# Coupling local and nonlocal equations with Neumann boundary conditions.

Gabriel Acosta, Francisco Bersetche, Julio D. Rossi  Departamento de
Matemática, FCEyN, Universidad de Buenos Aires & IMAS CONICET, Pabellón I,
Ciudad Universitaria (1428), Buenos Aires, Argentina<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

We introduce two different ways of coupling local and nonlocal equations with
Neumann boundary conditions in such a way that the resulting model is
naturally associated with an energy functional. For these two models we prove
that there is a minimizer of the resulting energy that is unique modulo adding
a constant.

###### Key words and phrases:

Local equations, nonlocal equations, Neumann boundary conditions
2020 Mathematics Subject Classification: 35R11, 45K05, 47G20,

G.A. partially supported by ANPCyT under grant PICT 2018 - 3017 (Argentina).
F.M.B. partially supported by ANPCyT under grant PICT 2018 - 3017 (Argentina).
J.D.R. partially supported by CONICET grant PIP GI No 11220150100036CO
(Argentina), PICT 2018 - 3183 (Argentina) and UBACyT grant 20020160100155BA
(Argentina).

##  1\. Introduction

Nonlocal models can be used to describe phenomena (including problems
characterized by long-range interactions and discontinuities) that can not be
well represented by classical Partial Differential Equations, PDE. For
instance, long-range interactions effectively describe anomalous diffusion and
crack formation results in material models. The fundamental difference between
nonlocal models and classical local models is the fact that the latter only
involve differential operators (local equations), whereas the former rely on
integral operators (nonlocal equations). For general references on nonlocal
models and their applications to elasticity, population dynamics, image
processing, etc, we refer to to [6, 7, 9, 10, 11, 12, 13, 18, 21, 25, 27, 33,
34, 35, 36, 37, 38] and the book [3].

It is often the case that nonlocal effects occur only in some parts of the
domain, whereas, in the remaining parts, the system can be accurately
described by a local equation. The goal of coupling local and nonlocal models
is to combine a local equation (a PDE) with a nonlocal one (an integral
equation) acting in different parts of the domain, under the assumption that
the spacial location of local and nonlocal effects can be identified in
advance. In this context, one of the challenges of a coupling strategy is to
provide a mathematically consistent formulation.

Along this work we consider an open bounded domain
$\Omega\subset\mathbb{R}^{N}$. In our models it is assumed that $\Omega$ is
divided into two disjoint subdomains; a local region that we will denote by
$\Omega_{\ell}$ and a nonlocal region, $\Omega_{n\ell}$. Thus we have
$\Omega_{\ell},\Omega_{n\ell}\subset\Omega\subset\mathbb{R}^{N}$ with
$\Omega=(\overline{\Omega}_{\ell}\cup\overline{\Omega}_{n\ell})^{\circ}$. Our
main goal in this paper is to introduce two different ways of coupling a local
classical PDE in $\Omega_{\ell}$ with a nonlocal equation in $\Omega_{n\ell}$
in such a way that the resulting problem is naturally associated with an
energy functional that is invariant under the addition of a constant (as is
the usual case for Neumann boundary conditions in the literature). This paper
is a continuation of [1] where we tackled the Dirichlet case.

Let us first recall some well-known facts: for the classical Laplacian (a
local operator) with homogeneous Neumann boundary conditions the model problem
read as

$\begin{cases}\displaystyle\Delta u(x)=f(x),\ &x\in\Omega_{\ell},\\\\[5.0pt]
\displaystyle\frac{\partial
u}{\partial\eta}(x)=0,\qquad&x\in\partial\Omega_{\ell}.\end{cases}$ (1.1)

Here, $u$ is the unknown and $f$ is an external source. Associated to the
problem we have the natural energy

$F_{\ell}(u)=\frac{1}{2}\int_{\Omega_{\ell}}|\nabla
u(x)|^{2}\,dx-\int_{\Omega_{\ell}}f(x)u(x)\,dx$ (1.2)

that is well posed in the space $H_{\ell}=\\{u\in
H^{1}(\Omega_{\ell}):\int_{\Omega_{\ell}}u(y)\,dy=0\\}$. Notice that we need
to impose that $\int_{\Omega_{\ell}}f(x)\,dx=0$ in order to have existence of
solutions to (1.1) and in this case we get existence and uniqueness for (1.1)
modulo an additive constant: there is a unique solution to (1.1) with
$\int_{\Omega_{\ell}}u(x)\,dx=0$ (that is obtained as a minimizer of the
energy $F_{\ell}$ in $H_{\ell}$) and any other solution can be obtained by
adding a constant. For the proofs we refer to the textbook [20].

For a nonlocal counterpart of (1.1) in $\Omega_{n\ell}$, we need to introduce
a nonnegative kernel $J:{\mathbb{R}}^{N}\mapsto{\mathbb{R}}$ and then we
consider as a nonlocal analogous to (1.1) the following equation,

$\displaystyle 2\int_{\Omega_{n\ell}}J(x-y)(u(y)-u(x))\,dy=f(x),\quad
x\in\Omega_{n\ell}.$ (1.3)

Assuming that the kernel is symmetric, $J(z)=J(-z)$, we have the associated
energy functional

$F_{n\ell}(u)=\frac{1}{2}\int_{\Omega_{n\ell}}\int_{\Omega_{n\ell}}J(x-y)|u(y)-u(x)|^{2}\,dy\,dx-\int_{\Omega_{n\ell}}f(x)u(x)\,dx.$
(1.4)

When the kernel is in $L^{1}$ this functional can be considered in the space
$H_{n\ell}=\\{u\in L^{2}(\Omega_{n\ell}):\int_{\Omega_{n\ell}}u(x)\,dx=0\\}$.
Like for the local case, we need to assume that
$\int_{\Omega_{n\ell}}f(x)\,dx=0$ and again we have existence and uniqueness
for solutions to (1.3) modulo a constant (there is a unique solution to (1.1)
with $\int_{\Omega_{n\ell}}u(x)\,dx=0$ and it is obtained as a minimizer of
$F_{n\ell}$ in $H_{n\ell}$), see [3].

Our main goal here is to present two different ways of coupling local (in
$\Omega_{\ell}$) and nonlocal models (in $\Omega_{n\ell}$ in in such a way
that the resulting problem (in the whole $\Omega$) has the same properties as
the previous two Neumann problems; the problem is naturally associated with an
energy functional and it has a unique solution (up to an additive constant)
when the external source is such that $\int_{\Omega}f(x)\,dx=0$.

###  1.1. First model. Volumetric couplings.

Let us present our first model coupling local and nonlocal equations in two
disjoint subdomains
$\Omega_{\ell},\Omega_{n\ell}\subset\Omega\subset\mathbb{R}^{N}$. For
$u:\Omega\mapsto\mathbb{R}$, we consider the local/nonlocal energy

$\begin{array}[]{l}\displaystyle E_{Neu}(u):=\int_{\Omega_{\ell}}\frac{|\nabla
u(x)|^{2}}{2}\,dx+\frac{1}{2}\int_{\Omega_{n\ell}}\int_{\Omega_{n\ell}}J(x-y)|u(y)-u(x)|^{2}\,dy\,dx\\\\[10.0pt]
\displaystyle\qquad\qquad\qquad+\frac{1}{2}\int_{\Omega_{n\ell}}\int_{\Omega_{\ell}}G(x,y)|u(y)-u(x)|^{2}\,dy\,dx-\int_{\Omega}f(x)u(x)\,dx,\end{array}$
(1.5)

and we look for critical points (minimizers) of this energy and the
corresponding equations that they satisfy.

In the functional (1.5) we can identify the local part of the energy in
$\Omega_{\ell}$,

$\int_{\Omega_{\ell}}\frac{|\nabla u(x)|^{2}}{2}\,dx,$ (1.6)

the nonlocal part, acting in $\Omega_{n\ell}$,

$\frac{1}{2}\int_{\Omega_{n\ell}}\int_{\Omega_{n\ell}}J(x-y)|u(y)-u(x)|^{2}\,dy\,dx,$
(1.7)

a coupling term, that involves integrals in $\Omega_{\ell}$ and in
$\Omega_{n\ell}$ and a different kernel $G(x,y)$,

$\frac{1}{2}\int_{\Omega_{n\ell}}\int_{\Omega_{\ell}}G(x,y)|u(y)-u(x)|^{2}\,dy\,dx$
(1.8)

(here $G:\Omega_{\ell}\times\Omega_{n\ell}\mapsto\mathbb{R}$ is assumed to be
nonnegative but not necessarily symmetric; notice that the two variables
belong to different sets) and, finally, the term that involves the external
source

$\int_{\Omega}f(x)u(x)\,dx.$ (1.9)

Now, let us state our hypothesis on the involved domains and kernels. With
$J:{\mathbb{R}}^{N}\mapsto{\mathbb{R}}$ we denote a nonnegative measurable
function such that

  * (J1)

_Visibility:_ there exist $\delta>0$ and $C>0$ such that $J(z)>C$ for all $z$
such that $\|z\|\leq 2\delta$.

  * (J2)

_Compactness:_ the convolution type operator
$T_{J}(u)(x)=\int_{\Omega_{n\ell}}J(x-y)u(y)\,dy$, defines a compact operator
in $L^{2}(\Omega_{n\ell})$.

In nonlocal models, $J$ is a kernel that encodes the effect of a general
_volumetric_ nonlocal interaction inside the nonlocal part of the domain.
Condition $(J1)$ guarantees the influence of nonlocality within an horizon of
size at least $2\delta$ while $(J2)$ is a technical requirement fulfilled, for
instance, by continuous kernels, characteristic functions, or even for $L^{2}$
kernels, (this holds since these kernels produce Hilbert-Schmidt operators of
the form $u\mapsto T(u)(x):=\int k(x,y)u(y){\rm d}y$ that are compact if $k\in
L^{2}$, see Chapter VI in [8]). We also need to introduce a connectivity
condition.

######  Definition 1.1.

We say that an open set $D\subset\mathbb{R}^{N}$ is $\delta-$connected , with
$\delta\geq 0$, if it can not be written as a disjoint union of two
(relatively) open nontrivial sets that are at distance greater or equal than
$\delta.$

Notice that if a set $D$ is $\delta$ connected then it is $\delta^{\prime}$
connected for any $\delta^{\prime}\geq\delta$. From Definition 1.1, we notice
that $0-$connectedness agrees with the classical notion of being connected (in
particular, open connected sets are $\delta-$connected). Definition 1.1 can be
written in an equivalent way: an open set $D$ is $\delta-$connected if given
two points $x,y\in D$, there exists a finite number of points
$x_{0},x_{1},...,x_{n}\in D$ such that $x_{0}=x$, $x_{n}=y$ and
$dist(x_{i},x_{i+1})<\delta$.

Informally, $\delta$-connectedness combined with $(J1)$ says that the effect
of nonlocality can travel beyond the horizon $2\delta$ through the whole
domain.

Now we can write the following assumptions on the local/nonlocal domains:

  1. (1)

$\Omega_{\ell}$ is connected and smooth ($\Omega_{\ell}$ has Lipschitz
boundary),

  2. (2)

$\Omega_{n\ell}$ is $\delta-$connected.

Concerning the kernel involved in the coupling term, $G$, that encodes the
interactions of $\Omega_{n\ell}$ with $\Omega_{\ell}$ we assume that it is
given by nonnegative and measurable function
$G:\Omega_{\ell}\times\Omega_{n\ell}\mapsto\mathbb{R},$ such that

  * $(G1)$

there exist $\delta>0$ and $C>0$ such that for any
$(x,y)\in\Omega_{\ell}\times\Omega_{n\ell}$, $G(x,y)>C$ if $\|x-y\|\leq
2\delta$.

Finally, in order to avoid trivial couplings, we impose that $\Omega_{\ell}$
and $\Omega_{n\ell}$ need to be closer than the horizon of the kernel involved
in the coupling, we assume that

  1. $(P1)$

$dist(\Omega_{\ell},\Omega_{n\ell})<\delta$.

######  Remark 1.2.

Our results are valid for more general domains. In fact, we assumed that
$\Omega_{\ell}$ is connected and that $\Omega_{n\ell}$ is $\delta-$connected
with $dist(\Omega_{\ell},\Omega_{n\ell})<\delta$, but we can also handle the
case in which $\Omega_{\ell}$ has several connected components and
$\Omega_{n\ell}$ has several $\delta-$connected components as long as they are
close between them. We prefer to state our results under conditions (1), (2),
$(G1)$ and $(P1)$ just to simplify the presentation.

