ref>), (<ref>), (<ref>) and (<ref>), it follows that

    M^(𝒵)(0)=(I+(e^-Ct))M^(𝒵_ξ)(0).

and also

    M^(𝒵)(S_j)=(I+(e^-Ct))M^(𝒵_ξ)(S_j),   j=1,2.





  §.§.§ Solvability of M^(2)_RH


In this part, we discuss the solvability of M^(2)_RH. 
We choose small discs U_j as the neighborhood of S_j. 
We rewrite M^(2)_RH in the form

    M^(2)_RH=
    		EM^(𝒵),   λ∈ℂ∖(Σ^(2)∪ U_1∪ U_2),
    
    		EM^(𝒵)M^(2,j)_loc,   λ∈ U_j∖Σ^(2),

where M^(𝒵) is the solution of RH problem <ref>, M^(2,j)_loc is the solution of RH problem <ref>, and E admits the RH problem <ref>.

	Find a 2×2 matrix-valued function M^(2,j)_loc such that
	
		
  * M^(2,j)_loc is analytic on ℂ∖Σ^(2,j).
		
  * M^(2,j)_loc∼ I+(λ^-1) as λ→∞. 
		
  * M^(2,j)_loc admits the jump condition on Σ^(2,j)
		
    (M^(2,j)_loc)_+=(M^(2,j)_loc)_-V^(2,j)_loc,    on Σ^(2,j).

		where
		
			
    V^(2,1)_loc=V^(2,1)_loc(λ)=[
    						1   0
    T_1^2r(S_1)ζ^-2ν_1e^ζ^2/2   1
    					]   λ∈Σ^(2,1)_1,
    [
    						1   T_1^-2r(S_1)/1+r(S_1)^2ζ^2ν_1e^-ζ^2/2
    0   1
    					]   λ∈Σ^(2,1)_2,
    [
    						1   0
    T_1^2r(S_1)/1+r(S_1)^2ζ^-2ν_1e^ζ^2/2   1
    					]   λ∈Σ^(2,1)_3,
    [
    						1   T_1^-2r(S_1)ζ^2ν_1e^-ζ^2/2
    0   1
    					]   λ∈Σ^(2,1)_4,
    				
       V^(2,2)_loc=V^(2,2)_loc(λ)=[
    						1   T_2^-2r(S_2)ζ^-2ν_2e^ζ^2/2
    0   1
    					]   λ∈Σ^(2,2)_1,
    [
    						1   0
    T_2^2r(S_2)/1+r(S_2)^2ζ^2ν_2e^-ζ^2/2   1
    					]   λ∈Σ^(2,2)_2,
    [
    						1   T_2^-2r(S_2)/1+r(S_2)^2ζ^-2ν_2e^ζ^2/2
    0   1
    					]   λ∈Σ^(2,2)_3,
    [
    						1   0
    T_2^2r(S_2)ζ^2ν_2e^-ζ^2/2   1
    					]   λ∈Σ^(2,2)_4,

		
		and 
		
    λ=β_jζ+S_j,  β_j=(-1)^j/√(2)(1-ξ^2)^-1/4t^-1/2S_j,
       T_j=∏_λ_l∈𝒵_ξ^+(S_j-λ_l/S_j-λ̅_l^-1)^α_l(-2√(2)(1-ξ^2)^3/4t^1/2)^(-1)^j-1ν_j( S_j)^(-1)^jν_j/e^ t[1+√(1-ξ^2)+ξ((1+(-1)^j)/4π+(-1)^j-1arcsinξ)]-α_j(S_j).

	


	Find a 2×2 matrix-valued function such that
	
		
  * E is analytic on Σ^E=∂ U_1∪∂ U_2∪(Σ^(2)).
		
  * As λ→∞, E(λ)∼ I+(λ^-1).
		
  * On Σ^E,
		
    E_+=E_-V^E,

		where 
		
    V^E=
    				M^(𝒵)V^(2)(M^(𝒵))^-1,   λ∈Σ^(2)∖(∂ U_1∪∂ U_2),
    
    				M^(𝒵)M^(2,j)_locV^(2)(V^(2,j)_loc)^-1(M^(𝒵)M^(2,j)_loc)^-1,   λ∈Σ^(2)∩ U_j,
    
    				M^(𝒵)M^(2,j)_loc(M^(𝒵))^-1,   λ∈∂ U_1∪∂ U_2.

	

Using (<ref>), we obtain the solution M^(2)_RH of RH problem <ref> by constructing M^(𝒵), M^(2,j)_loc and E one by one. 

To see the solvability of M^(2)_RH, we first investigate the part of M^(𝒵) and M^(2,j)_loc. 
On the one hand, recalling the discussion in Section <ref>, we see that M^(𝒵) is uniquely solved for t≫0.
On the other hand, M^(2,j)_loc can be represented by 

	
    M^(2,1)_loc(λ)=T_1^-σ_3M^(PC)(ζ,r(S_1))T_1^σ_3,
       M^(2,2)_loc(λ)=σ_1T_2^σ_3M^(PC)(ζ,r(S_2))T_2^-σ_3σ_1,


where M^(PC) the well-known as the model RH problem <ref>. 


Based on the solvability of M^(𝒵) and M^(2,j)_loc, we come to the error part of E. 
By (<ref>), (<ref>), (<ref>) and the sign table of ϕ, it is readily seen that for some positive constant ϵ

    V^(2)(λ)-I≤ e^-ϵ t,  λ∈Σ^(2)∖(U_1∪ U_2).

By (<ref>), (<ref>) and Proposition <ref>, we obtain that

    V^(2)(V^(2,j)_loc)^-1-I≲V^(2)-V^(2,j)_loc≲ t^-1/4ζ^1/2e^-ζ^2/2,
    	  λ∈Σ^(2)∩ U_j.

By (<ref>), (<ref>) and (<ref>), it follows that

    (M^(2,j)_loc)^-1-I≲ t^-1/2,  λ∈∂ U_j.

Considering (<ref>), (<ref>) and (<ref>), by the boundedness of M^(𝒵) on Σ^E, we conclude that

    V^E-I≲ t^-1/4,

that is, E is the solution of a small-norm RH problem.
Recalling that C_-^Σ^E is a bounded operation on L^2(Σ^E), by (<ref>), it is readily seen that the Cauchy-type integral operator

    C_V^E-I: C_V^E-If=C_-(f(V^E-I))

is consequently bounded on L^2(Σ^E), and 

    C_V^E-I_L^2(Σ^E)≲ t^-1/4.

(<ref>) indicates that as t→+∞, the operator decays, and for sufficiently large t, (-C_V^E-I)^-1 exists and is bounded on L^2(Σ^E).
By RH problem <ref>, it follows that the solution can be written in the integral form:

    E(λ,n,t)=I+∫_Σ^E[(-C_V^E-I)^-1I](s,n,t)(V^E(s,n,t)-I)/s-λ s/2π.


Now, we focus on the estimate of E(0)=E(0,n,t), and assert that as t→+∞,

    E(0,n,t)-I=   t^-1/2M^(𝒵_ξ)(S_1,n,t)/√(2)(1-ξ^2)^1/4[
    			0   -γ_1(r(S_1))/T_1^2
    γ_2(r(S_1))T_1^2   0
    		](M^(𝒵_ξ)(S_1,n,t))^-1
         +t^-1/2M^(𝒵_ξ)(S_2,n,t)/√(2)(1-ξ^2)^1/4[
    			0   -γ_2(r(S_2))/T_2^2
    γ_1(r(S_2))T_2^2   0
    		](M^(𝒵_ξ)(S_2,n,t))^-1+(t^-3/4).

Taking the value of (<ref>) at λ=0, 

    E(0)=I+∮_∂ U_1∪∂ U_2s^-1[(-C_V^E-I)^-1C_V^E-II](s)(V^E(s)-I) s/2π
         +∮_Σ^(2)s^-1[(-C_V^E-I)^-1I](s)(V^E(s)-I) s/2π+∮_∂ U_1∪∂ U_2s^-1(V^E(s)-I) s/2π.

In (<ref>), the first integral is dominated by (t^-3/4) because of (<ref>) and (<ref>), and using (<ref>) and (<ref>), it follows that the second integral is constrained by (t^-3/4).
Then we rewrite (<ref>) in the asymptotic form that as t→+∞,

    E(0,n,t)=I+∮_∂ U_1∪∂ U_2s^-1(V^E(s,n,t)-I) s/2π+(t^-3/4).

Using (<ref>), (<ref>), (<ref>), (<ref>) and the Residue Theorem, we formulate the integral term in (<ref>), and consequently obtain (<ref>).















	If we set the transformation on ℂ, j=1,2: λ=β_jζ+S_j, where β_j are defined as (<ref>) and S_j are the stationary phase points,
	then when ξ∈(-1,1), we have the following asymptotic properties that as λ→ S_j,
	
    R_1(λ)∼ r(S_j)+(t^-1/4ζ^1/2),   R_3(λ)∼r(S_j)/1+r(S_j)^2+(t^-1/4ζ^1/2),
       T^2(λ)e^ϕ(λ)∼ T_j^2ζ^(-1)^j2ν_je^-(-1)^jζ^2/2+(t^-1/4ζ^1/2e^-(-1)^jζ^2/2),

	


	On the one hand, recalling that r∈ H^1, we have that for λ on the neighborhood of S_j,
	
    R_1(λ)-r(S_j)=r(β_jζ+S_j/β_jζ+S_j)-r(S_j)≲β_jζ+S_j/β_jζ+S_j-S_j^1/2≲β_jζ^1/2,

	and we proved the result for R_1 in (<ref>).
	In addition, when r∈ H^1, it is readily seen that r/1+r^2∈ H^1, therefore, we similarly obtain R_3(λ)'s result in (<ref>) and complete the proof of (<ref>).
	On the other hand, the result (<ref>) is a consequence of (<ref>), (<ref>) and (<ref>).
	This completes the proof.


We have proved that M^(2)_RH is uniquely solved,
then we claim that M^(2)_RH is invertible.
It is readily seen that M^(2)_RH has no jump over ℂ and tends to 1 at the infinity.
In addition, utilizing the technique to prove Proposition <ref>, we can remove the pole of M^(2)_RH at λ=λ_l' by a triangular function with constant determinant, 
that is, the singularities of M^(2)_RH at λ=λ_l' are removable.
In the similar way, its singularities at λ=λ̅_l'^-1 are also removable. 
By Liouville's Theorem, M^(2)_RH is the constant 1 over ℂ, that is, M^(2)_RH is invertible.

Defining the function

    M^(2)_D=M^(2)(M^(2)_RH)^-1.

Since M^(2)_RH is invertible, the definition (<ref>) is well posed,
and comparing ∂̅-RH problem <ref> to RH problem <ref>, we find that M^(2)_D is the solution of ∂̅ problem <ref>. 



  §.§.§ Estimate on the ∂̅ problem

Since we have already proved the existence of M^(2)_RH, in this part, we concentrate on some estimates of M^(2)_D. 

That M^(2)_D solves ∂̅ problem <ref> is equivalent to the integral equation

    M^(2)_D(λ)=I+∬_ℂ[M^(2)_Dℛ̃](s)/s-λ L(s)/π,

where L(s) is the  Lebesgue's measure on the complex plane.
Rewrite (<ref>) as

    (-S)M^(2)_D=I,   Sf(λ)=∬_ℂ[fℛ̃](s)/s-λ L(s)/π.

The linear integral operator S is proved bounded on L^∞(ℂ) in Proposition <ref>, and the bound S_L^2(ℂ) decays as t→∞.
Therefore, (<ref>) is uniquely solved in L^∞(ℂ) for t≫0.


	Define the integral operator S as (<ref>).
	If r∈ H^1, then S is bounded on L^∞(ℂ), and as t→+∞, the bound of S on L^∞(ℂ) is dominated by t^-1/4, that is,
	
    S_L^∞(ℂ)≲ t^-1/4.



	Recalling that ℛ̃ is supported on ⋃_k=1,3,4,6Ω_k and the discrete spectrum of RH problem <ref> is away from ⋃_k=1,3,4,6Ω_k, we have that M^(2)_RH, (M^(2)_RH)^-1, T and T^-1 are all bounded on ⋃_k=1,3,4,6Ω_k, and consequently,
	
    Sf(λ)=∬_ℂ[fℛ̃](s)/s-λ L(s)/π=∬_⋃_k=1,3,4,6Ω_k[fℛ̃](s)/s-λ L(s)/π
         ≲f_∞∑_k=1,3,4,6∬_Ω_k∂̅ℛ(s)/s-λ L(s)/π
          =f_∞∑_k=1,3,4,6∬_Ω_k[∂̅R_ke^(-1)^[k/2]ϕ](s)/s-λ L(s)/π,

	where (-1)^[k/2]=
		1   k=1,4
-1   k=2,6.
	
	For the region Ω_1, we write s∈Ω_1 in the polar coordinates: s=ρ e^iθ, (ρ,θ)∈(1,ρ_0)×(θ_ρ,-π-θ_ρ), where ρ_0=1+ϵ_0e^π/4, and ρ e^θ_ρ, ρ e^-π-θ_ρ are on the upper boundaries of Ω_1: S_1(1+e^π/4(0,ϵ_0)), S_2(1+e^-π/4(0,ϵ_0)), respectively.
	Taking Taylor's expansion of (<ref>) at λ=S_1, we see that as ρ e^θ_ρ→ S_1, 
	
    ϕ(ρ e^θ_ρ)=   ϕ(ρ e^θ_ρ,n,t)
    			∼ -2t√(1-ξ^2)(ρ-1)^2+(ρ-1)^3.

	Utilizing (<ref>), (<ref>) and the boundedness of T on Ω_1, we estimate the integral on Ω_1 for λ0,
	
    ∬_Ω_1[∂̅R_1e^ϕ](s)/s-λ L(s)/π =∫_1^ρ_0∫_-π-θ_ρ^θ_ρρ^-1∂_θ(r(e^θ))e^ϕ(ρ e^θ)/2π(ρ e^θ-λ)ρθρ
         ≤∫_1^ρ_0∫_-3π/2^π/2∂_θ (r(e^θ))e^ϕ(ρ e^θ_ρ)/2πρ e^θ-λθρ
         ≤∫_1^ρ_0ρe^ϕ(ρ e^θ_ρ)/2πr_H^1(∫_-π^πρ e^θ-λ^-2θ)^1/2
         ≲∫_1^ρ_0ρe^ϕ(ρ e^θ_ρ)/2πρ-λ^1/2.

	By (<ref>) and (<ref>), since we have assumed ϵ_0 small enough, it is readily seen that
	
    ∬_Ω_1[∂̅R_1e^ϕ](s)/s-λ L(s)/π≤∫_0^ρ_0ρe^-t√(1-ξ^2)(ρ-1)^2/2πρ-λ^1/2≲ t^-1/4.

	
	For regions Ω_3, Ω_4, Ω_6, letting λ0, it admits the same estimates as (<ref>), i.e., for sufficient small ϵ_0 and k=1,3,4,6,
	
    ∬_Ω_k[∂̅R_ke^(-1)^[k/2]ϕ](s)/s-λ L(s)/π≲ t^-1/4.

	Utilizing (<ref>) and (<ref>), we finally prove the results.


Now, we focus on the estimate of M^(2)_D at λ=0.

    M^(2)_D(0)=I+∬_ℂ[M^(2)_Dℛ̃](s)s^-1 L(s)/π.

Again taking s=ρ e^θ∈Ω_1, by (<ref>) it is easy to check that for any fixed ξ∈(-1,1), there is a positive constant ϵ̃(ξ) such that

    sinθ-sinθ_ρ≤ϵ̃(ξ)(θ-θ_ρ)   θ∈(-π/2,θ_ρ),
    
    			-ϵ̃(ξ)(θ+π+θ_ρ)   θ∈(-π-θ_ρ,-π/2).

Utilizing the techniques applied in Proposition <ref> and the estimate (<ref>), we obtain that the following integral on Ω_1 is dominated by t^-3/4, that is,

    ∬_Ω_1[M^(2)_Dℛ̃](s)s^-1 L(s)/π≲∬_Ω_1∂̅R_1(s)e^ϕ(s)s^-1 L(s)/π
         =∫_1^ρ_0e^ϕ(ρ e^θ_ρ)/2πρρ∫_-π-θ_ρ^θ_ρ∂_θ(r(e^θ))e^ϕ(ρ e^θ)-ϕ(ρ e^θ_ρ)θ
         ≤r_H^1∫_1^ρ_0 e^(ρ e^θ_ρ)/2πρρ(∫_-π/2^θ_ρe^2t(ρ-ρ^-1)(θ-θ_ρ)cosθ_ρθ+∫_-π-θ_ρ^-π/2e^-2t(ρ-ρ^-1)(θ+π+θ_ρ)cosθ_ρθ)^1/2
         ≲∫_1^ρ_0e^-t√(1-ξ^2)(ρ-1)^2/2πρt^-1/2(ρ-ρ^-1)^-1/2ρ≲ t^-3/4.

Utilizing the same technique, the results still hold for Ω_3, Ω_4 and Ω_6:

    ∬_Ω_k[M^(2)_Dℛ̃](s)s^-1 L(s)/π≲ t^-3/4,   k=1,3,4,6.

Substituting (<ref>) into (<ref>), it is readily seen that as t→+∞,

    M^(2)_D(0)∼ I+(t^-3/4).






  §.§.§ The asymptotic formula


By (<ref>), (<ref>), (<ref>), (<ref>), (<ref>), (<ref>), (<ref>) and (<ref>), we obtain the asymptotic formula (<ref>).



§ LONG-TIME ASYMPTOTICS FOR THE SOLITON REGION

When ξ∈(-1,1), we have defined the scalar function T(λ) as (<ref>), otherwise, we set it as

    T(λ)=T(λ,ξ)=∏_λ_l'(λ-λ_l'/λ-λ̅_l'^-1)^α_l'   ξ≤-1,
    
    		e^-∫_Σln1+r(s)^2/s-λ s/2π∏_λ_l'(λ-λ_l'/λ-λ̅_l'^-1)^α_l'   ξ≥1.

In the following, we discuss the long-time asymptotics in the two sectors: ξ>1 and ξ<-1.




In the sector ξ<-1, the stationary phase points are off the jump contour Σ={λ=1}.
In this vein, set the positive number

    ϵ_2=min{λ_1-1,…,λ_l-1,√(ξ^2-1)+ξ-1}/2.

By the sign table shown in FIGURE <ref> and (<ref>), it is clear that

    Σ_ϵ_2={λ=1+ϵ_2}⊂{ϕ>0},
       Σ_ϵ_2^-1⊂{ϕ<0},

and there is no discrete spectrum in the region between Σ and Σ_ϵ_2.
By (<ref>) and (<ref>), it follows that there is a positive constant C such that the jump matrix define in (<ref>) satisfies

    V^(4)-I_L^∞(ℛ)≲ e^-C t,   t→+∞,

Define

    M^(4)=MT^σ_3[
    			1   0
    -R_1T^2e^ϕ   1
    		]   λ∈Ω_8,
    [
    			1   R_6T^-1e^-ϕ
    0   1
    		]   λ∈Ω_9,
    
    		I   λ∈Ω_7∪Ω_10.

It is clear that M^(4) solves the following ∂̅-RH problem uniquely, and one can refer to its jump contour Σ^(4)=Σ_ϵ_2∪Σ_ϵ_2^-1 in FIGURE <ref>.

	Find a 2×2 matrix-valued function M^(4) such that
	
		
  * M^(4) is continuous on ℂ∖{Σ^(4)∪𝒵∪𝒵^-1} and holomorphic on (Ω_7∪Ω_10)∖{𝒵∪𝒵^-1}.
		
  * As λ→∞, M^(4)∼ I+(λ^-1).
		
  * On Σ^(4),
		
    M^(4)_+=M^(4)_-V^(4),

		where
		
    V^(4)=[
    					1   0
    R_1T^2e^ϕ   1
    				]   λ∈Σ_ϵ_2,
    [
    					1   R_6T^-2e^-ϕ
    0   1
    				]   λ∈Σ_ϵ_2^-1.

		
  * On the poles on 𝒵∪𝒵^-1, 
		
			
    λ=λ_l'_-αM^(4)=∑_j=0^α_l'-α∂_λ^α_l'-α-jM^(4)(λ_l')/(α_l'-α-j)![
    						0   p^T_λ_l',je^-ϕ(λ_l')
    0   0
    					]    λ_j∈𝒵_ξ^+,
    ∑_j=0^α_l'-α∂_λ^α_l'-α-jM^(4)(λ_l')/(α_l'-α-j)![
    						0   0
    p^T_λ_l',je^ϕ(λ_l')   0
    					]    λ_j∈𝒵∖𝒵_ξ^+,
    				
       λ=λ̅_l'^-1_-αM^(4)=∑_j=0^α_l'-α∂_λ^α_l'-α-jM^(4)(λ̅_l'^-1)/(α_l'-α-j)![
    						0   0
    p^T_λ̅_l'^-1,je^ϕ(λ̅_l'^-1)   0
    					]   λ_j∈𝒵_ξ^+,
    ∑_j=0^α_l'-α∂_λ^α_l'-α-jM^(4)(λ̅_l'^-1)/(α_l'-α-j)![
    						0   p^T_λ̅_l'^-1,je^-ϕ(λ̅_l'^-1)
    0   0
    					]   λ_j∈𝒵∖𝒵_ξ^+.
 
