A simple characterization of $H$-convergence for a class of nonlocal problems

TL;DR

This paper provides an explicit characterization of $H$-convergence for nonlocal fractional $p$-Laplace operators, showing that weak-* convergence of coefficients is equivalent to $H$-convergence, highlighting differences from local cases.

Contribution

It establishes that weak-* convergence of coefficients characterizes $H$-convergence for nonlocal fractional $p$-Laplace operators, extending previous work.

Findings

Weak-* convergence of coefficients is equivalent to $H$-convergence.

The result leverages nonlocality, contrasting with local $p$-Laplacian cases.

Provides an explicit characterization of $H$-convergence for nonlocal operators.

Abstract

This is a follow-up of a paper by Fern\'andez-Bonder-Ritorto-Salort [8], where the classical concept of $H$-convergence was extended to fractional \(p\)-Laplace type operators. In this short paper we provide an explicit characterization of this notion by demonstrating that the weak-* convergence of the coefficients is an equivalent condition for $H$-convergence of the sequence of nonlocal operators. This result takes advantage of nonlocality and is in stark contrast to the local p-Laplacian case

Equations

$${\mathcal{A}}_{\lambda ,\Lambda }=\left\{a\in L^{\infty }({\mathbb{R}}^n\times {\mathbb{R}}^n):a(x,y)=a(y,x),\lambda \le a(x,y)\le \Lambda ,\text{a.e. in Rn×Rn}\right\},$$

$$L_{a}u(x)=p.v.{\int }_{{\mathbb{R}}^n}a(x,y)\frac{|u(x)-u(y){|}^{p-2}(u(x)-u(y))}{|x-y{|}^{n+sp}}dy.$$

$$\left\{\begin{array}[]{ll}\operatorname{\mathcal{L}}_{a}u=f&\mbox{in }\Omega,\\ u=0&\mbox{in }\mathbb{R}^{n}\backslash\Omega.\end{array}\right.$$

$$W_0^{s,p}(\Omega )=\left\{u\in W^{s,p}({\mathbb{R}}^n):u=0\text{mbox}a.e.in{\mathbb{R}}^n\backslash \Omega \right\},$$

$$W^{s,p}({\mathbb{R}}^n)=\left\{u\in L^p({\mathbb{R}}^n):D_{s,p}u(x,y)\in L^p({\mathbb{R}}^n\times {\mathbb{R}}^n)\right\},$$

$$D_{s,p}u(x,y)=\frac{u(x)-u(y)}{|x-y{|}^{\frac{n}{p}+s}}$$

$$\Vert u{\Vert }_{s,p}={\left(\Vert u{\Vert }_p^p+|u{|}_{s,p}^p\right)}^{\frac{1}{p}},$$

$$|u{|}_{s,p}={\left(\int {\int }_{{\mathbb{R}}^n\times {\mathbb{R}}^n}\frac{|u(x)-u(y){|}^p}{|x-y{|}^{n+sp}}dxdy\right)}^{\frac{1}{p}}={\left(\int {\int }_{{\mathbb{R}}^n\times {\mathbb{R}}^n}{\left|D_{s,p}u(x,y)\right|}^{p}dxdy\right)}^{\frac{1}{p}}$$

$$W_0^{s,p}(\Omega )={\overline{C_c^{\infty }(\Omega )}}^{\Vert \cdot {\Vert }_{s,p}},$$

$$W_0^{s,p}(\Omega )=\left\{u\in W^{s,p}({\mathbb{R}}^n):u=0\text{mbox}a.e.in{\mathbb{R}}^n\backslash \Omega \right\}.$$

$$\Vert u{\Vert }_p\le C|u{|}_{s,p},$$

$$\Vert f{\Vert }_{-s,p^{\prime }}=\sup \left\{⟨f,u⟩:u\in W_0^{s,p}(\Omega ),|u{|}_{s,p}=1\right\}.$$

$$\frac{1}{2}\int {\int }_{{\mathbb{R}}^n\times {\mathbb{R}}^n}a(x,y)\frac{|u(x)-u(y){|}^{p-2}(u(x)-u(y))(v(x)-v(y))}{|x-y{|}^{n+sp}}dxdy=⟨f,v⟩,$$

$$I_a(v)=\frac{1}{2p}\int {\int }_{{\mathbb{R}}^n\times {\mathbb{R}}^n}a(x,y)|D_{s,p}v(x,y){|}^{p}dxdy-⟨f,v⟩.$$