Now, with all these conditions at hand we go back to our energy functional
(1.5) and look for possible critical points (minimizers). Here, as usual in
Neumann problems, we have to assume that

$\int_{\Omega}f(x)\,dx=0,$ (1.10)

and look for minimizers in the natural function space

$H_{Neu}=\left\\{u\in L^{2}(\Omega):\,u|_{\Omega_{\ell}}\in
H^{1}(\Omega_{\ell}),\int_{\Omega}u(x)\,dx=0\right\\}.$

Let us state our first theorem.

######  Theorem 1.3.

Assume that the kernels and the domains verify $(J1)$, $(J2)$, $(1)$, $(2)$,
$(G1)$ and $(P1)$. Given $f\in L^{2}(\Omega)$ with $\int_{\Omega}f=0$ there
exists a unique minimizer of $E_{Neu}(u)$ in $H_{Neu}$. The minimizer of
$E_{Neu}(u)$ in $H_{Neu}$ is a weak solution to the problem,

$\begin{cases}\displaystyle-f(x)=\Delta
u(x)+\int_{\Omega_{n\ell}}G(x,y)(u(y)-u(x))\,dy,\
&x\in\Omega_{\ell},\\\\[10.0pt] \displaystyle\frac{\partial
u}{\partial\eta}(x)=0,\qquad&x\in\partial\Omega_{\ell},\end{cases}$ (1.11)

in the local domain $\Omega_{\ell}$ together with a nonlocal equation with a
source in $\Omega_{n\ell}$,

$\displaystyle-f(x)=2\int_{\Omega_{n\ell}}\\!\\!J(x-y)(u(y)-u(x))\,dy,+\int_{\Omega_{\ell}}G(x,y)(u(y)-u(x))\,dy,\qquad
x\in\Omega_{n\ell}.$ (1.12)

Notice that in the resulting equations the coupling terms between the local
and the nonlocal regions appear as source integral terms in the corresponding
equations.

The key to obtain this result will be to prove a Poincaré-Wirtinger type
inequality: there exists $c>0$ such that

$\begin{array}[]{l}\displaystyle\int_{\Omega_{\ell}}|\nabla
u|^{2}+\frac{1}{2}\int_{\Omega_{n\ell}}\int_{\Omega_{n\ell}}J(x-y)(u(y)-u(x))^{2}\,dy\,dx+\frac{1}{2}\int_{\Omega_{n\ell}}\int_{\Omega_{\ell}}G(x,y)|u(y)-u(x)|^{2}\,dy\,dx\\\\[10.0pt]
\qquad\qquad\qquad\displaystyle\geq\displaystyle
c\int_{\Omega}u^{2}(x)\,dx\end{array}$

for every function $u\in H_{Neu}$. Here the constant $c>0$ can be estimated in
terms of the parameters of problem (the geometry of the domains and the
kernels).

###  1.2. Second model. Mixed couplings.

Now, let us present our second model. In the first model we have a nonlocal
volumetric coupling between $\Omega_{\ell}$ and $\Omega_{n\ell}$. In the
second model we introduce _mixed_ couplings. In mixed couplings volumetric and
lower dimensional parts can interact with each other. The mixed couplings that
we introduce involve interactions of $\Omega_{n\ell}$ with a fixed smooth
hypersurface

$\Gamma\subset\overline{\Omega}_{\ell}.$

For $u:\Omega\mapsto\mathbb{R}$ we consider the energy

$\begin{array}[]{l}F_{Neu}(u)\displaystyle:=\frac{1}{2}\int_{\Omega_{l}}|\nabla
u(x)|^{2}dx+\frac{1}{2}\int_{\Omega_{nl}}\int_{\Omega_{nl}}J(x-y)\left(u(y)-u(x)\right)^{2}dy\,dx\\\\[10.0pt]
\displaystyle\quad\qquad\qquad\qquad\qquad+\frac{1}{2}\int_{\Omega_{nl}}\int_{\Gamma}G(x,z)\left(u(x)-u(z)\right)^{2}d\sigma(z)dx-\int_{\Omega}f(x)u(x)\,dx.\end{array}$
(1.13)

Here, we can identify again the local part of the energy acting in
$\Omega_{\ell}$, (1.6), the nonlocal part acting in $\Omega_{n\ell}$, (1.7),
and the external source (1.9); but now the coupling term is different (now it
involves integrals in $\Omega_{n\ell}$ and on the surface
$\Gamma\subset\overline{\Omega_{\ell}}$),

$\frac{1}{2}\int_{\Omega_{nl}}\int_{\Gamma}G(x,z)\left(u(x)-u(z)\right)^{2}\,d\sigma(z)\,dx.$

In this context, we assume the same conditions on $J$ as for the first model.
Concerning the coupling, a nonnegative and measurable function
$G:\Gamma\times\Omega_{n\ell}\mapsto\mathbb{R}$ plays the role of the
associated kernel. The following condition is analogous to the volumetric
counterpart

  * $(G2)$

there exist $\delta>0$ and $C>0$ such that for any
$(x,y)\in\Gamma\times\Omega_{n\ell}$, $G(x,y)>C$ if $\|x-y\|\leq 2\delta$.

Again, to avoid trivial couplings we assume that

  1. $(P2)$

$dist(\Gamma,\Omega_{n\ell})<\delta$.

Here again we have to assume that $\int_{\Omega}f(x)\,dx=0,$ and then we look
for minimizers in $H_{Neu}=\left\\{u\in L^{2}(\Omega):\,u|_{\Omega_{\ell}}\in
H^{1}(\Omega_{\ell}),\,\int_{\Omega}u(x)\,dx=0\right\\}.$ As before, we can
obtain existence and uniqueness of a minimizer of $F_{Neu}(u)$ in $H_{Neu}$.

######  Theorem 1.4.

Assume that the kernels and the domains verify $(J1)$, $(J2)$, $(1)$, $(2)$,
$(G2)$ and $(P2)$. Given $f\in L^{2}(\Omega)$ with $\int_{\Omega}f=0$ there
exists a unique minimizer of $F_{Neu}$ in $H_{Neu}$. The minimizer of
$F_{Neu}$ is a weak solution to

$\left\\{\begin{array}[]{ll}\displaystyle-f(x)=\Delta
u(x),&x\in\Omega_{\ell},\\\\[10.0pt] \displaystyle\frac{\partial
u}{\partial\eta}(x)=0,&x\in\partial\Omega_{\ell}\setminus\Gamma,\\\\[10.0pt]
\displaystyle\frac{\partial
u}{\partial\eta}(x)=\int_{\Omega_{nl}}G(y,x)(u(y)-u(x))dy,&x\in\Gamma,\end{array}\right.$
(1.14)

and

$-f(x)=2\int_{\Omega_{nl}}J(x-y)\left(u(y)-u(x)\right)dy-\int_{\Gamma}G(x,z)(u(x)-u(z))d\sigma(z),\
x\in\Omega_{nl}.$ (1.15)

Notice that now the coupling term between the local and the nonlocal regions
in the local part of the problem appears as a flux condition on $\Gamma$.

###  1.3. Singular kernels.

We can also deal with singular kernels related to the fractional Laplacian and
consider energies of the form

$\begin{array}[]{l}\displaystyle E_{Neu}(u):=\int_{\Omega_{\ell}}\frac{|\nabla
u(x)|^{2}}{2}\,dx+\frac{1}{2}\int_{\Omega_{n\ell}}\int_{\Omega_{n\ell}}\frac{C}{|x-y|^{N+2s}}|u(y)-u(x)|^{2}\,dy\,dx\\\\[10.0pt]
\displaystyle\qquad\qquad\qquad+\frac{1}{2}\int_{\Omega_{n\ell}}\int_{\Omega_{\ell}}G(x,y)|u(y)-u(x)|^{2}\,dy\,dx-\int_{\Omega}f(x)u(x)\,dx.\end{array}$
(1.16)

Here we look for minimizers in the space

$H_{Neu}=\left\\{u:u|_{\Omega_{\ell}}\in H^{1}(\Omega_{\ell}),\
u|_{\Omega_{n\ell}}\in H^{s}(\Omega_{n\ell}),\
\int_{\Omega}u(x)\,dx=0\right\\}.$

This case is much simpler since we have the compact embeddings
$H^{1}(\Omega_{\ell})\hookrightarrow L^{2}(\Omega_{\ell})$ and
$H^{s}(\Omega_{n\ell})\hookrightarrow L^{2}(\Omega_{n\ell})$. Also for these
energies we can show the following result (that is the fractional counterpart
to our previous results).

######  Theorem 1.5.

Given $f\in L^{2}(\Omega)$ with $\int_{\Omega}f=0$ there exists a unique
minimizer of $E_{Neu}(u)$ in $H_{Neu}$. The minimizer of $E_{Neu}(u)$ in
$H_{Neu}$ is a weak solution to the equation

$\left\\{\begin{array}[]{ll}\displaystyle-f(x)=\Delta
u(x)+\int_{\Omega_{n\ell}}G(x,y)(u(y)-u(x))\,dy,&x\in\Omega_{\ell},\\\\[10.0pt]
\displaystyle\frac{\partial
u}{\partial\eta}(x)=0,&x\in\partial\Omega_{\ell}\end{array}\right.$ (1.17)

and a nonlocal equation with a source in $\Omega_{n\ell}$,

$\displaystyle-f(x)=2\int_{\Omega_{n\ell}}\frac{C}{|x-y|^{N+2s}}(u(y)-u(x))\,dy+\int_{\Omega_{\ell}}G(x,y)(u(y)-u(x))\,dy,\quad
x\in\Omega_{n\ell}.$ (1.18)

We can also deal with mixed couplings and look for minimizers of

$\begin{array}[]{l}F_{Neu}(u)\displaystyle:=\frac{1}{2}\int_{\Omega_{l}}|\nabla
u(x)|^{2}dx+\frac{1}{2}\int_{\Omega_{nl}}\int_{\Omega_{nl}}\frac{C}{|x-y|^{N+2s}}\left(u(y)-u(x)\right)^{2}dydx\\\\[10.0pt]
\displaystyle\quad\qquad\qquad\qquad\qquad+\frac{1}{2}\int_{\Omega_{nl}}\int_{\Gamma}G(x,z)\left(u(x)-u(z)\right)^{2}d\sigma(z)dx-\int_{\Omega}f(x)u(x)\,dx.\end{array}$
(1.19)

######  Theorem 1.6.

Given $f\in L^{2}(\Omega)$ with $\int_{\Omega}f=0$ there exists a unique
minimizer of $F_{Neu}(u)$ in $H_{Neu}$. The minimizer of $F_{Neu}$ is a weak
solution to

$\left\\{\begin{array}[]{ll}\displaystyle-f(x)=\Delta
u(x),&x\in\Omega_{\ell},\\\\[10.0pt] \displaystyle\frac{\partial
u}{\partial\eta}(x)=0,&x\in\partial\Omega_{\ell}\setminus\Gamma,\\\\[10.0pt]
\displaystyle\frac{\partial
u}{\partial\eta}(x)=\int_{\Omega_{nl}}G(y,x)(u(y)-u(x))dy,&x\in\Gamma,\end{array}\right.$
(1.20)

and

$-f(x)=2\int_{\Omega_{n\ell}}\frac{C}{|x-y|^{N+2s}}(u(y)-u(x))\,dy-\int_{\Gamma}G(x,z)(u(x)-u(z))d\sigma(z),\qquad
x\in\Omega_{nl}.$ (1.21)

######  Remark 1.7.

The kernel $G$ can also be a singular kernel

$G(x,z)=\frac{c}{|x-z|^{N+2t}}.$

In this case one has to add the condition

$\int_{\Omega_{n\ell}}\int_{\Omega_{\ell}}G(x,y)|u(y)-u(x)|^{2}\,dy\,dx<+\infty$

in the space $H_{Neu}$. The proof that there is a minimizer is similar and
hence we leave the details to the reader.