			
		
		
  * ∂̅ condition: On Ω_8∪Ω_9,
		
    ∂̅M^(4)=M^(4)[
    					0   0
    -∂̅R_1T^2e^ϕ   0
    				]   λ∈Ω_8,
    [
    					0   ∂̅R_6T^-1e^-ϕ
    0   0
    				]   λ∈Ω_9.

	


As we have done in the case ξ∈(-1,1), one can factorize the solution M^(4) into a product of solutions of a ∂̅ problem and an RH problem below,
that is, M^(4) admits the following factorization:

    M^(4)=M^(4)_DM^(4)_RH.



	Find a 2×2 matrix-valued function M^(4)_RH solves ∂̅-RH problem <ref> with ∂̅M^(4)_RH≡0 over the complex plane ℂ.


	Find a 2×2 matrix-valued function M^(4)_D=M^(4)_D(λ,n,t) such that
	
		
  * M^(4)_D is continuous on ℂ.
		
  * As λ→∞, M^(4)_D∼ I+(λ^-1).
		
  * On Ω_7,…,Ω_10, it admits the ∂̅ condition:
		
    ∂̅M^(4)_D=M^(4)_Dℛ^(4),  ℛ^(4)=M^(4)_RH[
    					0   0
    -∂̅R_1T^2e^ϕ   0
    				](M^(4)_RH)^-1   λ∈Ω_8,
    [
    					0   ∂̅R_6T^-1e^-ϕ
    0   0
    				](M^(4)_RH)^-1   λ∈Ω_9.

		
	

On the one hand for M^(4)_RH, we claim that M^(4)_RH is well-defined as what ones have done in Section <ref>.
With a little abuse of notation, for ξ>0, denote M^(𝒵,^T) as the solution of RH problem <ref> with V^(4)≡ I, where (𝒵,^T) is the discrete spectral data.
Referring to the techniques of small norm RH problem theory in Section <ref>, it is clear that for some positive constant C,

    M^(𝒵)(0,n,t)=(I+(e^-Ct))M^(𝒵_ξ)(0,n,t),

where M^(𝒵_ξ)=M^(𝒵_ξ)(λ,n,t) is defined for ξ>0 as
	
    M^(𝒵_ξ)=
    			I   𝒵_ξ=∅,
    
    			M^(𝒵_ξ,^T_ξ)   𝒵_ξ∅,
    		  𝒵_ξ=𝒵∩{ϕ=0},

and M^(𝒵_ξ,^T_ξ)=M^(𝒵_ξ,^T_ξ)(λ,n,t) is as defined in Remark <ref> with the discrete spectral data (𝒵_ξ,^T_ξ) that is the restriction of (𝒵,^T) on 𝒵_ξ. 
Utilizing the techniques in Section <ref>, it is clear that M^(4)_RH admits the form

    M^(4)_RH=E^(4)M^(𝒵),

where E^(4) is the solution of the small-norm RH problem:

	Find a 2×2 matrix-valued function E^(4)=E^(4)(λ,n,t) such that
	
		
  * E^(4) is analytic on ℂ∖Σ^(4).
		
  * As λ→∞, E^(4)∼ I+(λ^-1).
		
  * On Σ^(4), M^(4)_+=M^(4)_-V^(4,E), where V^(4,E)=M^𝒵V^(4)(M^𝒵)^-1.
	

By the boundedness of M^𝒵 and (M^𝒵)^-1 on Σ^(4), recalling (<ref>), ones obtain that as t→+∞, 

    V^(4,E)-I_L^∞(Σ^(4))≲ e^-Ct.

Again utilizing the small norm theory for the RH problems, we obtain that RH problem <ref> is uniquely solved and as t→+∞, 

    E^(4)(λ,n,t)∼ I+(e^-Ct).

Substituting (<ref>) into (<ref>) and utilizing (<ref>), it follows that as t→+∞,

    M^(4)_RH(0,n,t)∼ M^(𝒵_ξ)(0,n,t)+(e^-Ct).
 

On the other hand for M^(4)_D, we assert that when t→+∞, M^(4)_D(0,n,t) decays to the identity matrix like that it have done for ξ∈(-1,1), but the decaying rate is (t^-1),

    M^(4)_D(0,n,t)∼ I+(t^-1).

Recalling ∂̅ problem <ref>, rewrite M^(4)_D in the form

    M^(4)_D=(-S^(4))^-1I,   Sf(λ)=∬_ℂ[fℛ^(4)](s)/s-λ L(s)/π.

The integral operator is bounded on L^∞(ℂ) and the bound decays to zero for t≫0, that is, 

    S^(4)_L^∞≲(t^-1/2).

For λ0 and f∈ L^∞(ℂ), by the boundedness of M^(𝒵), (M^(𝒵))^-1 and T on Σ^(4), it is similar to the estimates in (<ref>) and (<ref>) that

    ∬_Ω_8[fℛ^(4)](s)/s-λ s/π   ≲f_L^∞(ℂ)∬_Ω_8∂̅R_1e^ϕ(s)/s-λ L(s)/π
       =f_L^∞(ℂ)∫_1^1+ϵ_2∫_-π^π∂_θ(r(e^θ))e^t((ρ-ρ^-1)sinθ+2ξlnρ)/2πρ e^θ-λθρ
       ≤f_L^∞(ℂ)r_H^1∫_1^1+ϵ_2e^t((ρ-ρ^-1)+2ξlnρ)ρ/2π(∫_-π^πρ e^θ-λ^-2θ)^1/2
       ≲f_L^∞(ℂ)∫_1^1+ϵ_2e^-2t(ρ-1)((-ξ)ln(1+ϵ_2)/ϵ_2-1)/ρ-λ^1/2ρ≲ t^-1/2f_L^∞(ℂ).

and also

    ∬_Ω_8[fℛ^(4)](s)/s-λ s/π≲ t^-1/2f_L^∞(ℂ),

thus, (<ref>) follows. 
Utilizing (<ref>), we obtain that (<ref>) is well-posed, and at λ=0, similarly we have the estimate (<ref>).

Considering (<ref>), (<ref>), (<ref>), (<ref>) and (<ref>), we have that for ξ<-1

    q_n(t)=q_n^𝒵_ξ(t)+(t^-1),

where q_n^𝒵_ξ(t)=[M^(4)]_1,2(0,n+1,t).
By the same technique, it is readily seen that for ξ>1, the solution q_n(t) also satisfies (<ref>). 







§ PARABOLIC CYLINDER MODEL







Here, describe the parabolic cylinder model that is first introduced by <cit.> and further by <cit.>.
And it has been frequently applied to the literature of long-time asymptotics <cit.>.
Set Σ^(PC)=⋃_k=1^4Σ^PC_k, where Σ^PC_k is the ray e^π/4(2k-1) oriented as shown in FIGURE <ref>.
The regions Ω^PC are the sectors on ζ∈ℂ that is divided by Σ^(PC) and ζ.
Consider the parabolic-cylinder RH problem below. 

	Find a 2×2 matrix-valued function M^(PC)=M^(PC)(ζ,τ) on ζ∈ℂ, such that
	
		
  * M^(PC) is analytic on ℂ∖Σ^(PC).
		
  * As ζ→∞, M^(PC)∼ I+(ζ^-1).
		
  * On Σ^(PC), 
		
    M^(PC)_+=M^(PC)_-V^(PC),  
    			V^(PC)=V^(PC)(ζ,τ)=
    			[
    					1   0
    τζ^-2ν_je^ζ^2/2   1
    				]   ζ∈Σ^PC_1,
    [
    					1   τ/1+τ^2ζ^2ν_je^-ζ^2/2
    0   1
    				]   ζ∈Σ^PC_2,
    [
    					1   0
    τ/1+τ^2ζ^-2ν_je^ζ^2/2   1
    				]   ζ∈Σ^PC_3,
    [
    					1   τζ^2ν_je^-ζ^2/2
    0   1
    				]   ζ∈Σ^PC_4.

		
	

The solution of RH problem <ref> is explicitly solved by:

    M^(PC)(ζ,τ)=Φ^(PC)(ζ,τ)𝒬(ζ,τ)ζ^-νσ_3e^/4ζσ_3,
       Φ^(PC)(ζ,τ)=[
    			e^-3πν/4D_ν(e^-3π/4ζ)   -γ_1(τ)e^π(ν-)/4D_-ν-1(e^-π/4ζ)
    γ_2(τ)e^-3π(ν+)/4D_ν-1(e^-3π/4ζ)   e^πν/4D_-ν(e^-π/4ζ)
    		]   ζ∈{ζ>0},
    [
    			e^πν/4D_ν(e^π/4ζ)   -γ_1(τ)e^-3π(ν-)/4D_-ν-1(e^3π/4ζ)
    γ_2(τ)e^π(ν+)/4D_ν-1(e^π/4ζ)   e^-3πν/4D_-ν(e^3π/4ζ)
    		]   ζ∈{ζ<0},
    	
       𝒬(ζ,τ)=[
    			1   0
    -τ   1
    		]   ζ∈Ω^PC_1,
    [
    			1   -τ̅/1+τ^2
    0   1
    		]   ζ∈Ω^PC_3,
    [
    			1   0
    τ/1+τ^2   1
    		]   ζ∈Ω^PC_4,
    [
    			1   τ̅
    0   1
    		]   ζ∈Ω^PC_6,
    
    		I   ζ∈Ω_2∪Ω_5,

where D_a(λ) is the solution of the parabolic cylinder equation:

    (∂_λ^2+1/2-λ^2/2+a)D_a(λ)=0

and the complex constants γ_1, γ_2 and ν read

    γ_1=γ_1(τ)=√(2π)e^π/4-πν(τ)/2/τΓ(-ν(τ)),  γ_2=γ_2(τ)=√(2π)e^-π/4-πν(τ)/2/τ̅Γ(ν(τ)),  ν=ν(τ)=ln(1+τ^2)/2π.

It is well acknowledged in <cit.> that M^(PC) admits the asymptotic property at ζ→∞,

    M^(PC)(ζ,τ)∼ I+1/ζ[
    		0   -γ_1(τ)
    γ_2(τ)   0
    	]+(ζ^-2).








siam






 Paper Title: A Nečas-Lions inequality with symmetric gradients on star-shaped domains based on a first order Babuška-Aziz inequality.



Finding Defective Elements in Intelligent Reflecting Surface via Over-the-Air Measurements
    
   Ziyi Zhao1,
                     Zhaorui Wang2,
                     Shuowen Zhang1,
                     and Liang Liu1
   1
                    Department of Electrical and Electronic Engineering, The Hong Kong Polytechnic University, Hong Kong SAR, China,

                     E-mail: ziyii.zhao@connect.polyu.hk, {shuowen.zhang,liang-eie.liu}@polyu.edu.hk
   2
                     School of Science and Engineering, The Chinese University of Hong Kong, Shenzhen, Shenzhen, China 

                     E-mail: wangzhaorui@cuhk.edu.cn
 The work was supported in part by the National Key R&D Project of China under Grant No. 2022YFB2902800; by the National Natural Science Foundation of China with Grant No. 62101474; and by the Research Grants Council, Hong Kong, China, with Grant No. 15203222 and Grant No. 15230022.
    August 5, 2024
===============================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================



§ ABSTRACT 
We prove a Nečas-Lions inequality with symmetric gradients
on two and three dimensional domains of diameter R
that are star-shaped with respect
to a ball of radius ρ;
the constants in the inequality are explicit
with respect to R and ρ.
Crucial tools in deriving this inequality
are a first order Babuška-Aziz inequality
based on Bogovskiĭ's construction
of a right-inverse of the divergence
and Fourier transform techniques proposed by Durán.
As a byproduct, we derive arbitrary order estimates
in arbitrary dimension for that operator.

AMS subject classification:
35C05; 35C15; 35S05; 42B20; 42B37; 53A45

Keywords: 
Babuška-Aziz inequality;
Nečas-Lions inequality;
inf-sup condition;
Bogovskiĭ operator;
symmetric tensor.





§ INTRODUCTION
 


We derive a Nečas-Lions inequality with symmetric gradients
on star-shaped, Lipschitz domains in two and three dimensions;
first order Babuška-Aziz and (vector) Nečas-Lions inequalities are crucial tools
to show estimates that are explicit
with respect to certain geometric quantities of the domain.
As a byproduct, we also derive explicit arbitrary order
Babuška-Aziz inequalities in arbitrary dimension.




  
*Outline of the introduction.

After introducing the functional setting and the domains of interest,
we review the literature and the main concepts related to
the lowest order Babuška-Aziz and Nečas-Lions inequalities
in Sections <ref> and <ref>.
In Sections <ref> and <ref>,
we discuss the generalisation of these two results
to the first order case.
The Nečas-Lions inequality with symmetric gradients,
based on all the foregoing results,
is shown in Section <ref>.
Finally, we describe the outline of the remainder of the paper.




  
*Functional spaces and notation.

In what follows, ∇, ∇×, and ÷ denote the
gradient, curl, and divergence operators.
The operators  and  are
the symmetric and skew-symmetric parts of ∇:

    ∇ =  + .

We use standard notation <cit.>
for Sobolev spaces on Lipschitz domains Ω
with boundary ∂Ω.
The outward unit normal vector to ∂Ω is .
H^s(Ω) denotes the Sobolev space of order s≥0,
which we equip with inner product (·,·)_s,
seminorm ·_s,Ω,
and norm ·_s,Ω.
The case s=0 corresponds to H^0(Ω)=L^2(Ω).
The space of functions in L^2(Ω) with zero average
over Ω is denoted by L^2_0(Ω).

For s positive, we define H^s_0(Ω) as
the closure of (Ω) with respect to the H^s(Ω) norm.
In what follows, we shall particularly use the spaces H_0^1(Ω)
and H^2_0(Ω),
which coincide <cit.> with the spaces of functions with zero trace,
and functions with zero trace and whose gradients have zero trace
over ∂Ω,
respectively.

Negative order Sobolev spaces are defined by duality.
We introduce the spaces H^-1(Ω) := [H^1_0(Ω)]^*
and H^-2(Ω) := [H^2_0(Ω)]^*
equipped with the norms

    u_-1,Ω 
    : = sup_v∈ H^1_0(Ω)_-1⟨ u,v ⟩_1/v_1,Ω,
                u_-2,Ω 
    : = sup_v∈ H^2_0(Ω)_-2⟨ u,v ⟩_2/v_2,Ω,

where _-ℓ⟨·,·⟩_ℓ
is the duality pairing between H^-ℓ(Ω) and H^ℓ(Ω).

The definitions above extend to the case of
vector fields and tensors.
With an abuse of notation, the norms on scalar, vector fields,
and tensors are denoted with the same symbols.

For positive a and b,
by a ≲ b, we shall occasionally mean
that there exists a positive constant c
independent of relevant geometric parameters
such that a ≤ c  b.
An extra subscript makes it explicit
a hidden dependence on a parameter of interest.




  
*Domains of interest.

Henceforth, Ω in ^n
is a

    Lipschitz domain of diameter 
    that is star-shaped with respect to a ball of radius ρ.





 §.§ The lowest order Babuška-Aziz inequality
 

The standard, lowest order version of the Babuška-Aziz inequality
was proven as early as 1961 by Cattabriga <cit.>.
However, the name is associated to the authors
of the later work <cit.>
and was assigned by Horgan and Payne <cit.>;
see also <cit.>.

The inequality reads as follows:
for any q in L^2_0(Ω), there exist  in [H^1_0(Ω)]^n
and a positive constant  such that

    ÷ = q,
                _1,Ω≤q_0,Ω.

The subscript in  relates to Babuška-Aziz.
Explicit constructions of a vector field  as in (<ref>)
may be performed in different ways;
here, we shall follow the approach by
Bogovskiĭ in <cit.>,
where he showed a particular construction of the right inverse
of the divergence based on integral kernels.
Alternative avenues, which give less information on the constant 
and construct less smooth right-inverses of the divergence,
are based on solving curl-div, diffusion, or Stokes problems;
see, e.g., <cit.> and <cit.>;
as such, higher order estimates based on this approach
require extra regularity assumptions on the boundary of the domain,
which are instead not needed following Bogovskiĭ's approach.




  
*Minimal literature on the lowest order Babuška-Aziz inequality.
The literature associated with inequality (<ref>) is widespread.
We refer to <cit.>, <cit.>,
and <cit.> for a thorough historical review.

Here, we only mention that the divergence problem in (<ref>)
was raised as early as in 1961 by Cattabriga <cit.>;
see also the later works <cit.>.
Bogovskiĭ introduced an explicit representation for  solving (<ref>)
in <cit.>.
Several references discuss the validity of similar estimates;
for instance, implicit constants for general Lipschitz domains are available
in <cit.>.
The case of negative Sobolev norms is described in <cit.>.

The explicit dependence of the constant  on geometric parameters of Ω
is studied in fewer references;
see <cit.> for a list.
For R and ρ as in (<ref>),
Galdi <cit.> gives estimates of the form

    ≲_n (/ρ)^n+1.

The main tool in the analysis is the one
discussed originally by Bogovskiĭ <cit.>,
i.e., the Calderón-Zygmund singular integral operator theory.

Improved estimates of the form

    ≲_n R/ρ( Ω/| B|) ^n-2/2(n-1)( logΩ/| B |)^n/2(n-1)

were proven by Durán <cit.> based on the properties
of the Fourier transform.
In the same reference, it is shown that the estimates
are optimal up to the logarithmic factor.
More precisely, a 2D counterexample is exhibited showing that
the following holds true:

    ≳_n /ρ.

For the two dimensional case,
Costabel and Dauge <cit.> proved
that the logarithmic factor in Durán's estimates can be removed.




  
*Minimal literature on higher order Babuška-Aziz inequalities.
Higher order Babuška-Aziz inequalities are far less investigated.
They are stated in the original paper by Bogovskiĭ <cit.>
without mention on the behaviour of the constants.
Galdi <cit.>
claims that similar bounds to the lowest order case can be derived;
however, no explicit constants are given in that case sa well;
the analysis hinges upon the Calderón-Zygmund theory.
Costabel and McIntosh prove arbitrary order estimates <cit.>
without explicit dependence on the geometry on the domain.
Guzmán and Salgado <cit.>
prove an explicit first order generalised Poincaré inequality,
which is related to Bogovskiĭ's operator
without imposition of boundary conditions,
and give a road map on how to prove higher order explicit estimates;
tools as those in <cit.> are employed;
no estimates are given for Bogovskiĭ's operator.




 §.§ The lowest order Nečas-Lions inequality



The standard, lowest order Nečas-Lions inequality
is a very well known result in the theory of Sobolev spaces.
It is proven in the book by Nečas <cit.>;
the connection to the name of Lions is less clear,
and is probably due to <cit.>,
where the result is mentioned as a private communication
by Lions himself to Magenes and Stampacchia, yet without an explicit proof.

The inequality reads as follows:
given : L^1(Ω) → the average operator over Ω,

    q - q_0,Ω≤∇ q_-1,Ω        ∀ q ∈ L^2(Ω).

The subscript in  relates to Nečas-Lions.
An equivalent statement for (<ref>)
is that the following constants are bounded from above
and below, respectively:

    := sup_q ∈ L^2(Ω)q - q_0,Ω/∇ q_-1,Ω,
            ^-1:= inf_q ∈ L^2(Ω)∇ q_-1,Ω/q - q_0,Ω.

The constants  and 
in (<ref>) and (<ref>) are related
to constants in other relevant inequalities in Sobolev spaces
as well, including the standard inf-sup constant 
defined as

    inf_q∈ L^2_0(Ω)sup_∈ [H^1_0(Ω)]^n(÷,q)_0,Ω/_1,Ωq_0,Ω
            = .


 
Let , , and  be given
in (<ref>), (<ref>),
and (<ref>).
Then, the following holds true:

    ≥ = ^-1.




For all q in L^2_0(Ω), the definition
of negative norms in (<ref>) implies

    sup_∈ [H^1_0(Ω)]^n(÷,q)_0,Ω/_1,Ωq_0,Ω
    = sup_∈ [H^1_0(Ω)]^n_-1⟨∇ q, ⟩_1/_1,Ωq_0,Ω
    = ∇ q_-1,Ω/q_0,Ω .

We take the inf over all possible q in L^2_0(Ω),
use (<ref>),
and deduce that = ^-1.

On the other hand, for all q in L^2_0(Ω),
we can consider a specific  satisfying (<ref>),
which gives

    sup_∈ [H^1_0(Ω)]^n(÷,q)_0,Ω/_1,Ωq_0,Ω≥q_0,Ω^2/_1,Ωq_0,Ω≥^-1.

Taking the inf over all possible q in L^2_0(Ω)
and recalling the standard inf-sup condition (<ref>)
give ≥^-1.
The assertion follows.



A consequence of Proposition <ref>
is that if an upper bound on  is available,
which is explicit in terms of n, R, and ρ as in (<ref>),
then one has an upper bound for , viz., for ^-1.

The relation with constants appearing in other
inequalities is discussed in <cit.>.




 §.§ Main result 1: a first order Babuška-Aziz inequality
 

An important tool in the proof of Theorem <ref> below
is the proof of a Babuška-Aziz inequality,
based on first order estimates for Bogovskiĭ's construction
of the right-inverse of the divergence.
More precisely, given f in H^1_0(Ω) ∩ L^2_0(Ω),
we prove the existence of  in [H^2_0(Ω)]^n such that

    ÷ = f,
                    _2,Ω≤f_0,Ω 
            + f_1,Ω,

for positive constants  and  that are
explicit in terms of n, R, and ρ in (<ref>).

