# A homogenization result in finite plasticity

Elisa Davoli  ,  Chiara Gavioli  and  Valerio Pagliari

###### Abstract.

We carry out a variational study for integral functionals that model the
stored energy of a heterogeneous material governed by finite-strain
elastoplasticity with hardening. Assuming that the composite has a periodic
microscopic structure, we establish the $\Gamma$-convergence of the energies
in the limiting of vanishing periodicity. The constraint that plastic
deformations belong to $\mathsf{SL}(3)$ poses the biggest hurdle to the
analysis, and we address it by regarding $\mathsf{SL}(3)$ as a Finsler
manifold.

2020 Mathematics Subject Classification: 49J45; 74C15; 74E30; 74Q05.

Keywords and phrases: finite-strain elastoplasticity, $\Gamma$-convergence,
homogenization, Finsler manifold.

###### Contents

  1. 1 Introduction
  2. 2 Mathematical setting and results
  3. 3 Preliminaries
    1. 3.1 Localization and integral representation
    2. 3.2 Finsler structure on $\mathsf{SL}(3)$
  4. 4 Homogenization in finite plasticity
    1. 4.1 Integral representation
    2. 4.2 Characterization of the limiting energy density

##  1\. Introduction

This paper deals with the homogenization of some variational integrals related
to the theory of finite plasticity. Precisely, we focus on those materials
governed by finite-strain elastoplasticity with hardening. Starting from the
microscopic level, we aim at retrieving effective macroscopic descriptions by
means of a $\Gamma$-convergence approach.

Let $\Omega\subset^{3}$ be an open, bounded, connected set with Lipschitz
boundary, and let us denote by $\mathsf{SL}(3)$ the group of $3\times 3$ real
matrices with determinant equal to $1$. Our main result describes the limiting
behavior of the functionals

$\mathcal{F}_{\varepsilon}(y,P)\coloneqq\int_{\Omega}W\left(\frac{x}{\varepsilon},\nabla
y(x)P^{-1}(x)\right)\,\mathrm{d}x+\int_{\Omega}H\left(\frac{x}{\varepsilon},P(x)\right)\,\mathrm{d}x+\int_{\Omega}|\nabla
P(x)|^{q}\,\mathrm{d}x,$ (1.1)

where $y\in W^{1,2}(\Omega;^{3})$, $P\in W^{1,q}(\Omega;K)$ with $q>3$ and
$K\subset\mathsf{SL}(3)$ compact, and $W$ and $H$ encode, respectively, the
periodic nonlinear elastic energy density and the hardening of the
elastoplastic material sitting in $\Omega$. The precise statement is contained
in Theorem 2.2, and the exact mathematical setting of the analysis is detailed
in Section 2.

As we touched upon before, we work within the classical mathematical theory of
elastoplasticity at finite strains (see, e.g., [28]). We assume that the
elastic behavior of our sample $\Omega$ is independent of preexistent plastic
distortions. This can be rephrased as the assumption that the deformation
gradient $\nabla y$ associated with any deformation
$y\colon\Omega\to\mathbb{R}^{3}$ of the body decomposes into an elastic strain
and a plastic one. Within the framework of linearized elastoplasticity, the
decomposition would have an additive nature; in the setting of finite
plasticity, instead, a multiplicative structure is traditionally assumed [26,
27]. Following this approach, as in [29, 30], we suppose that for any
deformation $y\in W^{1,2}(\Omega;\mathbb{R}^{3})$ there exist an elastic
strain $F_{\mathrm{el}}\in L^{2}(\Omega;^{3\times 3})$ and a plastic strain
$P\in L^{2}(\Omega;{\rm SL}(3))$ such that

$\nabla y(x)=F_{\mathrm{el}}(x)P(x)\quad\text{for a.\,e. }x\in\Omega.$

Such multiplicative decomposition, recently justified in the setting of
dislocation systems and crystal plasticity in [10, 11], motivates the
definitions of the energy functionals in (1.1). Other constitutive models in
finite plasticity have been also taken into account in the mathematical
literature (see, e.g., [15, 20, 21, 32]). We point out that the stored energy
in (1.1) has already been considered in [31] in a variational setting. We
refer to [30] for more general models in finite plasticity including hardening
variables. Concerning the regularization via $\nabla P$ in (1.1), we note that
it is common in engineering models and prevents the formation of
microstructures, see [4, 7]. We also refer to [18] for a discussion of its
drawbacks and for alternative higher order regularizations.

In the small strain regime, the homogenization of the plasticity equations has
been undertaken in the series of works [33, 25, 24] both in the periodic and
in the aperiodic and stochastic settings. The framework of perfect plasticity
in the case of a linear elastic response has been completely characterized in
the seminal work [19]. See also [23] for a homogenization result in the small
strain regime with hardening. To the authors’ knowledge, the only available
results in the large strain setting are instead confined to the framework of
crystal plasticity for stratified composites. In particular, we refer to [8,
9] for the superlinear growth case, to [14] for the linear growth scenario,
and to [17] for some first homogenization analysis in the evolutionary
setting. Theorem 2.2 fills a gap in the study of elastoplastic microstructures
by providing a static homogenization result in the large strain regime. As an
application of our contribution, in [16] the analysis of high-contrast
elastoplastic materials is carried out.

We conclude this introduction with a few words on the proofs. The existence of
an integral $\Gamma$-limit is based on the localization technique in the
context of integral representation results. The biggest hurdle is devising a
version of the so-called fundamental estimate that complies with the
constraint that plastic deformations must belong to $\mathsf{SL}(3)$. For
technical reasons (cf. [31]), we will need to strengthen the constraint and
require that the plastic deformations actually belong to a compact
neighborhood of the identity. To accommodate for this, we will regard
$\mathsf{SL}(3)$ as a manifold endowed with a Finsler metric, which
additionally establishes a connection to the mechanism of plastic dissipation.
The problem of devising the fundamental estimate for functionals involving
manifold-valued Sobolev maps has already been considered in the literature,
see e.g. [2, 12]. The main difference between these references and ours is
that not only we constrain our maps to take values in a manifold, but we also
require them to range in a specific compact subset. We refer to Section 3 for
a review of the localization method and of the geometric properties of
$\mathsf{SL}(3)$. The characterization of the limiting energy density is
obtained by a perturbative argument that grounds on standard homogenization
results, see Theorem 3.1.

### Outline

The setup of our analysis and the main result, Theorem 2.2, is presented in
Section 2. Section 3 contains some reminders about the technical tools to be
used in the sequel. The proof of Theorem 2.2 is the subject of Section 4.

##  2\. Mathematical setting and results

Throughout the paper, $\Omega$ is an open, bounded, and connected set with
Lipschitz boundary in 3 (the analysis would not change significantly if we
settled the problem in d with $d=2$ or $d>3$). We use 3×3 and 3×3×3 to denote
real-valued $3\times 3$ and $3\times 3\times 3$ tensors, respectively. We use
the notation $I$ for the identity matrix. The symbol $|\,\cdot\,|$ is
indiscriminately adopted for the Euclidean norms in 3, 3×3 and 3×3×3. To deal
with plastic strains, we recall the classical notation

$\displaystyle\mathsf{SL}(3)\coloneqq\\{F\in^{3\times 3}:\det F=1\\}.$

In Subsection 3.2 we will endow $\mathsf{SL}(3)$ with a metric structure and
regard it as a Finsler manifold.

If $A\subset^{3}$ is a measurable set, we will denote by $\mathcal{L}^{3}(A)$
its three-dimensional Lebesgue measure.

The building block of our study is the following variational notion of
convergence:

######  Definition 2.1.

Let $X$ be a set endowed with a notion of convergence. We say that the family
$\\{\mathcal{G}_{\varepsilon}\\}$, with $\mathcal{G}_{\varepsilon}\colon
X\to[-\infty,+\infty]$, $\Gamma$-converges as $\varepsilon\to 0$ to
$\mathcal{G}\colon X\to[-\infty,+\infty]$ if for all $x\in X$ and all
infinitesimal sequences $\\{\varepsilon_{k}\\}_{k\in\mathbb{N}}$ the following
holds:

  1. (1)

for every sequence $\\{x_{k}\\}_{k\in\mathbb{N}}\subset X$ such that $x_{k}\to
x$, we have

$\mathcal{G}(x)\leq\liminf_{k\to+\infty}\mathcal{G}_{\varepsilon_{k}}(x_{k});$

  2. (2)

there exists a sequence $\\{x_{k}\\}_{k\in\mathbb{N}}\subset X$ such that
$x_{k}\to x$ and

$\limsup_{k\to+\infty}\mathcal{G}_{\varepsilon_{k}}(x_{k})\leq\mathcal{G}(x).$

When $X$ is equipped with a topology $\tau$, we sometimes use expressions such
as $\Gamma(\tau)$-convergence or $\Gamma(\tau)$-limit to emphasize the
underlying convergence for sequences in $X$. We elaborate on the connections
between $\Gamma$-convergence and homogenization in Subsection 3.1. In what
follows, for notational convenience, we indicate the dependence on
$\varepsilon_{k}$ by means of the subscript $k$ alone, e.g.
$\mathcal{F}_{k}\coloneqq\mathcal{F}_{\varepsilon_{k}}$.

We collect here the assumptions under which the $\Gamma$-convergence of the
functionals in (1.1) will be proved.

Let $Q\coloneqq[0,1)^{3}$ be the periodicity cell. We assume that the elastic
energy density $W\colon^{3}\times^{3\times 3}\to[0,+\infty]$ satisfies the
following:

  1. E1:

It is a Carathéodory function such that $W(\,\cdot\,,F)$ is $Q$-periodic for
every $F\in^{3\times 3}$.

  2. E2:

It is $2$-coercive and has at most quadratic growth, i.e., there exist
$0<c_{1}\leq c_{2}$ such that for a. e. $x\in^{3}$ and for all $F\in^{3\times
3}$

$c_{1}|F|^{2}\leq W(x,F)\leq c_{2}\left(|F|^{2}+1\right).$

  3. E3:

It is $2$-Lipschitz: there exists $c_{3}>0$ such that for a. e. $x\in^{3}$ and
for all $F_{1},F_{2}\in^{3\times 3}$

$|W(x,F_{1})-W(x,F_{2})|\leq
c_{3}\left(1+|F_{1}|+|F_{2}|\right)|F_{1}-F_{2}|.$

Let us compare E1–E3 with the common requirements for physically admissible
elastic energy densities. Setting

$\mathsf{GL}_{+}(3)\coloneqq\\{F\in^{3\times 3}:\det F>0\\},$

a couple of natural conditions are $W\equiv+\infty$ on ${}^{3\times
3}\setminus\mathsf{GL}_{+}(3)$, which rules out interpenetrations, and
$\lim_{\det F\to 0}W(F)=+\infty$, which means that it should take infinite
energy to squeeze a small block of material down to a point. Feasible elastic
densities are also required to be frame indifferent and to attain their
minimum, conventionally set to $0$, on the identity matrix $I$.

In our analysis, it would be particularly challenging to preserve the
constraint of positive determinant for deformation gradients. Therefore, we
work in a simplified setting and consider the whole space of $3\times 3$
matrices as domain for $W$. Similarly, the quadratic growth from above in E2
rules out the blow up of the energy densities, but it is a rather common
assumption in variational studies. Frame indifference, that is,
$W(x,RF)=W(x,F)$ for a. e. $x\in^{3}$, for all $F\in\mathsf{GL}_{+}(3)$ and
all rotations $R$, does not play any role in our analysis, thus we ignore it.
Finally, up to adding a constant to $W$, the growth condition E2 is compatible
with the requirement $\min W(x,\,\cdot\,)=W(x,I)=0$ for a. e. $x\in^{3}$.

As for the hardening functional, we assume that $H\colon\times^{3\times
3}\to[0,+\infty]$ meets the ensuing requirements:

  1. H1:

$H$ is a Carathéodory function such that $H(\,\cdot\,,F)$ is $Q$-periodic for
every $F\in^{3\times 3}$.

  2. H2:

Assume that a Finsler structure on $\mathsf{SL}(3)$ is assigned. For a.e.
$x\in\Omega$, $H(x,F)$ is finite if and only if $F\in K$, where
$K\subset\mathsf{SL}(3)$ is a geodesically convex, compact neighborhood of
$I$.

  3. H3:

The restriction of $H(x,\,\cdot\,)$ to $K$ is Lipschitz continuous, uniformly
in $x$.

Let us spend some words on H2. This rather strong hypothesis prescribes that
the effective domain of $H(x,\,\cdot\,)$, namely the set
$\left\\{F\in\mathsf{SL}(3):H(x,F)<+\infty\right\\}$ is an $x$-independent
compact set containing $I$. A consequence of technical advantage is that
uniform $L^{\infty}$-bounds on the plastic strains become available, provided
the latter give rise to finite hardening. Indeed, if $K$ is as in H2, then
there exists $c_{K}>0$ such that

$|F|+|F^{-1}|\leq c_{K}\quad\text{for every }F\in K,$ (2.1)

because $\mathsf{SL}(3)$ is by definition well separated from $0$. From such
bound we infer that for any $F\in K$ and $G\in^{3\times 3}$

$\left|G\right|=\left|GF^{-1}F\right|\leq c_{K}\left|GF^{-1}\right|.$ (2.2)

The requirement that the effective domain $K$ is geodesically convex (in the
Finsler sense we mentioned, see Subsection 3.2 for the details) roots in the
localization argument that we adopt to establish the $\Gamma$-convergence (see
Subsection 3.1). We recall that a subset of a Finsler manifold is said to be
geodesically convex if, for any couple of points in the set, there is a unique
shortest path contained in the set that joins those two points. The existence
of a compact set $K$ complying with H2 is ensured by Proposition 3.3 and
Remark 3.4 below.

We are now ready to state the main result of this paper, namely the
homogenization formula for the functionals in (1.1). We work in the space
$W^{1,2}(\Omega;^{3})\times W^{1,q}(\Omega;\mathsf{SL}(3))$ endowed with the
topology $\tau$ characterized by

$(y_{k},P_{k})\stackrel{{\scriptstyle\tau}}{{\to}}(y,P)\quad\text{if and only
if}\quad\begin{cases}y_{k}\to y&\text{strongly in
}L^{2}(\Omega;^{3}),\\\\[2.84526pt] P_{k}\to P&\text{uniformly}.\end{cases}$
(2.3)

######  Theorem 2.2.

Let $\mathcal{F}_{\varepsilon}$ be the functionals in (1.1), which we extend
by setting

$\mathcal{F}_{\varepsilon}(y,P)=+\infty\quad\text{on
}\big{[}L^{2}(\Omega;^{3})\times
L^{q}(\Omega;\mathsf{SL}(3))\big{]}\setminus\big{[}W^{1,2}(\Omega;^{3})\times
W^{1,q}(\Omega;K)\big{]}.$

If $W$ and $H$ satisfy E1–E3 and H1–H3, respectively, then for all $y\in
L^{2}(\Omega;^{3})$, $P\in L^{q}(\Omega;\mathsf{SL}(3))$ the $\Gamma$-limit

$\mathcal{F}(y,P)\coloneqq\Gamma(\tau)\mbox{-}\lim_{\varepsilon\to
0}\mathcal{F}_{\varepsilon}(y,P)$

exists. We also have that

$\mathcal{F}(y,P)=\left\\{\begin{aligned}
\displaystyle\int_{\Omega}\Big{(}W_{\mathrm{hom}}(\nabla
y(x),P(x))&+H_{\mathrm{hom}}(P(x))+|\nabla P(x)|^{q}\Big{)}\,\mathrm{d}x\\\
&\text{if }(y,P)\in W^{1,2}(\Omega;^{3})\times W^{1,q}(\Omega;K),\\\\[3.0pt]
+\infty\qquad\qquad\qquad\qquad&\text{otherwise in }L^{2}(\Omega;^{3})\times
L^{q}(\Omega;\mathsf{SL}(3)),\end{aligned}\right.$

where $W_{\mathrm{hom}}\colon^{3\times 3}\times K\to[0,+\infty)$ and
$H_{\mathrm{hom}}\colon K\to[0,+\infty)$ are defined as

$\displaystyle
W_{\mathrm{hom}}(F,G)\coloneqq\lim_{\lambda\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\to+\infty}\frac{1}{\lambda\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}^{3}}\inf\left\\{\int_{(0,\lambda\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0})^{3}}W\big{(}x,(F+\nabla
y(x))G^{-1}\big{)}\,\mathrm{d}x:y\in
W^{1,2}_{0}((0,\lambda\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0})^{3};^{3})\right\\},$
$\displaystyle H_{\mathrm{hom}}(F)\coloneqq\int_{Q}H(z,F)\,\mathrm{d}z.$

##  3\. Preliminaries

We gather in this section the technical tools to be employed in the sequel.