######  Remark 1.8.

When $s>1/2$ since we have a trace theorem for $H^{s}(\Omega_{n\ell})$ we can
also deal with surface to surface couplings and study

$\begin{array}[]{l}F_{Neu}(u)\displaystyle:=\frac{1}{2}\int_{\Omega_{l}}|\nabla
u|^{2}dx+\frac{1}{2}\int_{\Omega_{nl}}\int_{\Omega_{nl}}\frac{C}{|x-y|^{N+2s}}\left(u(y)-u(x)\right)^{2}dydx\\\\[10.0pt]
\displaystyle\quad\qquad\qquad\qquad\qquad+\frac{1}{2}\int_{\Gamma_{\ell}}\int_{\Gamma_{n\ell}}G(x,z)\left(u(x)-u(z)\right)^{2}d\sigma(z)d\sigma(x)-\int_{\Omega}f(x)u(x)\,dx.\end{array}$
(1.22)

Here $\Gamma_{\ell}$ is a smooth hypersurface in $\overline{\Omega_{\ell}}$
and $\Gamma_{n\ell}$ is a smooth hypersurface in $\overline{\Omega_{n\ell}}$.
Notice that now we are coupling the domains via interactions on hypersurfaces.
The hypothesis on the kernel $G$ that is needed to obtain a coupling between
the two domains is, as before, some strict positivity on pairs of points
belonging to the coupling surfaces.

Let us end the introduction with a brief description of previous references.
From a mathematical point of view, interesting problems arise from coupling
local and nonlocal models, see [4, 5, 14, 15, 19, 22, 23, 26, 29] and
references therein. As previous examples of coupling approaches between local
and nonlocal regions we refer the reader to [2, 4, 5, 14, 15, 16, 19, 22, 23,
24, 26, 29, 30, 31, 32] the survey [17] and references therein. Previous
strategies treat the coupling condition as an optimization problem (the goal
is to minimize the mismatch of the local and nonlocal solutions in a common
overlapping region). Another possible strategy for coupling relies on the
partitioned procedure as a general coupling strategy for heterogeneous
systems, the system is divided into sub-problems in their respective sub-
domains, which communicate with each other via transmission conditions.
Moreover, couplings between sets of different dimension are possible. In [7]
the effects of network transportation on enhancing biological invasion is
studied. The proposed mathematical model consists of one equation with
nonlocal diffusion in a one-dimensional domain coupled via the boundary
condition with a standard reaction-diffusion in a two-dimensional domain. In
[14], local and nonlocal problems were coupled through a prescribed region in
which both kinds of equations overlap (the value of the solution in the
nonlocal part of the domain is used as a Dirichlet boundary condition for the
local part and vice-versa). This kind of coupling gives continuity of the
solution in the overlapping region but does not preserve the total mass when
Neumann boundary conditions are imposed. In [14] and [19], numerical schemes
using local and nonlocal equations were developed and used to improve the
computational accuracy when approximating a purely nonlocal problem. In [23]
and [29] (see also [22, 26]), evolution problems related to energies closely
related to ours are studied (here we deal with stationary problems).

The paper is organized as follows: In Section 2 we deal with our first model
involving volumetric coupling and prove Theorem 1.3 while in Section 3 we
prove the results concerning the second model, Theorem 1.4. Finally, in
Section 4 we deal with singular kernels.

##  2\. Volumetric coupling

First model. Coupling local/nonlocal problems via source terms. Our aim is to
look for a minimizer of the energy

$\begin{array}[]{l}\displaystyle E_{Neu}(u):=\int_{\Omega_{\ell}}\frac{|\nabla
u|^{2}}{2}+\frac{1}{2}\int_{\Omega_{n\ell}}\int_{\Omega_{n\ell}}J(x-y)|u(y)-u(x)|^{2}\,dy\,dx\\\\[10.0pt]
\displaystyle\qquad\qquad\qquad+\frac{1}{2}\int_{\Omega_{n\ell}}\int_{\Omega_{\ell}}G(x,y)|u(y)-u(x)|^{2}\,dy\,dx-\int_{\Omega}f(x)u(x)\,dx\end{array}$

in the space

$H_{Neu}=\left\\{u\in L^{2}(\Omega),u|_{\Omega_{\ell}}\in
H^{1}(\Omega_{\ell}),\int_{\Omega}u=0\right\\},$

assuming

$\int_{\Omega}f(x)\,dx=0.$

Let us first prove an auxiliary lemma.

######  Lemma 2.1.

Let $D$ be an open $\delta$ connected set, and $u:D\to{\mathbb{R}}$. If

$\int_{D}\int_{D}J(x-y)|u(x)-u(y)|^{2}\,dx\,dy=0,$

then $u(x)=k$ a.e. $x\in D$.

###### Proof.

Pick $x_{0}\in D$ and a ball $B_{0}=B_{\delta}(x_{0})$, we have

$C\int_{D\cap B_{0}}\int_{D\cap B_{0}}(u(x)-u(y))^{2}\leq\int_{D\cap
B_{0}}\int_{D\cap B_{0}}J(x-y)(u(x)-u(y))^{2}=0,$

(since $J(x-y)>C$ for $x,y\in B_{0}$) and hence $u(x)=k_{0}$ a.e. $x\in D\cap
B_{0}$. In order to see that this property holds a.e. $x\in D$, let us
introduce the set $\mathcal{M}=\\{A\subset
D,A\,\,\mbox{open}:u(x)=k_{0}\,\mbox{a.e.}\,x\in A\\}$ with the partial order
given by $\subset$. Since $\mathcal{M}\neq\emptyset$ there exists a maximal
open set $M\in\mathcal{M}$. If $M\subsetneq\Omega$ then we consider the set
$\emptyset\neq D\setminus M$.

If $D\setminus M$ is open we necessarily have that $dist(M,D\setminus
M)<\delta$ (here we are using that $D$ is $\delta$ connected). If $D\setminus
M$ is not open, then $dist(M,D\setminus M)=0$ (since $D$ is open). Either
case, there exists a ball $B_{1}$ of radius $\delta$ such that $B_{1}\cap
D\setminus M\neq\emptyset$ and $B_{1}\cap M$ has positive measure (since both,
$B_{1}$ and $M$, are open sets). Arguing as before we see that $u(x)=k_{0}$
a.e. $x\in B_{1}\cap D$, a contradiction (since $M$ is maximal with that
property and we would have $M\subsetneq M\cup(B_{1}\cap D)=M$). We see that
$M=D$ and the proof is complete. ∎

######  Lemma 2.2.

Let $u_{n}:\Omega\to{\mathbb{R}}$ be a sequence such that $u_{n}\to 0$
strongly in $L^{2}(\Omega_{\ell})$ and weakly in $L^{2}(\Omega_{n\ell})$, if
in addition

$\lim_{n\to\infty}\int_{\Omega_{n\ell}}\int_{\Omega}J(x-y)(u_{n}(x)-u_{n}(y))^{2}dydx=0,$
(2.1)

then

$\lim_{n\to\infty}\int_{\Omega_{n\ell}}u_{n}(x)^{2}dx=0.$

###### Proof.

From (2.1), the convergence of $\\{u_{n}\\}$ and property (J2), we easily find
that

$\lim_{n\to\infty}\int_{\Omega_{n\ell}}\int_{\Omega_{\ell}}J(x-y)u_{n}(x)^{2}dydx=0.$
(2.2)

Let us define

$A^{0}_{\delta}=\Big{\\{}x\in\Omega_{n\ell}:dist(x,\Omega_{\ell})<\delta\Big{\\}}.$

Notice that thanks to property $(P1)$ and to the fact that $\Omega_{n\ell}$ is
open we see that $A^{0}_{\delta}$ is open and non empty. In particular it has
positive n-dimensional measure. For any $x\in\overline{A^{0}_{\delta}}$ we
consider the continuous and _strictly_ positive function
$g(x)=|B_{2\delta}(x)\cap\Omega_{\ell}|$. Since $\overline{A^{0}_{\delta}}$ is
a compact set, there exists a constant $m>0$ such that $g(x)\geq m$ for any
$x\in\overline{A^{0}_{\delta}}$. As a consequence

$\int_{\Omega_{n\ell}}\int_{\Omega_{\ell}}J(x-y)u_{n}(x)^{2}dydx\geq\int_{A^{0}_{\delta}}\int_{B_{2\delta}(x)\cap\Omega_{\ell}}J(x-y)u_{n}(x)^{2}dydx\geq
mC\int_{A^{0}_{\delta}}u_{n}(x)^{2}dx.$

and therefore, thanks to (2.2), $u_{n}\to 0$ in $L^{2}(A^{0}_{\delta})$. In
order to iterate this argument we notice that at this point we know that
$u_{n}\to 0$ strongly in $A^{0}_{\delta}$ and weakly in
$\Omega_{n\ell}\setminus\overline{A^{0}_{\delta}}$, hence again from (2.1) we
get

$\lim_{n\to\infty}\int_{\Omega_{n\ell}\setminus\overline{A^{0}_{\delta}}}\int_{A^{0}_{\delta}}J(x-y)u_{n}(x)^{2}dydx=0.$
(2.3)

Since $\Omega_{n\ell}$ is $\delta$ connected,
$dist(\Omega_{n\ell}\setminus\overline{A^{0}_{\delta}},A^{0}_{\delta})<\delta$.
Considering now

$A^{1}_{\delta}=\\{x\in\Omega_{n\ell}\setminus\overline{A^{0}_{\delta}}:dist(x,A^{0}_{\delta})<\delta\\},$

and proceeding as before, we obtain, from (2.3), that $u_{n}\to 0$ strongly in
$A^{1}_{\delta}$. This argument can be repeated and giving strong converge in
$L^{2}(A^{j}_{\delta})$ for

$A^{j}_{\delta}=\left\\{x\in\Omega_{n\ell}\setminus\overline{\bigcup_{0\leq
i<j}A^{i}_{\delta}}:dist\Big{(}x,\bigcup_{0\leq
i<j}A^{i}_{\delta}\Big{)}<\delta\right\\}.$

Since $\Omega_{n\ell}$ is bounded, we have, for a finite number
$J\in{\mathbb{N}}$,

$\Omega_{n\ell}=\bigcup_{0\leq i<J}A^{i}_{\delta}$

and therefore the proof is complete. ∎

As we have mentioned in the Introduction our goal is to show that given $f\in
L^{2}(\Omega)$ with $\int_{\Omega}f=0$ there exists a unique minimizer of the
energy functional $E_{Neu}(u)$ in the space $H_{Neu}$.

To use the direct method of calculus of variations to obtain the result we
have to show that $E_{Neu}(u)$ is coercive and weakly lower semicontinuous.

To this end we prove a key result, a Poincaré-Wirtinger type inequality holds.

######  Lemma 2.3.

There exists $c>0$ such that

$\begin{array}[]{l}\displaystyle\int_{\Omega_{\ell}}|\nabla
u|^{2}+\frac{1}{2}\int_{\Omega_{n\ell}}\int_{\Omega_{n\ell}}J(x-y)(u(y)-u(x))^{2}\,dy\,dx+\frac{1}{2}\int_{\Omega_{n\ell}}\int_{\Omega_{\ell}}G(x,y)|u(y)-u(x)|^{2}\,dy\,dx\\\\[10.0pt]
\qquad\qquad\qquad\displaystyle\geq\displaystyle
c\int_{\Omega}u^{2}(x)\,dx\end{array}$

for every function $u\in H_{Neu}$.

###### Proof.

First, let us point out that when $G(x,y)=J(x-y)$,
$\Omega_{\ell}\subset\Omega$ is a smooth convex subdomain we can use a result
from [23]. In this case the norm in $H_{Neu}$ bounds a pure nonlocal seminorm.
There exists $c>0$ such that,

$\displaystyle\int_{\Omega_{\ell}}|\nabla
u|^{2}+\frac{1}{2}\int_{\Omega_{n\ell}}\int_{\Omega}J(x-y)(u(y)-u(x))^{2}\,dy\,dx\geq\displaystyle
c\int_{\Omega}\int_{\Omega}J(x-y)(u(y)-u(x))^{2}\,dy\,dx,$ (2.4)

for every $u\in\mathcal{H}:=\\{u\in L^{2}(\Omega):u|_{\Omega_{\ell}}\in
H^{1}(\Omega_{\ell})\\}$. Then the desired Poincaré-Wirtinger inequality
follows from an analogous inequality for the purely nonlocal energy: there
exists $c>0$ such that

$\displaystyle\int_{\Omega}\int_{\Omega}J(x-y)(u(y)-u(x))^{2}\,dy\,dx\geq\displaystyle
c\int_{\Omega}u^{2}(x)\,dx$

for every function $u\in\mathcal{H}$ such that $\int_{\Omega}u=0$; see [3].