We state the result here and postpone its proof
to Section <ref> below.


 
Let  and f be as in (<ref>),
and Ω, , R, and ρ be as in (<ref>).
Then, the following inequality holds true:

    _2,Ω   ≲_n /ρ^2[ 1 + ( Ω/B)^n-2/2(n-1)( logΩ/B)^n/2(n-1)] f_0,Ω
                + /ρf_1,Ω.

In other words, the constants in (<ref>)
are bounded as follows:

    ≲R/ρ^2[ 1 + ( Ω/B)^n-2/2(n-1)( logΩ/B)^n/2(n-1)],
                ≲R/ρ.



We provide the reader with some comments on the optimality
of the constants in (<ref>)
in Section <ref> below.




 §.§ Main result 2: a first order Nečas-Lions inequality
 

Introduce the vector space [H^-1(Ω) / ]^n,
which is the space [H^-1(Ω)]^n equipped with the norm

    _[H^-1(Ω) / ]^n
    := inf_∈^n-_-1,Ω.

Recall the negative norms in (<ref>).
We discuss a first order Nečas-Lions inequality (for vectors):
there exists a positive constant  such that

    _[H^-1(Ω) / ]^n≤∇_-2,Ω        ∀∈ [H^-1(Ω)/ ]^n.

An equivalent statement for (<ref>)
is that the following constants are bounded from above
and below, respectively:

    = sup_∈ [H^-1(Ω)/ ]^n_[H^-1(Ω) / ]^n/∇_-2,Ω,
            ^-1 = inf_∈ [H^-1(Ω)/ ]^n∇_-2,Ω/_[H^-1(Ω) / ]^n.

The constants  and , and 
in (<ref>) and (<ref>) are related
to constants in other relevant inequalities in Sobolev spaces
as well, including the first order inf-sup constant 
defined as

    inf_∈ [H^-1(Ω) / ]^nsup_∈ [H^2_0(Ω)]^n× n_-1⟨, ⟩_1/_[H^-1(Ω) / ]^n_2,Ω
    =

and the Poincaré inequality

    _0,Ω≤_1,Ω        ∀ [H^1_0(Ω)]^n.

The constant  depends linearly on R
and is independent of ρ in (<ref>);
see, e.g., <cit.>.

The following result is the first order version of Proposition <ref>.

 
Let  and , , , and  be given
in (<ref>), (<ref>),
(<ref>), and (<ref>).
Then, the following holds true:

    +≥ = ^-1.




For all  in [H^-1(Ω) / ]^n,
an integration by parts and the definition
of negative Sobolev norms in (<ref>) imply

    sup_∈ [H^2_0(Ω)]^n× n_-1⟨, ⟩_1/_[H^-1(Ω) / ]^n_2,Ω
    = sup_∈ [H^2_0(Ω)]^n× n_-2⟨∇, ⟩_2/_[H^-1(Ω) / ]^n_2,Ω
    =: ∇_-2,Ω/_[H^-1(Ω) / ]^n.

We take the infimum over all such possible 
and exploit the identities

    (<ref>)=inf_∈ [H^-1(Ω) / ]^n∇_-2,Ω/_[H^-1(Ω) / ]^n
    = ( sup_∈ [H^-1(Ω) / ]^n∇_-2,Ω/_[H^-1(Ω) / ]^n)^-1(<ref>)=^-1,

which implies =^-1.

On the other hand, the tensor version of (<ref>)
guarantees for all 
in [H^1_0(Ω)]^n the existence of 
in [H^2_0(Ω)]^n× n such that

    ÷ = ,
            _2,Ω≤_0,Ω
       + _1,Ω.

This and the Poincaré inequality (<ref>) give

    sup_∈ [H^2_0(Ω)]^n× n_-1⟨, ⟩_1/_[H^-1(Ω) / ]^n_2,Ω≥sup_∈ [H^1_0(Ω)]^n_-1⟨, ⟩_1/_[H^-1(Ω) / ]^n
        [_0,Ω + _1,Ω]
       ≥sup_∈ [H^1_0(Ω)]^n_-1⟨, ⟩_1/(+)_[H^-1(Ω) / ]^n_1,Ω
    = (+)^-1.

We take the infimum over all  in [H^-1(Ω) / ]^n,
recall the first order inf-sup condition (<ref>),
and deduce ≥^-1.
The assertion follows.


A consequence of Proposition <ref>
is that if an upper bound on  and  is available,
which is explicit in terms of n, R, and ρ as in (<ref>),
then one can derive an upper bound for , viz., for ^-1.

For more general Nečas-Lions inequalities,
yet with unknown constants,
see <cit.> and the references therein.




 §.§ Main result 3: a Nečas-Lions inequality with symmetric gradients
 

The spaces (Ω) of rigid body motions
in two and three dimensions have
cardinality 3 and 6, and are given by

    (Ω)
    := {(x) =  + b (x_2,-x_1)^T for any ∈^2,  b ∈}    in 2D
    {(x) =  +× (x_1,x_2,x_3)^T for any , ∈^3 }    in 3D.

Let  denote the projection onto (Ω).
We further introduce the space of symmetric tensors

    :={∈ H(÷,Ω)  | is symmetric},

which we endow with the norm

    _^2
    :=  _0,Ω^2
        + ÷_0,Ω^2 .

Note that

    _-1⟨∇,⟩_1
    = _-1⟨,⟩_1 
            ∀∈ [L^2(Ω)]^2, ∈.

We state a Nečas-Lions inequality with symmetric gradients
on two and three dimensional domains,
which is explicit in terms of R, ρ, and n as in (<ref>).

We state the inequality here
and postpone its proof to Section <ref> below.    


 
There exists a positive constant 
depending only on n, R, and ρ as in (<ref>)
through  in (<ref>),
 and  in (<ref>),
and  in (<ref>),
such that

    -_0,Ω   ≤_-1,Ω
        := [ 1+√(2)(+) ] 
        _-1,Ω        ∀∈ [L^2(Ω)]^n.



Introduce the spaces

    := {∈|⟨ ,
            w⟩_∂Ω=0 
              ∀w∈ [H^1(Ω)]^n};
            
    :=  {∈[L^2(Ω)]^n| = 0}.

A consequence of Theorem <ref>
is an inf-sup condition,
which is of great importance in the analysis of the mixed
(Hellinger-Reissner) formulation
of linear elasticity problems:

    inf_∈ sup_∈(, )_0,Ω/__≥inf_∈ sup_∈(, )_0,Ω/_1,Ω_≥.


 
Let  and 
be given in (<ref>)
and (<ref>).
Then, the following inequality holds true:

    ≥ ()^-1.




We have

    inf_∈ sup_∈(, )_0,Ω/__≥inf_∈ sup_∈∩ [H^1_0(Ω)]^n× n(, )_0,Ω/__
       IBP=inf_∈sup_∈∩ [H^1_0(Ω)]^n× n_-1⟨∇, ⟩_1/__(<ref>)=inf_∈ sup_∈∩ [H^1_0(Ω)]^n× n_-1⟨, ⟩_1/__
       (<ref>)≥inf_∈ sup_∈∩ [H^1_0(Ω)]^n× n_-1⟨, ⟩_1/_1,Ω_
      =    inf_∈ sup_∈ [H^1_0(Ω)]^n× n_-1⟨, ⟩_1/_1,Ω_
       (<ref>)=inf_∈_-1,Ω/_(<ref>)≥
      ()^-1.








  
*Outline of the remainder of the paper.

In Section <ref>, we prove Theorem <ref>,
whereas in Section <ref>, we prove Theorem <ref>.
We also prove an arbitrary order version
of Theorem <ref>
in Appendix <ref>.




§ PROOF OF A FIRST ORDER BABUŠKA-AZIZ INEQUALITY
 

In this section, we prove Theorem <ref> in several steps
and further discuss the optimality of the bounds on the constants therein.
To this aim, we follow Bogovskiĭ's construction <cit.>
of a right-inverse  of the divergence
and generalise Durán's analysis <cit.> to the first order case.




  
*Explicit construction of Bogovskiĭ's right-inverse of the divergence.

Consider ω in (Ω) with

    ∫_Ωω(x)  = 1,
                (ω) ⊂ .

Given

    :Ω×Ω→^n,
             (x,y)
     := ∫_0^1 x-y/t ω( y + x-y/t) /t^n,

we define

    (x) := ∫_Ω(x,y) f(y) .






 §.§ Preliminary results
 
We recall basic properties of the Fourier transform.
Given f in L^1(), we define its Fourier transform as

    (ξ) 
    := ∫_ e^-2πı x ·ξ f(x) .

If f is in L^2(Ω), we have the isometry

    f_0, = _0,

and the following property on the derivatives of the Fourier transform:

    f (ξ)
        = 2πı(ξ)
                ∀ j=1,…,n.

We consider the following splitting of each component k,
k=1,…,n, of :

    :=-
        := ∫_0^1∫_( + -/t)  
            ω( y + x-y/t) f(y) /t^n
          - ∫_0^1∫_ω( y + x-y/t) f(y) /t^n .

In order to take derivatives of ,
it is convenient to take a limit
in the sense of distributions <cit.>:

    = lim_ε→ 0∫_ε^1∫_(- -/t)  
            ω( y + x-y/t) f(y) /t^n,
        = lim_ε→0∫_ε^1∫_ω( y + x-y/t) f(y) /t^n.

By doing that, we can interchange the derivative with the limit ε→ 0
and then pass with the derivative under the integral symbol;
in fact, the functions under the integral are in L^1 and admit L^1 derivatives.


We take the second derivatives of  and 
with respect to the j-th and ℓ-th directions
(without loss of generality we assume j different from ℓ),
and get

    ()(x)
    = [ (f(y)) +  ( f(y))](x).

In (<ref>), given  any of the two operators
 and , we let

    (g)(x)
    := lim_ε→0∫_ε^1 ∫_[ φ( y+ x-y/t) ]
            g(y) /t^n,

where, for all j,ℓ=1,…,n, we have set

    g(y) :=
    
    f(y)           if  = , 
    
    y_k f(y)       if  = ,
            φ(x) :=
    ω(x)      if  = , 
    ω(x)          if  = .

In the forthcoming sections,
we shall prove the continuity of the operators
in (<ref>).
To this aim, we henceforth fix j and ℓ, and consider the decomposition

    g
    :=  g +  g,

where

    g(x)
    := lim_ε→0∫_ε^1/2∫_[ φ( y+ x-y/t) ]
            g(y) /t^n

and

    g(x)
    := ∫_1/2^1 ∫_[ φ( y+ x-y/t) ]
            g(y) /t^n.

The continuity estimates will follow
summing over all j and ℓ.




 §.§ Continuity of 



We discuss the continuity of the operator in (<ref>).
We proceed in several steps.
First, we prove some properties of the Fourier transform
of (g).


 
If g in (<ref>) belongs to (),
then the following inequality is valid:

    g(ξ)
        = (2πı) 
       lim_ε→ 0[ ∫_ε^1/2( t  ξ)
         g(ξ) 
        + ∫_ε^1/2φ (t ξ)
        ((1-t)ξ) ]
        =:  g(ξ)
        +  g(ξ).




By definition, we write

    g(x) =
    lim_ε→ 0 g(x).

For all positive ε, we write

    g(ξ)
    = ∫_∫_ε^1/2∫_[ φ( y+ x-y/t) ]
            e^-2 πı x·ξ 
            g(y) /t^n.

Due to the regularity of g and φ,
the integral exists.
Therefore, we can change the order of the integrals,
integrate by parts twice,
use the change of variable z=y+(x-y)/t,
the definition of the Fourier transform (<ref>) twice,
and (<ref>),
recall that the support of φ is compact,
and arrive at

    g(ξ)  
     = (2πı) (2πı)
       ∫_ε^1/2∫_∫_φ( y+ x-y/t)
            g(y) e^-2 πı x·ξ/t^n
        = (2πı) (2πı)
       ∫_ε^1/2∫_∫_φ( z )
            e^-2 πı (t  z +(1-t) y)·ξ  g(y)
            
        = (2πı) (2πı)
       ∫_ε^1/2∫_( t  ξ)
            e^-2 πı (1-t) y ·ξ  g(y)
            
        = (2πı) (2πı)
       ∫_ε^1/2( t  ξ)
            ((1-t)ξ) 
        = (2πı) 
       ∫_ε^1/2( t  ξ)
        [2πı (1-t) ] ((1-t)ξ) 
        + (2πı) 
       ∫_ε^1/2( t  ξ)
        [2πı t ] ((1-t)ξ) 
        = (2πı) 
       ∫_ε^1/2( t  ξ)
         g((1-t)ξ) 
        + (2πı) 
       ∫_ε^1/2φ (t ξ)
        ((1-t)ξ)  .

The assertion follows taking the limit ε→ 0.



Next, we prove a technical result.

 
Let φ be any of the two options in (<ref>).
Then, the two following inequalities hold true:
 

    2π∫_0^∞(tξ)≤
    := ρ^-1φ_L^1()
      + ρφ_L^1()  ∀ j=1,…,n,



    2π∫_0^∞φ(tξ)≤
    := ρ^-1φ_L^1()
      + ρφ_L^1()  ∀ j=1,…,n .





The proof of (<ref>) is given in <cit.>
and is therefore omitted here.
On the other hand, inequality (<ref>)
may be shown as an application of (<ref>)
to φ.
For the sake of completeness,
we provide full details.

We begin by showing the splitting

    2π∫_0^∞φ(tξ)
     = 2π∫_0^1/2πρφ(tξ)
       + 2π∫_1/2πρ^∞φ(tξ) 
        = I + II.

The Hölder inequality and the definition
of the Fourier transform (<ref>) give

    I
    ≤ρ^-1φ_L^∞(Ω)≤ρ^-1φ_L^1(Ω) .

As for term II, we have

    II
        =  2π∫_1/2πρ^∞t^2 ^2 φ(tξ)/t^2 ≤ 2πsup_ξ∈^2
        φ(ξ)∫_1/2πρ^∞1/t^2 .

Observe that

    φ(ξ)
    = (2πı)(2πı)
    φ(ξ).

In other words, we have

    ^2 φ(ξ)
    = 1/4π^2φ(ξ).

Using the definition of the Fourier transform (<ref>),
we deduce

    II
       ≤2π/4π^2φ_L^∞()∫_1/2πρ^∞1/t^2 
      = 2π/4π^2φ_L^∞()2πρ/
        = ρφ_L^∞()≤ρφ_L^1().

The assertion follows combining the bounds on I and II.    



We are now in a position to prove the continuity
of the operator .

 
Let φ be any of the two options in (<ref>).
Then, for all g in H^1(^n),
the operator  defined in (<ref>)
satisfies the following continuity property:

    g_0,≤ 2^n-1/2[  g_0,
          + g_0,] .

If g vanishes outside Ω, we also have

    g_0,Ω≤ 2^n-1/2[  g_0,Ω
          + g_0,Ω] .




We only prove the second assertion
and focus on functions g in ^∞_0(Ω);
the general statement follows then from a density argument.

We consider splitting (<ref>)
and show separate bounds for the two terms on the right-hand side.
The first one can be handled as in <cit.>
and its proof is therefore omitted:

    g_0,≤ 2^n-1/2_0,.

Thus, we focus on the second term.
By the definition of g,
the Cauchy-Schwarz inequality implies

    g (ξ)^2
    ≤( ∫_0^1/2 2πφ(t ξ)) 
    ( ∫_0^1/2 2πφ(t ξ)((1-t)ξ)^2 ).

Using (<ref>), we deduce

    g (ξ)^2
    ≤∫_0^1/2 2πφ(t ξ)((1-t)ξ)^2  .

Integrating over ξ and employing
the change of variable η=(1-t)ξ give

    ∫_ g (ξ)^2 ≤∫_0^1/2∫_2π/(1-t)^n+1∂_η_ℓφ( t η/1-t)(η)^2

If we consider the change of variable s=t/(1-t), which entails

    = (1+s)^-2,    1/(1-t)^n+1 = 
    ( s/t)^n+1
    = (1+s)^n+1,

then we arrive at

    ∫_ g (ξ)^2 ≤ 2^n-1∫_( ∫_0^1 2π∂_η_ℓφ(sη))
     (η)^2   .

We apply again (<ref>):

    g_0,^2
    = ∫_ g (ξ)^2 ≤ 2^n-1^2
    ∫_(η)^2 
    = 2^n-1^2 _0,^2.

The assertion follows using the Fourier isometry (<ref>)
in (<ref>), identity (<ref>),
and the properties of ω detailed in Section <ref>.






 §.§ Continuity of 



We discuss the continuity of the operator in (<ref>).
As discussed in <cit.>,
a direct application of the Hölder and Cauchy-Schwarz inequalities
would end up with suboptimal estimates
as those in <cit.>.
Therefore, finer estimates are in order.
To this aim, we extend <cit.> to the first order case.

We begin with the following technical result.

 
Let g and  be as in (<ref>) and (<ref>).
Given 1≤ p < n/(n-1),
p' the conjugate index of p,
and g in L^p(), we have

    g_L^p()≤2^n/p'/1-n/p'φ_L^1()g_L^p().




The proof follows along the same lines of
<cit.>,
the only difference being the number of derivatives of φ.



Lemma <ref> is ancillary for the proof
of the continuity of the operator 
under the extra assumption that g vanishes outside Ω.


 
Let g and  be as in (<ref>) and (<ref>).
Assume that g belongs to L^2() and has support contained in Ω.
Given 1≤ p < n/(n-1)
and p' the conjugate index of p,
the following inequality holds true:

    g_0,Ω≤2^n/2/(1-n/p')^p/2Ω^1-p/2φ_L^1(Ω)^p/2φ_L^∞(Ω)^1-p/2g_0,Ω.




The proof follows along the same lines of
<cit.>,
the only difference being the number of derivatives of φ.






 §.§ Continuity of 



We prove the continuity of the operator in (<ref>).

 
Let g and  be as in (<ref>) and (<ref>).
Assume that g belongs to H^1_0(Ω).
Given 1≤ p < n/(n-1)
and p' the conjugate index of p,
the following inequality holds true:

    g_0,Ω   ≤ 2^n-1/2
         (ρ^-1φ_L^1(Ω)
            + ρφ_L^1(Ω))
          g_0,Ω
          + 2^n-1/2
        (ρ^-1φ_L^1(Ω)
            + ρφ_L^1(Ω))
        g_0,Ω
          + 2^n/2/(1-n/p')^p/2Ω^1-p/2φ_L^1(Ω)^p/2φ_L^∞(Ω)^1-p/2g_0,Ω.




The assertion follows combining
Lemmas <ref> and <ref>,
and the explicit representation of the constants
 and 
in (<ref>).
 


Next, we derive explicit constants with respect to  and ρ
for inequality (<ref>),
i.e., we are in a position for proving one of the main results of the manuscript.



For all j, ℓ=1,…,n,
we have to bound the two terms on the right-hand side
of splitting (<ref>).
We only prove bounds for ( f(y)),
as the bounds for (f(y)) are analogous.
For the term ( f(y)), we have that
φ(x) in (<ref>)
is given by ω(x),
which is supported, with integral 1,
in the ball  of radius ρ.
Without loss of generality, the ball can be centred at the origin.

We can write

    φ(x)
    = ρ^-nψ(ρ^-1 x),

where ψ has the same smoothness as φ,
is supported in the unitary ball 
with integral 1,
and is fixed once and for all.

The following properties of φ and its derivatives are valid:

    3           φ_L^1()≈ 1,        φ_L^∞()≈ρ^-n;
    
       φ(x) = ρ^-n-1φ(ρ^-1x),            φ_L^1()≈ρ^-1,        φ_L^∞()≈ρ^-n-1;
    
       φ(x) = ρ^-n-2φ(ρ^-1x),        φ_L^1()≈ρ^-2,        φ_L^∞()≈ρ^-n-2;
    
       φ(x) = ρ^-n-3φ(ρ^-1x),        φ_L^1()≈ρ^-3,        φ_L^∞()≈ρ^-n-3.

The definition of ( f),
the chain rule,
the inequality ||≤ R,
and the fact that ρ^-1≤ Rρ^-2 imply

    _2,Ω   ≲_n
      [ρ^-1 +ρ ρ^-2] f_1,Ω
      + [ρ^-1 + ρρ^-2] f_0,Ω
                   
        +  [ρ^-1ρ^-1 + ρρ^-3] f_0,Ω
        + 2^n/2/(1-n/p')^p/2Ω^1-p/2 (ρ^-2)^p/2
      (ρ^-n-2)^1-p/2f_0,Ω
       ≲[ ρ^-1f_1,Ω
            + ρ^-2f_0,Ω
            + 2^n/2/(1-n/p')^p/2Ω^1-p/2ρ^-2-n( 1-p/2)f_0,Ω]
       ≲_n
        /ρf_1,Ω
        + /ρ^2f_0,Ω[ 1 + (1-n/p')^-p/2Ω^1-p/2ρ^-n( 1-p/2)] .
We cope with part of the last coefficient on the right-hand side separately.
Using that

    B^1-p/2ρ^-n( 1-p/2)≈ 1,

we can write

    (1-n/p')^-p/2Ω^1-p/2ρ^-n( 1-p/2)
    = (1-n/p')^-p/2(Ω/B)^1-p/2B^1-p/2ρ^-n( 1-p/2)
       ≈(1-n/p')^-p/2(Ω/B)^n-2/2(n-1)(Ω/B)^1/2( n/n-1 - p ) .