###  3.1. Localization and integral representation

To the aim of laying the ground for the proof of Theorem 2.2, we briefly
outline the localization technique in the context of integral representation
results for $\Gamma$-limits. More detail and a thorough treatment of
$\Gamma$-convergence, which we introduced in Definition 2.1, may instead be
found in the monographs [6, 13, 5]. For definiteness, we report on a well-
known result (see e. g. [6, Theorem 14.5]) that is underpinned by the
localization method.

######  Theorem 3.1.

Let $g\colon^{3}\times^{3\times 3}\to[0,+\infty)$ be a Borel function that is
$Q$-periodic in its first argument and that satisfies standard $p$-growth
conditions for some $p\in(1,+\infty)$. For $\varepsilon>0$ and $y\in
L^{p}(\Omega;^{3})$ we define

$\mathcal{G}_{\varepsilon}(y)\coloneqq\begin{cases}\displaystyle{\int_{\Omega}g\bigg{(}\frac{x}{\varepsilon},\nabla
y(x)\bigg{)}\,\mathrm{d}x}&\text{if }y\in W^{1,p}(\Omega;^{3}),\\\\[3.0pt]
+\infty&\text{otherwise}.\end{cases}$

Then, we have

$\Gamma(L^{p})\text{-}\lim_{\varepsilon\to
0}\mathcal{G}_{\varepsilon}(y)=\begin{cases}\displaystyle{\int_{\Omega}g_{\mathrm{hom}}\big{(}\nabla
y(x)\big{)}\,\mathrm{d}x}&\text{if }y\in W^{1,p}(\Omega;^{3}),\\\\[3.0pt]
+\infty&\text{otherwise},\end{cases}$

where the $\Gamma$-limit is taken with respect to the strong
$L^{p}(\Omega;\mathbb{R}^{3})$-topology, and $g_{\mathrm{hom}}\colon^{3\times
3}\to[0,+\infty)$ is a quasiconvex function characterized by the asymptotic
homogenization formula

$g_{\mathrm{hom}}(F)=\lim_{\lambda\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\to+\infty}\frac{1}{\lambda\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}^{3}}\inf\left\\{\int_{(0,\lambda\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0})^{3}}g\big{(}x,F+\nabla
y(x)\big{)}\,\mathrm{d}x:y\in
W^{1,p}_{0}((0,\lambda\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0})^{3};^{3})\right\\}$

for all $F\in^{3\times 3}$.

As starting point to establish this theorem, one resorts to a general property
of $\Gamma$-convergence which ensures that if $X$ is a separable metric space,
then any family $\\{\mathcal{G}_{\varepsilon}\\}$ with
$\mathcal{G}_{\varepsilon}\colon X\to[-\infty,+\infty]$ has a
$\Gamma$-convergent subsequence (see e.g. [6, Proposition 7.9]). Such
$\Gamma$-compactness principle yields that, up to subsequences, an abstract
$\Gamma(L^{p})$-limit of $\\{\mathcal{G}_{\varepsilon}\\}$ exists. One is then
naturally led to wonder whether the limit is in turn a functional of integral
type. In order to show that this is in fact the case, a localization argument
is employed. This amounts to regard $\mathcal{G}_{\varepsilon}$ as a function
of the pair $(y,A)$, where $y\in L^{p}(\Omega;^{3})$ and
$A\in\mathcal{A}(\Omega)\coloneqq\\{\text{open subsets of }\Omega\\}$; more
precisely, setting

$\mathcal{G}_{\varepsilon}(y,A)\coloneqq\int_{A}g\bigg{(}\frac{x}{\varepsilon},\nabla
y(x)\bigg{)}\,\mathrm{d}x\qquad\text{if }y\in W^{1,p}(A;^{3})$

and $\mathcal{G}_{\varepsilon}(y,A)\coloneqq+\infty$ otherwise, the idea is to
focus on the properties of $\mathcal{G}_{\varepsilon}(y,\,\cdot\,)$ as a set
function. In this respect, we recall some terminology.

######  Definition 3.2.

Let $\alpha\colon\mathcal{A}(\Omega)\to[0,+\infty]$ be a set function. We say
that $\alpha$ is

  * •

an increasing set function if $\alpha(\emptyset)=0$ and
$\alpha(A)\leq\alpha(B)$ if $A\subseteq B$;

  * •

subadditive if $\alpha(A\cup B)\leq\alpha(A)+\alpha(B)$ for all $A,B$;

  * •

superadditive if $\alpha(A\cup B)\geq\alpha(A)+\alpha(B)$ for all $A,B$ such
that $A\cap B=\emptyset$;

  * •

inner regular if
$\alpha(A)=\sup\left\\{\alpha(B):B\in\mathcal{A}(\Omega),\,B\Subset
A\right\\}$.

De Giorgi-Letta criterion (see e.g. [6, Theorem 10.2]) states that an
increasing set function is a restriction to $\mathcal{A}(\Omega)$ of Borel
measure if and only if subadditive, superadditive and inner regular.
Therefore, since the $\Gamma(L^{p})$-limit of
$\\{\mathcal{G}_{\varepsilon}\\}$ must be an increasing set function and we
work under $p$-growth conditions, the integral representation boils down to
proving that $\Gamma(L^{p})\text{-}\lim\mathcal{G}_{\varepsilon}$ is
subadditive, superadditive and inner regular. The subadditivity and inner
regularity are the most delicate points, and they hinge in turn upon the so-
called fundamental estimate (see e.g. [6, Definition 11.2]). Roughly speaking,
given $A,B\in\mathcal{A}(\Omega)$, $y\in L^{p}(\Omega;^{3})$ and sequences
$\\{y_{\varepsilon}^{\prime}\\}$ and $\\{y_{\varepsilon}^{\prime\prime}\\}$
converging to $y$ in $L^{p}$ such that

$\Gamma(L^{p})\text{-}\lim\mathcal{G}_{\varepsilon}(y,A)=\lim_{\varepsilon\to
0}\mathcal{G}_{\varepsilon}(y_{\varepsilon}^{\prime},A)\qquad\text{and}\qquad\Gamma(L^{p})\text{-}\lim\mathcal{G}_{\varepsilon}(y,B)=\lim_{\varepsilon\to
0}\mathcal{G}_{\varepsilon}(y_{\varepsilon}^{\prime\prime},B),$

the fundamental estimate allows to construct a third sequence
$\\{y_{\varepsilon}\\}$ such that

$\mathcal{G}_{\varepsilon}(y_{\varepsilon},A\cup
B)\leq\mathcal{G}_{\varepsilon}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}(y_{\varepsilon}^{\prime},A)+\mathcal{G}_{\varepsilon}(y_{\varepsilon}^{\prime\prime},B)+R_{\varepsilon},\qquad\lim_{\varepsilon\to
0}R_{\varepsilon}=0,$

with $R_{\varepsilon}$ satisfying a precise estimate in terms of
$\|y_{\varepsilon}^{\prime}-y_{\varepsilon}^{\prime\prime}\|_{L^{p}(A\cap
B)}$. The exact form of the fundamental estimate to be used in our analysis is
contained in (4.3), while its application to establish subadditivity and inner
regularity is discussed in Proposition 4.6.

###  3.2. Finsler structure on $\mathsf{SL}(3)$

To the purpose of devising a form of the fundamental estimate that suits the
functionals under consideration, it is convenient to endow $\mathsf{SL}(3)$
with a metric structure. In order to link the latter to the physics of the
system we would like to model, we follow the approach in [29], which is
grounded on the concept of _plastic dissipation_.

We recall some basic facts about the geometry of $\mathsf{SL}(3)$, which is a
smooth manifold with respect to the topology induced by the inclusion in 3×3.
For every $F\in\mathsf{SL}(3)$ the tangent space at $F$ is characterized as

$\mathrm{T}_{F}\mathsf{SL}(3)=F{\sf sl}(3)\coloneqq\\{FM\in^{3\times
3}:\mathrm{tr}M=0\\}.$

In particular, $\mathrm{T}_{I}\mathsf{SL}(3)$ coincides with ${\sf
sl}(3)\coloneqq\\{M\in^{3\times 3}:\mathrm{tr}M=0\\}$. We equip
$\mathsf{SL}(3)$ with a Finsler structure starting from a $C^{2}$ function
$\Delta_{I}\colon{\sf sl}(3)\to[0,+\infty)$, on which we make the following
assumptions:

  1. D1:

It is positively $1$-homogeneous: $\Delta_{I}(cM)=c\Delta_{I}(M)$ for all
$c\geq 0$ and $M\in{\sf sl}(3)$;

  2. D2:

It is $1$-coercive and has at most linear growth: there exist $0<c_{4}\leq
c_{5}$ such that for all $M\in{\sf sl}(3)$

$c_{4}|M|\leq\Delta_{I}(M)\leq c_{5}|M|.$

  3. D3:

It is strictly convex.

We point out that we work under stronger regularity assumptions than the ones
in [29]. This is mainly due to the fact that we borrow results developed
within the context of differential geometry, where smoothness is customarily
required. As a consequence, some models, such as single crystal plasticity,
are not covered by our analysis; on the other hand, our assumptions encompass
Von Mises plasticity, see the considerations in [22, 29]. We also recall that
Finsler structures are known to appear as homogenized limits of periodic
Riemannian metrics [1].

Essentially, $\Delta_{I}$ is a Minkowski norm on ${\sf sl}(3)$, which we can
“translate” to the other tangent spaces by setting

$\begin{array}[]{rccc}\Delta\colon&\rm
T\mathsf{SL}(3)&\to&[0,+\infty)\\\\[1.0pt]
&(F,M)&\mapsto&\Delta_{I}(F^{-1}M),\end{array}$

where $\rm T\mathsf{SL}(3)$ is the tangent bundle to $\mathsf{SL}(3)$. Then,
it can be checked that $(\mathsf{SL}(3),\Delta)$ is a $C^{2}$ Finsler
manifold. We refer to the monograph [3] by Bao, Chern & Shen for an
introduction to Finsler geometry.

Let now $\mathcal{C}(F_{0},F_{1})$ be the family of piecewise $C^{2}$ curves
$\Phi\colon[0,1]\to\mathsf{SL}(3)$ such that $\Phi(0)=F_{0}$ and
$\Phi(1)=F_{1}$. We define a non-symmetric distance on $\mathsf{SL}(3)$ as
follows:

$D(F_{0},F_{1})\coloneqq\inf\left\\{\int_{0}^{1}\Delta\big{(}\Phi(t),\dot{\Phi}(t)\big{)}\\!\,\mathrm{d}t:\Phi\in\mathcal{C}(F_{0},F_{1})\right\\},$

where $\dot{\Phi}$ is the velocity of the curve. The function $D$ is positive,
attains $0$ if and only if it is evaluated on the diagonal of
$\mathsf{SL}(3)\times\mathsf{SL}(3)$, and fulfills the triangular inequality;
in general, however, $D(F_{0},F_{1})\neq D(F_{1},F_{0})$.

From a physical viewpoint, if $P_{0},P_{1}\colon\Omega\to\mathsf{SL}(3)$ are
admissible plastic strains, the integral of $D(P_{0},P_{1})$ over $\Omega$ is
interpreted as the minimum amount of energy that is dissipated when the system
moves from a plastic configuration to another.

An application of the direct method of the calculus of variations (cf. [29,
Theorem 5.1]) proves that for every $F_{0},F_{1}\in\mathsf{SL}(3)$ there
exists a curve $\Phi\in C^{1,1}([0,1];\mathsf{SL}(3))$ such that
$\Phi(0)=F_{0}$, $\Phi(1)=F_{1}$ and

$D(F_{0},F_{1})=\int_{0}^{1}\Delta\big{(}\Phi(t),\dot{\Phi}(t)\big{)}\,\mathrm{d}t.$

We call such $\Phi$ a shortest path between $F_{0}$ and $F_{1}$. The following
result, which summarizes the content of [3, Exercises 6.3.3], is crucial for
the proof of the fundamental estimate.

######  Proposition 3.3.

For any point $F$ in the Finsler manifold $\mathsf{SL}(3)$ there exists a
relatively compact neighborhood $U$ of $F$ such that for any $F_{0},F_{1}\in
U$ there exists a unique shortest path $\Phi$ joining $F_{0}$ and $F_{1}$, and
such path depends smoothly on its endpoints $F_{0}$ and $F_{1}$.

The uniqueness of the shortest path allows to characterize it in terms of the
exponential map. We recall that a path between $F_{0}$ and $F_{1}$ is called a
geodesic if it is a critical point of the length functional under variations
that do not change the endpoints, and that, given $(F,M)\in T\mathsf{SL}(3)$
such that $\Delta(F,M)$ is sufficiently small, there exists a unique geodesic
$\gamma_{F,M}\colon(-2,2)\to\mathsf{SL}(3)$ satisfying $\gamma_{F,M}(0)=F$ and
$\dot{\gamma}_{F,M}(0)=M$. Therefore, for $M$ in a neighborhood of $0\in
T_{F}\mathsf{SL}(3)$, the exponential map

$\exp(F,M)\coloneqq\begin{cases}F&\text{if }M=0,\\\
\gamma_{F,M}(1)&\text{otherwise}\end{cases}$

is well defined [3, p. 126] and $\exp(F,\,\cdot\,)$ is invertible. Denoting by
$\exp_{F}^{-1}$ the inverse, if $\Phi$ is the unique shortest path from
$F_{0}$ to $F_{1}$, one sees that necessarily

$\Phi(t)=\exp(F_{0},t\exp_{F_{0}}^{-1}(F_{1})).$ (3.1)

######  Remark 3.4.

Grounding on Proposition 3.3, we can show that there exists a compact $K$
meeting the requirements in H2. Let $U$ be a relatively compact neighborhood
of $I\in\mathsf{SL}(3)$ such that for any $F_{0},F_{1}\in U$ there is a unique
shortest path $\Phi$ joining $F_{0}$ and $F_{1}$. By a Finsler variant of a
theorem by Whitehead [3, Exercise 6.4.3], there exists an open neighborhood
$V$ of $I$ that is compactly contained in $U$ and geodesically convex. Since
$K\coloneqq\bar{V}\subset U$, there is a unique shortest path $\Phi$ from
$F_{0}$ to $F_{1}$ for any $F_{0},F_{1}\in K$. By (3.1), the path depends
smoothly on its endpoints. The fact that $K$ is geodesically convex as well
follows then by the same argument that proves that the closure of a convex set
is still convex.