To obtain the Poincaré-Wirtinger inequality in the general case we argue by
contradiction. Assume that there is a sequence $u_{n}$ such that

$\int_{\Omega_{\ell}}|\nabla u_{n}|^{2}(x)\,dx\to 0,$ (2.5)
$\frac{1}{2}\int_{\Omega_{n\ell}}\int_{\Omega_{n\ell}}J(x-y)|u_{n}(y)-u_{n}(x)|^{2}\,dy\,dx\to
0,$ (2.6)
$\frac{1}{2}\int_{\Omega_{n\ell}}\int_{\Omega_{\ell}}G(x,y)|u_{n}(y)-u_{n}(x)|^{2}\,dy\,dx\to
0,$ (2.7) $\int_{\Omega}|u_{n}|^{2}(x)\,dx=1$ (2.8)

and

$\int_{\Omega}u_{n}(x)\,dx=0.$ (2.9)

Since the $L^{2}$ norm is bounded we can extract a subsequence such that

$u_{n}\rightharpoonup u\qquad\mbox{weakly in }H^{1}(\Omega_{\ell}),$

then, from (2.5) we get that there is a constant $k_{1}$ such that

$u_{n}\to k_{1}\qquad\mbox{strongly in }H^{1}(\Omega_{\ell}).$

Now, from (2.6) (using again that the $L^{2}$ norm is bounded) we can refine
the subsequence to obtain that

$u_{n}\rightharpoonup k_{2}\qquad\mbox{weakly in }L^{2}(\Omega_{n\ell}),$

for some constant $k_{2}$.

From (2.7) we conclude that

$\frac{1}{2}\int_{\Omega_{n\ell}}\int_{\Omega_{\ell}}G(x,y)|k_{1}-k_{2}|^{2}\,dy\,dx\leq\liminf_{n\to\infty}\frac{1}{2}\int_{\Omega_{n\ell}}\int_{\Omega_{\ell}}G(x,y)|u_{n}(y)-u_{n}(x)|^{2}\,dy\,dx=0,$

and hence

$k_{1}=k_{2}.$

We have

$|\Omega_{\ell}|k_{1}=\lim_{n}\int_{\Omega_{\ell}}u_{n}(x)\,dx$

and

$|\Omega_{n\ell}|k_{2}=\lim_{n}\int_{\Omega_{n\ell}}u_{n}(x)\,dx.$

Then, adding the previous identities and using that $k_{1}=k_{2}$, from (2.9),
we obtain

$k_{1}=k_{2}=0.$

From the strong convergence in $H^{1}(\Omega_{\ell})$ we obtain

$\lim_{n}\int_{\Omega_{\ell}}|u_{n}|^{2}(x)\,dx=0$

and hence from (2.8) we get

$\int_{\Omega_{n\ell}}|u_{n}|^{2}(x)\,dx\to 1.$

This contradicts the result in [3] since there exists $c>0$ such that

$\displaystyle
0=\lim_{n}\int_{\Omega_{n\ell}}\int_{\Omega_{n\ell}}J(x-y)(u_{n}(y)-u_{n}(x))^{2}\,dy\,dx\geq\displaystyle
c\lim_{n}\int_{\Omega_{n\ell}}\left|u_{n}(x)-\,\ThisStyle{\ensurestackMath{\stackinset{c}{.2\LMpt}{c}{.5\LMpt}{\SavedStyle-}{\SavedStyle\phantom{\int}}}\kern-16.56937pt}\int_{\Omega_{n\ell}}u_{n}\right|^{2}\,dx=1.$

This contradiction finishes the proof. ∎

The proof of Lemma 2.3 is made by contradiction and therefore it does not
provide an estimate for the constant. In what follows we present an
alternative proof that provides an explicit constant in terms of the relevant
quantities, the geometries of the involved domains $\Omega_{\ell}$ and
$\Omega_{n\ell}$, and the kernels $J$ and $G$ (in fact the constant depends on
the parameter $\delta$ and $\min J$, $\min G$ for points $(x,y)$ such that
$|x-y|<2\delta$). In order to do so, we introduce the following remark.

######  Remark 2.4 (Geometric structure of bounded $\delta-connected$ sets).

Let $D$ a bounded $\delta-connected$ set. Boundedness implies that it is
possible to find a _finite_ collection $\mathcal{C}$ of open sets, each of
them of diameter less than or equal to $\delta/2$, such that

$D=\bigcup_{B\in\mathcal{C}}B.$

We introduce a tree structure in $\mathcal{C}$ in the following way: for a
fixed $B\in\mathcal{C}$, that will be called the root of the tree, we set
$B_{0}=B$. Now we pick $B_{1}\neq B_{0}$, $B_{1}\in\mathcal{C}$ such that
$dist(B_{0},B_{1})<\delta$ (if more than one set has this property we choose
any of them). Now we proceed in the same way, picking
$B_{2}\in\mathcal{C}\setminus\\{B_{0},B_{1}\\}$ such that
$dist(B_{1},B_{2})<\delta$ and repeat the procedure until we reach a set
$B_{k}$ such that either
$\emptyset=\mathcal{C}\setminus\\{B_{0},B_{1},\cdots,B_{k}\\}$ or
$\emptyset\neq\mathcal{C}\setminus\\{B_{0},B_{1},\cdots,B_{k}\\}$ and all the
elements in $\mathcal{C}\setminus\\{B_{0},B_{1},\cdots,B_{k}\\}$ are at
distance greater or equal than $\delta$ from $B_{k}$. In the first case we are
done and we introduce a (total) order in $\mathcal{C}$ given by $B_{i}<B_{j}$
if $B_{j}$ was chosen later than $B_{i}$. In the second case, we call the
obtained totally ordered set $\mathcal{C}^{0}_{1}$ a branch with root $B_{0}$.
Then we continue with further branches. First, repeating our procedure, we
check if with the remaining elements
$\mathcal{C}\setminus\mathcal{C}^{0}_{1}$, it is possible to get another
branch $\mathcal{C}^{0}_{2}$ with the same root $B_{0}$. When all branches
with root $B_{0}$ are exhausted, we look within the set of remaining elements
for potential branches with root $B_{1}$, $B_{2}$, etc. until we reach
$B_{k-1}$. At this point, and iteratively, all elements of all generated
branches are tested as roots of new branches. Since $\mathcal{C}$ is finite,
this procedure reaches an end in a finite number of steps yielding a partially
orderer set $\mathcal{T}$. Naturally $\mathcal{T}\subset\mathcal{C}$, we claim
that $\mathcal{T}=\mathcal{C}$. Otherwise, we consider the non empty open sets
$D_{1}=\cup_{B\in\mathcal{C}\setminus\mathcal{T}}B$ ,
$D_{2}=\cup_{B\in\mathcal{T}}B$ and then $D=D_{1}\cup D_{2}$ with
$dist(D_{1},D_{2})>\delta$ a contradiction since $D$ is $\delta-connected$.

In $\mathcal{T}$ we define the natural _partial_ order and $\mathcal{T}$ is
called a $\delta-tree$. Notice that for two _consecutive_ elements
$B<\tilde{B}$ we have $diam(B,\tilde{B})<2\delta$. In particular, from our
hypothesis on the kernel $J$, this implies that $J(x-y)>0$ for $x\in B$,
$y\in\tilde{B}$.

For further use we introduce the following notations: we let the degree of a
root as the number of branches emerging from that root while the degree of
$\mathcal{T}$ is the maximum degree of all the roots belonging to
$\mathcal{T}$. Finally, the length of a branch is the cardinal of that branch
excluding its root.

######  Remark 2.5.

In simple cases it is possible to easily characterize potential tree
structures on domains. In a square $D\subset{\mathbb{R}}^{2}$ (or a cube in
${\mathbb{R}}^{N}$) we can build $\mathcal{T}$ with a single branch. The
cardinal of $\mathcal{T}$ is bounded by $\frac{8^{2}|D|}{\delta^{2}}$ (or by
$\frac{8^{N}|D|}{\delta^{N}}$ in ${\mathbb{R}}^{N}$). Moreover, even rather
complicated structures can be covered by simple trees if the scale related to
$\delta$ is large enough as it is shown in Figures 1 and 2.

Figure 1. A tree with a single branch in a square $\Omega$. In this example a
branch is built with a collection of sets $\mathcal{C}=\\{B_{i}\\}$ given by
the intersection of $\Omega$ with balls of diameter $\delta/2$. A possible
ordering is given by the path along the red arrows.  Figure 2. A fractal like
domain with a simple tree structure with root $B_{0}$. Only a single branch
$\mathcal{C}_{0}^{1}$ is shown in the figure together with a few neighboring
elements of $\mathcal{C}_{1}^{0}$. Clearly, details smaller than the ”scale”
size $\delta$ are ignored by the tree structure.

With this structure at hand, let us prove the following lemma.

######  Lemma 2.6.

Let $B^{*}=\\{B_{0},B_{1},\cdots,B_{N}\\}$, a finite collection of open sets
$B_{i}\subset{\mathbb{R}}^{n}$, such that $diam(B_{i}\cup B_{i+1})<2\delta$.
Then, for any $u:\cup_{j}B_{j}\to{\mathbb{R}}$, it holds

$\begin{array}[]{l}\displaystyle\sum_{k=0}^{N}\int_{B_{k}}|u(x)|^{2}\,dx\leq\left(\sum_{k=0}^{N}2^{k}\frac{|B_{k}|}{|B_{0}|}\right)\int_{B_{0}}u^{2}\,dx\\\\[10.0pt]
\qquad\qquad\qquad\qquad\quad\displaystyle+\\!\frac{1}{m_{J}}\sum_{i=1}^{N}2^{i}\left(\sum_{k=i}^{N}\frac{|B_{k}|}{|B_{k-i}||B_{k-i+1}|}\int_{B_{k-i}}\int_{B_{k-i+1}}\\!\\!\\!\\!\\!\\!J(x-y)(u(x)-u(y))^{2}\,dx\,dy\right),\end{array}$

where

$0<m_{J}=\inf_{|z|<2\delta}J(z).$

###### Proof.

Since

$\int_{B_{1}}|u(x)|^{2}\,dx=\frac{1}{|B_{0}|}\int_{B_{0}}\int_{B_{1}}|u(x)|^{2}\,dx\,dy,$

we have

$\int_{B_{1}}|u(x)|^{2}\,dx\leq\frac{2}{|B_{0}|}\int_{B_{0}}\int_{B_{1}}|u(x)-u(y)|^{2}\,dx\,dy+2\frac{|B_{1}|}{|B_{0}|}\int_{B_{0}}|u(y)|^{2}\,dx\,dy,$

and similarly

$\int_{B_{k}}|u(x)|^{2}\,dx\leq\frac{2}{|B_{k-1}|}\int_{B_{k-1}}\int_{B_{k}}|u(x)-u(y)|^{2}\,dx\,dy+2\frac{|B_{k}|}{|B_{k-1}|}\int_{B_{k-1}}|u(y)|^{2}\,dx\,dy.$

As a consequence, calling $\Delta=u(x)-u(y)$, we get

$\begin{array}[]{l}\displaystyle\int_{B_{k}}|u(x)|^{2}\,dx\leq\sum_{i=1}^{k}\frac{2^{i}|B_{k}|}{|B_{k-i}||B_{k-i+1}|}\int_{B_{k-i}}\int_{B_{k-i+1}}\Delta^{2}\,dx\,dy+2^{k}\frac{|B_{k}|}{|B_{0}|}\int_{B_{0}}u^{2}\,dx,\end{array}$

and therefore, we obtain,

$\begin{array}[]{ll}\displaystyle\sum_{k=0}^{N}&\displaystyle\int_{B_{k}}|u(x)|^{2}\,dx\\\\[10.0pt]
&\displaystyle\leq\left(\sum_{k=0}^{N}2^{k}\frac{|B_{k}|}{|B_{0}|}\right)\int_{B_{0}}u^{2}\,dx+\sum_{k=1}^{N}\left(\sum_{i=1}^{k}\frac{2^{i}|B_{k}|}{|B_{k-i}||B_{k-i+1}|}\int_{B_{k-i}}\int_{B_{k-i+1}}\Delta^{2}\,dx\,dy\right)\\\\[10.0pt]
&\displaystyle=\left(\sum_{k=0}^{N}2^{k}\frac{|B_{k}|}{|B_{0}|}\right)\int_{B_{0}}u^{2}\,dx+\sum_{i=1}^{N}2^{i}\left(\sum_{k=i}^{N}\frac{|B_{k}|}{|B_{k-i}||B_{k-i+1}|}\int_{B_{k-i}}\int_{B_{k-i+1}}\Delta^{2}\,dx\,dy\right).\end{array}$

Hence, the Lemma follows by using that $1\leq J(x-y)/m_{J}$ for any $x,y$
belonging to consecutive sets $B_{i}$. ∎

The previous lemma in particular says that in a branch of a tree structure of
sets (as the ones described in Remark 2.4), the $L^{2}$ norm of a function
defined on that branch can be bounded in terms of the $L^{2}$ norm on the root
of the branch plus an energy involving the nonlocal operator $J$ along the
branch. Taking into account that in a tree $\mathcal{T}$ roots of branches are
members of previous branches, it is clear that the $L^{2}$ norm of a function
defined on $\mathcal{T}$ can be bounded in a similar fashion in terms of the
$L^{2}$ norm on the root of the tree and the nonlocal energy. Moreover, if
needed, explicit constants can be tracked back along the tree structure. For
the sake of clarity, we do not state the next result with such a generality,
although we provide below some simple examples with explicit constants.