For Ω/B sufficiently large
(the ball  is anyhow fixed once and for all in the reference framework),
we choose p such that

    1/2( n/n-1 - p ) = 1/log( Ω/B).

Equivalently, we pick p such that

    log( Ω/B)
    = 1/1/2( n/n-1 - p )  ⟹  Ω/B
    = e^1/1/2( n/n-1 - p )  ⟹  ( Ω/B)
    ^1/2( n/n-1 - p ) = e^1.

We plug the last identity in (<ref>) and deduce

    (1-n/p')^-p/2Ω^1-p/2ρ^-n( 1-p/2)≈(1-n/p')^-p/2(Ω/B)^n-2/2(n-1) e^1.

Going back to (<ref>), we write

    _2,Ω≲_n
     /ρf_1,Ω
        + /ρ^2f_0,Ω[ 1 + (1-n/p')^-p/2(Ω/B)^n-2/2(n-1)] .

We further note that

    1-n/p'
    = ( n-1/p) ( n/n-1 - p )
    = 2(n-1)/p log( Ω/B).

We combine the two above displays:

    _2,Ω≲_n
     /ρf_1,Ω
        + /ρ^2f_0,Ω[ 1 +  (Ω/B)^n-2/2(n-1)p^p/2/2(n-1)^p/2log( Ω/B)^p/2] .

We have that the following quantity is uniformly bounded in p and thus in n:

    p^p/2/2(n-1)^p/2 .

Moreover, we know that p/2 < n/(2(n-1)).
We deduce that

    _2,Ω≲_n
     /ρf_1,Ω
        + /ρ^2f_0,Ω[ 1 +  (Ω/B)^n-2/2(n-1)log( Ω/B)^n/2(n-1)] ,

which is the assertion.




Compared to the lowest order estimate (<ref>),
we have an extra term involving the gradient of f
and an extra ρ^-1 scaling factor for the term involving f.

 
The issue on whether estimates as in Theorem <ref>
can be extended to union of star-shaped domains
was addressed in <cit.>.
Their proof relies on partition of unity techniques;
this entails that estimates have constants that are not fully explicit
with respect to the shape of the domain <cit.>.
A simpler open problem is whether
one may be able to prove
arbitrary order Babuška-Aziz inequalities
with explicit constants
on the union of simpler star-shaped domains,
e.g., on simplicial patches.





 §.§ Zero boundary conditions
 

We prove that  in (<ref>)
satisfies the boundary conditions in (<ref>),
i.e., that  belongs to [H_0^2(Ω)]^n.
To this aim, we provide a detailed proof of
an alternative expression for the first derivatives of ,
which has been stated in <cit.>,
and then proceed along the same lines
as in <cit.>.

We start by showing a preliminary result,
which requires the definition
of an operator
_j:Ω×Ω→ given by

    _j(x,y)
    :=  ∫_0^1 x-y/t ∂_x_jω( y + x-y/t) /t^n    ∀ j =1,…, n.



Let  and _j,
j =1,…, n, be defined
as in (<ref>) and (<ref>).
Then,
for all positive ε,
the following identity holds true:

    ∂_x_j(x,y)
    = - ∂_y_j(x,y) + _j(x,y)
            ∀x-y>ε.




We fix two indices j,k =1,…, n
and show the assertion
on the k-th components of  and .
The fact that ω
belongs to ()
and direct computations reveal

    ∂_x_j_k(x,y) 
    = ∫_0^1 [ δ_kj ω( y + x-y/t)
      + (x-y)_k/t ∂_x_jω( y + x-y/t) 
      ]/t^n+1

and

    ∂_y_j_k(x,y)
       = ∫_0^1 [ -δ_kj/t ω( y + x-y/t)
       +(x-y)_k/t ∂_x_jω( y + x-y/t)
            (1-1/t) ] /t^n
       = - ∫_0^1 [ δ_kj ω( y + x-y/t)
        +(x-y)_k/t ∂_x_jω( y + x-y/t)]
            /t^n+1
        + (_j)_k(x,y).

A combination of the two previous displays gives the assertion.


Next, we prove an identity involving the first derivatives of .


Let  and _j,
j =1,…, n, be defined
as in (<ref>) and (<ref>).
Given f in H^1_0(Ω),
consider  as in (<ref>).
Then, the following identity holds true:

    ∂_x_j (x) 
    = ∫_Ω(x,y)  ∂_y_jf(y) dy 
        + ∫_Ω_j(x,y) f(y) dy
            ∀  j =1,…, n.




Without loss of generality, we assume
that f belongs to (Ω);
the general assertion follows from a density argument.
Moreover, we prove the assertion on the k-th components
of , , and .

For any ϕ in (Ω) and j,k =1,…, n, proceeding as in <cit.>,
we have

    -∫_Ω   _k(x) ∂_x_jϕ(x) dx 
     = -∫_Ω∫_Ω_k(x,y) f(y)∂_x_jϕ(x) dx dy 
        = -∫_Ω f(y) lim_ε→ 0(∫_|y-x|>ε_k(x,y)  ∂_x_jϕ(x) dx) dy 
        = ∫_Ω f(y) lim_ε→ 0(∫_|y-x|>ε∂_x_j_k(x,y)ϕ(x) dx 
            -∫_|y-ξ|=ε_k(ξ,y) ϕ(ξ)
            y_j-ξ_j/|y_j-ξ_j|dξ) dy.

Using (<ref>) and switching
the order of integration lead us to

    -∫_Ω   _k(x)∂_x_jϕ(x) dx 
        = -lim_ε→0∫_Ωϕ(x) 
        (∫_|y-x|>ε∂_y_j_k(x,y) 
            f(y) dy) dx 
        + ∫_Ω∫_Ω (_j)_k(x,y) 
            f(y) ϕ(x) dx dy
        -lim_ε→0∫_Ω∫_|y-ξ|=ε_k(ξ,y) ϕ(ξ) f(y) 
            y_j-ξ_j/|y_j-ξ_j|dξ dy 
        =: ℐ_1 + ℐ_2 + ℐ_3.

Integrating by parts with respect to the y variable
and recalling that f belongs to (Ω),
we obtain

    ℐ_1
        = lim_ε→0∫_Ωϕ(x)
            (∫_|x-y|>ε_k(x,y) ∂_y_jf(y) dy
        -  ∫_|x-ζ|=ε_k(x,ζ) f(ζ)
            x_j-ζ_j/|x_j-ζ_j|dζ)dx 
        = ∫_Ω∫_Ω_k(x,y) ∂_y_jf(y) ϕ(x) dy  dx 
        + lim_ε→0∫_Ω∫_|ζ-x|=ε_k(x,ζ) f(ζ) ϕ(x)
            ζ_j-x_j/|ζ_j-x_j|dx dζ
        = ∫_Ω∫_Ω_k(x,y) ∂_y_jf(y) ϕ(x) dy  dx - ℐ_3.

Combining the two above displays, we infer

    -∫_Ω_k(x) ∂_x_jϕ(x) dx =
    ∫_Ω(∫_Ω_k(x,y) ∂_y_jf(y) dy +
    ∫_Ω (_j)_k(x,y) f(y) dy) ϕ(x) dx,

which gives (<ref>) for any f
in (Ω).


We are in a position to prove that 
satisfies homogeneous boundary conditions.


Given f in H^1_0(Ω)∩ L^2_0(Ω),
consider  as in (<ref>).
Then,  belongs to [H^2_0(Ω)]^n.



Step 1: a decomposition of .
Proposition <ref> allows us to
express the first derivatives of 
as a sum of two contributions.
The first correspond to Bogovskiĭ's constructions
applied to the first derivatives of f;
the second are equivalent to (<ref>)
with ω in (<ref>)
replaced by its first derivatives,
which still belongs to (Ω).

In other words, we write =  +,
where the j-th columns, j =1, …, n, of 
and  are given by

    _j(x) =∫_Ω(x,y)  ∂_y_jf(y) dy ;
            _j(x) = ∫_Ω_j(x,y) f(y) dy.


Step 2: treating .
Let j =1,…, n be fixed.
Consider a sequence g_m in L^∞(Ω) such that
g_m→∂_y_jf in L^2(Ω) as m→∞.
We consider a sequence of tensors _m
whose j-th columns are given by

    (_m)_j (x) = ∫_Ω(x,y)  g_m (y) dy.

Applying <cit.>,
it follows that (_m)_j is continuous
and vanishes on ∂Ω;
Aa a result of <cit.>,
we obtain that (_m)_j
belongs to [W^1,∞_0(Ω)]^n.
Using <cit.>
[In the literature, this result is referred to
as generalised Poincaré inequality,
which differs from (<ref>)
as no zero average condition and zero boundary conditions
on f are imposed.]
applied to ∂_y_jf-g_m∈ L^2(Ω),
we get (_m)_j →_j in [H^1(Ω)]^n.
Since (_m)_j is in [W^1,∞_0(Ω)]^n for all m,
we deduce that _j belongs to [H^1_0(Ω)]^n.



Step 3: treating  and conclusion.
The proof that _j
belongs to [H^1_0(Ω)]^n
essentially boils down
to the proof that  in (<ref>)
vanishes on the boundary of Ω;
the only difference resides in the presence
of  in lieu of ,
which solely impacts the constant in (<ref>).
We conclude that  belongs
to [H^1_0(Ω)]^n× n.
In other words,  belongs to [H^2_0(Ω)]^n,
since (<ref>) already gives
that  is in [H^1_0(Ω)]^n.



 
Identity (<ref>) may allow us
to prove a first order Babuška-Aziz inequality
also in a non-Hilbertian setting,
namely we may substitute H^k-type spaces
by W^k,p-type spaces, p  2.
However, this would come at the price of suboptimal estimates
as those in (<ref>).
The reason for this is the use of the Calderón-Zygmund theory
instead of the Fourier transform approach by Durán
while handling the term 
in Section <ref>.





 §.§ On the optimality of the estimates in Theorem <ref>


We discuss the optimality of the estimates
in Theorem <ref>
based on a counterexample in <cit.>.
Introduce the domain

    := (-a,a) × (-ε, ε)

and the function

    f(x_1,x_2)=x_1.

Let  be the solution to the divergence problem (<ref>). We have

    x_1_0,^2
     = ∫_ x_1 ÷
      = -∫__1
      = -1/2∫_ x_2^2 ∂_x_2^2^2 _1
    ≤1/2x_2^2_0,∂_x_2^2^2 _1_0,.

We use estimates as in (<ref>) for the last term on the right-hand side:
there exists positive constants C_1 and C_2 depending
on =2a and ρ=ε such that

    x_1_0,^2
    ≤1/2x_2^2_0,[  x_1_0,
          + 1_0,] .

We have

    x_1_0,^2   
        = 4/3 a^3 ε ,
                x_2^2_0,^2
        = 4/5 a ε^5 ,
                1_0,^2     
     = 4 a ε.

Combining the two displays, we get

    a^3 ε≲ a^2ε^3
            +  a ε^3,

whence

    1
    ≲ a^-1ε^2 +  a^-2ε^2 
    ≈ρ^2/ + ρ^2/^2 .

This inequality implies that
at least one of the following must hold true:

    ≳/ρ^2,
                ≳^2/ρ^2.

Using (<ref>),
we deduce

    ^2/ρ^2≲≲/ρ,

which cannot be valid in general with hidden constants
independent of R and ρ as in (<ref>).
This entails that

    ≳/ρ^2,

i.e., the first bound in (<ref>)
is optimal up to a logarithmic factor.





§ PROOF OF A NEČAS-LIONS INEQUALITY WITH SYMMETRIC GRADIENTS
 

We prove Theorem <ref>.

For any generic  in the space
of rigid body motions (Ω)∩ [L^2_0(Ω)]^n
as discussed in Section <ref>,
we have

    -_0,Ω≤- - (-)_0,Ω        ∀∈ [L^2(Ω)]^n.

An immediate consequence of the standard,
lowest order Nečas-Lions
inequality (<ref>) is that

    -_0,Ω≤∇(-)_-1,Ω        ∀∈(Ω) ∩ [L^2_0(Ω)]^n.

We are left to prove the existence of a positive constant C
with explicit dependence on R and ρ as in (<ref>),
such that,
for a specific choice of ,
the following inequality is valid:

    ∇(-)_-1,Ω≤ C _-1.

Using splitting (<ref>)
of the gradient into symmetric and skew-symmetric parts,
and the triangle inequality entails

    ∇(-)_-1,Ω≤(-)_-1,Ω
         + (-)_-1,Ω.

Since  is a rigid body motion,
  is the zero tensor in ^n× n:
the first term on the right-hand side is equal to _-1,Ω.
As for the second term on the right-hand side,
we define A^n× n as the space of (n× n) skew-symmetric matrices,
n=2,3.
We take  such that

    (-)_-1,Ω
    : = inf_∈(Ω)∩[L^2_0(Ω)]^n× n(-)_-1,Ω
    = inf_∈ A^n× n- _-1,Ω .

With this at hand, elementary computations give

    (-)_-1,Ω
    = inf_∈ A^n× n-_-1,Ω≤1/√(2)inf_∈^2n-3∇× - _-1,Ω.

Note that the last two norms involve tensors and vectors, respectively.

An immediate consequence of (<ref>)
and (<ref>) is that

    inf_∈^2n-3∇× - _-1,Ω≤∇(∇×)_-2,Ω≤ (+) ∇(∇×)_-2,Ω.

Direct computations reveal that 

    inf_∈^2n-3∇× - _-1,Ω   ≤ 2 (+) 
        ∇×_-2,Ω
       ≤ 2 (+) _-1,Ω.

Combining (<ref>), (<ref>),
(<ref>), and (<ref>)
yields the assertion.




  
*Acknowledgements.
The Authors have been partially funded by the European Union
(ERC, NEMESIS, project number 101115663);
views and opinions expressed are however those of the author(s) only
and do not necessarily reflect those of the EU or the ERC Executive Agency.
LM has been partially funded by MUR (PRIN2022 research grant n. 202292JW3F).
The Authors are also members of the Gruppo Nazionale Calcolo Scientifico-Istituto
Nazionale di Alta Matematica (GNCS-INdAM).




plain





§ ARBITRARY ORDER BABUŠKA-AZIZ INEQUALITIES
ON STAR-SHAPED DOMAINS WITH EXPLICIT CONSTANTS



In this appendix, we provide a road-map
for proving arbitrary order Babuška-Aziz inequalities
that are explicit with respect
to R and ρ in (<ref>),
based on a generalisation of the analysis in Section <ref>.
In particular, we prove the following generalisation of Theorem <ref>.

 
Let Ω have a sufficiently smooth boundary
so as to be able to define H^d_0(Ω),
and R and ρ be as in (<ref>).
Assume that f is in H^d-1_0(Ω)∩ L^2_0(Ω),
d in  larger than or equal to 1.
Then, there exists  in [H^d_0(Ω)]^n such that
÷=f and

    _d,Ω   ≲_n ∑_ℓ=1^d-1/ρ^d-ℓf_ℓ,Ω 
            + /ρ^d[ 1 + ( Ω/B)^n-2/2(n-1)( logΩ/B)^n/2(n-1)] f_0,Ω.



We give some details on how to prove (<ref>).
Essentially, we have to prove the continuity of the counterpart 
of the operator in (<ref>)
involving all the derivatives of order d.
More precisely,
for each multi-index =(j_1, …,j_n)
in { 1 ,…, d }^n,
=d,
we study the continuity of the operator

    g
    :=  g +  g,

where

    g(x)
    := lim_ε→0∫_ε^1/2∫_[ φ( y+ x-y/t) ]
            g(y) /t^n

and

    f(x)
    := ∫_1/2^1 ∫_[ φ( y+ x-y/t) ]
            g(y) /t^n.

Above, g and φ are precisely as in (<ref>).
The general assertion then follows summing over
all possible multi-indices .
In the remainder of the appendix, 

    is fixed,            
    j_1 0.





 §.§ Continuity of 



We discuss here the continuity of the operator in (<ref>).
We have the two following technical results. 


 
Let  be as in (<ref>).
If g in (<ref>) belongs to (),
then the following identity is valid:

    g(ξ)
        = (2πı)
       lim_ε→ 0[
       ∑_,∈{ 1,…,d }^n ,+=(j_1-1,j_2,…,j_d) ∫_ε^1/2∂^_ξ_1^k_1,…,ξ_n^k_nφ (t ξ)
        ∂^_ξ_1^ℓ_1,…,ξ_n^ℓ_n g(ξ) ].


The proof is a modification of that of Lemma <ref>.
Since this result contains most of the differences
compared to the first order case,
we prove the assertion for the second order case,
i.e., we assume that =(,,)
with ++=3.
To further simplify the proof, we discuss the case j_1=j_2=j_3=1.
The general assertion is proven analogously.


In analogy to the proof of Lemma <ref>,
we can write

    g(ξ)  
     = (2πı) (2πı)(2πı)
       ∫_ε^1/2∫_∫_φ( y+ x-y/t)
            g(y) e^-2 πı x·ξ/t^n
        = (2πı) [
       ∫_ε^1/2( t  ξ)
        [2πı (1-t) ][2πı (1-t) ] ((1-t)ξ) . 
                     +   ∫_ε^1/2( t  ξ)
        [2πı t ][2πı (1-t) ] ((1-t)ξ) 
                    +    ∫_ε^1/2( t  ξ)
        [2πı (1-t) ][2πı t ] ((1-t)ξ) 
                   .   +   ∫_ε^1/2( t  ξ)
        [2πı t ][2πı t ] ((1-t)ξ) ] 
        = (2πı) [
       ∫_ε^1/2( t  ξ)
        ∂^2_ g((1-t)ξ) 
        + ∫_ε^1/2∂_φ (t ξ)
        ∂_g((1-t)ξ) . 
                   . + ∫_ε^1/2∂_φ( t  ξ)
        ∂_ g((1-t)ξ) 
        + ∫_ε^1/2∂^2_φ (t ξ)
        ((1-t)ξ) ] .

This yields the assertion for the case d=3, and j_1=j_2=j_3=1.


From Lemma <ref>, it is apparent that we
have to bound the norm of several derivatives of φ,
which is what we accomplish in the next result.


 
Let φ and  be as in (<ref>) and (<ref>).
Then, the following inequalities hold true:
for all multi-indices  in { 1,…,d }^n
such that ≤=d,

    2π∫_0^∞∂^_^k_1-1,…,^k_nφ(tξ)   ≤
        := ρ^-1∂^_x_1^k_1-1,x_2^k_2,…,x_n^k_nφ_L^1()
      + ρ∂^+2_x_1^k_1+1,x_2^k_2,…,x_n^k_nφ_L^1() .



The proof is a modification of that of Lemma <ref>.


The two above technical lemmas give the following result.


 
Let φ be any of the two options in (<ref>).
Then, for all g in H^d(^n),
the operator  defined in (<ref>)
satisfies the following continuity property:

    g_0,≤ 2^n-1/2∑_,∈{ 1,…,d }^n ,+=(j_1-1,…,j_d) [∂^_x_1^ℓ_1,…,x_n^ℓ_n g_0,],

where φ is any of the two options in (<ref>).


The proof is a modification of that of Proposition <ref>,
and further combines Lemmas <ref> and <ref>.





 §.§ Continuity of 



We discuss here the continuity of the operator in (<ref>).
We have the following result.


 
Let g, , and  be as
in (<ref>), (<ref>), and (<ref>).
Assume that g belongs to L^2() and has support contained in Ω.
Given 1≤ p < n/(n-1)
and p' the conjugate index of p,
the following inequality holds true:

    g_0,Ω≤2^n/2/(1-n/p')^p/2Ω^1-p/2∂^d_x_1^j_1… x_n^j_nφ_L^1(Ω)^p/2∂^d_x_1^j_1… x_n^j_nφ_L^∞(Ω)^1-p/2g_0,Ω.




The proof is a modification of that of Proposition <ref>.





 §.§ Continuity of 



We discuss here the continuity of the operator in (<ref>).
We have the following result.


 
Let g, , and 
be as in (<ref>), (<ref>), and (<ref>).
Assume that g belongs to H^d-1_0(Ω).
Given 1≤ p < n/(n-1)
and p' the conjugate index of p,
the following inequality holds true:

    g_0,Ω
       ≤ 2^n-1/2∑_,∈{ 1,…,d }^n ,+=(j_1-1,…,j_d) [
                (  ρ^-1∂^_x_1^k_1-1,x_2^k_2…,x_n^k_nφ_L^1() +
                ρ∂^+2_x_1^k_1+1,x_2^k_2,…,x_n^k_nφ_L^1())
                ∂^_x_1^ℓ_1,…,x_n^ℓ_n g_0,] 
          + 2^n/2/(1-n/p')^p/2Ω^1-p/2φ_L^1(Ω)^p/2φ_L^∞(Ω)^1-p/2g_0,Ω.



The assertion follows combining
Lemmas <ref> and <ref>,
and the explicit representation of the constants
 in (<ref>).
 


Theorem <ref> follows using
Theorem <ref>,
the chain rule, and proceeding as in the proof
of Theorem <ref>.