##  4\. Homogenization in finite plasticity

We devote this section to the proof of Theorem 2.2, that is, we exhibit the
$\Gamma(\tau)$-limit of the functionals in (1.1). We recall that the topology
$\tau$ was introduced in (2.3).

######  Remark 4.1.

The energy functionals at stake depend on the plastic strain $P$ through its
inverse $P^{-1}$. In this respect, it is useful to notice that if $P_{k}\to P$
uniformly, then $P_{k}^{-1}\to P^{-1}$ uniformly as well. Indeed, recalling
that for any $k\in\mathbb{N}$ we can write

$P_{k}^{-1}=\frac{\left({\rm cof}P_{k}\right)^{T}}{\det P_{k}}=\left({\rm
cof}P_{k}\right)^{T},$

we deduce convergence for the sequence of inverses.

Since $(W^{1,2}(\Omega;^{3})\times W^{1,q}(\Omega;\mathsf{SL}(3)),\tau)$ is a
separable metric space, we know from general properties of
$\Gamma$-convergence that, up to extraction of subsequences, the functionals
in (1.1) $\Gamma(\tau)$-converge. Our main task is therefore to show that the
limit is an integral functional, and we achieve this by a localization
approach as the one streamlined in Subsection 3.1. Precisely, denoting by
$\mathcal{A}(\Omega)$ the family of open subset of $\Omega$, with a slight
abuse of notation, for any triple $(y,P,A)\in L^{2}(A;^{3})\times
L^{q}(A;\mathsf{SL}(3))\times\mathcal{A}(\Omega)$ we set

$\mathcal{F}_{k}(y,P,A)\coloneqq\left\\{\begin{aligned}
\displaystyle\int_{A}W\left(\frac{x}{\varepsilon_{k}},\nabla
yP^{-1}\right)\,\mathrm{d}x&+\displaystyle\int_{A}H\left(\frac{x}{\varepsilon_{k}},P\right)\,\mathrm{d}x+\int_{A}|\nabla
P|^{q}\,\mathrm{d}x\\\\[6.0pt] &\text{if }(y,P)\in W^{1,2}(A;^{3})\times
W^{1,q}(A;K),\\\\[3.0pt] +\infty\quad\qquad\qquad\qquad\qquad&\text{otherwise
in }L^{2}(A;^{3})\times L^{q}(A;\mathsf{SL}(3)),\end{aligned}\right.$ (4.1)

where ${\varepsilon_{k}}$ is an infinitesimal sequence.

We first show that the limits of $\Gamma$-convergent subsequences are in turn
integral functionals. Second, we characterize the limiting energy densities,
proving as well that the whole family $\\{\mathcal{F}_{\varepsilon}\\}$
$\Gamma$-converges.

###  4.1. Integral representation

In this subsection we establish the following:

######  Theorem 4.2.

Let $\mathcal{F}_{k}$ be as in (4.1), where $W$ and $H$ satisfy E1–E3 and
H1–H3, respectively. Then, up to subsequences, $\\{\mathcal{F}_{k}\\}$
$\Gamma(\tau)$-converges and for all $y\in L^{2}(A;^{3})$ and $P\in
L^{q}(A;\mathsf{SL}(3))$

$\Gamma(\tau)\mbox{-}\lim_{k\to+\infty}\mathcal{F}_{k}(y,P,A)=\begin{cases}\displaystyle\int_{A}f(x)\,\mathrm{d}x&\text{if
}(y,P)\in W^{1,2}(A;^{3})\times W^{1,q}(A;K),\\\\[6.0pt]
+\infty&\text{otherwise in }L^{2}(A;^{3})\times
L^{q}(A;\mathsf{SL}(3)),\end{cases}$ (4.2)

for some $f\in L^{1}_{\rm loc}(^{3})$ (which depends on the subsequence).

As a first step, we introduce a version of the fundamental estimate fit for
the functionals in (4.1). We recall that, given
$A,A^{\prime}\in\mathcal{A}(\Omega)$ with $A^{\prime}\Subset A$ (i.e.
$\overline{A^{\prime}}$ is a compact set contained in $A$), we say that a
function $\varphi$ is a cut-off function between $A^{\prime}$ and $A$ if
$\varphi\in C^{\infty}_{0}(A)$, $0\leq\varphi\leq 1$ and $\varphi\equiv 1$ in
a neighborhood of $\overline{A^{\prime}}$.

######  Definition 4.3.

Let $\mathcal{C}$ be a class of functionals $\mathcal{F}\colon
W^{1,2}(\Omega;^{3})\times
W^{1,q}(\Omega;K)\times\mathcal{A}(\Omega)\to[0,+\infty]$. We say that
$\mathcal{C}$ satisfies uniformly the fundamental estimate if for every
$A,A^{\prime},B\in\mathcal{A}(\Omega)$ with $A^{\prime}\Subset A$ and for
every $\sigma>0$ there exists a constant $M_{\sigma}>0$ with the following
property: for all $\mathcal{F}\in\mathcal{C}$ and for every
$(y_{1},P_{1}),(y_{2},P_{2})\in W^{1,2}(\Omega;^{3})\times W^{1,q}(\Omega;K)$
there exist a cut-off function $\varphi$ between $A^{\prime}$ and $A$ and a
path $\gamma\colon[0,1]\times K\times K\to K$ satisfying $\gamma(0,F,G)=F$,
$\gamma(1,F,G)=G$ for all $F,G\in K$ such that

$\displaystyle\mathcal{F}\big{(}\varphi
y_{1}+(1-\varphi)y_{2},\gamma\circ(\varphi,P_{2},P_{1}),A^{\prime}\cup
B\big{)}$ (4.3)
$\displaystyle\leq(1+\sigma)\left(\mathcal{F}(y_{1},P_{1},A)+\mathcal{F}(y_{2},P_{2},B)\right)$
$\displaystyle\quad+M_{\sigma}\int_{(A\cap B)\setminus
A^{\prime}}\left(|y_{1}-y_{2}|^{2}+|P_{1}-P_{2}|^{q}\right)\,\mathrm{d}x+\sigma.$

Note that, as far as the functionals in (4.1) are concerned, it is not
restrictive to assume that plastic deformations range just in $K$ and not in
$\mathsf{SL}(3)$, because (4.3) is otherwise trivially satisfied.

The use of the path $\gamma$ in (4.3) is motivated by the simple observation
that the “convex combination” $\varphi P_{1}+(1-\varphi)P_{2}$ does not belong
to $\mathsf{SL}(3)$ in general. For example, if we let

$P_{1}=\left(\begin{array}[]{ccc}1&0&0\\\ 0&1&0\\\
0&0&1\end{array}\right),\quad P_{2}=\left(\begin{array}[]{ccc}-1&0&0\\\
0&-1&0\\\ 0&0&1\end{array}\right),$

we see that $\det P_{1}=\det P_{2}=1$, but $\det(P_{1}/2+P_{2}/2)=0$. However,
not any path $\gamma$ is appropriate for our purposes, because in general it
may not be completely contained in $K$, that is, in the domain of $H$.
Moreover, the regularization term $\nabla P$ calls for some regularity of
$\gamma$ with respect to its endpoints (see (4.5)) and for a specific bound on
the velocity $\dot{\gamma}$ (see (4.6)). To tackle such issues, in Proposition
4.4 we select $\gamma$ suitably.

The next statement grants that the class $\mathcal{C}=\\{\mathcal{F}_{k}\\}$
meets the definition above.

######  Proposition 4.4.

The sequence of functionals $\\{\mathcal{F}_{k}\\}_{k\in\mathbb{N}}$ defined
in (4.1) satisfies uniformly the fundamental estimate (4.3), upon choosing
$\gamma$ as the map that associates to $(t,F,G)\in[0,1]\times K\times K$ the
image at $t$ of the unique shortest path connecting $F$ and $G$.

###### Proof.

Fix $A,A^{\prime},B\in\mathcal{A}(\Omega)$ with $A^{\prime}\Subset A$ and
$\sigma>0$. Fix also $(y_{1},P_{1}),(y_{2},P_{2})\in
W^{1,2}(\Omega;^{3})\times W^{1,q}(\Omega;K)$. We need to choose suitably the
cut-off function $\varphi$.

Let $\delta\coloneqq{\rm dist}(A^{\prime},\partial A)>0$. For a fixed
$N\in\mathbb{N}$, $N>1$, we define

$\displaystyle C_{0}$ $\displaystyle\coloneqq A^{\prime},$ $\displaystyle
C_{j}$ $\displaystyle\coloneqq\left\\{x\in A:{\rm
dist}(x,A^{\prime})<\frac{j}{N}\delta\right\\},\quad j=1,\dots,N,$
$\displaystyle D_{j}$ $\displaystyle\coloneqq C_{j}\setminus C_{j-1}.$

Observe that for all $j$ we can construct a cut-off function $\varphi_{j}$
between $C_{j-1}$ and $C_{j}$ such that $|\nabla\varphi_{j}|\leq 2N/\delta$.
Now, setting $\gamma^{-1}(t,F,G)\coloneqq(\gamma(t,F,G))^{-1}$ for all
$t\in[0,1]$,

$\mathcal{F}_{k}\left(\varphi_{j}y_{1}+(1-\varphi_{j})y_{2},\gamma\circ(\varphi_{j},P_{2},P_{1}),A^{\prime}\cup
B\right)=I_{\rm{el}}+I_{\rm{hard}}+I_{\rm{reg}},$ (4.4)

where

$\displaystyle I_{\rm el}\coloneqq\int_{A^{\prime}\cup
B}W\left(\frac{x}{\varepsilon_{k}},\big{[}\varphi_{j}\nabla
y_{1}+(1-\varphi_{j})\nabla
y_{2}+\nabla\varphi_{j}\otimes(y_{1}-y_{2})\big{]}\big{(}\gamma^{-1}\circ(\varphi_{j},P_{2},P_{1})\big{)}\right)\\!\,\mathrm{d}x,$
$\displaystyle I_{\mathrm{h}}\coloneqq\int_{A^{\prime}\cup
B}H\left(\frac{x}{\varepsilon_{k}},\gamma\circ(\varphi_{j},P_{2},P_{1})\right)\\!\,\mathrm{d}x,$
$\displaystyle I_{\rm reg}\coloneqq\int_{A^{\prime}\cup
B}\left|\nabla\big{(}\gamma\circ(\varphi_{j},P_{2},P_{1})\big{)}\right|^{q}\\!\,\mathrm{d}x.$

Since $P_{1},P_{2}\in K$, (2.1) holds. We now choose $\gamma$ as the map such
that $\gamma(t,F,G)$ is the evaluation at $t$ of the unique shortest path
connecting $F$ and $G$ for all $(t,F,G)\in[0,1]\times K\times K$. By
Proposition 3.3 we know that such $\gamma$ exists, that it lies completely in
$K$ if $F,G\in K$ (see H2), and that it depends smoothly on its arguments.

Let us estimate the three summands above separately. For the elastic
contribution we have

$\displaystyle I_{\mathrm{el}}$ $\displaystyle=\int_{(A^{\prime}\cup B)\cap
C_{j-1}}W\left(\frac{x}{\varepsilon_{k}},\nabla
y_{1}P_{1}^{-1}\right)\,\mathrm{d}x+\int_{(A^{\prime}\cup B)\setminus
C_{j}}W\left(\frac{x}{\varepsilon_{k}},\nabla
y_{2}P_{2}^{-1}\right)\,\mathrm{d}x$ $\displaystyle\quad+\int_{(A^{\prime}\cup
B)\cap D_{j}}\\!\\!W\left(\frac{x}{\varepsilon_{k}},\big{[}\varphi_{j}\nabla
y_{1}+(1-\varphi_{j})\nabla
y_{2}+\nabla\varphi_{j}\otimes(y_{1}-y_{2})\big{]}\big{(}\gamma^{-1}\circ(\varphi_{j},P_{2},P_{1})\big{)}\right)\\!\,\mathrm{d}x$
$\displaystyle\leq\int_{A}W\left(\frac{x}{\varepsilon_{k}},\nabla
y_{1}P_{1}^{-1}\right)\,\mathrm{d}x+\int_{B}W\left(\frac{x}{\varepsilon_{k}},\nabla
y_{2}P_{2}^{-1}\right)\,\mathrm{d}x$ $\displaystyle\quad+c\int_{B\cap
D_{j}}\left(1+|\nabla y_{1}|^{2}+|\nabla
y_{2}|^{2}+\left(\frac{2N}{\delta}\right)^{2}|y_{1}-y_{2}|^{2}\right)\,\mathrm{d}x,$

where we used the fact that $(A^{\prime}\cup B)\cap C_{j-1}\subseteq
C_{j-1}\subseteq A$, $(A^{\prime}\cup B)\setminus C_{j}\subseteq B$,
$(A^{\prime}\cup B)\cap D_{j}=B\cap D_{j}$, the growth condition E2 and the
uniform bound (2.1). Analogously, for the second summand we find

$\displaystyle I_{\mathrm{h}}$ $\displaystyle=\int_{(A^{\prime}\cup B)\cap
C_{j-1}}H\left(\frac{x}{\varepsilon_{k}},P_{1}\right)\,\mathrm{d}x+\int_{(A^{\prime}\cup
B)\setminus C_{j}}H\left(\frac{x}{\varepsilon_{k}},P_{2}\right)\,\mathrm{d}x$
$\displaystyle\quad+\int_{(A^{\prime}\cup B)\cap
D_{j}}H\left(\frac{x}{\varepsilon_{k}},\gamma\circ(\varphi_{j},P_{2},P_{1})\right)\,\mathrm{d}x$
$\displaystyle\leq\int_{A}H\left(\frac{x}{\varepsilon_{k}},P_{1}\right)\,\mathrm{d}x+\int_{B}H\left(\frac{x}{\varepsilon_{k}},P_{2}\right)\,\mathrm{d}x+c\,\mathcal{L}^{3}(B\cap
D_{j}),$

where we exploited again the geodesic convexity of $K$ and the fact that $H$
is bounded on $K$, since it is Lipschitz.