$$d_{s,p}\varphi (x)=p.v.{\int }_{{\mathbb{R}}^n}\frac{\varphi (x,y)-\varphi (y,x)}{|x-y{|}^{\frac{n}{p}+s}}dy.$$

$$\int {\int }_{{\mathbb{R}}^n\times {\mathbb{R}}^n}\varphi D_{s,p}udxdy=⟨d_{s,p}\varphi ,u⟩;$$

$${\varphi }_k{D}_{s,p}{v}_k\to \varphi D_{s,p}v$$

$$a(x,y)\le \Lambda ,\text{a.e. in Rn×Rn},$$

$$\int {\int }_{{\mathbb{R}}^n\times {\mathbb{R}}^n}|q_k{|}^{p^{\prime }}\phi dxdy$$

$$\le {\Lambda }^{p^{\prime }-1}\int {\int }_{{\mathbb{R}}^n\times {\mathbb{R}}^n}\phi q_k{D}_{s,p}{u}_{k}dxdy.$$

$$\int {\int }_{{\mathbb{R}}^n\times {\mathbb{R}}^n}{a}^{p^{\prime }}|D_{s,p}u{|}^p\phi dxdy\le {\Lambda }^{p^{\prime }-1}\int {\int }_{{\mathbb{R}}^n\times {\mathbb{R}}^n}a|D_{s,p}u{|}^p\phi dxdy,$$

$$a^{p^{\prime }}|D_{s,p}u{|}^p\le {\Lambda }^{p^{\prime }-1}a|D_{s,p}u{|}^p,\text{a.e. in Rn×Rn}.$$

$$E(a)=\int {\int }_{{\mathbb{R}}^n\times {\mathbb{R}}^n}a|D_{s,p}u{|}^{p}dxdy=\int {\int }_{{\mathbb{R}}^n\times {\mathbb{R}}^n}q{D}_{s,p}udxdy,$$

$$u_k\rightharpoonup u,\text{mbox}weaklyin{W}^{s,p}(\Omega ),$$

$$\int {\int }_{{\mathbb{R}}^n\times {\mathbb{R}}^n}(a-\tilde{a})|D_{s,p}{u}^f{|}^{p}dxdy=0,$$

$$a=\tilde{a},\text{mbox}a.e.in{\mathbb{R}}^n\times {\mathbb{R}}^n.$$

$$\underset{k^{\prime \prime }\to \infty }{\mathrm{limsup}}{I}_{a_{k^{\prime \prime }}}(u_{k^{\prime \prime }})\le \underset{k^{\prime \prime }\to \infty }{\lim }{I}_{a_{k^{\prime \prime }}}(u)=I_a(u)\le I_a(\tilde{u}).$$

$${\nu }_k(E)=\int {\int }_E{a}_k(x,y)dxdy=\int {\int }_{{\mathbb{R}}^n\times {\mathbb{R}}^n}{\chi }_E(x,y)a_k(x,y)dxdy,k\ge 1,$$

$$\nu (E)=\int {\int }_{E}a(x,y)dxdy=\int {\int }_{{\mathbb{R}}^n\times {\mathbb{R}}^n}{\chi }_E(x,y)a(x,y)dxdy,$$

$$I_a(\tilde{u})$$

Full text

1 Introduction

For $p\in(1,\infty)$, $s\in(0,1)$ and a nonlocal conductivity $a(x,y)$ belonging to the class

[TABLE]

where $0<\lambda\leq\Lambda<\infty$ are given constants, let us consider the following nonlocal operator related to the fractional $p$-Laplacian:

[TABLE]

For a fixed bounded domain $\Omega\subset\mathbb{R}^{n}$ with Lipschitz boundary and $f\in L^{p^{\prime}}(\Omega)$, with $p^{\prime}=(p-1)/p$ being the conjugate exponent to $p$, we consider the nonlocal problem

[TABLE]

This problem is well-posed, with a unique solution found in the space

[TABLE]

where $W^{s,p}(\mathbb{R}^{n})$ is the classical fractional Sobolev space over $\mathbb{R}^{n}$, see [5, 1]:

[TABLE]

and

[TABLE]

is the $(s,p)$-nonlocal gradient of $u$. In view of existence and uniqueness of solutions we can employ the shorthand notation $(u,q)=\mathcal{S}_{a}f$ to denote the solution $u\in W^{s,p}_{0}(\Omega)$ to (1) corresponding to coefficients $a$ and the right hand side $f$, and the non-local flux $q=a|D_{s,p}u|^{p-2}D_{s,p}u\in L^{p^{\prime}}(\Omega\times\Omega)$.