An arbitrary order Nečas-Lions,
generalising the first order version in (<ref>),
may be shown based on Theorem <ref>
following the proof of Proposition <ref>;
for d in , it reads

    _[H^-d+1(Ω) / _d-1(Ω)]^n≤∇_-d,Ω        ∀∈ [H^-d(Ω)/ _d-1(Ω)]^n,

where [H^-d+1(Ω) / _d-1(Ω)]^n is the space [H^-d(Ω)]^n
equipped with the norm

    _[H^-d(Ω) / _d-1(Ω)]^n
    := inf_𝐪_d-1∈ [_d-1(Ω)]^n-𝐪_d-1_-1,Ω.

The constant  depends on the Babuška-Aziz constants of all orders up to d
and the Poincaré constant through <cit.>.



 Paper Title: A diffuse-interface Landau-de Gennes model for free-boundary nematic liquid crystals.


Cooperative Orbital Angular Momentum Wireless Communications
    Ruirui Chen, Wenchi Cheng, Senior Member, IEEE, Jinyang Lin, and Liping Liang

This work is supported in part by Natural Science Foundation of Jiangsu Province under Grant BK20200650, in part by National Natural Science Foundation of China under Grant 62071472, in part by China Postdoctoral Science Foundation under Grant 2019M660133, in part by Program for “Industrial IoT and Emergency Collaboration” Innovative Research Team in CUMT under Grant 2020ZY002, in part by Fundamental Research Funds for the Central Universities under Grant 2019QNB01 and 2020ZDPY0304. (Corresponding author: Ruirui Chen.)

Ruirui Chen and Jinyang Lin are with School of Information and Control Engineering, China University of Mining and Technology, Xuzhou, 221116, China (emails: rrchen, linjinyang@cumt.edu.cn). Wenchi Cheng and Liping Liang are with State Key Laboratory of Integrated Services Networks, Xidian University, Xi'an, 710071, China (e-mail: wccheng, liangliping@xidian.edu.cn).




    August 5, 2024
======================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================



§ ABSTRACT 

We introduce a diffuse-interface Landau-de Gennes free energy for free-boundary nematic liquid crystals (NLC) in three dimensions submerged in isotropic liquid, where a phase field is introduced to model the deformable interface. 
The energy we propose consists of the original Landau-de Gennes free energy, three penalty terms and a volume constraint.

We prove the existence and regularity of minimizers to the diffuse-interface energy functional. We also prove a uniform maximum principle of the minimizer under appropriate assumptions, together with a uniqueness result for small domains.
Then, we establish a sharp-interface limit where minimizers of the diffuse-interface energy converge to a minimizer of a sharp-interface energy under the framework of Γ-convergence.

Finally, we conduct numerical experiments with the diffuse-interface model, the findings of which are compared with existing works.





  
*Keywords. nematic liquid crystals, phase separation, Landau-de Gennes, Γ-convergence




    

  
*AMS subject classifications. 76A15, 49J27, 49K20, 35B50, 49J45, 35F30






§ INTRODUCTION



The liquid crystal (LC) is an intermediate state between solids and liquids, in which positional order of the crystal is partially or completely lost, but molecular anisotropy is still present <cit.>.
The simplest case of liquid crystals is the nematic phase in which the constituent molecules have no positional order, but prefer to align locally along certain directions known as directors.
Nematic liquid crystals (NLC) are endowed with direction-dependent physical, optical and rheological properties <cit.>.

NLC's exhibit intriguing morphology because of their molecular anisotropy, especially when a droplet of undetermined shape interacts with a different material, about which people have conducted many experiments.
NLC dispersed in polymer (often called polymer-dispersed liquid crystal in the community) forms droplets in various shapes and sizes, leading to peculiar optoelectric properties when sandwiched among planar electrodes <cit.>. 
A special droplet morphology of interest is the nematic tactoid, i.e. spindle-shaped droplets filled with NLC whose directors align with the surface, which tends to emerge during the nematic-isotropic phase transition induced by temperature <cit.> or around rod-shaped bacteria as protective sheaths <cit.>. 

The myriad phenomena of free-boundary NLC droplets have inspired widespread applications.
For instance, polymer-dispersed LC have been produced with simple and cost-effective phase separation methods, leading to huge advances in the optoelectric display industry <cit.>. 
Surface grating patterns can be carefully tailored to obtain bistable nematic states, promising the production of low-energy, easily switchable LC displays <cit.>. 
With the aid of NLC morphology, people have also accurately simulated polymerized nematic elastic ribbons with multiple stable shapes <cit.>, biomimetic nematic droplets with non-equilibrium oscillation <cit.> and the morphogenesis of living cells <cit.>. 

These examples demonstrate the vast potentials of free-boundary NLC.


To study free-boundary NLC mathematically, we need to construct a model in two aspects: one about free boundary and the other about NLCs.

For modelling NLCs, there is the Onsager (molecular) model based on local director distributions, the Ossen-Frank (vector) based on the unit director field, the Landau-de Gennes (LdG, tensor) model based on moments of the director distribution and many others <cit.>.
Each model has its strengths and weaknesses, but all of them have been successful in predicting the behaviour of NLC, see e.g. <cit.> (Onsager), <cit.> (Oseen-Frank) and <cit.> (LdG).

For modelling free boundary, we quote existing methods from the field of shape and topology optimization <cit.>.
Among the most popular are the surface mesh method <cit.> that discretizes the free boundary with a finite-element mesh known as the interface segregating different phases, 
and the diffuse-interface method <cit.> that assumes a continuous variation of the phase within the interfacial regions described by a smooth field (space-dependent function) known as the phase field. 

In recent decades, there has been a multitude of research in free-boundary NLCs with fairly diverse modelling approaches, both in free boundary and in NLC <cit.>.
The common practice is to design energy functionals with respect to the shape and the NLC order parameter, and find the stable configuration with energy minimization methods. 
For instance, in <cit.> the authors describe a deformable two-dimensional NLC confinement with a finite-element mesh, and their energy functional consists of the Landau-de Gennes free energy and a penalty function for Dirichlet boundary values; they discover that with a strong penalty, the shape of the confinement converged into a tactoid through numerical simulation, which is consistent with experimental findings. 
The authors of <cit.> study the same problem with the finite element method (FEM) as well, but they choose the Oseen-Frank (vector) free energy instead and further penalize the energy functional with the perimeter of the confinement; they also find tactoids when the penalty factors are large. 
In <cit.> the authors use a phase field ϕ to denote separation of NLC and a Newtonian fluid with ϕ=1 for NLC and ϕ=-1 for fluid, and their energy functional consists of a boundary anchoring with ∇ϕ substituted for the boundary normal, a mixing energy of van der Waals-Cahn-Hilliard form <cit.> to penalize the sharp phase separation (ϕ≈± 1) and smoothness of ϕ, and finally the Oseen-Frank free energy density masked by ϕ+1/2 so that it is only integrated over the NLC phase; numerical tests of their energy minimization find multiple stable states, including one radially symmetric state and one slightly elliptical bipolar state reminiscent of the tactoid. 
In general, the application of shape/topology optimization methods to free-boundary NLC has been highly successful in explaining their complex anisotropy-induced morphology, and promises an immense area of exploration both numerically and theoretically.




We propose a diffuse-interface model for a free-boundary nematic phase in three dimensions (3D) submerged in the isotropic phase, using the Landau-de Gennes free energy for the modelling of liquid crystals and a phase field for deformable shapes. Our approach is mainly inspired by <cit.>. The model is relatively new since NO work as far as we know combines LdG theory with diffuse-interface models to study free-boundary NLCs.


We choose the LdG model to describe NLC because the Q-tensor order parameter has fewer degrees of freedom than the actual director distribution, so the computation is less intensive than the Onsager model. 
The LdG model can also represent +1/2 and -1/2 defects in two dimensions (a discontinuity of the director field around which the director rotates +1/2 or -1/2 cycles, quite common in real-life NLCs <cit.>), where the director field is non-orientable and thus impossible for the Oseen-Frank model. 
Theoretical properties such as maximum principle and uniaxial symmetry are also available <cit.>, which mathematically guarantee the feasibility of the LdG model. 
With the LdG model, one can also produce a plethora of complicated energy landscapes over confinements of different shapes, see e.g. <cit.>. 
Therefore, the LdG model possesses huge potentials in the free boundary problem.
We choose the diffuse-interface model to describe the deformable shape because the phase field can be defined on a fixed grid <cit.> while a surface mesh requires a full finite-element mesh <cit.>. The degrees of freedom are therefore considerably lower for the phase field, since it only involves a scalar function, but the finite-element mesh requires the spatial coordinates of all nodes and the adjacency relations between them. 
Moreover, the boundary of the deformable shape is identified by variations of the phase field, so topological changes can be implemented, e.g. one region splitting into two or two regions merging into one <cit.>. On the contrary, the surface mesh cannot describe topological changes because the node adjacency relations are fixed.


Our work is presented in the following order.
In Section <ref>, we define a diffuse-interface energy functional E_ made up of the classic LdG free energy in 3D defined by the Q-tensor, penalty terms and a volume constraint on the phase field ϕ. There are three extra competing energy terms besides the LdG energy. The volume constraint ensures the mass conservation of the nematic phase. There is also a hyperparameter >0 that signifies the width of the diffuse interface.
In Section <ref>, we establish the solvability, i.e. existence of minimizers of E_, with classical methods of calculus of variation <cit.>.
In Section <ref>, we discuss the problem of minimizing E_ on a fixed phase field. We discuss the maximum principle with the general theory of elliptic equations, and then derive uniqueness of the minimizer based on a convexity argument from <cit.>. Our discoveries correspond to well-known results of the classical LdG model on a fixed region.
In Section <ref>, we present our main contribution — the limit of E_ as → 0, known as the sharp-interface limit, with Γ-convergence <cit.> as the primary tool. We prove under appropriate assumptions the convergence of minimizers of E_ to the minimizer of a sharp-interface functional E_0 defined over deformable regions that also models the separation of nematic and isotropic phases.

In Section <ref>, we conduct numerical experiments on the minimization of a reduced version of E_ on a two-dimensional domain. We adjust hyperparameters to observe changes in the minimizer configuration, and compare them with past findings.





§ DIFFUSE-INTERFACE LANDAU-DE GENNES MODEL
 

Let Ω⊂^3 be a region with Lipschitz boundary, where a blob of nematic liquid crystals is surrounded by isotropic liquid. The separating interface is called the nematic-isotropic (N-I) interface.
We propose a diffuse-interface LdG energy

    E_ = E^ LdG + ω_p E^ mix_ + ω_a E^ anch_ + ω_v E^ void,

where ω_p,ω_v,ω_a are positive weights of competitive energy terms and >0 is a positive parameter appearing in the second and third terms.
The units of ω_p,ω_v,ω_a are N· m^-1, N· m^-1 and N· m^-2 respectively.
The diffuse-interface LdG energy is composed of four types of free energy: (a) LdG free energy of nematic liquid crystals, (b) mixing energy of the N-I interface, (c) tangential anchoring energy of the liquid crystal molecules on the interface and (d) void energy (penalty) of the isotropic phase.



  
(a) LdG energy
To describe the director and corresponding orientational order of the nematic phase, we introduce the tensor variable Q=(Q^ij)_3× 3, a 3× 3 symmetric traceless matrix encoding information about nematic director distribution <cit.>.
Q= 0 represent the isotropic phase; if only two of the eigenvalues are the same, Q is uniaxial; if the eigenvalues are distinct, Q is biaxial.
A uniaxial state, which can be represented by

    Q=s(n⊗n-1/3 I),

with s∈ and n∈𝕊^2, is the most interesting because n recovers the nematic director, i.e. local preferred molecular alignment.
The term E^ LdG in (<ref>) is the classical LdG free energy

    E^ LdG[ Q]=∫_Ω (F_el + F_b) x̣,

with elastic energy density

    F_el = L/2 |Q|^2,

where L>0 is the material-dependent elastic constant, |Q|^2≜∑_i,j,k |Q^ij_x_k|^2, and bulk energy density

    F_b( Q) = A/2 Q^2 - B/3 Q^3 + C/4( Q^2 )^2,

where A<0 is a rescaled temperature that can be tuned to achieve the nematic-isotropic phase transition (we assume that A<0 to stabilize the nematic phase), and B,C>0 are material-dependent constants.
The units of A,B,C are N· m^-2, and the unit of L is N.
As a reminder, the elastic energy density (<ref>) adopts the simplest one-constant form, but more general forms exist <cit.>.
Under our setup, the minimizers of the functional F_b (<ref>) are uniaxial states Q^*=s_+(n⊗n-1/3I), with arbitrary unit vector n∈𝕊^2 and

    s_+=B+√(B^2-24AC)/4C.




  
(b) Mixing energy
The term E^ mix_ in (<ref>) is the mixing energy. It is the well-known van der Waals-Cahn-Hilliard energy functional <cit.> describing binary mixture of two phases, which in this case are nematic and isotropic liquid crystals.

    E^ mix_[ϕ]=∫_Ω[  |∇ϕ|^2 + ^-1ϕ^2(1-ϕ)^2 ] x̣,

where ϕ is a phase field.
Informally speaking, the continuous double-well function ϕ^2(1-ϕ)^2 prefers the separation of Ω into subdomains with ϕ≈ 1 for nematics and ϕ≈ 0 for isotropic, while the gradient term |∇ϕ|^2 smoothens out ϕ to form a diffuse interface, whose thickness is represented by >0, the capillary width with unit m (see Figure <ref>).
The van der Waals-Cahn-Hilliard energy is known to approximate the perimeter of the interface in the sense of Γ-convergence as → 0 (the sharp interface limit) <cit.>, so it models the interfacial tension, as is common in numerous physical models <cit.>. 





  
(c) Anchoring energy
The term E^ anch in (<ref>), aimed to enforce tangential anchoring <cit.> on the N-I interface, is given by

    E^ anch_[ Q,ϕ]=∫_Ω| (Q(x)+s_+/3 I) ∇ϕ|^2x̣,

with s_+ defined in (<ref>).
E^ anch_ is minimized when Q ∇ϕ=-s_+/3∇ϕ, i.e. ∇ϕ is an eigenvector of Q corresponding to the negative eigenvalue -s_+/3, indicating that the nematic director of Q is perpendicular to ∇ϕ, which approximates the normal direction within the diffuse interface.
To keep this energy at O(1) order in the sharp-interface limit → 0, the factor  is needed in (<ref>). One can verify this fact by checking from Figure <ref> that |∇ϕ|=O(^-1) and that the size of its support is O(), making the integral without  in (<ref>) approximately O(^-2)· O()=O(^-1).
This weak anchoring has been applied in works such as <cit.> and <cit.>.



  
(d) Void energy
The term E^ void in (<ref>), aimed to penalize the isotropic phase (ϕ≈ 0) surrounding the nematic phase, is given by

    E^ void[ Q,ϕ]= ∫_Ω1/2(1-ϕ)^2 | Q|^2 x̣,

where | Q|^2=∑_i,j |Q^ij|^2 is the Frobenius norm.
It is inspired by similar penalty terms in the studies of phase separation in fluids <cit.>.

We impose the volume constraint 

    ∫_Ωϕx̣ = V_0,

to conserve the mass of NLCs with 0<V_0 <|Ω| (|Ω| is the volume of Ω),
and impose homogeneous Dirichlet boundary conditions

    Q|_Ω= 0,  ϕ|_Ω=0,

to model the isotropic phase surrounding the nematic phase.


The energy (<ref>) can be nondimensionalized. Let x̅=λ^-1x for all x∈Ω. After substituting x̅ in the integrals and dividing the entire energy by λ L, we obtain

    E_[Q,ϕ]
           =E̅^ LdG+ω̅_p λ̅E̅_^ mix+ω̅_a λ̅E̅_^ anch+ω̅_p λ̅^2E̅^ void
       =∫_Ω^(0)[ 1/2 |Q|^2 + λ̅^2 F̅_b( Q) ] ̣̅x + ω̅_p λ̅∫_Ω^(0)[  |∇ϕ|^2+ 1/ϕ^2(1-ϕ)^2]̣̅x
          +ω̅_a λ̅∫_Ω^(0)| (Q+s_+/3 I)∇ϕ|^2 ̣̅x + ω̅_v λ̅^2 ∫_Ω^(0)1/2(1-ϕ)^2 | Q|^2 ̣̅x

with nondimensional domain Ω^(0) and rescaled nondimensional constants

    λ̅^2 ≜λ^2 C/L, ≜/λ, ω̅_p ≜ω_p/√(CL),ω̅_v ≜ω_v/C, ω̅_a ≜ω_a/√(CL),

and rescaled LdG bulk energy F̅_b≜1/C F_b, subject to the nondimensional volume constraint

    ∫_Ω^(0)ϕ̣̅x = V̅_0,
 
with V̅_0=λ^-3V_0 satisfying 0<V̅_0 < |Ω^(0)|.

From this point onwards, we will focus on the nondimensional functional E_ (<ref>), which is almost identical to (<ref>) except for the additional λ̅ in the leading coefficients and the bars above.
For notational simplicity, we rewrite Ω^(0) as Ω and drop bars over all constants from (<ref>) (note: the bar over F̅_b is kept in order to differ from the classical bulk energy F_b (<ref>)).
The four terms of (<ref>) are identified with the LdG energy E^ LdG (<ref>), the mixing energy E^ mix_ (<ref>), the anchoring energy E^ anch_ (<ref>) and the void energy E^ void (<ref>) respectively.
We also denote the admissible space of the order parameters ( Q,ϕ) by

    𝒜= ( Q,ϕ):
             Q ∈ H_0^1(Ω;𝒮_0), ϕ∈ H_0^1 (Ω),∫_Ωϕx̣=V_0

where we use H_0^1 to denote the Sobolev space <cit.>

    H_0^1(Ω)=u∈ L^2(Ω): ∫_Ω [u^2+|∇ u|^2] x̣<∞, u|_Ω=0,

H_0^1(Ω;V) to denote the Sobolev space with function values in a normed vector space V,
and also

    𝒮_0={ Q∈^3× 3:  Q^T= Q,  Q=0}

to denote the space of symmetric traceless 3× 3 matrices.


If ( Q,ϕ) is a critical point of the functional (<ref>), then they are weak solutions to their corresponding Euler–Lagrange equations:

    Q   = λ^2 [ (A/C+ Q^2)   Q - B/C𝒫_0( Q^2)]
            + ω_vλ^2 (1-ϕ)^2  Q 
         
            +ω_aλ𝒫_0 [(∇ϕ⊗∇ϕ )( Q+s_+/3 I) + ( Q+s_+/3 I)(∇ϕ⊗∇ϕ)],

and

    ϕ + ω_a/ω_p[( Q+s_+/3 I)^2 ∇ϕ] = 1/^2ϕ(ϕ-1)(2ϕ-1) + ω_vλ| Q|^2/2ω_p (ϕ-1) + ξ,

where ∇_𝒮_0 is the projected gradient operator on 𝒮_0 (<ref>), which maps the matrix gradient onto 𝒮_0 with the projection operator

    𝒫_0( A)= A- A/3 I

to maintain the traceless constraint on Q,
and ξ∈ is a Lagrangian multiplier originating from the volume constraint (<ref>).
For a detailed calculation of (<ref>) and (<ref>), please refer to Appendix <ref>.



§ EXISTENCE OF MINIMIZERS
 
We use the direct method in calculus of variations <cit.> to prove the existence of a minimizer to the diffuse-interface LdG functional.

 
The functional E_ (<ref>) is weak lower semi-continuous (w.l.s.c.) on the Banach space ( Q,ϕ)∈ H_0^1(Ω;𝒮_0) × H_0^1(Ω), which means that if Q_k ⇀ Q,ϕ_k⇀ϕ_k are weakly convergent sequences in H^1, 

    lim inf_k→∞ E_[ Q_k,ϕ_k] ≥ E_[ Q,ϕ].




As <cit.>, 
for an integral energy functional of the form

    ∫_Ω F(∇ u,  u, x)x̣

where u: Ω→^m and F:^m× n×^m×Ω→, 
if F is uniformly bounded below and convex in its first argument
the functional is w.l.s.c. with respect to u∈ H^1(Ω;^m).



We look at individual terms.

Examine the form (<ref>) where we take u=( Q,ϕ): Ω→𝒮_0×. We notice that the energy density is bounded below because all terms are positive except for the LdG bulk energy F̅_b=C^-1F_b (<ref>), which satisfies that

    F̅_b( Q)≥ m_b >-∞,   ∀ Q∈𝒮_0

for some number m_b∈ whose explicit expression can be found in <cit.>.
Moreover, the terms involving the gradients, which read

    1/2 |Q|^2 + ω_pλ|∇ϕ|^2+ω_aλ| (Q+s_+/3 I)∇ϕ|^2,

are convex with respect to ( Q,∇ϕ) since they are positive-defintie quadratic polynomials. Therefore, the energy density of E_ (<ref>) satisfies the criterion, and is w.l.s.c. over the space H_0^1(Ω;𝒮_0)× H_0^1(Ω).


 
The energy functional E_ (<ref>) has at least one global minimizer (Q^*_,ϕ^*_)∈𝒜, with the admissible set 𝒜 defined as in (<ref>).



The admissible set 𝒜 (<ref>) is obviously nonempty since there always exists smooth ϕ∈ H_0^1 such that ∫_Ωϕ = V_0.

Moreover, the fact that all terms in E_ are bounded below (recall (<ref>)) implies the following coercivity condition.

    E_[Q,ϕ]    ≥∫_Ω[ 1/2 |Q|^2+ ω_p |∇ϕ|^2 + λ^2 m_b ]x̣
       =L/2 Q_L^2^2 + ω_p∇ϕ_L^2^2 + λ^2 m_b |Ω|.