We now estimate $I_{\rm reg}$. The chain rule yields

$\nabla\big{(}\gamma\circ(\varphi_{j},P_{2},P_{1})\big{)}\\\
=\big{[}\dot{\gamma}\circ(\varphi_{j},P_{2},P_{1})\big{]}\otimes\nabla\varphi_{j}+\big{[}\partial_{F}\gamma\circ(\varphi_{j},P_{2},P_{1})\big{]}\nabla
P_{2}+\big{[}\partial_{G}\gamma\circ(\varphi_{j},P_{2},P_{1})\big{]}\nabla
P_{1},$ (4.5)

whence, observing that by Proposition 3.3, the two differentials
$\partial_{F}\gamma$ and $\partial_{G}\gamma$ are continuous functions
restricted to compact sets,

$\displaystyle I_{\rm reg}$ $\displaystyle=\int_{(A^{\prime}\cup B)\cap
C_{j-1}}|\nabla P_{1}|^{q}\,\mathrm{d}x+\int_{(A^{\prime}\cup B)\setminus
C_{j}}|\nabla P_{2}|^{q}\,\mathrm{d}x+\int_{(A^{\prime}\cup B)\cap
D_{j}}\left|\nabla\big{(}\gamma\circ(\varphi_{j},P_{2},P_{1})\big{)}\right|^{q}\,\mathrm{d}x$
$\displaystyle\leq\int_{A}|\nabla P_{1}|^{q}\,\mathrm{d}x+\int_{B}|\nabla
P_{2}|^{q}\,\mathrm{d}x$
$\displaystyle\quad+\left(\frac{2N}{\delta}\right)^{q}\int_{B\cap
D_{j}}|\dot{\gamma}\circ(\varphi_{j},P_{2},P_{1})|^{q}\,\mathrm{d}x+c\int_{B\cap
D_{j}}\big{(}|\nabla P_{1}|^{q}+|\nabla P_{2}|^{q}\big{)}\,\mathrm{d}x.$

We now resort to the explicit expression of $\gamma$ in terms of the
exponential map, see (3.1). From (2.1), D2 and D3 it follows

$|\dot{\gamma}(t,F,G)|=|\exp\big{(}F,t\exp_{F}^{-1}(G)\big{)}\exp_{F}^{-1}(G)|\leq
c\,\Delta_{I}(F^{-1}\exp_{F}^{-1}(G))\leq c\,|G-F|,$ (4.6)

so that

$\displaystyle I_{\rm reg}$ $\displaystyle\leq\int_{A}|\nabla
P_{1}|^{q}\,\mathrm{d}x+\int_{B}|\nabla P_{2}|^{q}\,\mathrm{d}x$ (4.7)
$\displaystyle\quad+c\left[\left(\frac{2N}{\delta}\right)^{q}\int_{B\cap
D_{j}}|P_{1}-P_{2}|^{q}\,\mathrm{d}x+\int_{B\cap D_{j}}\big{(}|\nabla
P_{1}|^{q}+|\nabla P_{2}|^{q}\big{)}\,\mathrm{d}x\right].$

By gathering (4.4)–(4.7) we obtain

$\displaystyle\mathcal{F}_{k}\left(\varphi_{j}y_{1}+(1-\varphi_{j})y_{2},\gamma\circ(\varphi_{j},P_{2},P_{1}),A^{\prime}\cup
B\right)$
$\displaystyle\leq\mathcal{F}_{k}(y_{1},P_{1},A)+\mathcal{F}_{k}(y_{2},P_{2},B)$
$\displaystyle\quad+c\int_{B\cap D_{j}}\left(1+|\nabla y_{1}|^{2}+|\nabla
y_{2}|^{2}+\left(\frac{2N}{\delta}\right)^{2}|y_{1}-y_{2}|^{2}\right)\,\mathrm{d}x$
$\displaystyle\quad+c\left[\mathcal{L}^{3}(B\cap
D_{j})+\left(\frac{2N}{\delta}\right)^{q}\int_{B\cap
D_{j}}|P_{1}-P_{2}|^{q}\,\mathrm{d}x+\int_{B\cap D_{j}}\big{(}|\nabla
P_{1}|^{q}+|\nabla P_{2}|^{q}\big{)}\,\mathrm{d}x\right].$

Now, note that

$\displaystyle\sum_{j=1}^{N}\int_{B\cap D_{j}}\left(1+|\nabla
y_{1}|^{2}+|\nabla y_{2}|^{2}+|\nabla P_{1}|^{q}+|\nabla
P_{2}|^{q}\right)\,\mathrm{d}x$ $\displaystyle\leq\int_{(A\setminus
A^{\prime})\cap B}\left(1+|\nabla y_{1}|^{2}+|\nabla y_{2}|^{2}+|\nabla
P_{1}|^{q}+|\nabla P_{2}|^{q}\right)\,\mathrm{d}x,$

thus there certainly exists $\ell\in{1,\dots,N}$ such that

$\displaystyle\int_{B\cap D_{\ell}}\left(1+|\nabla y_{1}|^{2}+|\nabla
y_{2}|^{2}+|\nabla P_{1}|^{q}+|\nabla P_{2}|^{q}\right)\,\mathrm{d}x$
$\displaystyle\leq\frac{1}{N}\int_{(A\setminus A^{\prime})\cap
B}\left(1+|\nabla y_{1}|^{2}+|\nabla y_{2}|^{2}+|\nabla P_{1}|^{q}+|\nabla
P_{2}|^{q}\right)\,\mathrm{d}x$
$\displaystyle\leq\frac{1}{N}\mathcal{L}^{3}((A\setminus A^{\prime})\cap
B)+\frac{c}{N}\Big{(}\mathcal{F}_{k}(y_{1},P_{1},A)+\mathcal{F}_{k}(y_{2},P_{2},B)\Big{)}$

where in the last inequality we exploited the fact that the set $(A\setminus
A^{\prime})\cap B$ is contained in both $A$ and $B$, together with the growth
condition from below in E2. Therefore, we obtain

$\displaystyle\mathcal{F}_{k}\left(\varphi_{\ell}y_{1}+(1-\varphi_{\ell})y_{2},\gamma\circ(\varphi_{\ell},P_{2},P_{1}),A^{\prime}\cup
B\right)$
$\displaystyle\quad\leq\left(1+\frac{c}{N}\right)\Big{(}\mathcal{F}_{k}(y_{1},P_{1},A)+\mathcal{F}_{k}(y_{2},P_{2},B)\Big{)}$
$\displaystyle\quad\quad+c\int_{(A\cap B)\setminus
A^{\prime}}\left[\left(\frac{2N}{\delta}\right)^{2}|y_{1}-y_{2}|^{2}+\left(\frac{2N}{\delta}\right)^{q}|P_{1}-P_{2}|^{q}\right]\\!\,\mathrm{d}x+\frac{2c}{N}\mathcal{L}^{3}((A\setminus
A^{\prime})\cap B).$

Finally, choosing $N$ such that

$\begin{cases}\displaystyle\frac{c}{N}<\sigma,\\\\[11.38109pt]
\displaystyle\frac{2c\,\mathcal{L}^{3}((A\setminus A^{\prime})\cap
B)}{N}<\sigma,\end{cases}$

and letting

$M_{\sigma}\coloneqq
c\left[\left(\frac{2N}{\delta}\right)^{2}+\left(\frac{2N}{\delta}\right)^{q}\right],$

we see (4.3) is satisfied, and the proof is complete. ∎

Still following the strategy outlined in Subsection 3.1, we analyze the
properties of the $\Gamma(\tau)$-lower and $\Gamma(\tau)$-upper limit of our
sequence $\mathcal{F}_{k}$ when regarded as set functions. We recall that (see
e.g. [6, Chapter 7]), if $\\{\varepsilon_{k}\\}_{k\in\mathbb{N}}$ is such that
$\varepsilon_{k}\to 0^{+}$ and if $(y,P,A)\in W^{1,2}(\Omega;^{3})\times
W^{1,q}(\Omega;\mathsf{SL}(3))\times\mathcal{A}(\Omega)$, we have

$\begin{split}\mathcal{F}^{\prime}(y,P,A)&\coloneqq\Gamma(\tau)\mbox{-}\liminf_{k\to+\infty}\mathcal{F}_{k}(y,P,A)\\\
&\coloneqq\inf\left\\{\liminf_{k\to+\infty}\mathcal{F}_{k}(y_{k},P_{k},A):(y_{k},P_{k})\stackrel{{\scriptstyle\tau}}{{\to}}(y,P)\right\\},\\\
\mathcal{F}^{\prime\prime}(y,P,A)&\coloneqq\Gamma(\tau)\mbox{-}\limsup_{k\to+\infty}\mathcal{F}_{k}(y,P,A)\\\
&\coloneqq\inf\left\\{\limsup_{k\to+\infty}\mathcal{F}_{k}(y_{k},P_{k},A):(y_{k},P_{k})\stackrel{{\scriptstyle\tau}}{{\to}}(y,P)\right\\},\end{split}$
(4.8)

and that $\\{\mathcal{F}_{k}\\}$ $\Gamma(\tau)$-converges to $\mathcal{F}$ if
and only if $\mathcal{F}=\mathcal{F}^{\prime}=\mathcal{F}^{\prime\prime}$. A
key-step is the following “almost subadditivity” result, which is a
modification of [6, Proposition 11.5] or [13, Proposition 18.3]. It is a
consequence of the fundamental estimate, but it does not depend on the
explicit form of the functionals at stake.

######  Proposition 4.5.

Let $\\{\mathcal{F}_{k}\\}_{k\in\mathbb{N}}$, $\mathcal{F}^{\prime}$ and
$\mathcal{F}^{\prime\prime}$ be as above. Then for all
$A,A^{\prime},B\in\mathcal{A}(\Omega)$ with $A^{\prime}\Subset A$ and for all
$(y,P)\in W^{1,2}(\Omega;^{3})\times W^{1,q}(\Omega;K)$

$\displaystyle\mathcal{F}^{\prime}(y,P,A^{\prime}\cup B)$
$\displaystyle\leq\mathcal{F}^{\prime}(y,P,A)+\mathcal{F}^{\prime\prime}(y,P,B),$
$\displaystyle\mathcal{F}^{\prime\prime}(y,P,A^{\prime}\cup B)$
$\displaystyle\leq\mathcal{F}^{\prime\prime}(y,P,A)+\mathcal{F}^{\prime\prime}(y,P,B).$

###### Proof.

We need to prove the inequalities only when $\mathcal{F}^{\prime}(y,P,A)$,
$\mathcal{F}^{\prime\prime}(y,P,A)$ and $\mathcal{F}^{\prime\prime}(y,P,B)$
are finite. By the definitions in (4.8), there exist sequences
$\\{y^{\prime}_{k}\\}_{k}$, $\\{P^{\prime}_{k}\\}_{k}$,
$\\{y^{\prime\prime}_{k}\\}_{k}$, $\\{P^{\prime\prime}_{k}\\}_{k}$ such that

$\displaystyle(y^{\prime}_{k},P^{\prime}_{k})\stackrel{{\scriptstyle\tau}}{{\to}}(y,P),\quad(y^{\prime\prime}_{k},P^{\prime\prime}_{k})\stackrel{{\scriptstyle\tau}}{{\to}}(y,P),$
$\displaystyle\mathcal{F}^{\prime}(y,P,A)=\liminf_{k\to+\infty}\mathcal{F}_{k}(y^{\prime}_{k},P^{\prime}_{k},A),\quad\mathcal{F}^{\prime\prime}(y,P,B)=\limsup_{k\to+\infty}\mathcal{F}_{k}(y^{\prime\prime}_{k},P^{\prime\prime}_{k},B).$

Let us fix $\sigma>0$. The fundamental estimate (4.3) gives a constant
$M_{\sigma}$, a sequence $\\{\varphi_{k}\\}_{k}$ of cut-off functions between
$A^{\prime}$ and $A$, and a sequence $\\{\gamma_{k}\\}_{k}$ of shortest paths
from $P^{\prime\prime}_{k}$ to $P^{\prime}_{k}$ such that

$\displaystyle\mathcal{F}_{k}\left(\varphi_{k}y^{\prime}_{k}+(1-\varphi_{k})y^{\prime\prime}_{k},\gamma_{k}\circ(\varphi_{k},P_{k}^{\prime\prime},P_{k}^{\prime}),A^{\prime}\cup
B\right)$ (4.9)
$\displaystyle\leq(1+\sigma)\big{(}\mathcal{F}_{k}(y^{\prime}_{k},P^{\prime}_{k},A)+\mathcal{F}_{k}(y^{\prime\prime}_{k},P^{\prime\prime}_{k},B)\big{)}$
$\displaystyle\quad+M_{\sigma}\int_{(A\cap B)\setminus
A^{\prime}}\left(|y^{\prime}_{k}-y^{\prime\prime}_{k}|^{2}+|P^{\prime}_{k}-P^{\prime\prime}_{k}|^{q}\right)\,\mathrm{d}x+\sigma.$

Recalling (3.1), we find

$\begin{array}[]{rcll}\varphi_{k}y^{\prime}_{k}+(1-\varphi_{k})y^{\prime\prime}_{k}&\to&y&\text{strongly
in }L^{2}(A^{\prime}\cup B;^{3}),\\\\[5.69054pt]
\gamma_{k}\circ(\varphi_{k},P_{k}^{\prime\prime},P_{k}^{\prime})&\to&P&\text{uniformly.}\end{array}$

Taking the lower limit in (4.9) we obtain

$\displaystyle\mathcal{F}^{\prime}(y,P,A^{\prime}\cup B)$
$\displaystyle\leq\liminf_{k\to+\infty}\mathcal{F}_{k}\left(\varphi_{k}y^{\prime}_{k}+(1-\varphi_{k})y^{\prime\prime}_{k},\gamma_{k}\circ(\varphi_{k},P_{k}^{\prime\prime},P_{k}^{\prime}),A^{\prime}\cup
B\right)$
$\displaystyle\leq(1+\sigma)\left(\liminf_{k\to+\infty}\mathcal{F}_{k}(y^{\prime}_{k},P^{\prime}_{k},A)+\limsup_{k\to+\infty}\mathcal{F}_{k}(y^{\prime\prime}_{k},P^{\prime\prime}_{k},B)\right)+\sigma$
$\displaystyle=(1+\sigma)\left(\mathcal{F}^{\prime}(y,P,A)+\mathcal{F}^{\prime\prime}(y,P,B)\right)+\sigma,$

and the first inequality in the statement follows by letting $\sigma\to 0$.
The second one is obtained by taking the upper limit in (4.9) and arguing in a
similar way. ∎

Building on Proposition 4.5, we next establish subadditivity and inner
regularity. We exploit the fact that $\mathcal{F}^{\prime}(y,P,\,\cdot\,)$ and
$\mathcal{F}^{\prime\prime}(y,P,\,\cdot\,)$ are increasing set functions, as
well as the growth condition

$\mathcal{F}^{\prime\prime}(y,P,A)\leq c\int_{A}\left(1+|\nabla
yP^{-1}|^{2}+|\nabla P|^{q}\right)\,\mathrm{d}x.$ (4.10)

######  Proposition 4.6 (Subadditivity and inner regularity).

Let $\\{\mathcal{F}_{k}\\}_{k\in\mathbb{N}}$, $\mathcal{F}^{\prime}$ and
$\mathcal{F}^{\prime\prime}$ be as before. For all $(y,P)\in
W^{1,2}(\Omega;^{3})\times W^{1,q}(\Omega;K)$

  1. (1)

$\mathcal{F}^{\prime}(y,P,\,\cdot\,)$ and
$\mathcal{F}^{\prime\prime}(y,P,\,\cdot\,)$ are inner regular,

  2. (2)

$\mathcal{F}^{\prime\prime}(y,P,\,\cdot\,)$ is subadditive.

###### Proof.

We prove the two statements separately.