######  Corollary 2.7.

For any $u\in L^{2}(\Omega_{n\ell})$ we consider a tree structure
$\mathcal{T}$ in $\Omega_{n\ell}$ with root $B$, then there exists a constant
$C=C(\mathcal{T},m_{J})$ such that

$\int_{\Omega_{n\ell}}u^{2}\,dx\leq
C\left(\int_{B}u^{2}\,dx+\int_{\Omega_{n\ell}}\int_{\Omega_{n\ell}}J(x-y)(u(x)-u(y))^{2}\,dx\,dy\right).$

###### Proof.

Immediate from Lemma 2.6. ∎

######  Remark 2.8 (Explicit Constants).

Notice that in some cases an explicit expression for $C(\mathcal{T},m_{J})$
can be obtained. If $\Omega_{n\ell}$ is a cube in ${\mathbb{R}}^{N}$ (see
Remark 2.5) then we can take $\mathcal{T}$ with a single branch of length
bounded by $\frac{8^{N}|\Omega_{n\ell}|}{\delta^{N}}$. In this case, using
Lemma 2.6 and taking into account that
$|B_{i}|=|B|=c_{N}\frac{\delta^{N}}{2^{N}}$ is constant (here $c_{N}$ the
measure of the unit ball in $\mathbb{R}^{N}$), together with the fact that

$\sum_{k=i}^{N}2^{k}=2^{i}\sum_{k=0}^{N-i}2^{k}=2^{i}(2^{N-i+1}-1)\leq
2^{N+1},$

we can obtain the bound

$\int_{\Omega_{n\ell}}u^{2}\,dx\leq
2^{\frac{8^{N}|\Omega_{n\ell}|}{\delta^{N}}+1}\left(\int_{B}u^{2}\,dx+\frac{2^{N}}{c_{N}\delta^{N}m_{J}}\int_{\Omega_{n\ell}}\int_{\Omega_{n\ell}}J(x-y)(u(x)-u(y))^{2}\,dx\,dy\right).$

Notice that the obtained constant deteriorates as $\delta\to 0$ or $m_{J}\to
0$.

Now, we are ready to present a proof of the Poincaré-Wirtinger inequality
where the constant can be tracked.

######  Lemma 2.9.

The optimal constant $C^{*}>0$ such that

$\begin{array}[]{l}\displaystyle\int_{\Omega_{\ell}}|\nabla
u|^{2}+\frac{1}{2}\int_{\Omega_{n\ell}}\int_{\Omega_{n\ell}}J(x-y)(u(y)-u(x))^{2}\,dy\,dx+\frac{1}{2}\int_{\Omega_{n\ell}}\int_{\Omega_{\ell}}G(x,y)|u(y)-u(x)|^{2}\,dy\,dx\\\\[10.0pt]
\qquad\qquad\qquad\displaystyle\geq\displaystyle
C^{*}\int_{\Omega}u^{2}(x)\,dx\end{array}$

for every function $u\in H_{Neu}$ (that is, for $u\in\mathcal{H}$ with
$\int_{\Omega}u=0$) can be estimated as

$C^{*}\geq c^{*}(J,G,\Omega_{\ell},\mathcal{T},\delta,A,B)$

with $A,B$ two open sets $A\subset\Omega_{\ell}$, $B\subset\Omega_{n\ell}$
such that $diam(A\cup B)<2\delta$ and $\mathcal{T}$ a $\delta-tree$ structure
of $\Omega_{n\ell}$ with root $B$.

###### Proof.

We aim to obtain a bound of the form

$\begin{array}[]{l}\displaystyle\inf_{r\in\mathbb{R}}\int_{\Omega}|u-r|^{2}(x)\,dx\leq\frac{1}{c^{*}}\left[\int_{\Omega_{\ell}}|\nabla
u|^{2}+\frac{1}{2}\int_{\Omega_{n\ell}}\int_{\Omega_{n\ell}}J(x-y)(u(y)-u(x))^{2}\,dy\,dx\right.\\\\[10.0pt]
\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\displaystyle\left.+\frac{1}{2}\int_{\Omega_{n\ell}}\int_{\Omega_{\ell}}G(x,y)|u(y)-u(x)|^{2}\,dy\,dx\right]\end{array}$

for functions $u\in L^{2}(\Omega)$ such that $u|_{\Omega_{\ell}}\in
H^{1}(\Omega_{\ell})$.

From our hypothesis on $G$ and the assumption
$\mbox{dist}(\Omega_{\ell},\Omega_{n\ell})<\delta$, there exist two sets
$A\subset\Omega_{\ell}$ and $B\subset\Omega_{n\ell}$ such that

$|x-y|\leq 2\delta,\qquad\forall x\in A,\,y\in B$

and therefore

$G(x,y)\geq m_{G}=\min_{|x-y|<2\delta}G(x,y),\qquad\forall x\in A,y\in B.$

Moreover, reducing the size of $B$ if necessary, we may assume that
$diam(B)<\delta/2$ and we define $\mathcal{T}$ a $\delta-tree$ on
$\Omega_{n\ell}$ with root $B$.

Let us take $r\in\mathbb{R}$ such that

$\int_{A}(u(x)-r)\,dx=0.$

Then, we have, for $v=u-r$

$\begin{array}[]{l}\displaystyle\frac{1}{2}\int_{\Omega_{n\ell}}\int_{\Omega_{\ell}}G(x,y)|v(y)-v(x)|^{2}\,dy\,dx\\\\[10.0pt]
\displaystyle\geq\frac{1}{2}\int_{B}\int_{A}G(x,y)|v(y)-v(x)|^{2}\,dy\,dx\\\\[10.0pt]
\displaystyle\geq m_{G}\frac{|A|}{2}\int_{B}|v(x)|^{2}\,dx\end{array}$ (2.10)

where we are using that

$\int_{B}|v(x)|^{2}\,dx=\frac{1}{|A|^{2}}\int_{B}\left|\int_{A}(v(x)-v(y))\,dy\right|^{2}\,dx\leq\frac{1}{|A|}\int_{B}\int_{A}|v(x)-v(y)|^{2}\,dy\,dx.$

Now, since

$0<\sigma(\Omega_{\ell},A)=\inf_{v\in
H^{1}(\Omega_{\ell}),\int_{A}v\,dx=0}\frac{\displaystyle\int_{\Omega_{\ell}}|\nabla
v|^{2}(x)\,dx}{\displaystyle\int_{\Omega_{\ell}}v^{2}(x)\,dx},$

for a suitable Poincaré constant $\sigma(\Omega_{\ell},A)$, we have for the
local region,

$\int_{\Omega_{\ell}}v^{2}(x)\,dx\leq\frac{1}{\sigma(\Omega_{\ell},A)}\int_{\Omega_{\ell}}|\nabla
v|^{2}(x)\,dx.$

On the other hand, Corollary 2.7 says that

$\int_{\Omega_{n\ell}}v^{2}\,dx\leq
C(\mathcal{T},m_{J})\left(\int_{B}v^{2}\,dx+\int_{\Omega_{n\ell}}\int_{\Omega_{n\ell}}J(x-y)|v(x)-v(y)|^{2}\,dx\,dy\right),$

and the lemma follows by collecting the last two inequalities together with
(2.10). ∎

######  Remark 2.10.

The constant $\sigma(\Omega_{\ell},A)$ can be difficult to characterize even
in the case $A=\Omega_{\ell}$ (for which we just write
$\sigma(\Omega_{\ell})$). For some elementary domains, such as cubes or balls
in ${\mathbb{R}}^{N}$, explicit computations of $\sigma(\Omega_{\ell})$ can be
done. Moreover, in particular circumstances and for simple geometries some
bounds are easy to find. Notably, for convex domains it is known that

$\frac{\pi^{2}}{\mbox{diam}^{2}(\Omega)}\leq\sigma(\Omega_{\ell}),$

see [28]. On the other hand, a simple argument relates
$\sigma(\Omega_{\ell},A)$ with $\sigma(\Omega_{\ell})$ as follows: assume that
$u\in H^{1}(\Omega_{\ell})$, $\frac{1}{|A|}\int_{A}u(x)\,dx=0$ and write

$\bar{u}_{\Omega_{\ell}}=\frac{1}{|\Omega_{\ell}|}\int_{\Omega_{\ell}}u(x)\,dx=\int_{\Omega_{\ell}}u(x)\left(\frac{1}{|\Omega_{\ell}|}-\frac{\chi_{A}(x)}{|A|}\right)\,dx=\int_{\Omega_{\ell}}(u(x)-\bar{u}_{\Omega_{\ell}})\left(\frac{1}{|\Omega_{\ell}|}-\frac{\chi_{A}(x)}{|A|}\right)\,dx$

then, Schwartz’s inequality yields

$\bar{u}_{\Omega_{\ell}}^{2}\leq\frac{|\Omega_{\ell}|}{|A|^{2}}\int_{\Omega_{\ell}}(u(x)-\bar{u}_{\Omega_{\ell}})^{2}\,dx\leq\frac{|\Omega_{\ell}|}{|A|^{2}\sigma(\Omega_{\ell})}\int_{\Omega_{\ell}}|\nabla
u|^{2}(x)\,dx.$

Hence, we have

$\int_{\Omega_{\ell}}u(x)^{2}\,dx\leq
2\int_{\Omega_{\ell}}(u(x)-\bar{u}_{\Omega_{\ell}})^{2}\,dx+2\int_{\Omega_{\ell}}\bar{u}_{\Omega_{\ell}}^{2}\,dx\leq
2\frac{1+\left(\frac{|\Omega_{\ell}|}{|A|}\right)^{2}}{\sigma(\Omega_{\ell})}\int_{\Omega_{\ell}}|\nabla
u|^{2}(x)\,dx,$

and, as a consequence, we conclude that

$\frac{\sigma(\Omega_{\ell})}{2\left(1+\left(\displaystyle\frac{|\Omega_{\ell}|}{|A|}\right)^{2}\right)}\leq\sigma(\Omega_{\ell},A).$

######  Remark 2.11.

For two adjacent square domains $\Omega_{n\ell}=[-1,0]\times[0,1]$,
$\Omega_{\ell}=[0,1]\times[0,1]$ of unitary side, Remarks 2.8 and 2.10 give an
explicit estimate of the constant.

First, we have

$\int_{\Omega_{\ell}}v^{2}(x)\,dx\leq\frac{1}{\sigma(\Omega_{\ell},A)}\int_{\Omega_{\ell}}|\nabla
v|^{2}(x)\,dx\leq\frac{[\mbox{diam}(\Omega)]^{2}}{2\pi^{2}\left(1+\left(\displaystyle\frac{|\Omega_{\ell}|}{|A|}\right)^{2}\right)}\int_{\Omega_{\ell}}|\nabla
v|^{2}(x)\,dx.$

On the other hand, from Remark 2.8 we get

$\int_{\Omega_{n\ell}}u^{2}\,dx\leq
2^{\frac{8^{N}|\Omega_{n\ell}|}{\delta^{N}}+1}\left(\int_{B}u^{2}\,dx+\frac{2^{N}}{c_{N}\delta^{N}m_{J}}\int_{\Omega_{n\ell}}\int_{\Omega_{n\ell}}J(x-y)(u(x)-u(y))^{2}\,dx\,dy\right).$

Therefore, for the particular case of two adjacent unit squares we have that
the best constant in the Poincare type inequality

$\begin{array}[]{l}\displaystyle\int_{\Omega_{\ell}}|\nabla
u|^{2}+\frac{1}{2}\int_{\Omega_{n\ell}}\int_{\Omega_{n\ell}}J(x-y)(u(y)-u(x))^{2}\,dy\,dx+\frac{1}{2}\int_{\Omega_{n\ell}}\int_{\Omega_{\ell}}G(x,y)|u(y)-u(x)|^{2}\,dy\,dx\\\\[10.0pt]
\qquad\qquad\qquad\displaystyle\geq\displaystyle
C^{*}\int_{\Omega}u^{2}(x)\,dx,\qquad\forall u\in H_{Neu},\end{array}$

can be estimated as

$C^{*}\geq
c\,\delta^{N}\,e^{c\,\delta^{-N}}\min\Big{\\{}m_{J};m_{G}\Big{\\}}.$

for $\delta$ small enough (here $c$ is a constant independent of $\delta$).