The weak lower semi-continuity of E_ over the feasible set is guaranteed by Lemma <ref> since the admissible set 𝒜 is a closed subspace of H_0^1(Ω;𝒮_0)× H_0^1(Ω). 
Therefore, the existence of a global minimizer follows from the direct method in calculus of variations.


 
For each fixed ϕ∈ H_0^1, the w.l.s.c. and coercivity argument still applies with respect to the variable Q. Hence, there exists a Q-minimizer Q_∈ H_0^1(Ω;𝒮_0) such that

    Q_= Q_(ϕ)=_ Q∈ H_0^1(Ω; 𝒮_0) E_[ Q,ϕ].





§ MAXIMUM PRINCIPLE ON A FIXED DOMAIN
 

In this section, we prove that under a fixed ϕ, the critical points of E_ with respect to Q (referred to as Q-critical points) are bounded in L^∞ uniformly as → 0. The uniqueness of Q-critical points follows immediately.


The maximum principle and uniqueness results of the LdG model on a fixed region have been studied thoroughly <cit.>, but results for the diffuse-interface model (<ref>) involve different techniques, especially when dealing with the anchoring penalty (<ref>).



 §.§ Uniform maximum principle

Before introducing the uniform maximum principle, we improve the regularity of Q_ with the L^p theory of second-order elliptic equations <cit.>. We denote by

    W^m,p(Ω)=u∈ L^p(Ω): u_W^m,p^p = ∑_k=0^m ∫_Ω |∇^k u|^p <∞

the general Sobolev spaces.

 
If ϕ∈ W^1,∞(Ω), then a Q-critical point Q_∈ H_0^1(Ω;𝒮_0) of E_[·,ϕ] (<ref>) is in W^2,p for all p<∞ and C^1,α for all α<1.
Since Q_ is a weak solution, continuity means that Q_ is equal to a C^1,α function a.e..


By definition, the critical point Q_ satisfies the equation (<ref>).
Examining its right-hand side, we find that it is a cubic polynomial of Q (because of the bulk energy derivative), and its coefficients, excluding constants, are related to ϕ,∇ϕ, which are uniformly bounded by assumption.

If Q_∈ L^p(Ω), then the RHS (right-hand side) of (<ref>) as a whole is in L^p/3(Ω).
Therefore, by the L^p theory of elliptic equations <cit.>, Q_∈ W^2,p/3 if p>3.
This enables us to improve the regularity of Q_ through a chain of Sobolev embeddings.














We have the a priori estimate Q_∈ H_0^1, which indicates Q_∈ L^6 by the Sobolev embedding theorem in 3D. 
Carrying out the chain of embeddings, we eventually reach Q_∈ W^2,p for all p<∞, and with one extra step of embedding we get Q_∈ C^1,α for all α<1.

 
Under stronger smoothness assumptions, we prove that Q-critical points of the nondimensional energy (<ref>) is uniformly bounded. The bound is uniform in the sense that it is independent of the capillary width .

 
Let ϕ_∈ W^1,∞(Ω) be a phase field, with the following extra assumptions.


  * ϕ_∈[0,1] for all x.

  * Over the diffuse interface Γ_={x∈Ω: 0<ϕ(x)<1}, the diffuse unit normal vector n_, defined by

    n_=∇ϕ_|∇ϕ_|,    ∇ϕ_≠ 0, 
     0,    ∇ϕ_=0,

is part of a C^2-smooth vector field on Ω such that

    | n_|≤ 1, 
     n__C^2(Ω)≤ M_N <∞

for all . In other words, the normal vector in the diffuse interface Γ is part of a uniform C^2 vector field.

Suppose Q_ is a Q-critical point of E_[·,ϕ_] (<ref>).
Then with any fixed λ_0>0, there exists a uniform constant M determined by A,B,C,ω_v,λ_0 and the bound M_N, but independent of , such that

    sup_x∈Ω | Q_| ≤ M λ^-2/3.

The bound is given in the a.e. sense.



In the following proof we drop the subscript  for notational simplicity.

Take the dot product of Q+αn⊗n with the Euler-Lagrange equation (<ref>), where α is a number yet to be determined. Recall the operator 𝒫_0 as defined in (<ref>).

    ( Q+αn⊗n): Q
       = λ^2 ( Q+αn⊗n):[ (A/C+ Q^2)   Q - B/C𝒫_0( Q^2)]_(I)
          
        + ω_vλ^2 (1-ϕ)^2 ( Q+αn⊗n): Q_(II)
         
        +ω_aλ( Q+αn⊗n):𝒫_0 [(∇ϕ⊗∇ϕ )( Q+s_+/3 I) + ( Q+s_+/3 I)(∇ϕ⊗∇ϕ)]_(III).


For the LHS (left-hand side) of (<ref>), we compute

    ( Q+αn⊗n): Q
           =( Q+αn⊗n): ( Q+αn⊗n) - ( Q+αn⊗n):(αn⊗n)
       =1/2(| Q+αn⊗n|^2) - |∇( Q+αn⊗n)|^2 - α( Q+αn⊗n):(n⊗n) .

We have used the elementary equality A: A=1/2(| A|^2) - | A|^2 (A= Q+αn⊗n). It is well-defined in the weak sense because Q+αn⊗n∈ C^0∩ H^1 by the regularity estimate from Lemma <ref>.

For the RHS of (<ref>), we compute the three terms separately.
The first term equals

    (I)    =( Q+αn⊗n):[(A/C+ Q^2 )  Q -B/C( Q^2- Q^2/3 I)] 
       =(A/C+ Q^2)( Q^2+α n·Qn) - B/C( Q^3+α|Qn|^2-α Q^2/3).

The second term equals

    (II)=  Q^2 + α(n⊗n Q)
    =  Q^2 + α n·Qn.

We have extensively used the equality AB=BA.

Suppose first that x∈Γ, so by the assumption (<ref>) we have ∇ϕ(x)=|∇ϕ(x)|  n(x). Then we write

    (III)
       =|∇ϕ|^2 ( Q+αn⊗n):𝒫_0 [(n⊗n) ( Q+s_+/3 I) + ( Q+s_+/3 I)(n⊗n)] 
       =2|∇ϕ|^2 ( Q+αn⊗n):𝒫_0 [(n⊗n) ( Q+s_+/3 I)]

using n⊗n=1/|∇ϕ|^2∇ϕ⊗∇ϕ and that Q is symmetric. 
Notice that since the image of 𝒫_0 has zero trace, the product would be identical if one added s_+/3 I to Q+αn⊗n.
Denote by Q̃= Q+s_+/3 I (so Q̃=s_+), and we get

    (III)
           =2|∇ϕ|^2(Q̃+αn⊗n): 𝒫_0((n⊗n)Q̃) 
       =2|∇ϕ|^2(Q̃+αn⊗n):[(n⊗n)Q̃-((n⊗n)Q̃)/3 I] 
    
           =2|∇ϕ|^2 [ |Q̃ n|^2 +2α-s_+/3 ( n·Q̃ n)].

Setting α=s_+/2 eliminates the term 2α-s_+/3 ( n·Q̃ n), which leads us to

    (III)=2 |( Q+s_+/3 I)∇ϕ|^2,   x∈Γ.

If x∉Γ, then ϕ is a constant in a neighbourhood of x, so ∇ϕ(x)≡ 0 and the third term equals zero. Therefore, it holds for all x∈Ω that

    (III) ≥ 0.


Then, substituting the results above in (<ref>), and removing all positive terms on the RHS, we get the following inequality satisfied by the function | Q+αn⊗n|^2 over the entire Ω, known as a subsolution condition.



    -(| Q+αn⊗n|^2) + 2T( Q) ≤ 0.

The function T( Q) is defined as

    T( Q)   =T^(1)( Q)+λ^2 T^(2)( Q), 
    
        T^(1)( Q)   =α ( Q+αn⊗n):(n⊗n) 
    
        T^(2)( Q)
        
        
        
           =( Q^2)^2 - B/C Q^3 + α ( Q^2)( n·Qn) 
        +3A+Bα/3C Q^2 -B/Cα|Qn|^2 
          + α(A/C+ω_v(1-ϕ)^2) n·Qn.


Despite the complexity of its expression, T( Q) is basically a fourth order polynomial in terms of Q, with uniformly bounded coefficients and a strictly positive leading term λ^2 ( Q^2)^2.
Therefore, there exists a number R_0 depending on A,B,C,λ,ω_v and the C^2 norm M_N from assumption (ii), such that

    T( Q)≥ 0,   ∀ | Q|≥ R_0.

Note that α=s_+/2 is also determined by A,B,C according to (<ref>).

Finally, we use the standard technique in the maximum principle estimate of weak solutions. One can consult <cit.> for an example.
Define the cut-off 

    ψ_k = max | Q+αn⊗n|^2-k, 0,

such that ψ_k∈ H_0^1(Ω) for k>α^2 because Q|_Ω≡ 0 and that |αn⊗n|^2≤α^2.
By the chain rule, we also have

    ∇ψ_k=∇(| Q+αn⊗n|^2),     | Q+αn⊗n|^2>k, 
    
            0,     | Q+αn⊗n|^2≤ k,

Multiplying both sides of (<ref>) with ψ_k, integrating by parts and eliminating positive terms, we get an inequality

    ∫_Ω∇(| Q+αn⊗n|^2)·∇ψ_kx̣ + 2∫_Ωψ_k T( Q)x̣≤ 0,   k>α^2.

By (<ref>), the first term of the LHS of (<ref>) is simply ∫_Ω |∇ψ_k|^2.
The second term of (<ref>) is supported on the region where ψ_k≠ 0, i.e. | Q+αn⊗n|^2>k.
By the triangle inequality of the Frobenius norm,

    | Q| ≥ | Q+αn⊗n| - |αn⊗n| = | Q+αn⊗n| - α,

so choosing k>(R_0+α)^2 is sufficient to ensure | Q|>R_0 whenever ψ_k is nonzero. Thus, according to (<ref>), T( Q) is positive on the support of ψ_k, and we get

    ∫_Ω |∇ψ_k|^2x̣≤ 0 ⇒ψ_k≡ 0,

which means that | Q+αn⊗n| is uniformly bounded by k for all k>(R_0+α)^2. Therefore, using the triangle inequality on Q and αn⊗n again, we get

    sup_x∈Ω | Q|≤ R_0+α,

which is the uniform bound we desired.

Evidently, the bound (<ref>) depends on the choice of R_0, which ensures the positivity of T( Q) for all | Q|>R_0. Hence, to complete the proof, we also need to study the dependence of R_0 (<ref>) on λ.
Closely observing the expressions of T^(1) and T^(2) (<ref>), we find that their coefficients are controlled by a uniform bound dependent on A,B,C,ω_v,M_N but not on λ. We then get

    T^(1)( Q) ≥ - C_1 (| Q|+1), 
    
        T^(2)( Q) ≥1/2 | Q|^4 - C_2,

with constants C_1,C_2>0 depending on A,B,C,ω_v,M_N, since T^(1) is affine in Q and T^(2) is degree-4 in Q with leading term ( Q^2)^2.
In order that T( Q)≥ 0, it suffices to force the lower bound to be positive, i.e.

    T^(1)+λ^2 T^(2)≥λ^2(1/2 | Q|^4 - C_2) - C_1 (| Q|+1)≥ 0.

Multiplying both sides by λ^2/3, it reduces to checking that

    1/2|λ^2/3 Q|^4 - C_1 |λ^2/3 Q| ≥ C_1λ^2/3+C_2λ^8/3.

Suppose that λ_0>0 is given, so the RHS is bounded for all λ<λ_0, whose bound depends on C_1,C_2,λ_0.
Notice also that the LHS is a degree-4 polynomial of |λ^2/3 Q| as a whole. By the elementary property of polynomial functions, there exists C_3>0 depending on C_1,C_2,λ_0 such that the inequality holds whenever |λ^2/3 Q|>C_3 i.e. | Q|>C_3 λ^-2/3.
Therefore, we take R_0= C_3λ^-2/3 in the uniform bound (<ref>) and get 

    sup_x∈Ω | Q|≤ C_3 λ^-2/3+α≤ M λ^-2/3,  ∀λ<λ_0

with M=C_3+αλ_0^2/3 dependent on A,B,C,L,ω_v,λ_0,M_N as desired.




 §.§ Uniqueness on small domains

An immediate result of the maximum principle is the uniqueness of the critical points when λ is small.
 
Under the assumptions of Theorem <ref>, there exists a λ_1>0 depending on Ω,A,B,C,L,ω_v and the bound M_N,
such that for all λ<λ_1, the Q-critical point of E_[·,ϕ] (<ref>) is unique.


Following the approach of Lemma 8.2 from <cit.>, we only need to prove the strict convexity of the energy functional E_ on the closed convex set

    X_λ = { Q∈ H_0^1(Ω;𝒮_0): sup_x∈Ω | Q| ≤ Mλ^-2/3},

where M=M(A,B,C,L,ω_v,λ_0,M_N) is the constant stated in Theorem <ref>, with λ_0=1. By definition, X_λ contains all critical points of the energy. We highlight the dependence on λ since the bound changes as λ→ 0.
It suffices to check that

    E_[ Q+ h,ϕ] + E_[ Q- h,ϕ]/2≥ E_[ Q,ϕ]

if Q± h∈ X_λ and that the strict inequality holds when h≠ 0.
Since the void energy and the anchoring energy are all positive-definite quadratic polynomials with respect to Q, they are both convex, and we only need to check the LdG energy

    E^ LdG[ Q]=∫_Ω[1/2 |∇ Q|^2 + λ^2F̅_b( Q)]x̣.


By the convexity of X_λ, Q± h∈ X_λ implies that Q∈ X_λ.
Then, strong convexity of E^ LdG is equivalent to the positivity of the second-order central difference

    Δ( Q, h)=E^ LdG[ Q+ h] +  E^ LdG[ Q- h]-2  E^ LdG[ Q]

whenever h≠ 0. From the expression of E^ LdG we compute

    Δ( Q, h)=∫_Ω | h|^2
        +λ^2∫_Ω[A/C h^2
        -2B/C(Qh^2) + 2(Qh)^2+ Q^2  h^2+1/2( h^2)^2].

Since | Q|≤ Mλ^-2/3, we can estimate a lower bound of Δ( Q, h) by removing all positive terms and amplifying Q to its largest norm. We get

    Δ( Q, h)    ≥∫_Ω| h|^2x̣ - λ^2∫_Ω[|A|/C h^2 + 2B/C· Mλ^-2/3 h^2]x̣
       ≥∫_Ω| h|^2 x̣- M' λ^4/3∫_Ω | h|^2x̣,

where the constant M' is scaled from M by a constant related to |A|/C,B/C. 
We note that the factor λ^2 manages to absorb the unbounded λ^-2/3 from the uniform maximum value introduced in Theorem <ref>.

Poincaré's inequality <cit.> implies that

    ∫_Ω| h|^2 x̣≥1/C_4(Ω)∫_Ω| h|^2x̣,

so there exists a λ_1=min{λ_0,(C_4 M')^-3/4} such that when λ<λ_1, the quantity Δ( Q, h) is always strictly positive for all h≠ 0.
Therefore, E̅^LdG is strictly convex on X_λ, and the uniqueness of Q-critical points follows naturally.




§ SHARP-INTERFACE LIMIT
 
The limiting behaviour of E_[ Q,ϕ] (<ref>) when the capillary width → 0 is known as the sharp-interface limit.
A famous result of the sharp-interface limit is about van der Waals-Cahn-Hilliard functional (as used in our mixing energy (<ref>)), which states that it converges to the interface perimeter in the sense of Γ-limit <cit.>.
We are interested in the sharp-interface limit → 0 because it guarantees that solving the continuous diffuse-interface functional (e.g. the van der Waals-Cahn-Hilliard functional) approximates difficult-to-compute sharp-interface problems (e.g. interface perimeter).

For our diffuse-interface LdG energy E_ (<ref>), we also assert such a sharp-interface limit.
 
Let ( Q_^*,ϕ_^*)∈𝒜 be a minimizer of E_ (<ref>) in its admissible space (<ref>). Assume further that as → 0,

    
  * {ϕ_^*} converges to the indicator function I_D_0=
        1,     x∈ D_0, 
 0     x∉ D_0 in L^1, where D_0⊂⊂Ω (abbreviation for D_0⊂Ω and D_0⊂Ω, used throughout the rest of the text) such that |D_0|=V_0;
    
  * { Q_^*} converges uniformly to a continuous function Q_0∈ C(Ω;𝒮_0).
    

Then, ( Q_0, D_0) minimizes the energy functional

    E_0[ Q,D]
           =∫_Ω[ L/2 |Q|^2 + λ^2 F̅_b( Q) ] x̣
        +ω_vλ^2 ∫_D^c | Q|^2 x̣
         
        +ω_pλ/3∫_ D[1+ω_a/ω_p| ( Q+s_+/3 I)ν|^2 ]^1/2Ṣ(x)

among all matrix functions Q∈ C^0∩ H^1 and regions D⊂⊂Ω with smooth boundary satisfying |D|=V_0, where ν represents the unit normal vector of D.


The sharp-interface limit (<ref>) also comprises the LdG free energy and penalty factors on the void, the perimeter and the weak anchoring on D. 
The first integral is identical to E^ LdG[ Q], and the second is identical to E^ void[ Q,I_D] (<ref>) where I_D is the indicator function of D.
With an elementary inequality 1+|x|/√(2)≤√(1+|x|^2)≤ 1+ |x|, we find that

    1≤(D) + √(ω_a/ω_p)∫_ D|( Q+s_+/3 I)ν|
    /∫_ D[1+ω_a/ω_p|( Q+s_+/3 I)ν|^2 ]^1/2Ṣ(x)
    ≤√(2).

That is, the boundary integral is equivalent to the sum of the perimeter and an L^1 anchoring penalty.




 §.§ Proof by Γ-convergence


Theorem <ref> is proven in the framework of Γ-convergence, which is composed of an upper bound estimate and a lower bound estimate <cit.>.

We start by rewriting the diffuse-interface functional (<ref>) as

    E_[ Q,ϕ]= E^ LdG[ Q] + ω_vλ^2 E^ void[ Q,ϕ] + ω_pλ∫_Ω[ a_ Q(∇ϕ) + ^-1W(ϕ)]x̣

with W(s)=s^2(1-s)^2 denoting the double-well function,
and the sharp-interface functional (<ref>) as

    E_0[ Q,D]=E^ LdG[ Q]+ω_vλ^2 E^ void[ Q,I_D] + ω_pλ/3∫_ D |a_ Q(ν)|^1/2Ṣ(x).

Note that the expressions (<ref>) and (<ref>) share the LdG and void energy, as well as a quadratic form a_ Q(·):^3→ defined by

    a_ Q(ξ) = |ξ|^2 + ω_a/ω_p|( Q+s_+/3 I)ξ|^2,  ξ∈^3,

which strongly implies the connection between E_ and E_0.













Then, we state the Γ-convergence of E_ to E_0.
 
Work under the same assumptions as Theorem <ref>. Then, the following properties hold for all Q∈ C^0∩ H^1 and smooth region D⊂⊂Ω satisfying |D|=V_0.


  * For all sequences ϕ_→ I_D (I_D is the indicator function of D) in L^1, Q_→ Q uniformly,

    lim inf_→ 0 E_[ Q_,ϕ_] ≥ E_0[ Q,D];


  * There exists a sequence ϕ_→ I_D in L^1 satisfying the volume constraint (<ref>), Q_→ Q uniformly, such that 

    lim sup_→ 0 E_[ Q_,ϕ_] ≤ E_0[ Q,D].




Proposition <ref> relies on the following claims. We work under the same assumptions as Proposition <ref> throughout the claims.


  * Claim (I): For all sequences ϕ_→ I_D in L^1 and Q_→ Q uniformly,

    lim inf_→ 0( E^ LdG[ Q_] + ω_vλ^2 E^ void[ Q_,ϕ_]) ≥ 
    E^ LdG[ Q] 
    +ω_vλ^2 E^ void[ Q,I_D]


  * Claim (II): For all ϕ_→ I_D a.e. with 0≤ϕ_≤ 1, it holds that

    lim_→ 0( E^ LdG[ Q] + ω_vλ^2 E^ void[ Q,ϕ_]) = E^ LdG[ Q] +ω_vλ^2 E^ void[ Q,I_D]


  * Claim (III): For all sequences ϕ_→ I_D in L^1, Q_→ Q uniformly,

    lim inf_→ 0∫_Ω[ a_ Q_(∇ϕ_) + ^-1W(ϕ_)] x̣≥1/3∫_ D |a_ Q(ν)|^1/2Ṣ(x).


  * Claim (IV): There exists a sequence ϕ_→ I_D a.e. with 0≤ϕ_≤ 1 and the volume constraint (<ref>), such that

    lim sup_→ 0∫_Ω[ a_ Q(∇ϕ_) + ^-1W(ϕ_)] x̣≤1/3∫_ D |a_ Q(ν)|^1/2Ṣ(x).


Obviously, Claims (I) and (II) concern the convergence of the first two terms of (<ref>) and (<ref>), while Claims (III) and (IV) are about the final term.
For technical reasons, we note that the sequences in Claim (II) and (IV) have Q fixed, and that the sequences in Claim (IV) satisfy Claim (II).