(1) Let us focus on $\mathcal{F}^{\prime}$. According to Definition 3.2 we
need to prove that

$\mathcal{F}^{\prime}(y,P,A)=\sup\left\\{\mathcal{F}^{\prime}(y,P,B):B\in\mathcal{A}(\Omega),\,B\Subset
A\right\\}.$

Since $\mathcal{F}^{\prime}(y,P,\,\cdot\,)$ is an increasing set function, it
is trivial that $\mathcal{F}^{\prime}(y,P,A)$ is larger than the supremum in
the formula above. As for the reverse inequality, let us fix
$A\in\mathcal{A}(\Omega)$ and let $C\subset A$ be compact. There exist
$B,B^{\prime}\in\mathcal{A}(\Omega)$ such that $C\subset B^{\prime}\Subset
B\Subset A$. Then $B^{\prime}\cup(A\setminus C)=A$. We now apply Proposition
4.5:

$\displaystyle\mathcal{F}^{\prime}(y,P,A)$
$\displaystyle=\mathcal{F}^{\prime}(y,P,B^{\prime}\cup(A\setminus C))$
$\displaystyle\leq\mathcal{F}^{\prime}(y,P,B)+\mathcal{F}^{\prime\prime}(y,P,A\setminus
C)$
$\displaystyle\leq\sup\left\\{\mathcal{F}^{\prime}(y,P,B):B\in\mathcal{A}(\Omega),\,B\Subset
A\right\\}+c\int_{A\setminus C}\left(1+|\nabla yP^{-1}|^{2}+|\nabla
P|^{q}\right)\,\mathrm{d}x.$

By the arbitrariness of $C$, we can let $C$ invade $A$ and we obtain

$\mathcal{F}^{\prime}(y,P,A)\leq\sup\left\\{\mathcal{F}^{\prime}(y,P,B):B\in\mathcal{A}(\Omega),\,B\Subset
A\right\\}.$

The same argument applies to $\mathcal{F}^{\prime\prime}$.

(2) Fix $A,B\in\mathcal{A}(\Omega)$ and let $C\in\mathcal{A}(\Omega)$ satisfy
$C\Subset A\cup B$. We find $A^{\prime}\in\mathcal{A}(\Omega)$ such that
$A^{\prime}\Subset A$ and that $C\subset A^{\prime}\cup B$. Note that
$\overline{C}\setminus B$ is compact and $\overline{C}\setminus B\subset A$.
Now, by the monotonicity of $\mathcal{F}^{\prime\prime}(y,P,\,\cdot\,)$ and
Proposition 4.5,

$\mathcal{F}^{\prime\prime}(y,P,C)\leq\mathcal{F}^{\prime\prime}(y,P,A^{\prime}\cup
B)\leq\mathcal{F}^{\prime\prime}(y,P,A)+\mathcal{F}^{\prime\prime}(y,P,B).$

By taking the supremum over $C$, inner regularity yields

$\mathcal{F}^{\prime\prime}(y,P,A\cup
B)\leq\mathcal{F}^{\prime\prime}(y,P,A)+\mathcal{F}^{\prime\prime}(y,P,B).$

∎

By gathering the previous results, we infer the existence of a $\Gamma$-limit
admitting an integral representation.

###### Proof of Theorem 4.2.

Let $\\{\varepsilon_{k}\\}_{k\in\mathbb{N}}$ be such that $\varepsilon_{k}\to
0^{+}$, and let $\mathcal{F}^{\prime}$ and $\mathcal{F}^{\prime\prime}$ be as
in (4.8). It follows from the definitions that they are increasing set
functions, and Proposition 4.6 yields inner regularity for both and
subadditivity for $\mathcal{F}^{\prime\prime}$. By standard arguments (see
e.g. [6, Theorem 10.3]), we deduce that, upon extraction of a subsequences,

$\mathcal{F}(y,P,A)\coloneqq\Gamma(\tau)\mbox{-}\lim_{k\to+\infty}\mathcal{F}_{k}(y,P,A)$

exists for all triple $(y,P,A)$. From the definition of $\Gamma$-limit and
from the fact that $\mathcal{F}_{k}(y,P,\,\cdot\,)$ is a measure for each $k$,
it follows that $\mathcal{F}(y,P,\,\cdot\,)$ is superadditive. We are then in
a position to apply De Giorgi-Letta criterion (see Subsection 3.1), which
ensures that $\mathcal{F}(y,P,\,\cdot\,)$ is the restriction of a Borel
measure to $\mathcal{A}(\Omega)$ for $(y,P)\in W^{1,2}(A;^{3})\times
W^{1,q}(A;K)$. Thanks to (4.10) we infer also that it is absolutely continuous
with respect to the Lebesgue measure.

Thus, by Radon-Nikodym theorem there exists $f\in L^{1}_{\rm loc}(^{3})$ such
that

$\mathcal{F}(y,P,A)=\int_{A}f(x)\,\mathrm{d}x.$

∎

###  4.2. Characterization of the limiting energy density

So far, we have proved a $\Gamma(\tau)$-compactness result for the functionals
(4.1), and we have shown that the limit is actually an integral functional. We
conclude by providing an explicit formula for the limiting energy density.

###### Proof of Theorem 2.2.

Let $y\in W^{1,2}(\Omega;^{3})$, $P\in W^{1,q}(\Omega;K)$. Our aim is to
identify $f$ in (4.2) as

$f(x)=W_{\mathrm{hom}}(\nabla y(x),P(x))+H_{\mathrm{hom}}(P(x))+|\nabla
P(x)|^{q}\quad\text{ for a.\,e. }x\in\Omega.$ (4.11)

This also entails (see [6, Proposition 7.11]) that the whole family
$\\{\mathcal{F}_{\varepsilon}\\}$ $\Gamma(\tau)$-converges to $\mathcal{F}$ as
in the statement of the theorem

For all $x\in^{3}$, $F\in^{3\times 3}$ and $G\in K$ let us define

$\tilde{W}(x,F,G)\coloneqq W(x,FG^{-1}).$

From E1 and E2 we deduce that

  1. (1)

$\tilde{W}$ is a Borel function and $\tilde{W}(\,\cdot\,,F,G)$ is
$Q$-periodic;

  2. (2)

$\tilde{W}(x,\,\cdot\,,G)$ is $2$-coercive and has at most quadratic growth,
uniformly in $x$ and $G$: there exist $0<\tilde{c}_{1}\leq\tilde{c}_{2}$ such
that for a. e. $x\in^{3}$, for all $F\in^{3\times 3}$ and for all $G\in K$

$\tilde{c}_{1}|F|^{2}\leq\tilde{W}(x,F,G)\leq\tilde{c}_{2}\left(|F|^{2}+1\right).$

Note that the independence of $\tilde{c}_{1}$ and $\tilde{c}_{2}$ from $G$ is
a consequence of (2.2) and (2.1), respectively.

For $A\in\mathcal{A}(\Omega)$, we now introduce the functional

$\mathcal{G}_{k}(y,G,A)\coloneqq\int_{A}\left[\tilde{W}\left(\frac{x}{\varepsilon_{k}},\nabla
y(x),G\right)+H\left(\frac{x}{\varepsilon_{k}},G\right)\right]\\!\,\mathrm{d}x$

where $G\in K$ is fixed, and we observe that

$\mathcal{F}_{k}(y,P,A)=\int_{A}\left[\tilde{W}\left(\frac{x}{\varepsilon_{k}},\nabla
y(x),P(x)\right)+H\left(\frac{x}{\varepsilon_{k}},P(x)\right)+|\nabla
P(x)|^{q}\right]\\!\\!\,\mathrm{d}x.$

From Theorem 4.2 we know that

$\mathcal{F}(y,P,A)\coloneqq\Gamma(\tau)\mbox{-}\lim_{k\to+\infty}\mathcal{F}_{k}(y,P,A)=\int_{A}f(x)\,\mathrm{d}x,$

while, if $x_{0}\in A$ and $P_{0}\coloneqq P(x_{0})$, Theorem 3.1 and Riemann-
Lebesgue lemma on rapidly oscillating functions yield

$\displaystyle\mathcal{G}(y,P_{0},A)$
$\displaystyle\coloneqq\Gamma(L^{2})\mbox{ -
}\lim_{k\to+\infty}\mathcal{G}_{k}(y,P_{0},A)$ (4.12)
$\displaystyle=\int_{A}\Big{(}W_{\mathrm{hom}}\big{(}\nabla
y(x),P(x_{0})\big{)}+H_{\mathrm{hom}}(P(x_{0}))\Big{)}\,\mathrm{d}x.$

We next show that (4.11) holds by exploiting these two convergence results.

From now on, let $x_{0}\in\Omega$ be a Lebesgue point of the functions $f$,
$\nabla y$ and $\nabla P$. Notice that almost every $x_{0}\in\Omega$ has such
property. Let $A\in\mathcal{A}(\Omega)$ contain $x_{0}$, and let
$\\{(y_{k},P_{k})\\}\subset W^{1,2}(A;^{3})\times W^{1,q}(A;K)$ be a generic
sequence. We use E3 and (2.1) to estimate the difference between the elastic
contribution in $\mathcal{F}_{k}(y_{k},P_{k},A)$ and
$\mathcal{G}_{k}(y_{k},P_{0},A)$:

$\displaystyle\int_{A}\left|\tilde{W}\left(\frac{x}{\varepsilon_{k}},\nabla
y_{k},P_{k}\right)-\tilde{W}\left(\frac{x}{\varepsilon_{k}},\nabla
y_{k},P_{0}\right)\right|\\!\\!\,\mathrm{d}x$ $\displaystyle\qquad\leq
c_{3}\int_{A}\Big{(}1+|\nabla y_{k}P_{k}^{-1}|+|\nabla
y_{k}P^{-1}_{0}|\Big{)}|\nabla
y_{k}|\left|P_{k}^{-1}-P^{-1}_{0}\right|\\!\\!\,\mathrm{d}x$
$\displaystyle\qquad\leq c_{3}\int_{A}\big{(}1+2c_{K}|\nabla
y_{k}|\big{)}|\nabla
y_{k}|\Big{(}\left|P_{k}^{-1}-P^{-1}\right|+\left|P^{-1}-P^{-1}_{0})\right|\Big{)}\\!\,\mathrm{d}x$
$\displaystyle\qquad\leq c\,\Big{(}\mathcal{L}^{3}(A)+\|\nabla
y_{k}\|^{2}_{L^{2}(A)}\Big{)}\Big{(}\|P_{k}^{-1}-P^{-1}\|_{L^{\infty}(A)}+\|P^{-1}-P^{-1}_{0}\|_{L^{\infty}(A)}\Big{)},$

where $c$ is a constant independent of $k$ and $A$. We now use H3 to estimate
the difference between the hardenings in $\mathcal{F}_{k}(y_{k},P_{k},A)$ and
$\mathcal{G}_{k}(y_{k},P_{0},A)$:

$\displaystyle\int_{A}\left|H\left(\frac{x}{\varepsilon_{k}},P_{k}\right)-H\left(\frac{x}{\varepsilon_{k}},P_{0}\right)\right|\,\mathrm{d}x\leq
c\,\mathcal{L}^{3}(A)\Big{(}\|P_{k}-P\|_{L^{\infty}(A)}+\|P-P_{0}\|_{L^{\infty}(A)}\Big{)}.$

By the definitions of $\mathcal{F}_{k}$ and $\mathcal{G}_{k}$ we obtain

$\begin{split}&\left|\mathcal{F}_{k}(y_{k},P_{k},A)-\int_{A}|\nabla
P_{k}|^{q}\,\mathrm{d}x-\mathcal{G}_{k}(y_{k},P_{0},A)\right|\\\ &\qquad\leq
c\,\Big{(}\mathcal{L}^{3}(A)+\|\nabla
y_{k}\|^{2}_{L^{2}(A)}\Big{)}\Big{(}\|P_{k}^{-1}-P^{-1}\|_{L^{\infty}(A)}+\|P^{-1}-P^{-1}_{0}\|_{L^{\infty}(A)}\Big{)}\\\
&\qquad\quad+c\,\mathcal{L}^{3}(A)\Big{(}\|P_{k}-P\|_{L^{\infty}(A)}+\|P-P_{0}\|_{L^{\infty}(A)}\Big{)}.\end{split}$
(4.13)

We now prove that

$f(x)\geq W_{\mathrm{hom}}\big{(}\nabla
y(x),P(x)\big{)}+H_{\mathrm{hom}}(P(x))+|\nabla P(x)|^{q}\quad\text{ for
a.\,e. }x\in\Omega.$ (4.14)

To this aim, we select a recovery sequence $\\{(y_{k},P_{k})\\}\subset
W^{1,2}(A;^{3})\times W^{1,q}(A;K)$ for $\mathcal{F}(y,P,A)$, namely

$\displaystyle(y_{k},P_{k})\stackrel{{\scriptstyle\tau}}{{\to}}(y,P),\quad\lim_{k\to+\infty}\mathcal{F}_{k}(y_{k},P_{k},A)=\mathcal{F}(y,P,A).$
(4.15)

Owing to the growth assumptions on the energy densities, we can without loss
of generality suppose that

$\displaystyle y_{k}\rightharpoonup y\quad\text{weakly in
}W^{1,2}(A;^{3}),\qquad P_{k}\rightharpoonup P\quad\text{weakly in
}W^{1,q}(A;K).$

From (4.13) coupled with the $2$-coercivity of $\tilde{W}$, recalling Remark
4.1 about the convergence of inverses, (4.12) and (4.15), we infer

$\displaystyle\mathcal{F}(y,P,A)$
$\displaystyle=\lim_{k\to+\infty}\mathcal{F}_{k}(y_{k},P_{k},A)$
$\displaystyle\geq\liminf_{k\to+\infty}\mathcal{G}_{k}(y_{k},P_{0},A)+\liminf_{k\to+\infty}\int_{A}|\nabla
P_{k}|^{q}\,\mathrm{d}x$
$\displaystyle\quad-c\,\|P^{-1}-P^{-1}_{0}\|_{L^{\infty}(A)}\Big{(}\mathcal{L}^{3}(A)+\lim_{k\to+\infty}\mathcal{F}_{k}(y_{k},P_{k},A)\Big{)}-c\,\mathcal{L}^{3}(A)\,\|P-P_{0}\|_{L^{\infty}(A)}$
$\displaystyle\geq\mathcal{G}(y,P_{0},A)+\int_{A}|\nabla P|^{q}\,\mathrm{d}x$
$\displaystyle\quad-c\,\|P^{-1}-P^{-1}_{0}\|_{L^{\infty}(A)}\Big{(}\mathcal{L}^{3}(A)+\mathcal{F}(y,P,A)\Big{)}-c\,\mathcal{L}^{3}(A)\,\|P-P_{0}\|_{L^{\infty}(A)}.$

Observe that the lower bound on $\mathcal{F}(y,P,A)$ that we have just proved
holds for any set $A$ containing $x_{0}$. In particular, we select
$A=B_{r}(x_{0})$ and divide both sides of the corresponding estimates by
$\mathcal{L}^{3}(B_{r}(x_{0}))$. By letting $r\to 0$, we apply Lebesgue
differentiation theorem to deduce (4.14) for $x=x_{0}$. Indeed,

$\lim_{r\to
0}\|P^{-1}-P^{-1}_{0}\|_{L^{\infty}(B_{r}(x_{0}))}\left(1+\frac{1}{\mathcal{L}^{3}(B_{r}(x_{0}))}\mathcal{F}(y,P,B_{r}(x_{0}))\right)=0,$

because (4.10) grants

$\mathcal{F}(y,P,A)\leq c\int_{A}\left(1+|\nabla yP^{-1}|^{2}+|\nabla
P|^{q}\right)\,\mathrm{d}x.$