Now we are ready to proceed with the proof of the existence and uniqueness of
minimizers of $E_{Neu}(u)$ in $H_{Neu}$.

######  Theorem 2.12.

Given $f\in L^{2}(\Omega)$ with $\int_{\Omega}f=0$ there exists a unique
minimizer of $E_{Neu}(u)$ in $H_{Neu}$.

###### Proof of Theorem 2.12.

From the previous Poincaré-Wirtinger type inequality we obtain

$\begin{array}[]{l}\displaystyle E_{Neu}(u)=\int_{\Omega_{\ell}}\frac{|\nabla
u(x)|^{2}}{2}\,dx+\frac{1}{2}\int_{\Omega_{n\ell}}\int_{\Omega}J(x-y)|u(y)-u(x)|^{2}\,dy\,dx\\\\[10.0pt]
\qquad\qquad\qquad\displaystyle+\frac{1}{2}\int_{\Omega_{n\ell}}\int_{\Omega_{\ell}}G(x,y)|u(y)-u(x)|^{2}\,dy\,dx-\int_{\Omega}f(x)u(x)\,dx\\\\[10.0pt]
\qquad\qquad\displaystyle\geq
c\int_{\Omega}u^{2}(x)\,dx-\int_{\Omega}f(x)u(x)\,dx\end{array}$

from where it follows that $E_{Neu}(u)$ is bounded below and coercive in
$H_{Neu}$. Hence, existence of a minimizer follows by the direct method of
calculus of variations. Just take a minimizing sequence $u_{n}$ and extract a
subsequence that converges weakly in $L^{2}(\Omega)$. Then, we have

$0=\lim_{n\to\infty}\int_{\Omega}u_{n}(x)\,dx=\int_{\Omega}u(x)\,dx,$
$\lim_{n\to\infty}\int_{\Omega}f(x)u_{n}(x)\,dx=\int_{\Omega}f(x)u(x)\,dx$

and

$\begin{array}[]{l}\displaystyle\lim_{n\to\infty}\int_{\Omega_{\ell}}\frac{|\nabla
u(x)|^{2}}{2}\,dx+\frac{1}{2}\int_{\Omega_{n\ell}}\int_{\Omega}J(x-y)|u(y)-u(x)|^{2}\,dy\,dx\\\\[10.0pt]
\qquad\qquad\qquad\displaystyle+\frac{1}{2}\int_{\Omega_{n\ell}}\int_{\Omega_{\ell}}G(x,y)|u(y)-u(x)|^{2}\,dy\,dx\\\\[10.0pt]
\qquad\displaystyle\geq\int_{\Omega_{\ell}}|\nabla
u(x)|^{2}\,dx+\frac{\alpha}{2}\int_{\Omega_{n\ell}}\int_{\Omega_{n\ell}}J(x-y)(u(y)-u(x))^{2}\,dy\,dx\\\\[10.0pt]
\qquad\qquad\qquad\displaystyle+\frac{1}{2}\int_{\Omega_{n\ell}}\int_{\Omega_{\ell}}G(x,y)|u(y)-u(x)|^{2}\,dy\,dx.\end{array}$

Then, we obtain that $u\in H_{Neu}$ and that $u$ is a minimizer,

$E_{Neu}(u)=\min_{v\in H}E_{Neu}(v).$

Uniqueness of minimizers follows from the strict convexity of the functional
$E_{Neu}(u)$. ∎

Associated with this energy we have an equation in $\Omega$.

######  Lemma 2.13.

The minimizer of $E_{Neu}(u)$ in $H_{Neu}$ is a weak solution to the following
problem: a local equation with a source in $\Omega_{\ell}$,

$\begin{cases}\displaystyle-f(x)=\Delta
u(x)+\int_{\Omega_{n\ell}}G(x,y)(u(y)-u(x))\,dy,\ &x\in\Omega_{\ell},\
t>0,\\\\[10.0pt] \displaystyle\frac{\partial
u}{\partial\eta}(x)=0,\qquad&x\in\partial\Omega_{\ell},,\end{cases}$ (2.11)

and a nonlocal equation with a source in $\Omega_{n\ell}$,

$\displaystyle-f(x)=2\int_{\Omega_{n\ell}}\\!\\!J(x-y)(u(y)-u(x))\,dy+\int_{\Omega_{\ell}}G(x,y)(u(y)-u(x))\,dy,\quad
x\in\Omega_{n\ell}.$ (2.12)

###### Proof.

Let $u$ be the minimizer of $E_{Neu}(u)$ in $H_{Neu}$, then for every smooth
$\varphi$ with $\int_{\Omega}\varphi=0$ and every $t\in\mathbb{R}$ we have

$E_{Neu}(u+t\varphi)-E_{Neu}(u)\geq 0.$

Therefore, we have that

$\frac{\partial}{\partial t}E_{Neu}(u+t\varphi)|_{t=0}=0,$

that is,

$\begin{array}[]{l}\displaystyle\int_{\Omega}f\varphi=\int_{\Omega_{\ell}}\nabla
u\nabla\varphi+\int_{\Omega_{n\ell}}\int_{\Omega_{n\ell}}J(x-y)(u(y)-u(x))(\varphi(y)-\varphi(x))\,dy\,dx\\\\[10.0pt]
\displaystyle\qquad\qquad+\int_{\Omega_{n\ell}}\int_{\Omega_{\ell}}G(x,y)(u(y)-u(x))(\varphi(y)-\varphi(x))\,dy\,dx.\end{array}$

Using that the kernel $J$ is symmetric and Fubini’s Theorem we obtain

$\begin{array}[]{l}\displaystyle\int_{\Omega}f\varphi=\int_{\Omega_{\ell}}\nabla
u\nabla\varphi-2\int_{\Omega_{n\ell}}\int_{\Omega_{n\ell}}J(x-y)(u(y)-u(x))\,dy\,\varphi(x)\,dx\\\\[10.0pt]
\displaystyle\qquad\qquad+\int_{\Omega_{n\ell}}\int_{\Omega_{\ell}}G(x,y)(u(y)-u(x))\,dx\,\varphi(y)\,dy\\\\[10.0pt]
\displaystyle\qquad\qquad-\int_{\Omega_{n\ell}}\int_{\Omega_{\ell}}G(x,y)(u(y)-u(x))\,dy\,\varphi(x)\,dx.\end{array}$

From where if follows that $u$ is a weak solution to (2.11) and (2.12). ∎

Connections with probability theory. A probabilistic interpretation of this
model in terms of particle systems runs as follows: take an exponential clock
that controls the jumps of the particles, in the local region $\Omega_{\ell}$
the particles move according to Brownian motion (with a reflexion on the
boundary) and when the clock rings a new position is sorted according to the
kernel $G(x,\cdot)$, then they jump if the new position is in the nonlocal
region $\Omega_{n\ell}$ (if the sorted position lies in $\Omega_{\ell}$ then
just continue moving by Brownian motion ignoring the ring of the clock) while
in the nonlocal region $\Omega_{n\ell}$ the particles stay still until the
clock rings and then they jump using the kernel $J(x-\cdot)$ or the kernel
$G(x,\cdot)$ to select the new position in the whole $\Omega$. Notice that the
total number of particles inside the domain $\Omega$ remains constant in time.

The minimizer to our functional $E_{Neu}(u)$ gives the stationary distribution
of particles provided that there is an external source $f$ (that adds
particles where $f>0$ and remove particles where $f<0$).

##  3\. Mixed coupling

Second model. Coupling local/nonlocal problems via flux terms. Our aim now is
to look for a scalar problem with an energy that combines local and nonlocal
terms acting in different subdomains, $\Omega_{\ell}$ and $\Omega_{n\ell}$, of
$\Omega$, but now the coupling is made balancing the fluxes across a
prescribed hypersurface, $\Gamma$.

Recall from the Introduction that we look for a minimizer of the energy

$\begin{array}[]{l}F_{Neu}(u)\displaystyle:=\frac{1}{2}\int_{\Omega_{l}}|\nabla
u(x)|^{2}\,dx+\frac{1}{2}\int_{\Omega_{nl}}\int_{\Omega_{nl}}J(x-y)\left(u(y)-u(x)\right)^{2}\,dy\,dx\\\\[10.0pt]
\displaystyle\quad\qquad\qquad\qquad\qquad+\frac{1}{2}\int_{\Omega_{nl}}\int_{\Gamma}G(x,z)\left(u(x)-u(z)\right)^{2}\,d\sigma(z)\,dx-\int_{\Omega}f(x)u(x)\,dx\end{array}$
(3.1)

in

$H_{Neu}=\left\\{u\in L^{2}(\Omega)\ :\ u\in H^{1}(\Omega_{\ell}),\
\int_{\Omega}u(x)\,dx=0\right\\},$

assuming that

$\int_{\Omega}f(x)\,dx=0.$

As before we first prove a lemma that gives coercivity for our functional.

######  Lemma 3.1.

There exists a constant $C$ such that

$\begin{array}[]{l}\displaystyle\int_{\Omega_{\ell}}\frac{|\nabla
u(x)|^{2}}{2}\,dx+\frac{1}{2}\int_{\Omega_{n\ell}}\int_{\Omega_{n\ell}}J(x-y)(u(y)-u(x))^{2}\,dy\,dx\\\\[10.0pt]
\displaystyle\qquad\qquad\qquad\qquad\qquad+\frac{1}{2}\int_{\Omega_{nl}}\int_{\Gamma}G(x,z)\left(u(x)-u(z)\right)^{2}d\sigma(z)dx\geq
C\int_{\Omega}|u(x)|^{2}\,dx,\end{array}$

for every $u$ in $H_{Neu}$.

###### Proof.

We argue by contradiction, then we assume that there is a sequence $u_{n}\in
H_{Neu}$ such that

$\int_{\Omega}|u_{n}(x)|^{2}\,dx=1$

and

$\begin{array}[]{l}\displaystyle\int_{\Omega_{\ell}}\frac{|\nabla
u_{n}(x)|^{2}}{2}\,dx+\frac{1}{2}\int_{\Omega_{n\ell}}\int_{\Omega_{n\ell}}J(x-y)(u_{n}(y)-u_{n}(x))^{2}\,dy\,dx\\\\[10.0pt]
\qquad\displaystyle+\frac{1}{2}\int_{\Omega_{nl}}\int_{\Gamma}G(x,z)\left(u_{n}(x)-u_{n}(z)\right)^{2}\,d\sigma(z)\,dx\to
0.\end{array}$

Then we have that

$\int_{\Omega_{\ell}}|\nabla u_{n}(x)|^{2}\,dx\to 0,$
$\int_{\Omega_{nl}}\int_{\Gamma}G(x,z)\left(u_{n}(x)-u_{n}(z)\right)^{2}\,d\sigma(z)\,dx\to
0,$

and

$\int_{\Omega_{n\ell}}\int_{\Omega_{n\ell}}J(x-y)(u_{n}(y)-v_{n}(x))^{2}\,dy\,dx\to
0.$

Since

$\int_{\Omega}|u_{n}(x)|^{2}\,dx=1$

we obtain that $u_{n}$ is bounded in $L^{2}(\Omega_{\ell})$ and then, using
that,

$\int_{\Omega_{\ell}}\frac{|\nabla u_{n}(x)|^{2}}{2}\,dx\to 0$

we get that there exists a constant $k_{1}$, such that, along a subsequence,

$u_{n}\to k_{1}$

strongly in $H^{1}(\Omega_{\ell})$. Hence, using the trace theorem on
$\Gamma$, $H^{1}(\Omega_{\ell})\hookrightarrow L^{2}(\Gamma)$, we obtain

$u_{n}\to k_{1}$

strongly $L^{2}(\Gamma)$.