In this paper we are concerned with $H$-convergence of the nonlocal operators $\operatorname{\mathcal{L}}_{a_{k}}$ for a given sequence of coefficients $a_{k}$.

Definition 1**.**

Given the sequence of coefficients $\{a_{k}\}_{k=1}^{\infty}\subset\mathcal{A}_{\lambda,\Lambda}$, we say that $\operatorname{\mathcal{L}}_{a_{k}}$ $H$-converges to $\operatorname{\mathcal{L}}_{a}$ if for any $f\in W^{-s,p}(\Omega)$ (the dual space of $W^{s,p}_{0}(\Omega)$) the following conditions are satisfied:

convergence of states: $u_{k}\rightharpoonup u$, weakly in $W_{0}^{s,p}(\Omega)$; 2. 2.

convergence of non-local fluxes: $q_{k}\rightharpoonup q$, weakly in $L^{p^{\prime}}(\Omega\times\Omega)$,

where $(u_{k},q_{k})=\mathcal{S}_{a_{k}}f$, and $(u,q)=\mathcal{S}_{a}f$.

In [8] is was shown that for any sequence $\{a_{k}\}_{k=1}^{\infty}\subset\mathcal{A}_{\lambda,\Lambda}$ there exists a subsequence $\{a_{k^{\prime}}\}$ and a function $a\in\mathcal{A}_{\lambda,\frac{\Lambda^{p^{\prime}}}{\lambda}}$ such that $\operatorname{\mathcal{L}}_{a_{k^{\prime}}}$ $H$-converges to $\operatorname{\mathcal{L}}_{a}$. We show that in fact a $a$ belongs to the same class $\mathcal{A}_{\lambda,\Lambda}$. Furthermore, our main result establishes that weak-$*$ convergence of the sequence of coefficients is a necessary and sufficient condition for $H$-convergence in the considered case.

Theorem 2**.**

If $a_{k},a\in\mathcal{A}_{\lambda,\Lambda}$, then $a_{k}\rightharpoonup a$ weakly-$*$ in $L^{\infty}(\mathbb{R}^{n}\times\mathbb{R}^{n})$ if and only if $\operatorname{\mathcal{L}}_{a_{k}}$ $H$-converges to $\operatorname{\mathcal{L}}_{a}$.

This result generalizes, giving a simpler proof, previous results for a related nonlocal situation in the linear case [3, Th. 6]. We also refer to [15], where an abstract, general setting for nonlocal $H$-convergence is analyzed.

The outline of the paper is as follows. Section 2 is devoted to setting the functional analysis framework of this work, and to stating the main results from [8], which are the starting point of the investigation presented here. Section 3 deals with the relation of the nonlocal $H$-convergence notion introduced in Definition 1 with the weaker notion of $G$-convergence. We show that the two notions are equivalent. Additionally, we establish the uniqueness of $H$-limit. Finally, Section 4 contains the proof of Theorem 2.

2 Preliminaries

In this section we set the functional analysis framework in which problems are set and recall the main results from [8].

We start by recalling some fundamental facts about fractional Sobolev spaces. The space $W^{s,p}(\mathbb{R}^{n})$, previously defined, is equipped with the norm

[TABLE]

where $\|u\|_{p}$ is the usual norm of $u$ in $L^{p}(\mathbb{R}^{n})$ and

[TABLE]

is the Gagliardo seminorm [5, 1]. With this definition $W^{s,p}(\mathbb{R}^{n})$ is a separable and reflexive Banach space for $1<p<\infty$. $W^{s,p}_{0}(\Omega)$ is usually defined as

[TABLE]

and in the case $\Omega$ has a Lipschitz boundary the following identification holds

[TABLE]

An important mathematical fact is that, for $\Omega$ bounded, $W^{s,p}_{0}(\Omega)$ embeds continuously into $L^{p}(\Omega)$, thanks to the Poincaré’s inquality in this fractional situation: there exists $C=C(n,s,|\Omega|)>0$ such that