Eventually, we are only left to prove that

$f(x)\leq W_{\mathrm{hom}}\big{(}\nabla
y(x),P(x)\big{)}+H_{\mathrm{hom}}(P(x))+|\nabla P(x)|^{q}\quad\text{ for
a.\,e. }x\in\Omega.$ (4.16)

To establish the estimate, we let $\\{y_{k}\\}\subset W^{1,2}(A;^{3})$ be a
recovery sequence for $\mathcal{G}(y,P_{0},A)$, that is,

$\lim_{k\to+\infty}\mathcal{G}_{k}(y_{k},P_{0},A)=\mathcal{G}(y,P_{0},A),$

with $y_{k}\rightharpoonup y$ weakly in $W^{1,2}(A;^{3})$. We consider
estimate (4.13) for this particular $\\{y_{k}\\}$ and the constant sequence
given by $P_{k}=P$ for all $k$. Exploiting this time the $2$-coercivity of
$\mathcal{G}_{k}$ we obtain

$\displaystyle\mathcal{F}(y,P,A)$
$\displaystyle\leq\limsup_{k\to+\infty}\mathcal{F}_{k}\left(y_{k},P,A\right)$
$\displaystyle\leq\lim_{k\to+\infty}\mathcal{G}_{k}\big{(}y_{k},P_{0},A\big{)}+\int_{A}|\nabla
P|^{q}\,\mathrm{d}x$
$\displaystyle\quad+c\,\|P^{-1}-P^{-1}_{0}\|_{L^{\infty}(A)}\Big{(}\mathcal{L}^{3}(A)+\lim_{k\to+\infty}\mathcal{G}_{k}(y_{k},P_{0},A)\Big{)}+c\,\mathcal{L}^{3}(A)\,\|P-P_{0}\|_{L^{\infty}(A)}$
$\displaystyle\leq\mathcal{G}(y,P_{0},A)+\int_{A}|\nabla P|^{q}\,\mathrm{d}x$
$\displaystyle\quad+c\,\|P^{-1}-P^{-1}_{0}\|_{L^{\infty}(A)}\Big{(}\mathcal{L}^{3}(A)+\mathcal{G}\big{(}y,P_{0},A\big{)}\Big{)}+c\,\mathcal{L}^{3}(A)\,\|P-P_{0}\|_{L^{\infty}(A)}.$

Arguing as above with $A=B_{r}(x_{0})$ and exploiting the 2-growth conditions
of $W_{\mathrm{hom}}$, we conclude that (4.16) holds for $x=x_{0}$.

Since we can select almost every point in $\Omega$ as $x_{0}$, the conclusion
follows from (4.14) and (4.16). ∎

## Acknowledgements

We acknowledge support from the Austrian Science Fund (FWF) projects F65,
V662, Y1292, from the FWF-GAČR project I 4052/19-29646L, and from the OeAD-WTZ
project CZ04/2019 (MŠMTČR 8J19AT013).

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E. Davoli, C. Gavioli, V. Pagliari.

Institute of Analysis and Scientific Computing, TU Wien,

Wiedner Hauptstraße 8-10, 1040 Vienna, Austria.

E-mails<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>


Polynomial tau-functions of the $n$-th Sawada-Kotera hierarchy

Victor Kac

Department of Mathematics,

Massachusetts Institute of Technology,

Cambridge, Massachusetts 02139, U.S.A

e-mail<EMAIL_ADDRESS>
Johan van de Leur

Mathematical Institute,

Utrecht University,

P.O. Box 80010, 3508 TA Utrecht,

The Netherlands

e-mail<EMAIL_ADDRESS>
###### Abstract

We find all polynomial tau-functions of the $n$-th reduced BKP hierarchy
(=$n$-th Sawada-Kotera hierarchy). The name comes from the fact that for $n=3$
the simplest equation of the hierarchy is the famous Sawada-Kotera equation.

##  1 Introduction

The three most famous hierarchies of Lax equations on one function $u$ are the
Korteweg-de Vries (KdV) hierarchy, the Kaup-Kupershmidt hierarchy, and the
Sawada-Kotera hierarchy. The Lax operators are, respectively,

$\displaystyle{\cal L}=\partial^{2}+u,$ (1) $\displaystyle{\cal
L}=\partial^{3}+u\partial+\frac{1}{2}u^{\prime},$ (2) $\displaystyle{\cal
L}=\partial^{3}+u\partial.$ (3)

Let $t=(t_{1},t_{2},t_{3},\ldots)$ and $\tilde{t}=(t_{1},t_{3},t_{5},\ldots)$.
Recall that the KP hierarchy is the following hierarchy of Lax equations on
the pseudodifferential operator
$L(t,\partial)=\partial+u_{1}(t)\partial^{-1}+u_{2}(t)\partial^{-2}+\cdots$ in
$\partial=\frac{\partial}{\partial t_{1}}$ [13]:

$\frac{\partial L(t,\partial)}{\partial
t_{k}}=\left[\left(L(t,\partial)^{k}\right)_{\geq
0},L(t,\partial)\right],\quad k=1,2,\ldots,$ (4)

The KdV hierarchy is the second reduced KP hierarchy, meaning that one imposes
the following constraint on $L(t,\partial)$:

${\cal L}(t,\partial)=L(t,\partial)^{2}\quad\mbox{is a differential
operator.}$ (5)

Then this operator is given by (1), with $u(t)=2u_{1}(t)$, and the KP
hierarchy (4) reduces to the KdV hierarchy

$\frac{\partial{\cal L}(t,\partial)}{\partial t_{k}}=\left[\left({\cal
L}(t,\partial)^{\frac{k}{2}}\right)_{\geq 0},{\cal L}(t,\partial)\right],\quad
k=1,3,5,\ldots.$ (6)

For even $k$ this equation is trivial, ${\cal L}(t,\partial)$ is given by (1),
and equation (6) for $k=3$ is the KdV equation [2].

Recall that, in order to construct solutions of the KP hierarchy and the
reduced KP hierarchies one introduces the tau-function $\tau(t)$, defined by
[13], [2]:

$L(t,\partial)=P(t,\partial)\circ\partial\circ P(t,\partial)^{-1},$ (7)

where $P(t,\partial)$ is a pseudodifferential operator, with the symbol

$P(t,z)=\frac{1}{\tau(t)}\exp-\sum_{k=1}^{\infty}\frac{z^{-k}}{k}\frac{\partial}{\partial
t_{k}}\tau(t).$ (8)

The tau-function has a geometric meaning as a point on an infinite-dimensional
Grassmannian, and in [13] Sato showed that all Schur polynomials are tau-
functions of the KP hierarchy. Recently, all polynomial tau-functions of the
KP hierarchy and its $n$-reductions have been constructed in [8] (see also
[11]).

The CKP hierarchy can be constructed by making use of the KP hierarchy, and
assuming the additional constraint
$L(\tilde{t},\partial)^{*}=-L(\tilde{t},\partial)$ (see e.g. [10] for
details). Its 3-reduction is defined by the constraint ${\cal
L}(\tilde{t},\partial)=L(\tilde{t},\partial)^{3}$ is a differential operator,
and the corresponding hierarchy is

$\frac{\partial{\cal L}(\tilde{t},\partial)}{\partial t_{k}}=\left[\left({\cal
L}(\tilde{t},\partial)^{\frac{k}{3}}\right)_{\geq 0},{\cal
L}(\tilde{t},\partial)\right],\quad k=1,3,5,\ldots,k\not\in 3\mathbb{Z},$ (9)

where $\cal L$ is given by (2). The simplest non-trivial of these equations
arizes for $k=5$, and it is the Kaup-Kupershmidt equation. All polynomial tau-
functions of (9) (and all $n$ reductions of the CKP hierarchy) have been
constructed in [10].

In the present paper we construct all polynomial tau-functions of the
$n$-reduced BKP hierarchies. These are hierarchies of Lax equations on the
differential operator

${\cal
L}(\tilde{t},\partial)=\partial^{n}+u_{n-2}(\tilde{t})\partial^{n-2}+\cdots+u_{1}(\tilde{t})\partial,$
(10)

satisfying the constraint

${\cal L}(\tilde{t},\partial)^{*}=(-1)^{n}\partial^{-1}{\cal
L}(\tilde{t},\partial)\partial.$ (11)

The $n$-th reduced BKP hierarchy is

$\frac{\partial{\cal L}(\tilde{t},\partial)}{\partial t_{k}}=\left[\left({\cal
L}(\tilde{t},\partial)^{\frac{k}{n}}\right)_{\geq 0},{\cal
L}(\tilde{t},\partial)\right],\quad k=1,3,5,\ldots,k\not\in n\mathbb{Z}.$ (12)

We call it the $n$-th Sawada-Kotera hierarchy, since for $n=3$, $\cal L$ is
given by (3), and for $k=5$, equation (12) is the Sawada-Kotera equation [14]
(see equation (31)).

In the present paper, using the description of polynomial tau-functions of the
BKP hierarchy [9], [11] (see Theorem 2), we find all polynomial tau-functions
for the $n$-th Sawada-Kotera hierarchies (see Theorem 8), and, in particular,
for the Sawada-Kotera hierarchy (see Corollary 9).

##  2 The BKP hierarchy and its polynomial tau-functions

In this section we recall the construction of the BKP hierarchy [3] and
description of its polynomial tau-functions from [9] and [11].

Following Date, Jimbo, Kashiwara and Miwa [3] (see also [9] for details) we
introduce the BKP hierarchy in terms of the so called twisted neutral fermions
$\phi_{i}$, $i\in\mathbb{Z}$, which are generators of a Clifford algebra over
$\mathbb{C}$, satisfying the following anti-commutation relatons

$\phi_{i}\phi_{j}+\phi_{j}\phi_{i}=(-1)^{i}\delta_{i,-j}.$ (13)

Define a right (resp. left) irreducible module $F=F_{r}$ (resp. $F_{l}$) over
this algebra by the following action on the vacuum vector $|0\rangle$ (resp.
$\langle 0|$)

$\phi_{0}|0\rangle=\frac{1}{\sqrt{2}}|0\rangle,\quad\phi_{j}|0\rangle=0\qquad\left(\mbox{resp.
}\langle 0|\phi_{0}=\frac{1}{\sqrt{2}}\langle 0|,\quad\langle
0|\phi_{-j}=0\right),\quad\mbox{for }j>0.$ (14)

The quadratic elements

$\phi_{j}\phi_{k}-\phi_{k}\phi_{j}\ \mbox{for }j,k\in\mathbb{Z},\ j>k,$

form a basis of the infinite-dimensional Lie algebra $so_{\infty,odd}$ over
$\mathbb{C}$. Let $SO_{\infty,odd}$ be the corresponding Lie group. We proved
in [7, Theorem 1.2a] that a non-zero element $\tau\in F$ lies in this Lie
group orbit of the vacuum vector $|0\rangle$ if and only if it satisfies the
BKP hierarchy in the fermionic picture, i.e., the following equation in
$F\otimes F$:

$\sum_{j\in\mathbb{Z}}(-1)^{j}\phi_{j}\tau\otimes\phi_{-j}\tau=\frac{1}{2}\tau\otimes\tau.$
(15)

Non-zero elements of $F$, satisfying (15), are called tau-functions of the BKP
hierarchy in the fermionic picture,.

The group $SO_{\infty,odd}$ consists of elements $G$ leaving the symmetric
bilinear form

$(\phi_{j},\phi_{k})=(-1)^{j}\delta_{j,-k}$ (16)

on $F$ invariant, i.e.

$(G\phi_{j},G\phi_{k})=(\phi_{j},\phi_{k}).$ (17)

Stated differently,

$G\phi_{k}G^{-1}=\sum_{j_{\in}\mathbb{Z}}a_{jk}\phi_{k}\ \mbox{(finite sum)
with }\sum_{j\in\mathbb{Z}}(-1)^{j}a_{jk}a_{-j\ell}=(-1)^{k}\delta_{k,-\ell}.$
(18)

The group orbit of the vacuum vector is the disjoint union of Schubert cells
(see Section 3 of [9] for details). These cells are parametrized by the strict
partitions $\lambda=(\lambda_{1},\lambda_{2},\ldots,\lambda_{k})$, with
$\lambda_{1}>\lambda_{2}>.\cdots>\lambda_{k}>0$. Namely, the cell, attached to
the partition $\lambda$ is

$C_{\lambda}=\\{v_{1}v_{2}\cdots
v_{k-1}v_{k}|0\rangle|\,v_{j}=\sum_{i\geq-\lambda_{j}}a_{ij}\phi_{i}\
\mbox{(finite sum) with }a_{-\lambda_{j},j}\neq 0\\}.$ (19)

An element $\tau\in C_{\lambda}$ corresponds to the following point in the
maximal isotropic Grassmannian (i.e. a maximal isotropic subspace of
$V=\bigoplus_{j\in\mathbb{Z}}{\mathbb{C}}\phi_{j}$):

${\rm Ann}\,\tau=\\{v\in V|\,vv_{1}v_{2}\cdots v_{k-1}v_{k}|0\rangle=0\\}.$
(20)

For instance ${\rm Ann}\,|0\rangle={\rm span}\\{\phi_{1},\phi_{2},\ldots\\}.$

Using the bosonization of the equation (15), one obtains a hierarchy of
differential equations on $\tau$ ([3], [4], [11] or [9, Section 3]). This
bosonization is an isomorphism $\sigma$ between the spin module $F$ and the
polynomial algebra
$B=\mathbb{C}[\tilde{t}]=\mathbb{C}[t_{1},t_{3},t_{5},\ldots]$. Explicitly,
introduce the twisted neutral fermionic field

$\phi(z)=\sum_{j\in\mathbb{Z}}\phi_{j}z^{-j},$

and the bosonic field

$\alpha(z)=\sum_{j\in\mathbb{Z}}\alpha_{2j+1}z^{-2j-1}=\frac{1}{2}:\phi(z)\phi(-z):,$

where the normal ordering $:\quad:$ is defined by

$:\phi_{j}\phi_{k}:=\phi_{j}\phi_{k}-\langle 0|\phi_{j}\phi_{k}|0\rangle;$

equivalently: $:\phi_{j}\phi_{k}:=\phi_{j}\phi_{k}$ if $j\leq k$ and
$=-\phi_{k}\phi_{j}$ if $j>k$, except when $j=k=0$, then
$:\phi_{0}\phi_{0}:=0$. The operators $\alpha_{j}$ satisfy the commutation
relations of the Heisenberg Lie algebra

$[\alpha_{j},\alpha_{k}]=\frac{j}{2}\delta_{j-k},\quad\alpha_{i}|0\rangle=\langle
0|\alpha_{-i}=0,\quad\mbox{for }i>0,$ (21)

and its representation on $F$ is irreducible [7, Theorem 3.2]. Using this, we
obtain a vector space isomorphism $\sigma:F\to B$, uniquely defined by the
following relations:

$\sigma(|0\rangle)=1,\quad\sigma\alpha_{j}\sigma^{-1}=\frac{\partial}{\partial
t_{j}},\quad\sigma\alpha_{-j}\sigma^{-1}=\frac{j}{2}t_{j},\quad\mbox{for }j>0\
\mbox{odd}.$

Explicitly [7, Section 3.2] :