 §.§ Detailed proof


We start by proving Claims (I)–(IV).
Claims (I) and (II) can be obtained through basic calculus.

Assume w.l.o.g. that 

    lim inf_→ 0∫_Ω | Q_|^2x̣<∞,

for otherwise the conclusion is trivial.
Due to the weak compactness of H^1, we extract a subsequence of { Q_}, denoted by { Q__k} such that

    lim_k→∞∫_Ω | Q__k|^2x̣ = lim inf_→ 0∫_Ω | Q_|^2x̣ and  Q__k⇀ Q  in H^1.

Hence, we have by the w.l.s.c. property <cit.> that

    lim inf_→ 0∫_Ω | Q_|^2x̣ = lim_k→∞∫_Ω| Q__k|^2x̣≥∫_Ω | Q|^2x̣.

The uniform convergence Q_→ Q implies that F̅_b( Q_)→F̅_b( Q) uniformly as well, since the bulk energy F̅_b (<ref>) is a continuous polynomial function. Thus,

    lim_→ 0∫_ΩF̅_b( Q_)x̣ =∫_ΩF̅_b( Q)x̣.

For the void penalty E^ void (<ref>), we once again extract a subsequence {ϕ__k}→ I_D (w.l.o.g. the same subsequence as before) such that

    lim_k→∞ E^ void[ Q__k,ϕ__k] = lim inf_→ 0 E^ void[ Q_,ϕ_] and ϕ__k→ I_D,a.e..

Recall the expression (<ref>) of E^ void. It follows from Fatou's lemma <cit.> that

    lim inf_→ 0 E^ void[ Q_,ϕ_]
           = lim_k→∞∫_Ω1/2 (1-ϕ__k)^2 | Q__k|^2x̣
       ≥∫_Ωlim inf_k→∞1/2(1-ϕ__k)^2 | Q__k|^2x̣
        =E^ void[ Q,I_D].

Summing up the formulae above gives (<ref>).


The LdG energy is the same on both sides of (<ref>), so we only have to check the void energy. Since ϕ_ and Q∈ C^0 are uniformly bounded and since ϕ_→ I_D a.e., it follows from the dominated convergence theorem that

    lim_→ 0∫_Ω1/2(1-ϕ_)^2| Q|^2x̣ = ∫_Ω1/2 (1-I_D)^2 | Q|^2x̣,

which leads to (<ref>).


The proof of Claims (III) and (IV) involves discussion on a generalized form of the van der Waals-Cahn-Hilliard functional (<ref>).
We briefly digress from the main thread and present a general result, which are of critical importance to the proof. The notations used in the following two Lemmas are independent of the rest of the paper. 
 
Work on a bounded Lipschitz domain Ω⊂^n in general dimensions n.
Define the generalized van der Waals-Cahn-Hilliard functional L_^a: H^1(Ω) →∪{+∞} as

    L_^a(ϕ) = ∫_Ω[^-1 W(ϕ) +  a(x,∇ϕ)]x̣,

with the following assumptions.


  * a(x,p) = a^ij(x) p_ip_j with Einstein's summation convention applied. The symmetric-matrix-valued function A=((a^ij)):Ω→^n× n is continuous, and satisfies the uniform elliptic condition

    λI≼ A(x) ≼Λ I,   ∀ x.

The notation A≼ B means that B- A is positive definite, where A, B are symmetric matrices.

  * W(s) is a nonnegative continuous double-well potential with strict minima at s=α,β (α<β), and W(α)=W(β)=0.

  * D⊂⊂Ω is a region with C^2 boundary.

Let ϕ_0=α I_D + β I_D^c be the separation of the phases α and β.
Also, denote the perimeter of D by (D)=ℋ_n-1( D) (the (n-1)-dimensional Hausdorff measure <cit.>) and introduce the constant

    c_0=∫_α^β√(W(s))ṣ.

Then, for any sequence ϕ_→ϕ_0 in L^1(Ω), it holds that

    lim inf_→ 0 L_^a(ϕ_) ≥ 2c_0∫_ D |a(x,ν)|^1/2Ṣ(x),

where ν:  D →𝕊^n-1 is the unit outward normal vector.

 
Under the same assumptions as Lemma <ref>, suppose further that ϕ_0=α I_D+β I_D^c satisfies the volume constraint

    ∫_Ωϕx̣ = m

where α|Ω|<m<β|Ω|, or equivalently |D|=β|Ω|-m/β-α.
Then, there exists a sequence ϕ_∈ W^1,∞(Ω) satisfying the constraint (<ref>), α≤ϕ_≤β, ϕ_→ϕ_0 a.e., such that

    lim sup_→ 0 L_^a(ϕ_) ≤ 2c_0 ∫_ D |a(x,ν)|^1/2Ṣ(x).



Remarks on the results of Lemmas <ref> and <ref>:


  * These lemmas are stated in the exact same fashion as <cit.>, where the same properties are derived for the degenerate case a^ij(x,p)≡ |p|^2 and (<ref>) is reduced to the classical van der Waals-Cahn-Hilliard energy.


  * The theory of Lemmas <ref> and <ref> can apply to our model with α=1,β=0,W(s)=s^2(1-s)^2, and ϕ_0=I_D. The reason is that when α>β, one can set W̃(s)=W(-s) and apply the theory to {-ϕ_} with W̃ substituted for W.


For the sake of continuity, the proof of Lemmas <ref> and <ref> will be left for Appendices <ref> and <ref>, respectively. Assuming their correctness for the time being, we can proceed to prove Claims (III) and (IV).


Evidently, the quadratic form a_ Q(x) (<ref>) satisfies the uniform elliptic condition (<ref>) as long as Q:Ω→𝒮_0 is continuous.
Since Q_→ Q uniformly, ( Q_+s_+/3 I)^2→ ( Q+s_+/3 I)^2 uniformly as well. Therefore, for any δ>0, there exists _0 such that when <_0,

    a_ Q_(p) ≥ a_ Q(p) - δ|p|^2 ,  ∀ x∈Ω, p∈^3.

As a_ Q(p)≥ |p|^2 by definition (see (<ref>)), the quadratic form a_ Q(p)-δ|p|^2 is also uniformly elliptic when δ is sufficiently small (e.g. δ<1/2).
Therefore, applying Lemma <ref> to the sequence {ϕ_} with quadratic form a(x,p)=a_ Q(x)(p)-δ|p|^2, phases α=1,β=0 and region D shows that

    LHS of (<ref>)   ≥lim inf_→ 0∫_Ω[ (a_ Q(∇ϕ_)-δ|∇ϕ_|^2) + ^-1 W(ϕ_)] x̣
       ≥ 2 c_0 ∫_ D |a_ Q(ν)-δ|^1/2Ṣ.

Since W(s)=s^2(1-s)^2, the constant c_0=∫_0^1√(W(s))ṣ=∫_0^1 s(1-s)ṣ=1/6, so the coefficient is 2 c_0=1/3.
Taking the limit δ→ 0 the boundary integral, we immediately obtain

    LHS of (<ref>)≥1/3∫_ D |a_ Q(ν)|^1/2Ṣ=RHS of (<ref>)

as desired.



Let ϕ_ be the sequence guaranteed by Lemma <ref>, with a(x,p)=a_ Q(x)(p) and α=1,β=0, which satisfies ϕ_→ I_D a.e., 0≤ϕ≤ 1 and that the volume constraint (<ref>) holds.
Then, the conclusion of the lemma shows that

    LHS of (<ref>)   =lim sup_→ 0∫_Ω[  a_ Q(∇ϕ_) + ^-1 W(ϕ_)]x̣
       ≤ 2 c_0 ∫_ D |a_ Q(ν)|^1/2Ṣ = RHS of (<ref>)

as desired, since c_0=1/6.


Having established the claims, we proceed to prove Proposition <ref>.

i. Lower bound proof. For the lower bound (<ref>), we add up (<ref>) and (<ref>) for all sequences ϕ_→ I_D in L^1 and Q_→ Q uniformly. Using Claims (I) and (III), we conduct the following computation to obtain (<ref>).

    lim inf_→ 0 E_[ Q_,ϕ_]
           ≥lim inf_→ 0( E^ LdG[ Q_] +ω_vλ^2  E^ void[ Q_,ϕ_]) 
          + ω_pλlim inf_→ 0∫_Ω[ a_ Q_(∇ϕ_) + ^-1W(ϕ_)] x̣
       ≥ E^ LdG[ Q]
        +ω_vλ^2 E^ void[ Q,I_D]
        +ω_pλ/3∫_ D |a_ Q(ν)|^1/2Ṣ(x) 
       =E_0[ Q,D].


ii. Upper bound proof. Take the sequence guaranteed by the Claim (IV), where ϕ_→ I_D a.e. with 0≤ϕ_≤ 1 (so {ϕ_} also satisfies Claim (II)) with the volume constraint (<ref>). We also let Q_≡ Q be constant.
Therefore, (<ref>) and (<ref>) both hold for the sequence {(ϕ_, Q)}. Adding them together leads to (<ref>).

    lim sup_→ 0 E_[ Q,ϕ_]
           = lim_→ 0( E^ LdG[ Q] + ω_vλ^2 E^ void[ Q,ϕ_]) 
          + ω_pλlim sup_→ 0∫_Ω[ a_ Q(∇ϕ_) + ^-1W(ϕ_)] x̣
       ≤ E^ LdG[ Q]
        +ω_vλ^2 E^ void[ Q,I_D]
        +ω_pλ/3∫_ D |a_ Q(ν)|^1/2Ṣ(x) 
       =E_0[ Q,D].

We have proven the assertions of Proposition <ref> as desired.


Finally, the proof of Theorem <ref> follows from Proposition <ref> by a standard argument.


For any ( Q,D) satisfying Q∈ C^0∩ H^1 and D⊂⊂Ω, take the sequence {( Q_,ϕ_)} guaranteed by Proposition <ref>(b) to get

    E_0[ Q,D]    ≥lim sup_→ 0 E_[ Q_,ϕ_] 
       ≥lim inf_→ 0 E_[ Q_^*, ϕ_^*]≥ E_0[ Q_0,D_0].

The second inequality comes from the assumption that ( Q_^*,ϕ_^*) minimizes E_, and the third inequality utilizes Proposition <ref>(a) at ( Q_0,D_0).
Hence, by definition ( Q_0,D_0) minimizes E_0 among all ( Q,D) of concern.




§ NUMERICAL RESULTS
 
In this section, we study the numerical behaviour of the diffuse-interface LdG model.

To simplify implementation and demonstration, and to facilitate further comparison with the numerical results in literature, we use the 2D reduced LdG model on 2D Ω̃. 
The reduced LdG model is the thin film limit of the LdG model in 3D domain Ω = Ω̃× [0,h], as h→ 0, under certain boundary conditions <cit.>. 
Under the special reduced temperature A=-B^2/3C, the critical state Q of the LdG free energy (<ref>) can be related to the reduced 2× 2 symmetric traceless tensor P as follows <cit.>

    Q≏[ P+s_+/6I_2          0;          0     -s_+/3 ].

From now on, we return to the energy E_ (<ref>) and recommence the discrimination between quantities with and without bars.
The corresponding reduced energy is then obtained through shifting E_ (<ref>) by an additive constant.

    E_ [P,ϕ]
           =∫_Ω̃[ 1/2 |P|^2 + λ̅^2 (-B^2/4C^2P^2 + 1/4 (P^2)^2)] x̣
         +ω̅_pλ̅∫_Ω̃[ |∇ϕ|^2+^-1ϕ^2(1-ϕ)^2]x̣ 
        +ω̅_a λ̅∫_Ω̃| (P+s_+/2)∇ϕ|^2 x̣
          +ω̅_vλ̅^2 ∫_Ω̃1/2(1-ϕ)^2 | P|^2 x̣, 
    s.t.     ∫_Ω̃ϕx̣ = V̅_0.

Recall that =/λ is the relative capillary width, and the rescaled constants are defined by

    λ̅^2 ≜λ^2 C/L, ω̅_p = ω_p/√(CL), ω̅_v = ω_v/C, ω̅_a = ω_a/√(CL),

so the coefficients of the trailing integrals in (<ref>) are respectively ω_pλ/L,ω_aλ/L,ω_vλ^2/L.
We take B = [0.64× 10^4]N· m^-2, C = [0.35 × 10^4]N· m^-2, and L = [4× 10^-11]N in the following numerical simulations, which are the physical constants of the commonly used nematic liquid crystal material MBBA <cit.>. We also fix the following hyperparameters

    =/λ=0.005,  V̅_0=0.09, ω_p/L=[3× 10^7]m^-1, ω_v/L=[6× 10^14]m^-2.


Using the finite difference method, we discretize the 2D domain Ω̃=[0,1]× [0,1] with an N× N grid where N=128, and minimize E_ numerically following the gradient flow with respect with Q and ϕ. We investigate the minimizers of the diffuse-interface LdG energy under various domain sizes λ and anchoring strengths ω_a in Figure <ref>. 

The radial and polar states on a 2D disc with tangential boundary condition have been studied in <cit.>. The radial state is radially symmetric with a central +1 defect, and the polar state has two +1/2 defects along the diameter.
People have also found the tactoid, which is a spindle-shaped nematic droplet within which directors are almost parallel to each other, and with either defects near the pointy ends of the spindle or none at all <cit.>.

In Figure <ref>, when λ is small enough and ω_a is small, the minimizer is a radial state with circular N-I interface and a +1 defect in the centre of the nematic phase.
As ω_a increases, the minimizer becomes a polar state with an ellipsoidal interface and the two +1/2 defects along the long axis, and larger w_a, i.e. stronger tangential anchoring, causes defects to move towards the centre.
When λ is large, the minimizer is tactoidal state where nematic directors are almost parallel to each other and there are no defects.





§ CONCLUSION AND DISCUSSIONS



In this paper, we propose a diffuse-interface LdG energy functional E_ (<ref>) for the free boundary between a nematic droplet and an isotropic surrounding. The model incorporates a LdG Q-tensor and a phase field ϕ as order parameters, with an extra hyperparameter  introduced to describe the width of interface.



We establish the existence (Proposition <ref>) of global minimizers of E_.
With fixed ϕ, we have also shown that the Q-critical points satisfy the maximum principle (Theorem <ref>) and uniqueness (Proposition <ref>) under certain conditions.
The existence, uniqueness and regularity properties as a whole ensure the solvability of the diffuse-interface functional E_. The model is therefore applicable in the computation of free-boundary NLCs.
We also prove the sharp-interface limit of E_ when → 0 in the sense of Γ-convergence (Theorem <ref> and Proposition <ref>), i.e. convergence of minimizers to minimizers.
The sharp-interface limit is a functional with respect to the nematic region D and the Q-tensor order parameter that is also made up of LdG, void, perimeter and anchoring energies.


The numerical tests we conduct produce physically interpretable results, which are also consistent with existing observations in other free-boundary NLC models. In particular, we observe the transition form radial to polar and tactoidal states when adjusting the scale λ under different hyperparameters.




The paper also motivates research in the future.
For example, there is some room for improvement for our theoretical results such as the maximum principle and uniqueness, which is only proven in the fixed-ϕ case under smoothness assumptions. It could be generalized to global minimizers under weaker conditions.

The numerical results in Figure <ref> have demonstrated exceedingly complex structures within the minimizers of E_ as parameters change. Therefore, it is necessary to further investigate into the structure of minimizers under different parameters. One possible approach is to study its energy landscape, i.e. multiple minimizers and the links between them, using the method of saddle dynamics <cit.>.
The mesh width of the finite difference method has also limited our investigation into the sharp-interface limit, so more advanced numerical schemes, such as the spectral method, should also be implemented in the future. They will also be more useful for 3D problems.

theoremsection


§ CALCULATION OF EULER-LAGRANGE EQUATION
 
Consider a perturbation δ Q to the function Q∈ H_0^1(Ω;𝒮_0), and the variational derivative δ E_/δ Q is an 𝒮_0-valued function that satisfies 

    δ E_ = ∫_Ωδ E_/δ Q :δ Qx̣ + o(δ Q_L^2),

where A: B is the matrix inner product.
Taking the variational derivative of E_ with respect to Q results in the following Euler-Lagrange equation.

    δ E^ LdG/δ Q
        +ω_vλ^2 δ E^ void/δ Q
        +ω_aλ^2 δ E_^ anch/δ Q=0.

The mixing energy (<ref>) is ignored because it does not involve Q.

The variational derivative of the LdG energy (<ref>) has been computed in <cit.> as follows (notation adapted to our form).

    δ E^ LdG/δ Q=- Q + λ^2 ∇_𝒮_0F̅_b( Q),

where the Laplacian operator  is defined by f=∑_i=1^3 f_x_ix_i, and

    ∇_𝒮_0F̅_b( Q)=𝒫_0 [(A/C+ Q^2) Q-B/C Q^2]
    =(A/C+ Q^2) Q-B/C( Q^2- Q^2/3 I ).

In the formulae above, ∇_𝒮_0 is the projected gradient operator on 𝒮_0, which maps the matrix gradient with the projection operator 𝒫_0 (<ref>).
The projection is necessary because of the traceless constraint on Q.

With the help of ∇_𝒮_0, we get that the variational derivative of the void penalty (<ref>) equals

    δ E^ void/δ Q=ω_v(1-ϕ)^2 ∇_𝒮_0(1/2 | Q|^2),

and that the variational derivative of the anchoring penalty (<ref>) equals

    δ E_^ anch/δ Q=ω_a∇_𝒮_0[ | ( Q+s_+/3 I)∇ϕ|^2].


We can then compute

    δ E^ void/δ Q
        =ω_v(1-ϕ)^2  Q

and

    δ E_^ anch/δ Q
        =ω_a𝒫_0[(∇ϕ⊗∇ϕ)( Q+s_+/3 I) + ( Q+s_+/3 I)(∇ϕ⊗∇ϕ)],

where the matrices ∇ϕ⊗∇ϕ,  Q+s_+/3 I are multiplied with the (non-commutative) matrix multiplication. During the computation of (<ref>), we have used the transcription of |( Q+s_+/3 I)∇ϕ|^2 into a tensor contraction

    |( Q+s_+/3 I)∇ϕ|^2=(∇ϕ⊗∇ϕ): (  Q+s_+/3 I )^2,

and an obvious result on the matrix gradient of polynomial functions.

If f( Q)= A: Q^k where k∈ℕ, then the matrix gradient of f is

    ∇ f( Q)=∑_j=0^k-1 Q^j A  Q^k-1-j.


Summing up (<ref>), (<ref>) and (<ref>), we get the equation (<ref>).

Taking the first variation of E_ with respect to ϕ, we get

    δ E_^ mix/δϕ+ω_a/ω_pδ E_^ anch/δϕ+ω_vλ/ω_pδ E^ void/δϕ+ξ=0,

where ξ is a Lagrangian multiplier from the volume constraint (<ref>).
The LdG energy E^ LdG (<ref>) is ignored because it does not involve ϕ.
We can then directly compute that

    δ E_^ mix/δϕ   =-2ϕ + 2^-1ϕ(ϕ-1)(2ϕ-1), 
    δ E_^ anch/δϕ   =-2[( Q+s_+/3 I)^2 ∇ϕ], 
    δ E^ void/δϕ   = (ϕ-1) | Q|^2.

Summing up these formulae, and dividing the entire equation by 2, we get the equation (<ref>). Note that the Lagrangian multiplier ξ has been rescaled as well, but we still write it as ξ to simplify notation.



§ PROOF OF LEMMA <REF>
 
The proof of the lower bound in Lemma <ref> follows exactly <cit.>, which employs the elementary inequality a^2+b^2≥ 2ab and the lower semi-continuity of the total variation. In fact, we prove that the lower bound holds for any L^1-convergent sequence {ϕ_} regardless of the volume constraint (<ref>).

First, we assume w.l.o.g. that α≤ϕ_≤β, for otherwise we can define the truncated function ϕ̃_ (x)= τ_1(ϕ_), where

    τ_1(x)=max{min{x,β},α}=α,     x<α, 
    
        x,    α≤ x≤β, 
    β,     x>β.

It is obvious that |ϕ̃_ - ϕ_0| ≤ |ϕ_-ϕ_0|, so ϕ̃_-ϕ_0_L^1→ 0, i.e. {ϕ̃_} converges in L^1 as well.
In addition, it follows from the property of W and the chain rule that

    W(ϕ̃_) ≤ W(ϕ_),   ∇ϕ̃_ = I_{α<ϕ_ < β}∇ϕ_,

so

    ∫_Ω [^-1 W(ϕ̃_)+ a(x,∇ϕ̃_)]x̣≤∫_Ω [^-1 W(ϕ_)+ a(x,∇ϕ_)]x̣,

i.e. L^a_ (ϕ̃_) ≤ L^a_(ϕ_). Therefore, in order to prove Lemma <ref>, it suffices to check that

    lim inf_→ 0 L^a_ (ϕ̃_) ≥ 2c_0 ∫_ D |a(x,ν)|^1/2Ṣ,

where the sequence {ϕ̃_} also has the L^1 convergence, and comes with the extra assumption that α≤ϕ̃_≤β.

Assume that α≤ϕ_≤β and discard the volume constraint. We can estimate the LHS of Lemma <ref> with the elementary inequality a^2+b^2≥ 2ab. This step cancels out .

    ∫_Ω[^-1W(ϕ_) +  a(x,∇ϕ_) ]x̣≥ 2 ∫_Ω[ W(ϕ_) a(x,∇ϕ_) ]^1/2x̣.