[TABLE]

for all $u\in W_{0}^{s,p}(\Omega)$. Furthermore, the Rellich-Kondrachov theorem can be extended to fractional Sobolev spaces, and the embedding of $W^{s,p}_{0}(\Omega)$ into $L^{p}(\Omega)$ is compact. Proofs of these results can be found, for instance, in [5]. The dual space of $W^{s,p}_{0}(\Omega)$ is denoted by $W^{-s,p^{\prime}}(\Omega)$, and its norm is given by

[TABLE]

We now focus on the precise statement of the problem (1), which should be understood in the weak sense. Thus, we require that $\operatorname{\mathcal{L}}_{a}u=f$ holds in the sense of distributions, and we say that $u\in W_{0}^{s,p}(\Omega)$ is a solution to (1) if

[TABLE]

for all test functions $v\in C_{0}^{\infty}(\Omega)$. The first result documents the well-posedness of the problem (1) [8, Prop. 2.2, Cor. 2.4].

Proposition 3**.**

For each $a\in\mathcal{A}_{\lambda,\Lambda}$ and $f\in W^{-p,s}(\Omega)$, the problem (1) admits a unique solution $u\in W_{0}^{s,p}(\Omega)$. Furthermore, this solution is also the unique minimizer in $W_{0}^{s,p}(\Omega)$ of functional

[TABLE]

In addition to the non-local gradient (2), for each $\phi\in L^{p^{\prime}}(\mathbb{R}^{n}\times\mathbb{R}^{n})$ it will be convenient to define the corresponding $(s,p)$-divergence operator by

[TABLE]

The following result summarizes several properties of these operators [8, Th. 3.1., Lem. 3.3].

Theorem 4**.**

The following assertions hold:

Integration by parts:* For each $\phi\in L^{p^{\prime}}(\mathbb{R}^{n}\times\mathbb{R}^{n})$ and $u\in W^{s,p}(\mathbb{R}^{n})$ we have the inclusion $d_{sp}\phi\in W^{-s,p^{\prime}}(\mathbb{R}^{n})$, the dual of $W^{s,p}(\mathbb{R}^{n})$, and the integration by parts formula*

[TABLE] 2. 2.

Nonlocal div-curl lemma:* given $\phi_{k},\phi\in L^{p^{\prime}}(\mathbb{R}^{n}\times\mathbb{R}^{n})$ and $v_{k},\,v\in W^{s,p}(\mathbb{R}^{n})$, $k=1,2,\dots$ such that $v_{k}\rightharpoonup v$, weakly in $W^{s,p}(\mathbb{R}^{n})$, $\phi_{k}\rightharpoonup\phi$, weakly in $L^{p^{\prime}}(\mathbb{R}^{n}\times\mathbb{R}^{n})$ and $d_{s,p}\phi_{k}\rightarrow d_{s,p}\phi$, strongly in $W_{loc}^{-s,p^{\prime}}(\mathbb{R}^{n})$ then*

[TABLE]

in the sense of distributions.

Note that owing to the integration by parts identity and the assumed symmetry of the conductivity $a(x,y)=a(y,x)$, equation (3) can be equivalently understood as $d_{s,p}q=2f$, where $(u,q)=\mathcal{S}_{a}f$.

There are several other results in the literature, which are related to the previous one. In other works dealing with nonlocal or fractional problems a nonlocal vector calculus has been developed in order deal with the involved operators. References including integration by parts formulas are [6, 9, 10, 4]. Regarding the div-curl lemma, this is a very interesting compensated compactness-type result in the nonlocal context. In [14] a general analytic perspective for div-curl lemma that includes the nonlocal situation is considered. It is interesting to refer to [4], where in a very related situation to the one analyzed here, the weak convergence of any minor of the Riesz fractional gradient of vector fields has been shown by means of a nonlocal Piola identity.

Now we are prepared to state our point of departure, [8, Theorem 4.6], which establishes that $\mathcal{A}_{\lambda,\Lambda}$ is sequentially relatively compact with respect to $H$-convergence.

Theorem 5**.**

$0<\lambda\leq\Lambda$. For any sequence $\{a_{k}\}\subset\mathcal{A}_{\lambda,\Lambda}$, there exists a subsequence $\{a_{k^{\prime}}\}$ and $a\in\mathcal{A}_{\lambda,\Lambda}$ such that $\operatorname{\mathcal{L}}_{a_{k}}$ $H$-converges to $\operatorname{\mathcal{L}}_{a}$.