Now we argue in the nonlocal part $\Omega_{n\ell}$. Since $v_{n}$ is bounded
in $L^{2}(\Omega_{n\ell})$ and

$\int_{\Omega_{n\ell}}\int_{\Omega_{n\ell}}J(x-y)(u_{n}(y)-u_{n}(x))^{2}\,dy\,dx\to
0$

we have that (extracting another subsequence if necessary)

$u_{n}\rightharpoonup u$

weakly in $L^{2}(\Omega_{n\ell})$ and therefore,

$\int_{\Omega_{n\ell}}\int_{\Omega_{n\ell}}J(x-y)(u(y)-u(x))^{2}\,dy\,dx\leq\lim_{n\to\infty}\frac{1}{2}\int_{\Omega_{n\ell}}\int_{\Omega_{n\ell}}J(x-y)(u_{n}(y)-u_{n}(x))^{2}\,dy\,dx=0.$

Hence, we get that

$u\equiv k_{2}$

in $\Omega_{n\ell}$ (here we need to assume that $\Omega_{n\ell}$ is
$\delta-$connected).

From the weak convergence of $u_{n}$ to $u$ in $L^{2}(\Omega_{n\ell})$ and the
strong convergence of $u_{n}$ to $u$ in $L^{2}(\Gamma)$ we obtain

$\int_{\Omega_{nl}}\int_{\Gamma}G(x,z)\left(k_{2}-k_{1}\right)^{2}\,d\sigma(z)\,dx\leq\lim_{n\to\infty}\int_{\Omega_{nl}}\int_{\Gamma}G(x,z)\left(u_{n}(x)-u_{n}(z)\right)^{2}\,d\sigma(z)\,dx=0.$

Hence, from the fact that

$\int_{\Omega_{nl}}\int_{\Gamma}G(x,z)\,d\sigma(z)\,dx>0,$

we obtain

$k_{1}=k_{2}.$

Now, from the fact that $u_{n}\in H_{Neu}$, we have that

$\int_{\Omega}u_{n}(x)\,dx=0,$

and then, passing to the limit, we obtain that

$\int_{\Omega_{\ell}}k_{1}+\int_{\Omega_{n\ell}}k_{2}=0.$

Hence, as $k_{1}=k_{2}$ we conclude that

$k_{1}=k_{2}=0.$

Up to now we have that $u_{n}\to 0$ strongly in $H^{1}(\Omega_{\ell})$ and
$u_{n}\to 0$ weakly in $L^{2}(\Omega_{n\ell})$. Then, as

$1=\int_{\Omega}|u_{n}(x)|^{2}\,dx=\int_{\Omega_{\ell}}|u_{n}(x)|^{2}\,dx+\int_{\Omega_{n\ell}}|u_{n}(x)|^{2}\,dx$

we get that

$\int_{\Omega_{n\ell}}|u_{n}(x)|^{2}\,dx\to 1.$

Now, we go back to

$\int_{\Omega_{nl}}\int_{\Gamma}G(x,z)\left(u_{n}(x)-u_{n}(z)\right)^{2}\,d\sigma(z)\,dx\to
0$

and we obtain

$\begin{array}[]{l}\displaystyle
0=\lim_{n\to\infty}\int_{\Omega_{nl}}\int_{\Gamma}G(x,z)\left(u_{n}(x)-u_{n}(z)\right)^{2}\,d\sigma(z)\,dx\\\\[10.0pt]
\qquad\displaystyle=\lim_{n\to\infty}\int_{\Omega_{nl}}\int_{\Gamma}G(x,z)|u_{n}(x)|^{2}\,d\sigma(z)\,dx\\\\[10.0pt]
\qquad\displaystyle\qquad+\lim_{n\to\infty}\int_{\Omega_{nl}}\int_{\Gamma}G(x,z)|u_{n}(z)|^{2}\,d\sigma(z)\,dx\\\\[10.0pt]
\qquad\displaystyle\qquad-\lim_{n\to\infty}2\int_{\Omega_{nl}}\int_{\Gamma}G(x,z)u_{n}(x)u_{n}(z)\,d\sigma(z)\,dx.\end{array}$

Since $u_{n}\to 0$ strongly in $L^{2}(\Gamma)$, we have that

$\lim_{n\to\infty}\int_{\Omega_{nl}}\int_{\Gamma}G(x,z)|u_{n}(z)|^{2}\,d\sigma(z)\,dx=0$

and

$\lim_{n\to\infty}\int_{\Omega_{nl}}\int_{\Gamma}G(x,z)u_{n}(x)u_{n}(z)\,d\sigma(z)\,dx=0.$

Therefore, we obtain that

$\lim_{n\to\infty}\int_{\Omega_{n\ell}}|u_{n}(x)|^{2}\int_{\Gamma}G(x,z)\,dy\,dx=0.$
(3.2)

On the other hand we have

$\begin{array}[]{l}\displaystyle
0=\lim_{n\to\infty}\int_{\Omega_{n\ell}}\int_{\Omega_{n\ell}}J(x-y)(u_{n}(y)-u_{n}(x))^{2}\,dy\,dx\\\\[10.0pt]
\qquad\displaystyle=\lim_{n\to\infty}2\int_{\Omega_{n\ell}}\int_{\Omega_{n\ell}}J(x-y)|u_{n}(x)|^{2}\,dy\,dx\\\\[10.0pt]
\qquad\displaystyle\qquad-\lim_{n\to\infty}2\int_{\Omega_{n\ell}}\int_{\Omega_{n\ell}}J(x-y)u_{n}(y)u_{n}(x)\,dy\,dx.\end{array}$

Now we use that $v_{n}\rightharpoonup 0$ weakly in $L^{2}(\Omega_{n\ell})$ and
that $J$ is continuous to obtain

$\int_{\Omega_{n\ell}}\int_{\Omega_{n\ell}}J(x-y)u_{n}(y)dy\to 0$

strongly in $L^{2}(\Omega_{n\ell})$ and then we obtain

$-\lim_{n\to\infty}\int_{\Omega_{n\ell}}\int_{\Omega_{n\ell}}J(x-y)u_{n}(y)u_{n}(x)\,dy\,dx=0.$

Hence, we get that

$\lim_{n\to\infty}\int_{\Omega_{n\ell}}|v_{n}(x)|^{2}\int_{\Omega_{n\ell}}J(x-y)\,dy\,dx=0.$
(3.3)

From (3.2) and (3.3) and the fact that there exists a constant $c$ such that

$\Big{(}\int_{\Omega_{n\ell}}J(x-y)\,dy+\int_{\Gamma}G(x,z)d\sigma(z)\Big{)}\geq
c>0,$

we conclude that

$\begin{array}[]{l}\displaystyle
0<c=c\lim_{n\to\infty}\int_{\Omega_{n\ell}}|u_{n}(x)|^{2}dx\\\\[10.0pt]
\displaystyle\qquad\leq\lim_{n\to\infty}\int_{\Omega_{n\ell}}|u_{n}(x)|^{2}\Big{(}\int_{\Omega_{n\ell}}J(x-y)\,dy+\int_{\Gamma}G(x,z)d\sigma(z)\Big{)}dx\\\\[10.0pt]
\displaystyle\qquad=\lim_{n\to\infty}\int_{\Omega_{n\ell}}|u_{n}(x)|^{2}\int_{\Omega_{n\ell}}J(x-y)\,dy\,dx\\\\[10.0pt]
\displaystyle\qquad\qquad+\lim_{n\to\infty}\int_{\Omega_{n\ell}}|u_{n}(x)|^{2}\int_{\Gamma}G(x,z)d\sigma(z)dx=0\end{array}$

a contradiction. ∎

Similarly to Lemma 2.3, Lemma 3.1 does not provide information regarding the
constant $c$.

######  Lemma 3.2.

The optimal constant $c>0$ such that

$\begin{array}[]{l}\displaystyle\int_{\Omega_{\ell}}\frac{|\nabla
u(x)|^{2}}{2}\,dx+\frac{1}{2}\int_{\Omega_{n\ell}}\int_{\Omega_{n\ell}}J(x-y)(u(y)-u(x))^{2}\,dy\,dx\\\\[10.0pt]
\displaystyle\qquad\qquad\qquad\qquad\qquad+\frac{1}{2}\int_{\Omega_{nl}}\int_{\Gamma}G(x,z)\left(u(x)-u(z)\right)^{2}d\sigma(z)dx\geq
c\int_{\Omega}|u(x)|^{2}\,dx,\end{array}$

for every function $u\in H_{Neu}$ (that is, for $u$ in the natural space with
$\int_{\Omega}u=0$) can be estimated as

$c\geq c^{*}(J,G,\Omega_{\ell},\mathcal{T},\delta,\Gamma_{A},B)$

with $\Gamma_{A}\subset\Omega_{\ell}$, $B\subset\Omega_{n\ell}$ such that
$diam(\Gamma_{A}\cup B)<2\delta$ and $\mathcal{T}$ a $\delta-tree$ structure
of $\Omega_{n\ell}$ with root at $B$.

###### Proof.

The proof uses the same ideas as the ones of Lemma 2.9. The only significant
modifications are the following: First, from our hypothesis on $G$ and the
assumption $\mbox{dist}(\Gamma_{A},\Omega_{n\ell})<\delta$, there exist two
sets $\Gamma_{A}\subset\Gamma$ and $B\subset\Omega_{n\ell}$ such that

$|x-y|\leq 2\delta,\qquad\forall x\in\Gamma_{A},\,\forall y\in B.$

Take $r\in\mathbb{R}$ such that

$\int_{\Gamma_{A}}(u(z)-r)\,d\sigma(z)=0.$

Then, we have, for $v=u-r$ and calling $|\Gamma_{A}|$ the $N-1$ dimensional
measure of $\Gamma_{A}$ the following variant of (2.10),

$\begin{array}[]{l}\displaystyle\frac{1}{2}\int_{\Omega_{nl}}\int_{\Gamma}G(x,z)\left(u(x)-u(z)\right)^{2}d\sigma(z)dx\\\\[10.0pt]
\displaystyle\geq\frac{1}{2}\int_{B}\int_{\Gamma_{A}}G(x,z)\left(u(x)-u(z)\right)^{2}d\sigma(z)dx\\\\[10.0pt]
\displaystyle\geq
m_{G}\frac{|\Gamma_{A}|}{2}\int_{B}|v(x)|^{2}\,dx\end{array}$ (3.4)

where we are using that

$\int_{B}|v(x)|^{2}\,dx=\frac{1}{|\Gamma_{A}|^{2}}\int_{B}\left|\int_{\Gamma_{A}}(v(x)-v(z))\,d\sigma(z)\right|^{2}\,dx\leq\frac{1}{|A|}\int_{B}\int_{\Gamma_{A}}|v(x)-v(z)|^{2}\,d\sigma(z)\,dx.$

Using now that

$0<\sigma(\Omega_{\ell},\Gamma_{A})=\inf_{v\in
H^{1}(\Omega_{\ell}),\int_{\Gamma_{A}}v\,d\sigma(z)=0}\frac{\displaystyle\int_{\Omega_{\ell}}|\nabla
v|^{2}(x)\,dx}{\displaystyle\int_{\Omega_{\ell}}v^{2}(x)\,dx},$

for a suitable Poincaré constant $\sigma(\Omega_{\ell},\Gamma_{A})$, the proof
follows exactly as in Lemma 2.9. ∎

As before, we are ready to obtain existence and uniqueness of a minimizer of
$F_{Neu}(u)$ in $H_{Neu}$.

######  Theorem 3.3.

Given $f\in L^{2}(\Omega)$ with $\int_{\Omega}f=0$ there exists a unique
minimizer of $F_{Neu}(u)$ in $H_{Neu}$.

###### Proof of Theorem 3.3.

The previous lemma implies that $F_{Neu}(u)$ is bounded below and coercive.
Hence, existence of a minimizer follows just considering a minimizing sequence
$(u_{n})$ and extracting a subsequence that converges weakly in
$H^{1}(\Omega_{\ell})\cap L^{2}(\Omega_{n\ell})$. Uniqueness of a minimizer
follows from the strict convexity of the functional $F_{Neu}(u)$. ∎

Now, we find the equations verified by the minimizer.