Define Φ(s)=∫_α^s √(W(s))ṣ, so Φ(α)=0, Φ(β)=c_0 and Φ is strictly increasing on (α,β). Then by the chain rule we have ∇Φ(ϕ_) = Φ'(ϕ_) ∇ϕ_ = √(W(ϕ_))∇ϕ_, which enables us to rewrite

    ∫_Ω[ W(ϕ_) a(x,∇ϕ_)]^1/2x̣   =∫_Ω a(x, √(W(ϕ_))∇ϕ_)^1/2x̣
       = ∫_Ω a(x, ∇Φ(ϕ_))^1/2x̣.

Hence, we get the inequality

    ∫_Ω[^-1W(ϕ_) +  a(x,∇ϕ_) ]x̣≥ 2 ∫_Ω a(x, ∇Φ(ϕ_))^1/2x̣.


Assume for the moment that a^ij∈ C^1(Ω). We define the following functional inspired by the total variation <cit.>.

    V() ≜sup{∫_D ϕ_i(a^ij g_j)x̣:  g=[g_j]∈ C_0^1(Ω; ^n), sup_x a(x, g)≤ 1 },

where Einstein's convention is used for _i(a^ijg_j).
Obviously the set on the RHS is non-empty as g≡ 0 is an element. Thus, V() is nonnegative (equals +∞ if the set is unbounded).

We consider the following properties obvious, as the proof is identical to corresponding properties of the total variation.
 
We claim the following properties regarding the functional V: L^1(Ω) →_≥ 0∪{+∞}.


  * V() is sublinear and homogeneous, i.e. V(a+ψ)≤ |a| V()+V(ψ) for all a∈,,ψ∈ L^1(Ω).

  * V()=0 if ≡const., and as a consequence, V(+C)=V() for all ∈ L^1 and constant C.

  * V() is weakly lower semi-continuous on L^1(Ω), i.e. for any sequence _k→ in L^1,

    lim inf_k→∞ V(_k) ≥ V().


  * V()=∫_Ω |a(x,∇)|^1/2x̣ whenever ∈ W^1,1(Ω).

  * V()=∫_ D |a(x,ν)|^1/2Ṣ whenever = I_D where D⊂⊂ D is Lipschitz.



Since Φ is Lipschitz continuous on [α,β] (by Φ'=√(W)) and that ϕ_→ϕ_0 in L^1, Φ(ϕ_)→Φ(ϕ_0) in L^1 as well.
As ϕ_0=α I_D + β I_D^c by definition, we get that Φ(ϕ_0)=c_0(1-I_D).
We then make the following deduction based on Lemma <ref>.

    lim inf_→ 0∫_Ω |a(x,∇Φ(ϕ_))|^1/2x̣    (d)=lim inf_→ 0 V(Φ(ϕ_)) 
        (c)≥ V(c_0-c_0 I_D)
             (b)=V(-c_0 I_D) 
        (a)= c_0 V(I_D)
             (e)= c_0∫_ D |a(x,ν)|^1/2Ṣ.

Explanation of (<ref>): the first equality uses claim (d) on differentiable ; the second inequality uses claim (c) regarding w.l.s.c. property; the third equality uses claim (b) by shifting with a constant; the fourth equality uses the homogeneity from claim (a); the final equality uses claim (e) on indicator functions.

The result of Lemma <ref> then follows by connecting the inequalities (<ref>) and (<ref>).

    lim inf_→ 0∫_Ω[^-1W(ϕ_) +  a(x,∇ϕ_)]x̣≥ 2lim inf_→ 0 V(Φ(ϕ_))
    ≥ 2c_0 ∫_ D |a(x,ν)|^1/2Ṣ.


We have assumed D(x)∈ C^1 thus far. As for more general cases where A(x)∈ C^0, we consider a uniform approximation A_k(x) → A(x) “from below”, that is, satisfying A_k≼ A everywhere. Then by the property of positive-definite matrices,

    a_k(x,p) ≤ a(x,p),  ∀ x∈Ω,p∈^n,

where a_k(x,p)=a_k^ij(x)p_ip_j is the quadratic form defined by A_k(x).
Because of the uniform ellipticity (<ref>), construction of such an approximation is simple and hence omitted.

Applying the previous argument to the sequence A_k, we then get

    lim inf_→ 0∫_Ω[^-1W(ϕ_) +  a(x,∇ϕ_)]x̣   ≥lim inf_→ 0∫_Ω[^-1W(ϕ_) +  a_k(x,∇ϕ_)]x̣
       ≥ 2c_0 ∫_ D |a_k(x,ν)| Ṣ(x).

Let k→∞ in the inequality above, and the boundary integral on the RHS converges uniformly to ∫_ D |a(x,ν)|^1/2, concluding the proof.



§ PROOF OF LEMMA <REF>
 
The proof of the upper bound follows <cit.>, which uses an explicit construction of {ϕ_} aided by the signed distance function (SDF) <cit.>.

If the upper limit were to hold, then the limit of L_^a(ϕ_) should be exactly equal to the RHS because of the lower bound estimates by Lemma <ref>. Therefore, inequalities in the proof of the lower bound must approximate equalities.
Especially, for the elementary inequality in (<ref>) to become an equality, it must hold almost everywhere that

    ^-1W(ϕ_) ≈ a(x,∇ϕ_).

As a result, the integrations of the two terms are approximately equal,

    ∫_Ω^-1W(ϕ_) ≈∫_Ω a(x,∇ϕ_).

This property is known as the equipartition of energy <cit.>.

For the generalized model (<ref>), we propose the following motivations for our proof.


  * By constructing a function χ_:→[α,β] with χ_' ≈^-1√(W(χ_)) and a function h^a∈ W^1,∞, one obtains with the chain rule that

    a(x,∇χ_(h^a)) ≈·^-2 W(χ_(h^a)) a(x,∇ h^a) = ^-1 W(χ_(h^a)),

if a(x,∇ h^a)≡ 1.

  * In order to fulfil a(x,∇ h^a)≡ 1, we solve a nonlinear first-order partial differential equation with the method of characteristics. The solution we get is a generalized SDF.

  * The level sets of h^a are locally smooth mappings of the boundary D near h^a=0, so we evaluate L_^a(ϕ_) with the coarea formula, and complete the proof.




 §.§ Approximate equipartition of energy


We briefly reproduce the derivation of the function χ_ by Modica <cit.>. The function χ_ is intended to approximate the relation χ_'≈^-1√(W(χ_)), to fulfil the equipartition condition.

We propose for χ_:→ as the solution to the following differential equation.

    χ_'(t) = √(W(χ_(t))+), 
    χ_(0)=α.

χ_ is intended to approximate the standing-wave solution q, which happens to be the minimizer of the one-dimensional van der Waals-Cahn-Hilliard energy satisfying ^2 q”=1/2 W(q) <cit.>.











The following properties of χ_ are evident.
 
The function χ_ satisfies that:

    
  * χ_ is strictly increasing.
    
  * There exists an interval [0,η_], such that χ_(0)=α, χ_(η_)=β.
    
  * η_→ 0 as → 0.





 §.§ Method of characteristics


The next step is to construct (locally) a Lipschitz function h^a satisfying

    a(x,∇ h^a)=1,   a.e..

h^a is essentially the solution to the following first-order nonlinear partial differential equation (PDE)

    a^ij(x) u_x_iu_x_j = 1,     x∈Ω
    
            u=0,     x∈ D,

which can be solved with the method of characteristics. 
Assume for the moment that a^ij(x)∈ C^∞(Ω). Then by <cit.> (with F(x,u,p)=a(x,p)-1), the solution to (<ref>) is given by the characteristic curves ( X(s), U(s),  P(s)) satisfying the dynamical system


    Ẋ_i = -_p_i F( X,U, P)=2a^ij( X)P_j, 
    U̇ =∇_p F( X,U, P)· P= 2a^ij( X)P_i P_j, 
    Ṗ_i =-_x_i F( X,U, P)-P_i _uF( X,U, P) = - a^jk_x_i( X) P_j P_k,
          ((̇*̇)̇ = (*)/ s)

(Einstein's summation convention is used) with initial values

    X(0)=y∈ D, 
    
            U(0)=0, 
     P(0)=|a(y,ν)|^-1/2ν.

The curve X(s) originates from the manifold of boundary conditions, which in this case is D, and the functions U(s), P(s) picture respectively the evolution of h^a(x) and ∇ h^a(x) along the curves.
The initial values (<ref>) are designed to be compatible with the boundary condition h^a=0 and a(x,∇ h^a)=1.







With the general theory of first-order nonlinear PDE's, we assert that the system (<ref>) solves (<ref>) correctly in a neighbourhood of D, with U(t),  P(t) standing for the respective value of h^a, ∇ h^a at the point X(t).
 
Assume that D is compact and C^2 smooth.
Then, there exists an open interval I containing 0 such that: 


    
  * For each y∈ D and s∈ I, a unique C^1 characteristic curve ( X(s),U(s), P(s)) exists with the initial conditions (<ref>).
    
  * The mapping
    
    I× D   →Ω××^n 
    
            (s,y)   ↦( X,U, P)

    is C^1 with respect to s∈ I and y∈ D.
    
  * The mapping (s,y)↦ X(s;y) for s∈ I,y∈ D is bijective, and its inverse is also C^1, i.e. there exists C^1 inverse mappings s=s(x),y=y(x).
    
  * On the open region containing D defined by
    
    V={ X(s;y): s∈ I, y∈ D},

    the function
    
    h^a(x) = U(s(x); y(x))

    is a C^1 classical solution to the equation (<ref>), with ∇ h^a(x)= P(s(x);y(x)).



We can also make the following observations on the solution to (<ref>) on the neighbourhood V.
It states that the sign of h^a(x) determines which side of D the point x is on, a property shared by the classical SDF.
 
Assume the same as Lemma <ref>, and let the solution to (<ref>) defined by characteristic curves be u(x).
Then for any x∈ V, the relative position of x to D is determined by the sign of h^a(x). Specifically,

    x∈ D,     h^a(x)<0, 
    
        x∈ D,     h^a(x)=0, 
    
        x∈ (D)^c,    h^a(x)>0.



Since U(s)=h^a( X(s)),  P(s)=∇ h^a( X(s)) solves (<ref>) by Lemma <ref>,

    a( X(s), P(s))≡ 1

for all s∈ I.
Therefore, we notice from the characteristic equation (<ref>) that U̇(s)=2a( X, P)=2,
which indicates 

    h^a(x)=U(s(x))=2s.


By (<ref>), the initial values satisfy Ẋ(0)=2a^ij(y)P_j(0),  P(0)=c ν (c>0), so

    Ẋ(0)·ν =c 2a^ij(y) ν_iν_j>0

since a^ij satisfies the elliptic condition (<ref>). As ν is the exterior normal vector, the characteristic curve points outside the region D as s increases. Therefore, s>0 corresponds to the outside of D and s<0 corresponds to the inside of D.
By (<ref>), the signs of h^a and s(x) are identical, and so we yield Lemma <ref>.


Level sets of h^a are local continuous isomorphisms of the boundary D.
 
Use the same assumptions and notations as Lemma <ref>.
The level sets Σ_t={h^a=t} of h^a(x) satisfies the following properties.

    
  * For all t with t/2∈ I, Σ_t is a C^1 hypersurface.
    
  * For any continuous function g(x):Ω→, the integral
    
    ∫_Σ_t g(x)Ṣ(x)

    is continuous with respect to t. Specifically, when g(x)≡ 1, we get the continuity of ℋ_n-1(Σ_t) with respect to t.



Recall from (<ref>) that U(s)=2s, so the level set h^a=t is expressed equivalently as

    h^a(x)=t ⇔ s(x)=t/2.

That is, Σ_t is a section of the C^1 bijective mapping (s,y)↦ X(s;y) (by Lemma <ref>(c)) at s=t/2.

    Σ_t = { X(t/2;y): y∈ D},   ∀t/2∈ I.

Therefore, s=t/2 is a C^1 parametrization of the surface family {Σ_t}, and the assertions of Corollary <ref> follow immediately.

















 §.§ Integration with coarea formula


Work on the interval I determined by Lemma <ref>. W.l.o.g., assume that I=(-η_*,η_*).







Since η_→ 0 as → 0 by Lemma <ref>, we choose  small enough such that η_<η_*. Then according to Corollary <ref>, for all t∈[-η_,η_], the level set Σ_t resides within the neighbourhood V. We define on V

    ϕ_(x) = χ_(h^a(x)+δ_),     -δ_≤ h^a(x)≤η_-δ_, 
    α,     h^a< -δ_, 
    β,     h^a> η_-δ_,

where δ_∈[0,η_] is yet to be determined.
Effectively, we have applied a truncated function ϕ_=χ̃_(h^a+δ_), where

    χ̃_(t)=min{β, max{α, χ_(t)}}=α,     t<0, 
    χ_(t),     t∈[0,η_], 
    β,     t>η_

is an increasing and Lipschitz continuous function from →[α,β], so ϕ_ is also Lipschitz continuous on V.


We choose δ_∈[0,η_] to meet the volume constraint (<ref>).
 
There exists δ_∈[0,η_] such that 

    ∫_V ϕ_(x)x̣ = ∫_Vϕ_0(x)x̣,

where ϕ_0=α I_D + β I_D^c.


Define χ_0=α,     t<0 
β,     t>0, which satisfies

    χ̃_(t)≤χ_0(t)≤χ̃_(t+η_),  ∀ t∈

according to Figure <ref>.
According to Lemma <ref>(c), the sign of h^a(x)=U(s)=2s is determined by x∈ D^c or x∈ D, so

    χ_0(h^a(x))=α,     x∈ D 
    β,     x∈ D^c
    =ϕ_0(x),

where ϕ_0=α I_D+β I_D^c is the sharp-interface limit in Lemmas <ref> and <ref>.
Since the integral

    ∫_V ϕ_(x)x̣ = ∫_V χ̃_(h^a(x)+δ)x̣

is continuous and monotonic with respect to δ, and since

    ∫_V χ̃_(h^a(x))x̣≤∫_V χ_0(h^a(x))x̣=m ≤∫_Vχ_(h^a(x)+η_)x̣,

there exists δ_∈ [0,η_] such that ∫_V χ̃_(h^a+δ_)x̣=m by the mean value theorem.


We then extend ϕ_ to the entire Ω. Define

    ϕ_(x)=α,     x∈ D∩ V^c, 
    β,     x∈ D^c∩ V^c.

We claim that:

    
  * The extension (<ref>) is a Lipschitz function over Ω.
    
  * ϕ_ satisfies the volume constraint (<ref>).
    
  * ϕ_→ϕ_0 a.e., where ϕ_0=α I_D+β I_D^c.


To prove claim (i), we just need to prove the differentiability near the boundary V.
Since V is defined by images of the C^1 mapping X(s;y) with s∈ (-η_*,η_*), the boundary V equals

    V = { X(s;y): s=±η_*, y∈ D},

which is composed of two patches of level set surfaces Σ_± 2η_*={h^a=±2η_*}.
It holds on either of the patches that |h^a(s)|>η_ in a neighbourhood, so for all x∈ V in that neighbourhood, ϕ_(x)=χ̃_(h^a) is constantly α or β according as h^a<0 or h^a>0.
On the other hand, the extension (<ref>) on x∈ V^c also equals α or β according as x∈ D or x∈ D^c, which is equivalent to x∈ D or x∈ D^c according to Lemma <ref>.
Therefore, the extended ϕ_ is constant near V, and is obviously Lipschitz continuous. 

To prove claim (ii), we notice that ϕ_0 is also constantly α or β according as x∈ D or x∈ D^c satisfies the relations in (<ref>), which ensures

    ∫_V^cϕ_(x)x̣ = ∫_V^cϕ_0(x)x̣.

Meanwhile, Lemma <ref> ensures that

    ∫_V ϕ_(x)x̣=∫_V ϕ_0(x)x̣.

Adding them together, we get the volume constraint (<ref>) on the entire Ω.

To prove claim (iii), we observe that ϕ_=ϕ_0 whenever h^a∉(-δ_, η_-δ_). Therefore, for each x∉ D, either x∉ V⇒ϕ_≡ϕ_0 or h^a(x)≠ 0 ⇒ when  is sufficiently small, h^a∉(-δ_, η_-δ_) ⇒ϕ_(x)=ϕ_0(x). This means that ϕ_→ϕ_0 a.e..

Finally, we estimate the functional (<ref>) with the aid of ϕ_ and the following coarea formula for a Lipschitz function h∈ W^1,∞(Ω) and an integrable function f∈ L^1(Ω) <cit.>, which translates volume integrals to surface integrals.

    ∫_Ω f(x) |∇ h|x̣=∫_-∞^∞ṛ∫_{h=r}∩Ω f(x)Ṣ(x).

The following steps are identical to those of Modica <cit.>.
When h^a≥η_-δ_ or h^a≤ -δ_, ϕ_ is constantly α or β, so both W(ϕ_) and ∇ϕ_ are zero. Therefore, the integral (<ref>) is supported on the “strip” {-δ_ < h^a < η_-δ_}, or equivalently, {α<ϕ_<β}.
Using the differential equation (<ref>) and the chain rule, we get

    a(x,∇ϕ_) = ^-1(W(ϕ_)+) a(x,∇ h^a)=^-1(W(ϕ_)+).

Thus, we transcribe the integral into

    L_^a(ϕ_)   =∫_{α<ϕ_<β}^-1(2W(ϕ_)+)x̣
       =∫_-δ_^η_-δ_ṭ∫_Σ_t^-1/|∇ h^a| (2W(ϕ_)+) Ṣ(x),

where the coarea formula (<ref>) is applied to h^a.
Note that because of (<ref>), 1=|a(x,∇ h^a)|≤Λ |∇ h^a|^2 ⇒ |∇ h^a| ≥Λ^-1/2, and thus it can be placed on the denominator.
As ϕ_=χ̃_(h^a), we can extract the factor W(ϕ_)+ from the surface integral.

    L_^a(ϕ_)
           = ∫_-δ_^η_-δ_2W(χ̃_(t+δ_))+/ṭ∫_Σ_t1/|∇ h^a|Ṣ(x) 
       = ∫_0^η_2W(χ_(t))+/ṭ∫_Σ_t-δ_1/|∇ h^a|Ṣ(x).

As χ_(t) is strictly increasing by Lemma <ref>(a), we make the change of variable r=χ_(t)∈(α,β), whose substitution formula is ṛ=χ_'(t)ṭ=^-1√(W(r)+)ṭ. Then,

    L_^a(ϕ_) = ∫_α^β2W(r)+/√(W(r)+)ṛ∫_Σ_χ_^-1(r)-δ_1/|∇ h^a|Ṣ(x),

where χ_^-1(r)∈(-δ_, η_-δ_) is the inverse function of χ_.
By Lemma <ref>, η_→ 0 and δ_→ 0 as → 0, so χ_^-1(r)-δ_→ 0 uniformly for all r∈(α,β). 
1/|∇ h^a| is also continuous over V because h^a∈ C^1 and |∇ h^a|≥Λ^-1/2.
Thus, by Lemma <ref>(b) we can assert that

    lim_→ 0∫_Σ_χ_^-1(r)-δ_1/|∇ h^a|Ṣ(x) = ∫_Σ_01/|∇ h^a|Ṣ(x)

uniformly for all r∈(α,β).
Moreover, 2W(r)+/√(W(r)+)≤ 2√(W(r)+) is uniformly bounded above by a continuous function for all r∈[α,β].
Therefore, we apply the dominated convergence theorem on (<ref>) by taking the limit → 0 under the integral sign, and get

    lim_→ 0 L_^a(ϕ_)
    = ∫_α^β 2√(W(r))ṛ∫_Σ_01/|∇ h^a|Ṣ(x).

Since Σ_0 is just D, and since |∇ h^a(x)|=| P(x;0)|=|a(x,ν)|^-1/2 (x∈ D) according to (<ref>),

    lim_→ 0 L_^a(ϕ_) = 2c_0 ∫_ D |a(x,ν)|^1/2Ṣ(x),

matching the form in Lemma <ref>.

All previous derivations are based on the assumption that a^ij(x)∈ C^∞. For the general case a^ij∈ C^0, we only need to prove the existence of a subsequence, i.e. {_k}→ 0 and ϕ_k→ϕ_0 a.e. such that 

    lim sup_k→∞ L__k^a(ϕ_k) ≤ 2c_0 ∫_ D|a(x,ν)|^1/2Ṣ(x).

Choose a sequence of smooth matrix functions A_k(x)∈ C^∞ such that A_k→ A uniformly and that A_k(x)≽ A(x),∀ x.
Let {ϕ^k_} be the approximation sequence satisfying Lemma <ref> with respect to A_k. For each k, choose _k<min{1/k,_k-1} such that

    L_^a_k(ϕ^k__k) ≤ 2c_0 ∫_ D|a_k(x,ν)|^1/2Ṣ(x) +1/k,

and denote by ϕ_k=ϕ^k__k. It is then obvious that

    lim sup_k→∞ L__k^a(ϕ_k)
    ≤lim sup_k→∞ 2c_0 ∫_ D|a_k(x,ν)|^1/2Ṣ(x) = 2c_0 ∫_ D|a(x,ν)|^1/2Ṣ(x),

as desired.



§ ACKNOWLEDGEMENTS


Dawei Wu and Baoming Shi appreciate the University of Strathclyde for their support and hospitality when work on this paper was undertaken.

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