To be precise this is not the exact statement of [8, Theorem 4.6], as it differs in the upper bound on the coefficients of the $H$-limiting problem. In  [8, Theorem 4.6] it is claimed that the $H$-limit $a\in\mathcal{A}_{\lambda,\frac{\Lambda^{p^{\prime}}}{\lambda}}$. However, we will show that the more natural upper bound

[TABLE]

holds in this case. Indeed, let us assume that $\operatorname{\mathcal{L}}_{a_{k}}$ $H$-converges to $a$. Let us further fix an arbitrary $f\in L^{p^{\prime}}\setminus\{0\}$, and put $(u_{k},q_{k})=\mathcal{S}_{a_{k}}f$, and $(u,q)=\mathcal{S}_{a}f$. Since $a_{k}\in\mathcal{A}_{\lambda,\Lambda}$, for any $\varphi\in C_{c}^{\infty}(\mathbb{R}^{n}\times\mathbb{R}^{n})$, $\varphi\geq 0$ we have

[TABLE]

The term on the left is weakly lower semicontinuous with respect to the fluxes, which converge weakly owing to the $H$-convergence assumption. The term on the right converges owing to the non-local div-curl lemma. Passing to the limit we therefore arrive at the inequality

[TABLE]

and as $\varphi$ is nonnegative but otherwise arbitrary,

[TABLE]

Additionally, since $f$ is arbitrary, $u$ is also arbitrary, and (5) holds for any $u\in W^{s,p}_{0}(\Omega)$, and hence the inequality (4) holds.

3 $G$-convergence

In the local case, $H$-convergence was proposed by Murat and Tartar, see for example [11], as an extension of the previously proposed $G$-convergence concept [13]. $G$-convergence was formulated for linear elliptic equations with symmetric coefficients in divergence form and only requires weak convergence of the states. $H$-convergence, on the other hand, requires convergence of both states and fluxes, and has been formulated for problems with non-symmetric coefficients. In the case of elliptic PDEs with symmetric coefficients both notion are known to coincide. In the more general case of non-symmetric coefficients $G$-convergence is less useful in the sense that the $G$-limit is not guaranteed to be unique [2, Section 1.3.2].

The previous definition of nonlocal $H$-convergence requires both convergence of the states and convergence of the fluxes. As a consequence of this, owing to the div-curl lemma, the associated energy

[TABLE]

with $(u,q)=\mathcal{S}_{a}f$ is continuous with respect to $H$-convergence. This is a remarkable and a desirable property, especially when dealing with optimal design problems. In this section, we show that in the considered nonlocal situation of scalar and symmetric coefficients in $\mathcal{A}_{\lambda,\Lambda}$, the requirement on flux convergence in the definition of $H$-convergence is unnecessary. In other words, nonlocal $G$-convergence implies $H$-convergence, precisely as in the local case. First of all we make rigorous the definition of nonlocal $G$-convergence.

Definition 6**.**

Given the sequence of coefficients $\{a_{k}\}_{k=1}^{\infty}\subset\mathcal{A}_{\lambda,\Lambda}$, we say that $\operatorname{\mathcal{L}}_{a_{k}}$ $G$-converges to $\operatorname{\mathcal{L}}_{a}$ if for any $f\in W^{-s,p}(\Omega)$ we have

[TABLE]

where $(u_{k},q_{k})=\mathcal{S}_{a_{k}}f$ and $(u,q)=\mathcal{S}_{a}f$.

Proposition 7**.**

Consider a sequence $\{a_{k}\}_{k=0}^{\infty}\subset\mathcal{A}_{\lambda,\Lambda}$. Then $\operatorname{\mathcal{L}}_{a_{k}}$ $H$-converges to $\operatorname{\mathcal{L}}_{a}$ if and only if $\operatorname{\mathcal{L}}_{a_{k}}$ $G$-converges to $\operatorname{\mathcal{L}}_{a}$, with $a\in\mathcal{A}_{\lambda,\Lambda}$.

Proof.