######  Lemma 3.4.

The minimizer of $F_{Neu}$ is a weak solution to

$\left\\{\begin{array}[]{ll}\displaystyle f(x)=\Delta
u(x),&x\in\Omega_{\ell},\\\\[10.0pt] \displaystyle\frac{\partial
u}{\partial\eta}(x)=0,&x\in\partial\Omega_{l}\cap\partial\Omega,\\\\[10.0pt]
\displaystyle\frac{\partial
u}{\partial\eta}(x)=\int_{\Omega_{nl}}G(y,x)(u(y)-u(x))dy,&x\in\Gamma,\end{array}\right.$
(3.5)

and

$f(x)=2\int_{\Omega_{nl}}J(x-y)\left(u(y)-u(x)\right)dy-\int_{\Gamma}G(x,z)(u(x)-u(z))d\sigma(z),\qquad
x\in\Omega_{n\ell}.$ (3.6)

###### Proof.

Let $u$ be the minimizer of $F_{Neu}(u)$ in $H_{Neu}$, then for every smooth
$\varphi$ with $\int_{\Omega}\varphi=0$ and every $t\in\mathbb{R}$ we have

$F_{Neu}(u+t\varphi)-F_{Neu}(u)\geq 0.$

Therefore, we get that

$\frac{\partial}{\partial t}F_{Neu}(u+t\varphi)|_{t=0}=0,$

that is,

$\begin{array}[]{l}\displaystyle\int_{\Omega}f\varphi=\int_{\Omega_{\ell}}\nabla
u\nabla\varphi+\int_{\Omega_{n\ell}}\int_{\Omega_{n\ell}}J(x-y)(u(y)-u(x))(\varphi(y)-\varphi(x))\,dy\,dx\\\\[10.0pt]
\displaystyle\qquad\qquad+\int_{\Omega_{n\ell}}\int_{\Gamma}G(x,y)(u(y)-u(x))\varphi(y)\,d\sigma(y)\,dx\\\\[10.0pt]
\displaystyle\qquad\qquad-\int_{\Omega_{n\ell}}\int_{\Gamma}G(x,y)(u(y)-u(x))\varphi(x)\,d\sigma(y)\,dx.\end{array}$

Using that the kernel $J$ is symmetric we obtain

$\begin{array}[]{l}\displaystyle\int_{\Omega}f\varphi=\int_{\Omega_{\ell}}\nabla
u\nabla\varphi-2\int_{\Omega_{n\ell}}\int_{\Omega_{n\ell}}J(x-y)(u(y)-u(x))\,dy\,\varphi(x)\,dx\\\\[10.0pt]
\displaystyle\qquad\qquad-\int_{\Gamma}\int_{\Omega_{n\ell}}G(y,x)(u(y)-u(x))\,dy\varphi(x)\,d\sigma(x)\\\\[10.0pt]
\displaystyle\qquad\qquad-\int_{\Omega_{n\ell}}\int_{\Gamma}G(x,y)(u(y)-u(x))\,d\sigma(y)\,\varphi(x)\,dx,\end{array}$

that is, $u$ is a weak solution to (3.5) and (3.6). ∎

Probability interpretation. A probabilistic interpretation of this model in
terms of particle systems runs as follows: in the local region $\Omega_{\ell}$
the particles move according to Brownian motion (with a reflexion on the
boundary of $\Omega_{\ell}$, $\partial\Omega_{\ell}$) and when they arrive to
the surface $\Gamma$ they pass to the nonlocal region to a point selected
using the kernel $G$; in the nonlocal region take an exponential clock that
controls the jumps of the particles, and when the clock rings a new position
is sorted according to the kernel $J(x-\cdot)$ (for the movements inside
$\Omega_{n\ell}$) or according to $G(x,\cdot)$ for jumping back to the local
region entering at a point in the surface, $\Gamma$. The total number of
particles inside the domain $\Omega$ remains constant in time.

The minimizer to our functional $F_{Neu}(u)$ gives the stationary distribution
of particles ($u$ for the particles inside the local region $\Omega_{\ell}$
and $v$ for particles in the nonlocal part $\Omega_{n\ell}$) provided that
there is an external source $f$ (that adds particles where $f>0$ and remove
particles where $f<0$).

Notice that here the coupling term appears as a flux boundary condition for
the local part of the problem.

##  4\. Singular kernels

First, let us deal with

$\begin{array}[]{l}\displaystyle E_{Neu}(u):=\int_{\Omega_{\ell}}\frac{|\nabla
u(x)|^{2}}{2}\,dx+\frac{1}{2}\int_{\Omega_{n\ell}}\int_{\Omega_{n\ell}}\frac{C}{|x-y|^{N+2s}}|u(y)-u(x)|^{2}\,dy\,dx\\\\[10.0pt]
\displaystyle\qquad\qquad\qquad+\frac{1}{2}\int_{\Omega_{n\ell}}\int_{\Omega_{\ell}}G(x,y)|u(y)-u(x)|^{2}\,dy\,dx-\int_{\Omega}f(x)u(x)\,dx.\end{array}$
(4.1)

We look for minimizers in the space

$H_{Neu}=\left\\{u:\,u|_{\Omega_{\ell}}\in
H^{1}(\Omega_{\ell}),u|_{\Omega_{n\ell}}\in H^{s}(\Omega_{n\ell}),\
\int_{\Omega}u=0\right\\}.$

In fact, now the key lemma (the Poincaré-Wirtinger type inequality) is
simpler.

######  Lemma 4.1.

There exists $c>0$ such that

$\begin{array}[]{l}\displaystyle\int_{\Omega_{\ell}}|\nabla
u|^{2}+\frac{1}{2}\int_{\Omega_{n\ell}}\int_{\Omega_{n\ell}}\frac{C}{|x-y|^{N+2s}}(u(y)-u(x))^{2}\,dy\,dx+\frac{1}{2}\int_{\Omega_{n\ell}}\int_{\Omega_{\ell}}G(x,y)|u(y)-u(x)|^{2}\,dy\,dx\\\\[10.0pt]
\qquad\qquad\qquad\displaystyle\geq\displaystyle
c\int_{\Omega}u^{2}(x)\,dx\end{array}$

for every function $u\in H_{Neu}$.

###### Proof.

As before, we argue by contradiction. Assume that there is a sequence $u_{n}$
such that

$\int_{\Omega_{\ell}}|\nabla u_{n}|^{2}(x)\,dx\to 0$ (4.2)
$\frac{1}{2}\int_{\Omega_{n\ell}}\int_{\Omega_{n\ell}}\frac{C}{|x-y|^{N+2s}}|u_{n}(y)-u_{n}(x)|^{2}\,dy\,dx\to
0,$ (4.3)
$\frac{1}{2}\int_{\Omega_{n\ell}}\int_{\Omega_{\ell}}G(x,y)|u_{n}(y)-u_{n}(x)|^{2}\,dy\,dx\to
0,$ (4.4) $\int_{\Omega}|u_{n}|^{2}(x)\,dx=1$ (4.5)

and

$\int_{\Omega}u_{n}(x)\,dx=0.$ (4.6)

Since the $L^{2}$ norm is bounded we can extract a subsequence such that

$u_{n}\rightharpoonup u\qquad\mbox{weakly in }H^{1}(\Omega_{\ell}),$

and

$u_{n}\rightharpoonup u\qquad\mbox{weakly in }H^{s}(\Omega_{n\ell}).$

then, from the compact embeddings $H^{1}(\Omega_{\ell})\hookrightarrow
L^{2}(\Omega_{\ell})$ and $H^{s}(\Omega_{\ell})\hookrightarrow
L^{2}(\Omega_{\ell})$ and (2.5) and (4.3) we get that there is are constants
$k_{1}$, $k_{2}$ such that

$u_{n}\to k_{1}\qquad\mbox{strongly in }L^{2}(\Omega_{\ell})$

and

$u_{n}\to k_{2}\qquad\mbox{strongly in }L^{2}(\Omega_{n\ell}).$

From (2.7) we conclude that

$\frac{1}{2}\int_{\Omega_{n\ell}}\int_{\Omega_{\ell}}G(x,y)|k_{1}-k_{2}|^{2}\,dy\,dx\leq\lim_{n\to\infty}\frac{1}{2}\int_{\Omega_{n\ell}}\int_{\Omega_{\ell}}G(x,y)|u_{n}(y)-u_{n}(x)|^{2}\,dy\,dx=0,$

and hence

$k_{1}=k_{2}.$

Now, from (2.9), we obtain

$k_{1}=k_{2}=0.$

Hence we get

$\int_{\Omega}|u_{n}|^{2}(x)\,dx=\int_{\Omega_{\ell}}|u_{n}|^{2}(x)\,dx+\int_{\Omega_{n\ell}}|u_{n}|^{2}(x)\,dx\to
0.$

This contradicts that

$\int_{\Omega}|u_{n}|^{2}(x)\,dx\to 1,$

and ends the proof. ∎

Now we are ready to proceed with the proof of Theorem 1.5.

###### Proof of Theorem 1.5.

From the previous Poincaré-Wirtinger type inequality we obtain

$E_{Neu}(u)\geq c\int_{\Omega}u^{2}(x)\,dx-\int_{\Omega}f(x)u(x)\,dx$

from where it follows that $E_{Neu}(u)$ is bounded below and coercive in $H$.
Hence, existence of a minimizer follows by the direct method of calculus of
variations. Just take a minimizing sequence $u_{n}$ and extract a subsequence
that converges weakly in $L^{2}(\Omega)$. Then, using the lower semicontinuity
of the seminorms we obtain that $u\in H_{Neu}$ and that $u$ is a minimizer,

$E_{Neu}(u)=\min_{v\in H}E_{Neu}(v).$

Uniqueness of minimizers follows from the strict convexity of the functional
$E_{Neu}(u)$.

Associated with this energy we have an equation in $\Omega$ that can be
obtained exactly as before. ∎

Now let us move to mixed couplings,

$\begin{array}[]{l}F_{Neu}(u)\displaystyle:=\frac{1}{2}\int_{\Omega_{l}}|\nabla
u|^{2}dx+\frac{1}{2}\int_{\Omega_{nl}}\int_{\Omega_{nl}}\frac{C}{|x-y|^{N+2s}}\left(u(y)-u(x)\right)^{2}dydx\\\\[10.0pt]
\displaystyle\quad\qquad\qquad\qquad\qquad+\frac{1}{2}\int_{\Omega_{nl}}\int_{\Gamma}G(x,z)\left(u(x)-u(z)\right)^{2}d\sigma(z)dx-\int_{\Omega}f(x)u(x)\,dx.\end{array}$
(4.7)

The key lemma follows as before.

######  Lemma 4.2.

There exists $c>0$ such that

$\begin{array}[]{l}\displaystyle\int_{\Omega_{\ell}}|\nabla
u|^{2}+\frac{1}{2}\int_{\Omega_{n\ell}}\int_{\Omega_{n\ell}}\frac{C}{|x-y|^{N+2s}}(u(y)-u(x))^{2}dydx+\frac{1}{2}\int_{\Omega_{n\ell}}\int_{\Gamma}G(x,y)|u(y)-u(x)|^{2}d\sigma(y)dx\\\\[10.0pt]
\qquad\qquad\qquad\displaystyle\geq\displaystyle
c\int_{\Omega}u^{2}(x)\,dx\end{array}$

for every function $u\in H_{Neu}$.

###### Proof.

The proof runs as before, arguing by contradiction. The only minor point is
that, after proving that there is a subsequence $u_{n}$ such that

$u_{n}\to k_{1}\qquad\mbox{strongly in }L^{2}(\Omega_{\ell})$

and

$u_{n}\to k_{2}\qquad\mbox{strongly in }L^{2}(\Omega_{n\ell})$

we have to use that

$\frac{1}{2}\int_{\Omega_{n\ell}}\int_{\Gamma}G(x,y)|k_{1}-k_{2}|^{2}\,d\sigma(y)\,dx\leq\lim_{n\to\infty}\frac{1}{2}\int_{\Omega_{n\ell}}\int_{\Gamma}G(x,y)|u_{n}(y)-u_{n}(x)|^{2}\,d\sigma(y)\,dx=0,$

to obtain

$k_{1}=k_{2}.$

The rest of the arguments until reaching a contradiction are exactly as
before. ∎

Now we are ready to proceed with the proof of Theorem 2.12.

###### Proof of Theorem 2.12.

From the previous Poincaré-Wirtinger type inequality we obtain

$E_{Neu}(u)\geq c\int_{\Omega}u^{2}(x)\,dx-\int_{\Omega}f(x)u(x)\,dx$

from where it follows that $E_{Neu}(u)$ is bounded below and coercive in $H$.
Hence, existence of a minimizer follows by the direct method of calculus of
variations. Just take a minimizing sequence $u_{n}$ and extract a subsequence
that converges weakly in $L^{2}(\Omega)$. Then, using the lower semicontinuity
of the seminorms we obtain that $u\in H_{Neu}$ and that $u$ is a minimizer,

$E_{Neu}(u)=\min_{v\in H}E_{Neu}(v).$

Uniqueness of minimizers follows from the strict convexity of the functional
$E_{Neu}(u)$.

Associated with this energy we have an equation in $\Omega$ that can be
obtained exactly as before. ∎

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