$\sigma\phi(z)\sigma^{-1}=\frac{1}{\sqrt{2}}\exp\sum_{j=1}^{\infty}t_{2j-1}z^{2j-1}\exp\sum_{j=1}^{\infty}-2\frac{\partial}{\partial
t_{2j-1}}\frac{z^{-2j+1}}{2j-1}.$

Since (15) can be rewritten as

${\rm
Res}_{z}\,\phi(z)\tau\otimes\phi(-z)\tau\frac{dz}{z}=\frac{1}{2}\tau\otimes\tau,$

under the isomorphism $\sigma$ the equation (15) turns into:

${\rm
Res}_{z}e^{\sum_{j=1}^{\infty}(t_{2j-1}-t^{\prime}_{2j-1})z^{2j-1}}e^{\sum_{j=1}^{\infty}2\left(\frac{\partial}{\partial
t^{\prime}_{2j-1}}-\frac{\partial}{\partial
t_{2j-1}}\right)\frac{z^{-2j+1}}{2j-1}}\tau(\tilde{t})\tau(\tilde{t}^{\prime})\frac{dz}{z}=\tau(\tilde{t})\tau(\tilde{t}^{\prime}),$
(22)

where $\tilde{t}=(t_{1},t_{3},t_{5},\ldots)$ and
$\tilde{t}^{\prime}=(t_{1}^{\prime},t_{3}^{\prime},t_{5}^{\prime},\ldots)$.
Therefore, $\tau(\tilde{t})$ is the vacuum expectation value

$\tau(\tilde{t})=\sigma\tau\sigma^{-1}=\langle
0|e^{\sum_{j=1}^{\infty}t_{2j-1}\alpha_{2j-1}}\tau.$ (23)

Furthermore, by making a change of variables, as in [4, page 972], viz.
$t_{2k-1}=x_{2k-1}-y_{2k-1}$ and
$t^{\prime}_{2k-1}=x^{\prime}_{2k-1}-y^{\prime}_{2k-1}$, and using the
elementary Schur polynomials $s_{j}(r)$, which are defined by

$\exp\sum_{k=1}^{\infty}r_{k}z^{k}=\sum_{j=0}^{\infty}s_{j}(r)z^{j},$ (24)

we can rewrite (22), where we assume $x_{2k}=y_{2k}=0$:

$\sum_{j=1}^{\infty}s_{j}(-2\tilde{y})s_{j}(2\tilde{\partial}_{y})\tau(\tilde{x}-\tilde{y})\tau(\tilde{x}+\tilde{y})=0,$
(25)

where $\tilde{y}=(y_{1},0,y_{3},0,\ldots)$ and
$\tilde{\partial}_{y}=(\frac{\partial}{\partial
y_{1}},0,\frac{1}{3}\frac{\partial}{\partial
y_{3}},0,\frac{1}{5}\frac{\partial}{\partial y_{5}},\ldots)$. Using Taylor’s
formula we thus obtain the BKP hierarchy of Hirota bilinear equations [4, page
972],:

$\sum_{j=1}^{\infty}s_{j}(-2\tilde{y})s_{j}(2\tilde{\partial}_{u})\exp\sum_{j=1}^{\infty}y_{2j-1}\frac{\partial}{\partial
u_{2j-1}}\tau(\tilde{x}-\tilde{u})\tau(\tilde{x}+\tilde{u})\Big{|}_{\tilde{u}=0}=0.$
(26)

Using the notation $p(D)f\cdot g=p(\frac{\partial}{\partial
u_{1}},\frac{\partial}{\partial
u_{2}},\ldots)f(\tilde{x}+\tilde{u})g(\tilde{x}-\tilde{u})\Big{|}_{\tilde{u}=0}$,
this turns into

$\sum_{j=1}^{\infty}s_{j}(-2\tilde{y})s_{j}(2\tilde{D})e^{\sum_{j=1}^{\infty}y_{2j-1}D_{2j-1}}\tau\cdot\tau=0.$
(27)

The simplest equation in this hierarchy is [4, Appendix 3]:

$(D_{1}^{6}-5D_{1}^{3}D_{3}-5D_{3}^{2}+9D_{1}D_{5})\tau\cdot\tau=0.$ (28)

If we assume that our tau-function does not depend on $t_{3}$, then this gives

$(D_{1}^{6}+9D_{1}D_{5})\tau\cdot\tau=0.$ (29)

Letting $x=t_{1}$, $t=\frac{1}{9}t_{5}$, and

$u(x,t)=2\frac{\partial^{2}\log\tau(x,t)}{\partial x^{2}},$ (30)

and viewing the remaining $t_{j}$ as parameters, equation (29) turns into the
famous Sawada-Kotera equation [14]:

$u_{t}+15(uu_{xxx}+u_{x}u_{xx}+3u^{2}u_{x})+u_{xxxxx}=0.$ (31)

Another approach is by using the wave function, see [3, page 345],

$\displaystyle w(\tilde{t},z)$
$\displaystyle=\frac{1}{\tau(t)}\exp\sum_{j=1}^{\infty}t_{2j-1}z^{2j-1}\exp{-\sum_{j=1}^{\infty}2\frac{\partial}{\partial
t_{2j-1}}\frac{z^{-2j+1}}{2j-1}}\tau(t)$ (32)
$\displaystyle=P(\tilde{t},z)e^{\sum_{j=1}^{\infty}(t_{2j-1})z^{2j-1}},$

where $P(\tilde{t},z)=1+\sum_{j=1}^{\infty}p_{j}(\tilde{t})z^{-j}$, and in
particular

$p_{1}(\tilde{t})=-2\frac{\partial\log\tau(\tilde{t})}{\partial t_{1}}.$ (33)

Letting $P(\tilde{t},\partial)$ be the pseudodifferential operator in
$\partial=\frac{\partial}{\partial t_{1}}$ with the symbol $P(\tilde{t},z)$,
equation (22) turns into

${\rm
Res}_{z}P(\tilde{t},\partial)P(\tilde{t}^{\prime},\partial^{\prime})e^{\sum_{j=1}^{\infty}(t_{2j-1}-t^{\prime}_{2j-1})z^{2j-1}}\frac{dz}{z}=1.$
(34)

Now, using the fundamental lemma, Lemma 1.1 of [2] or Lemma 4.1 of [6], we
deduce from (34):

$\displaystyle
P(\tilde{t},\partial)\partial^{-1}P(\tilde{t},\partial)^{*}\partial=1,$ (35)
$\displaystyle\frac{\partial P(\tilde{t},\partial)}{\partial
t_{2j-1}}=-(P(\tilde{t},\partial)\partial^{2j-1}P(\tilde{t},\partial)^{-1}\partial^{-1})_{-}\partial
P(\tilde{t},\partial),\quad j=1,2,\ldots.$

Next, introducing the Lax operator

$L(\tilde{t},\partial)=P(\tilde{t},\partial)\partial
P(\tilde{t},\partial)^{-1}=\partial+u_{1}(\tilde{t})\partial^{-1}+u_{2}(\tilde{t})\partial^{-2}+\cdots,$

we deduce from (35) that $L$ satisfies [3]

$\displaystyle L(\tilde{t},\partial)^{*}$
$\displaystyle=-\partial^{-1}L(\tilde{t},\partial)\partial,$ (36)
$\displaystyle\frac{\partial L(\tilde{t},\partial)}{\partial t_{2j-1}}$
$\displaystyle=\left[\left(L(\tilde{t},\partial))^{2j-1}\right)_{\geq
0},L(\tilde{t},\partial)\right],\quad j=1,2,\ldots.$

Note that, since $u_{1}(\tilde{t})=-\frac{\partial p_{1}(\tilde{t})}{\partial
t_{1}}$ and the fact that $p_{1}(t)$ is given by (33), we find that

$u_{1}(\tilde{t})=2\frac{\partial^{2}\log\tau(\tilde{t})}{\partial
t_{1}^{2}},$ (37)

which explains the choice (30) of $u(x,t)$ to obtain the Sawada-Kotera
equation from the Hirota bilinear equation (29).

To obtain the second equation of (36), we use (35) and the first equation of
(36), which is equivalent (see [3], page 356) to the fact that
$L(\tilde{t},\partial)^{2j-1}$, for $j=1,2,\ldots$, has zero constant term.
Let us prove that the first equation of (36) indeed implies this fact. We have

$L^{k}\partial^{-1}=(-\partial^{-1}L^{*}\partial)^{k}\partial^{-1}=(-1)^{k}\partial^{-1}L^{*k}=(-1)^{k+1}(L^{k}\partial^{-1})^{*}.$

Now, using the fact that the constant term of $L^{k}$ is equal to

${\rm Res}_{\partial}L^{k}\partial^{-1}=-{\rm
Res}_{\partial}(L^{k}\partial^{-1})^{*}={\rm
Res}_{\partial}(-1)^{k+1}(L^{k}\partial^{-1})^{*}=(-1)^{k}{\rm
Res}_{\partial}L^{k}\partial^{-1},$

we find that the constant term of $L^{k}$ is zero whenever $k$ is odd.

###### Remark 1

Note that this also means that we can replace the second equation of (36) by,
cf. [12],

$\frac{\partial L(\tilde{t},\partial)}{\partial
t_{2j-1}}=\left[\left(L(\tilde{t},\partial))^{2j-1}\right)_{\geq
1},L(\tilde{t},\partial)\right],\quad j=1,2,\ldots.$

In the formulation of Kupershmidt [12], this means that $L$ satisfies not only
the KP equation for the odd times, but also his formulation of the modified KP
hierachy (only for the odd times).

Next, we describe polynomial tau-functions $\tau(t_{1},t_{3},\ldots)$ of the
BKP hierarchy obtained in [9, Theorem 6] (see also [11]). For that, given
integers $a$ and $b$, $a>b\geq 0$, let

$\displaystyle\chi_{a,b}(t,t^{\prime})$
$\displaystyle=\frac{1}{2}s_{a}(t^{\prime})s_{b}(t)+\sum_{j=1}^{b}(-1)^{j}s_{a+j}(t^{\prime})s_{b-j}(t),$
(38) $\displaystyle\chi_{b,a}(t,t^{\prime})$
$\displaystyle=-\chi_{a,b}(t,t^{\prime}),\quad\chi_{a,a}(t,t^{\prime})=0,$

and let $\chi_{a,b}(t,t^{\prime})=0$ if $b<0$. Then

###### Theorem 2

[9, Theorem 6] All polynomial tau-functions of the BKP hierarchy (22), up to a
scalar multiple, are equal to

$\tau_{\lambda}(\tilde{t})=Pf\left(\chi_{\lambda_{i},\lambda_{j}}(\tilde{t}+c_{i},\tilde{t}+c_{j})\right)_{1\leq
i,j\leq 2n},$ (39)

where $Pf$ is the Pfaffiann of a skew-symmetric matrix,
$\lambda=(\lambda_{1},\lambda_{2},\ldots,\lambda_{2n})$ is an extended strict
partition, i.e. $\lambda_{1}>\lambda_{2}>\cdots>\lambda_{2n}\geq 0$,
$\tilde{t}=(t_{1},0,t_{3},0,\ldots)$, $c_{i}=(c_{1i},c_{2i},c_{3i},\ldots)$,
$c_{ij}\in\mathbb{C}$.

###### Remark 3

The connection between the set of strict partitions and the extended strict
partitions is as follows. If
$\lambda=(\lambda_{1},\lambda_{2},\ldots,\lambda_{k})$ is a strict partition
and $k$ is even, then this partition is equal to the extended strict partition
$\lambda$. However, if $k$ is odd, the Pfaffian of a $k\times k$ anti-
symmetric matrix is equal to $0$, hence in that case we extend $\lambda$ by
the element $0$, i.e., the corresponding extended strict partition is then
$(\lambda_{1},\lambda_{2},\ldots,\lambda_{k},0)$.

##  3 The $n$-th Sawada-Kotera hierarchy and its polynomial tau-functions

As we have seen in Section 2, a necessary condition for a tau-function to give
a solution of the Sawada-Kotera equation is that
$\frac{\partial\tau(\tilde{t})}{\partial t_{3}}=0$. This means that the tau-
function lies in a smaller group orbit of the vacuum vector $|0\rangle$.
Instead of the $SO_{\infty,odd}$ orbit of the vacuum vector $|0\rangle$, we
consider the twisted loop group $G_{3}^{(2)}$, corresponding to the affine Lie
algebra $sl_{3}^{(2)}$, to obtain the 3-reduced BKP hierarchy [3]. More
generally (see also [3]), when $n=2k+1>1$ is odd, the $2k+1$-reduced hierarchy
is related to the twisted loop group $G_{2k+1}^{(2)}$ corresponding to the
twisted affine Lie algebra $sl_{2k+1}^{(2)}$. When $n=2k>2$ is even one has
the twisted loop group $G_{2k}^{(2)}$ corresponding to the affine Lie algebra
$so_{2k}^{(2)}$ [1], [7] (see [5, Chapter 7] for the construction of these Lie
algebras). Elements $G$ in this twisted loop group not only satisfy (18),
which implies
$\sum_{j\in\mathbb{Z}}(-1)^{j}a_{kj}a_{\ell,-j}=(-1)^{k}\delta_{k,-\ell}$, but
also the $n$-periodicity condition $a_{i+n,j+n}=a_{ij}$. This means that these
group elements also commute with the operator

$\sum_{i\in\mathbb{Z}}(-1)^{pn-i}\phi_{i}\otimes\phi_{pn-i},\quad\mbox{for
}p=1,2,3,\ldots,$

namely

$\displaystyle(G\otimes
G)\sum_{i\in\mathbb{Z}}(-1)^{pn-i}\phi_{i}\otimes\phi_{pn-i}$
$\displaystyle=\sum_{i\in\mathbb{Z}}(-1)^{pn-i}G\phi_{i}G^{-1}\otimes
G\phi_{pn-i}G^{-1}G$
$\displaystyle=\sum_{i,j,k\in\mathbb{Z}}(-1)^{pn-i}a_{ji}a_{k,pn-i}\phi_{j}G\otimes\phi_{k}G$
$\displaystyle=\sum_{i,j,k\in\mathbb{Z}}(-1)^{pn-i}a_{ji}a_{k-pn,-i}\phi_{j}G\otimes\phi_{k}G$
$\displaystyle=\sum_{j\in\mathbb{Z}}(-1)^{pn-j}\phi_{j}G\otimes\phi_{pn-j}G$
$\displaystyle=\sum_{j\in\mathbb{Z}}(-1)^{pn-j}\phi_{j}\otimes\phi_{pn-j}(G\otimes
G).$

Since
$\sum_{i\in\mathbb{Z}}(-1)^{pn-i}\phi_{i}|0\rangle\otimes\phi_{pn-i}|0\rangle=0$,
we find that the elements $\tau$ in the orbit of the vacuum vector of this
twisted loop not only satisfy (15), but also satisfy the conditions

$\sum_{j\in\mathbb{Z}}(-1)^{pn-j}\phi_{j}\tau\otimes\phi_{pn-j}\tau=0,\quad
p=1,2,\ldots.$ (40)

This means that $\tau(\tilde{t})=\sigma(\tau)$ not only satisfies (22), but
also the conditions

${\rm
Res}_{z}z^{pn-1}e^{\sum_{j=1}^{\infty}(t_{2j-1}-t^{\prime}_{2j-1})z^{2j-1}}e^{\sum_{j=1}^{\infty}2\left(\frac{\partial}{\partial
t^{\prime}_{2j-1}}-\frac{\partial}{\partial
t_{2j-1}}\right)\frac{z^{-2j+1}}{2j-1}}\tau(\tilde{t})\tau(\tilde{t}^{\prime})dz=0,\quad
p=1,2,\ldots.$ (41)

From (41) one deduces, using the fundamental Lemma [6, Lemma 4.1] and the
first equation of (35), that

$(P(\tilde{t},\partial)\partial^{pn-1}P(\tilde{t},\partial)^{*})_{<0}=(P(\tilde{t},\partial)\partial^{pn}P(\tilde{t},\partial)^{-1}\partial^{-1})_{<0}=0.$

Thus, the Lax operator $L(\tilde{t},\partial)$ satisfies

$\left(L(\tilde{t},\partial)^{pn}\right)_{\leq 0}=0,\quad p=1,2,\ldots.$ (42)

Hence ${\cal L}(\tilde{t},\partial)={L}(\tilde{t},\partial)^{n}$ is a monic
differential operator with zero constant term. Moreover, ${\cal
L}(\tilde{t},\partial)$ is equal to (10), and, by the first formula of (35),
we have the relation (11).