Obviously $H$-convergence implies $G$-convergence. For the sake of contradiction, let us now assume that we have $G$-convergence, but not $H$-convergence. For an arbitrary $f\in W^{-s,p}(\Omega)$ let $(u^{f}_{k},q^{f}_{k})=\mathcal{S}_{a_{k}}f$ and $(u^{f},q^{f})=\mathcal{S}_{a}f$. Owing to the assumed $G$-convergence we have $u^{f}_{k}\rightharpoonup u^{f}$ in $W^{s,p}_{0}(\Omega)$. The assumed lack of $H$-convergence is equivalent to saying that for some $\hat{f}\in W^{-s,p}(\Omega)$ there is a subsequence $a_{k^{\prime}}$ and a $L^{p^{\prime}}(\mathbb{R}^{n}\times\mathbb{R}^{n})$-weakly open neighbourhood $\hat{N}$ of $q^{\hat{f}}$, such that $q^{\hat{f}}_{k^{\prime}}\not\in\hat{N}$, for all $k^{\prime}$. Owing to Theorem 5, the set $\{\operatorname{\mathcal{L}}_{a_{k^{\prime}}}\}$ is relatively sequentially compact with respect to $H$-convergence. Therefore there is a further subsequence $a_{k^{\prime\prime}}$ and $\tilde{a}\in\mathcal{A}_{\lambda,\Lambda}$ such that $\operatorname{\mathcal{L}}_{a_{k^{\prime\prime}}}$ $H$-converges to $\operatorname{\mathcal{L}}_{\tilde{a}}$. Consequently, if we put $(\tilde{u}^{f},\tilde{q}^{f})=\mathcal{S}_{\tilde{a}}f$, then $u^{f}=\tilde{u}^{f}$ owing to the assumed $G$-convergence and the uniqueness of the weak limit, and the uniqueness of solutions to (3). Therefore we have the equality

[TABLE]

which holds for an arbitrary $f$. Since $f$ is arbitrary, $u^{f}\in W^{s,p}_{0}(\Omega)$ is also arbitrary. Applying [7, Proposition 17], which characterizes null nonlocal functionals, we obtain that necessarily $a=\tilde{a}$, a.e. in $\mathbb{R}^{n}\times\mathbb{R}^{n}$. But then $\tilde{q}^{f}=q^{f}$, $q_{k^{\prime\prime}}^{f}\rightharpoonup q^{f}$ in $L^{p^{\prime}}(\mathbb{R}^{n}\times\mathbb{R}^{n})$, and consequently $q^{\hat{f}}_{k^{\prime\prime}}\in\hat{N}$ for all large $k^{\prime\prime}$, which is a contradiction. ∎

Another important point is the uniqueness of the $G$-limit. This is established in the next result.

Proposition 8**.**

The $G$-limit of a sequence in $\mathcal{A}_{\lambda,\Lambda}$ is unique.

Proof.

The proof follows the lines of the second part of proof of Proposition 7. Let us assume that the sequence $\{a_{n}\}\subset\mathcal{A}_{\lambda,\Lambda}$ $G$-converges to both $a$ and $\tilde{a}$. Then, arguing as in the proof of Proposition 7, we have that

[TABLE]

for any $f$, where $u^{f}$ is the solution of the Dirichlet problem (1) for both coefficients $a$ and $\tilde{a}$. The same argument as above yields the conclusion

[TABLE]

∎

4 Proof of Theorem 2

This section is devoted to the proof Theorem 2.

We claim that weak-* convergence is sufficient for $H$-convergence

Indeed, let us assume that $\{a_{k}\}_{k=1}^{\infty}\in\mathcal{A}_{\lambda,\Lambda}$ be such that $a_{k}\rightharpoonup a\in\mathcal{A}_{\lambda,\Lambda}$, weak-* in $L^{\infty}(\mathbb{R}^{n}\times\mathbb{R}^{n})$. Owing to Theorem 5 there exists a subsequence $\{a_{k^{\prime}}\}$ and $\tilde{a}\in\mathcal{A}_{\lambda,\Lambda}$ such that $\operatorname{\mathcal{L}}_{a_{k^{\prime}}}$ $H$-converges to $\operatorname{\mathcal{L}}_{\tilde{a}}$. Let $f\in W^{-s,p}(\Omega)$ be fixed but arbitrary, and let $(u_{k},q_{k})=\mathcal{S}_{a_{k}}f$, $(u,q)=\mathcal{S}_{a}f$, and $(\tilde{u},\tilde{q})=\mathcal{S}_{\tilde{a}}f$. Owing to $H$-convergence, $u_{k^{\prime}}\rightharpoonup\tilde{u}$, weakly in $W^{s,p}_{0}(\Omega)$. Recalling that $u_{k^{\prime}}=0$ in $\mathbb{R}^{n}\backslash\Omega$ and the compact embedding of $W^{s,p}_{0}(\Omega)$ into $L^{p}(\Omega)$, $\|u_{k^{\prime}}-\tilde{u}\|_{L^{p}(\mathbb{R}^{n})}\to 0$. In particular, there is a further subsequence $u_{k^{\prime\prime}}$ of $u_{k^{\prime}}$, such that $u_{k^{\prime\prime}}(x)\to\tilde{u}(x)$, for almost all $x\in\mathbb{R}^{n}$.

Owing to the variational characterization of solutions to (3) given in Proposition 3, we have the inequality $I_{a_{k}}(u_{k})\leq I_{a_{k}}(u)$, $\forall\>k=1,2,\dots$ Taking into account the facts that $|D_{s,p}u|^{p}\in L^{1}(\mathbb{R}^{n}\times\mathbb{R}^{n})$ and $a_{k}\rightharpoonup a$ weak-$*$ in $L^{\infty}(\mathbb{R}^{n}\times\mathbb{R}^{n})$, we obtain the inequality

[TABLE]

On the other hand, let us define the measures

[TABLE]

and

[TABLE]

where $E\subset\mathbb{R}^{n}\times\mathbb{R}^{n}$ is an arbitrary Lebesgue measurable set. Weak-$*$ convergence of $a_{k}$ to $a$ implies the strong convergence of these measures, that is, $\lim_{k\rightarrow\infty}\nu_{k}(E)=\nu(E)$ for any measurable set $E\subset\mathbb{R}^{n}\times\mathbb{R}^{n}$. Since $u_{k^{\prime\prime}}(x)\to\tilde{u}(x)$ for almost all $x\in\mathbb{R}^{n}$, it follows that $|D_{s,p}u_{k^{\prime\prime}}(x,y)|^{p}\rightarrow|D_{s,p}\tilde{u}(x,y)|^{p}$, for almost all $(x,y)\in\mathbb{R}^{n}\times\mathbb{R}^{n}$. These facts together with the upper bound (6) allow us to apply the generalized Fatou’s lemma [12, Proposition 17, p. 269] to get the inequality

[TABLE]

Therefore $I_{a}(u)=I_{a}(\tilde{u})$, whence $u=\tilde{u}$, owing to the uniqueness of solutions to (3) and their variational characterization. Arguing further as in the proof of Proposition 7, we conclude that $a=\tilde{a}$, almost everywhere in $\mathbb{R}^{n}\times\mathbb{R}^{n}$. Finally, since from every subsequence of $\operatorname{\mathcal{L}}_{a_{k}}$ we can extract a further subsequence, which $H$-converges to $\operatorname{\mathcal{L}}_{a}$, the whole sequence must converge to $\operatorname{\mathcal{L}}_{a}$.

We now claim that weak-* convergence is also necessary for $H$-convergence

Assume that $\operatorname{\mathcal{L}}_{a_{k}}$ $H$-converges to $\operatorname{\mathcal{L}}_{a}$, but for some weak-* open neighbourhood $N$ of $a\in\mathcal{A}_{\lambda,\Lambda}$ and a subsequence $k^{\prime}$ we have $a_{k^{\prime}}\not\in N$. Since $\{a_{k^{\prime}}\}_{k^{\prime}=1}^{\infty}\subset\mathcal{A}_{\lambda,\Lambda}$ and is thus bounded in $L^{\infty}(\mathbb{R}^{n}\times\mathbb{R}^{n})$, it has a non-empty set of weak-* limit points. Suppose that $a_{k^{\prime\prime}}\rightharpoonup\tilde{a}\in\mathcal{A}_{\lambda,\Lambda}$ for some further subsequence $k^{\prime\prime}=1,2,\dots$ By the already established implication, $\operatorname{\mathcal{L}}_{a_{k^{\prime\prime}}}$ $H$-converges to $\operatorname{\mathcal{L}}_{\tilde{a}}$. Owing to Propositions 7, 8, and [7, Proposition 17] we necessarily have $\tilde{a}=a$. But then $a_{k^{\prime\prime}}\in N$, for all large enough $k^{\prime\prime}$, which is a contradiction. This completes the proof.

Acknowledgements

AE’s research is financially supported by the Villum Fonden through the Villum Investigator Project InnoTop. The work of JCB is funded by FEDER EU and Ministerio de Economía y Competitividad (Spain) through grant MTM2017-83740-P.