Now, if $n$ is odd, one can use the the Sato-Wilson equation, i.e. the second
equation of (35), to find that

$\frac{\partial P(\tilde{t},\partial)}{\partial t_{(2j-1)n}}=0,\quad
j=1,2,\ldots.$

From this we find that the tau-function satisfies

$\frac{\partial\tau(\tilde{t})}{\partial
t_{(2j-1)n}}=\lambda_{j}\tau(\tilde{t}),\quad\lambda_{j}\in\mathbb{C},\quad\mbox{for
}j=1,2,\ldots.$ (43)

Since we consider only polynomial tau-functions, we find that for odd $n$:

$\frac{\partial\tau(\tilde{t})}{\partial t_{(2j-1)n}}=0,\quad j=1,2,\ldots.$
(44)

If $n$ is even there is no such restriction, because the Sato-Wilson equation
(35) is only defined for odd flows. However, the additional equations (41)
still hold and give additional constraints on the tau-function.

###### Proposition 4

For $n$ odd, equation (44) for $j=1$ and the BKP hierarchy (22) on
$\tau(\tilde{t})$ are equivalent to (22) and (41).

Proof. We only have to show that (44) for $j=1$ and (22) imply (41). For this
differntiate (22) by $t_{n}$ and use (44), this gives Equation (41) for $p=1$.
Next differentiate (41) for $p=1$ again by $t_{n}$ and use again (44); this
gives (41) for $p=2$, etc. $\square$

Since $L$ satisfies the BKP hierarchy ${\cal L}=L^{n}$ also satisfies the BKP
hierarchy. For $n=3$, assuming the constraint that $\cal L$ is a differential
operator, ${\cal L}$ is given by (3) and $\frac{\partial{\cal L}}{\partial
t_{3}}=[({\cal L}^{\frac{5}{3}})_{\geq 0},{\cal L}]$ is the Sawada-Kotera
equation (31). This leads to the following definition.

###### Definition 5

Let ${\cal L}=\partial^{n}+u_{n-2}\partial^{n-2}+\cdots+u_{1}\partial$ be a
differential operator, satisfying (11). The system of Lax equations

$\frac{\partial{\cal L}(\tilde{t},\partial)}{\partial
t_{2j-1}}=\left[\left({\cal
L}(\tilde{t},\partial)^{\frac{2j-1}{n}}\right)_{\geq 0},{\cal
L}(\tilde{t},\partial)\right],\quad j=1,2,\ldots,$ (45)

is called the $n$-th reduced BKP hierarchy or the $n$-th Sawada-Kotera
hierarchy. For $n=3$ it is called the Sawada-Kotera hierarchy.

The geometric meaning of equation (40) is that the space ${\rm Ann}\,\tau$ is
invariant under the shift $\Lambda_{n}$, where

$\Lambda_{n}(\phi_{i})=\phi_{i+n}.$ (46)

As in the $SO_{\infty,odd}$ case, all polynomial tau-functions in this
$n$-reduced case lie in some Schubert cell. Such a Schubert cell has a
“lowest” element $w_{\lambda}$, for $\lambda$ a certain strict partition. This
element can be obtained from the vacuum vector by the action of the Weyl group
corresponding to $G_{n}^{(2)}$. The element

$w_{\lambda}=\phi_{-\lambda_{1}}\phi_{-\lambda_{2}}\cdots\phi_{-\lambda_{k}}|0\rangle$
(47)

lies in the Weyl group orbit of $|0\rangle$, corresponding to
$SO_{\infty,odd}$, see [15], however not all such elements lie in the Weyl
group orbit of $|0\rangle$ for $G_{n}^{(2)}$. For this, consider

${\rm Ann}\,w_{\lambda}={\rm
span}\\{\phi_{-\lambda_{1}},\phi_{-\lambda_{2}},\ldots,\phi_{-\lambda_{k}}\\}\oplus{\rm
span}\\{\phi_{i}|\,i>0,i\neq\lambda_{j},\ j=1,\ldots,k\\}.$ (48)

The element $w_{\lambda}$ lies in the $G_{n}^{(2)}$ Weyl group orbit if and
only if ${\rm Ann}\,w_{\lambda}$ is invariant under the action of
$\Lambda_{n}$, which means that the ($\lambda_{1}+1$ shifted) set

$V_{\lambda}=\\{\lambda_{1}+\lambda_{i}+1|,\
i=1,\ldots,k\\}\cup\\{\lambda_{1}-j+1|\,0<j<\lambda_{1},\ j\neq\lambda_{i}\
\mbox{for }i=1,\ldots,k\\},$ (49)

must satisfy the $-n$ shift condition, i.e.,

$\mbox{if }\mu_{j}\in V_{\lambda},\ \mbox{then }\mu_{j}-n\in V_{\lambda}\
\mbox{ or }\mu_{j}-n\leq 0.$ (50)

Only the elements $w_{\lambda}$, for which the corresponding $V_{\lambda}$
satisfies condition (50) lie in the $G_{n}^{(2)}$ group orbit.

###### Example 6

(a) For $n=2$ the only strict partition $\lambda$ that satisfies condition
(50) is $\lambda=\emptyset$.
(b)For $n=3$, the only strict partitions $\lambda$ that satisfy condition (50)
are

$(3m+1,3m-2,3m-5,\ldots,4,1)\ \mbox{and }(3m+2,3m-1,3m-4,\ldots,5,2),\quad
m\in\mathbb{Z}_{\geq 0}.$

###### Remark 7

Note that (50) means that $\lambda$ is a strict partition which is the union
of strict partitions $(nm+a_{i},n(m-1)+a_{i},,\ldots,n+a_{i},a_{i})$, with
$1\leq a_{i}<n$ and $1\leq i<n$, such that $a_{j}-n\neq-a_{\ell}$. In other
words $a_{j}+a_{\ell}\neq n$. Hence there are at most
$\left[\frac{n}{2}-1\right]$ such $a_{i}$.

To a strict partition $\lambda$, that satisfies condition (50), the
corresponding Schubert cell is then obtained through the action on a
$w_{\lambda}$ by an upper-triangular matrix in the group $G_{n}^{(2)}$. This
produces, up to a constant factor, elements

$v_{\lambda}=v_{1}v_{2}\cdots v_{k}|0\rangle,\ \mbox{where
}v_{j}=\phi_{-\lambda_{j}}+\sum_{i\geq 1-\lambda_{j}}a_{ij}\phi_{i}\
\mbox{(finite sum)},$ (51)

and

$(v_{j},v_{\ell})=0,\ \mbox{for }j,\ell=1,\ldots,k,\ \mbox{and if
}\lambda_{i}=\lambda_{j}-n,\ \mbox{then }v_{i}=\Lambda_{n}(v_{j}).$ (52)

Next, we use the following fact:

$w_{1}w_{2}\cdots w_{k}|0\rangle=(w_{1}+x_{1})\cdots(w_{k}+x_{k})|0\rangle,\
\mbox{for }x_{j}\in{\rm Ann}\,w_{j+1}w_{j+2}\cdots w_{k}|0\rangle.$

Hence we may replace each $v_{j}$ in (51) by

$v_{j}=\phi_{-\lambda_{j}}+\sum_{i=1-\lambda_{j}}^{\lambda_{j-1}}a_{ij}\phi_{i}+\sum_{i\geq\lambda_{j-1}+1}b_{ij}\phi_{i},$

for arbitrary vectors of the form
$\sum_{i\geq.\lambda_{j-1}+1}b_{ij}\phi_{i}$. This means that $\lambda$ should
still satisfy the $-n$ shift condition (50), but that we can drop the the
first condition of (52), viz. that the vectors $v_{j}$, $j=1,,\ldots,k$, span
an isotropic subspace. Hence, we may assume that the vectors $v_{j}$ are given
as in (51), but that the condition (52) gets replaced only by

$\mbox{if }\lambda_{i}=\lambda_{j}-n,\ \mbox{then }v_{i}=\Lambda_{n}(v_{j}).$
(53)

We can express the $a_{ij}$ in terms of other constants by letting

$a_{ij}=s_{i+\lambda_{j}}(c_{\overline{\lambda}_{j}}),\ \mbox{ where the
$s_{i}$ are elementary Schur polynomials}.$

Thus

$v_{j}=\phi_{-\lambda_{j}}+\sum_{i>-\lambda_{j}}s_{i+\lambda_{j}}(c_{\overline{\lambda}_{j}})\phi_{i},$
(54)

and
$c_{\overline{\lambda}_{j}}=(c_{1,\overline{\lambda}_{j}},c_{2,\overline{\lambda}_{j}},c_{3,\overline{\lambda}_{j}},\ldots)$.
Here $\overline{\lambda}_{j}={\lambda}_{j}\mod n$, which means that there are
at most $\left[\frac{n}{2}-1\right]$ of such infinite series of constants (see
Remark 7) and the $v_{j}$ satisfy the condition (53). We can now use the
isomorphism $\sigma$ to calculate the bosonization of elements $v_{\lambda}$.
For this we use formula (23) and apply this to $v_{\lambda}$ (which is given
by (51) with $v_{j}$ given by (54)). Now using formula (17) of [9], i.e.

$e^{\sum_{j=1}^{\infty}t_{2j-1}\alpha_{2j-1}}\phi(z)e^{-\sum_{j=1}^{\infty}t_{2j-1}\alpha_{2j-1}}=e^{\sum_{j=1}^{\infty}t_{2j-1}z^{2j-1}}\phi(z),$

we find that

$\displaystyle v_{j}(\tilde{t}):=$ $\displaystyle
e^{\sum_{j=1}^{\infty}t_{2j-1}\alpha_{2j-1}}v_{j}e^{-\sum_{j=1}^{\infty}t_{2j-1}\alpha_{2j-1}}$
$\displaystyle=$ $\displaystyle
e^{\sum_{j=1}^{\infty}t_{2j-1}\alpha_{2j-1}}(\phi_{-\lambda_{j}}+\sum_{i\geq
1-\lambda_{j}}s_{i+\lambda_{j}}(c_{\overline{\lambda}_{j}})\phi_{i})e^{-\sum_{j=1}^{\infty}t_{2j-1}\alpha_{2j-1}}$
$\displaystyle=$ $\displaystyle{\rm
Res}\,\sum_{\ell=0}^{\infty}s_{\ell}(c_{\overline{\lambda}_{j}})z^{\ell-\lambda_{j}}\phi(z^{\sum_{j=1}^{\infty}t_{2j-1}z^{2j-1}}\frac{dz}{z}$
$\displaystyle=$ $\displaystyle{\rm
Res}\,z^{-\lambda_{j}}e^{\sum_{i=1}^{\infty}c_{i,\overline{\lambda}_{j}}z^{i}+\sum_{j=1}^{\infty}t_{2j-1}z^{2j-1}}\phi(z)\frac{dz}{z}$
$\displaystyle=$ $\displaystyle\phi_{-\lambda_{j}}+\sum_{i\geq
1-\lambda_{j}}s_{i+\lambda_{j}}(\tilde{t}+c_{\overline{\lambda}_{j}})\phi_{i}.$

Since $e^{\sum_{j=1}^{\infty}t_{2j-1}\alpha_{2j-1}}|0\rangle=0$, we find that
the corresponding tau-function is equal to the vacuum expectation value

$\tau(\tilde{t})=\langle 0|v_{1}(\tilde{t})v_{2}(\tilde{t})\cdots
v_{k}(\tilde{t})|0\rangle.$ (55)

If $k=2m$, then this is the Pfaffian of a $2m\times 2m$ skew-symmetric matrix.
If $k=2m-1$, we use the fact hat

$\langle 0|v_{1}(\tilde{t})v_{2}(\tilde{t})\cdots
v_{k}(\tilde{t})|0\rangle=2\langle 0|v_{1}(\tilde{t})v_{2}(\tilde{t})\cdots
v_{k}(\tilde{t})\phi_{0}|0\rangle$

and again we find a Pfaffian. This gives the following

###### Theorem 8

All polynomial tau-functions of the $n$-th Sawada-Kotera hierarchy are, up to
a scalar factor, equal to the Pfaffian

$\tau_{\lambda}(\tilde{t})=Pf\left(\chi_{\lambda_{i},\lambda_{j}}(\tilde{t}+c_{\overline{\lambda}_{i}},\tilde{t}+c_{\overline{\lambda}_{j}})\right)_{1\leq
i,j\leq 2m},$ (56)

where $\lambda=(\lambda_{1},\lambda_{2},\ldots,\lambda_{2m})$, $m=0,1,\ldots$,
is an extended strict partition, which satisfies the $-n$ shift condition (50)
for (49). Here, as before, $\tilde{t}=(t_{1},0,t_{3},0,\ldots)$, and
$c_{\overline{\lambda}_{i}}=(c_{1,\overline{\lambda}_{i}},c_{2,\overline{\lambda}_{i}},c_{3,\overline{\lambda}_{i}},\ldots)$
are arbitrary constants and the polynomials $\chi_{a,b}$ are given by (38).

For $n=3$, see Example 6(b), we therefore find:

###### Corollary 9

All polynomial tau-functions of the Sawada-Kotera hierarchy are, up to a non-
zero constant factor,

$\tau_{\lambda}(\tilde{t})=Pf\left(\chi_{\lambda_{i},\lambda_{j}}(\tilde{t}+c,\tilde{t}+c)\right)_{1\leq
i,j\leq 2m},$ (57)

where $\lambda$ is one of the following extended strict partitions:

  1. 1.

$(6m+1,6m-2,6m-5,\ldots,4,1)$;

  2. 2.

$(6m-2,6m-5,6m-8,\ldots,4,1,0)$;

  3. 3.

$(6m+2,6m-1,6m-4,\ldots,5,2)$;

  4. 4.

$(6m-1,6m-4,6m-7,\ldots,5,2,0)$,

where $m=0,1,\ldots$, and $c=(c_{1},c_{2},c_{3},\ldots)$ are arbitrary
constants.

###### Remark 10

If we choose $c=\tilde{c}=(c_{1},0,c3,0,c_{5},\ldots)$, then the corresponding
Sawada-Kotera tau-function $\tau_{\lambda}(\tilde{t})$ is equal, up to a non-
zero constant factor, to a Schur $Q$-function
$Q_{\lambda}(\tilde{t}+\tilde{c})$ (cf. [15]).

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