G -convergence in a semilinear problem with a W 0 1,1 solution

Abstract

We study the stability, with respect to the G-convergence, of the $W_{0}^{1, 1}$ distributional solutions of a degenerate elliptic equation.

1. Introduction

In [4] (see also [5]) is proved the existence of a $W_{0}^{1, 1}$ (not only in $BV$ ) distributional solution for the following boundary value problem: $\{\begin{matrix} - div (\frac{M_{ε} (x) \nabla u_{ε}}{{(1 + b (x) | u_{ε} |)}^{2}}) + u_{ε} = f & in Ω, \\ u_{ε} = 0 & on \partial Ω, \end{matrix}$ (1) where Ω is a bounded, open subset of $R^{N}$ , with $N > 2$ , $M_{ε} (x)$ , $b (x)$ are measurable functions such that $α I ⩽ M_{ε} (x) ⩽ β I, 0 ⩽ b (x) ⩽ B,$ (2) with $α, β \in R^{+}$ , $B \in R^{+}$ and $f (x) belongs to L^{2} (Ω) .$ (3) Problems like (1) have been extensively studied in the past. In [6], existence (in some Sobolev space $W_{0}^{1, q} (Ω)$ , $q = q (θ, m) ⩽ 2$ ) and regularity results were obtained for $\{\begin{matrix} - div (\frac{M_{ε} (x) \nabla u_{ε}}{{(1 + | u_{ε} |)}^{θ}}) = f & in Ω, \\ u_{ε} = 0 & on \partial Ω, \end{matrix}$ (4) where $0 < θ ⩽ 1$ and f belongs to $L^{m} (Ω)$ for some $m ⩾ 1$ . Moreover, a nonexistence result is proved in [1], if $θ > 1$ , and a borderline case in studied in [2].

As pointed out in [3], existence of solutions can be recovered for any value of $θ > 0$ , by adding a lower order term of order zero. If we consider the problem $\{\begin{matrix} - div (\frac{M_{ε} (x) \nabla u_{ε}}{{(1 + | u_{ε} |)}^{2}}) + u_{ε} = f & in Ω, \\ u = 0 & on \partial Ω, \end{matrix}$ (5) with f in $L^{m} (Ω)$ , then the following results can be proved (see [3] and [8]):

$m > N$ , then there exists a weak solution in $W_{0}^{1, 2} (Ω) \cap L^{\infty} (Ω)$ ;

$m ⩾ 4$ , then there exists a weak solution in $W_{0}^{1, 2} (Ω) \cap L^{m} (Ω)$ ;

if $2 < m < 4$ , then there exists a distributional solution in $W_{0}^{1, \frac{2 m}{2 + 2}} (Ω) \cap L^{m} (Ω)$ ;

the borderline case $m = 2$ is studied in [4] and [5] and it is proved that there exists a distributional solution in $W_{0}^{1, 1} (Ω) \cap L^{2} (Ω)$ , that is $\{\begin{matrix} u \in W_{0}^{1, 1} (Ω) \cap L^{2} (Ω) : \int_{Ω} \frac{M (x) \nabla u \nabla v}{{(1 + b (x) | u |)}^{2}} + \int_{Ω} u v = \int_{Ω} f v, \\ \forall v \in W_{0}^{1, \infty} (Ω) . \end{matrix}$

On the other hand, it is very easy to say something more about the test functions used in [4,5].

Remark 1.

It is possible to prove the existence of solutions in the following sense $\{\begin{matrix} u \in W_{0}^{1, 1} (Ω) \cap L^{2} (Ω) : \int_{Ω} \frac{M (x) \nabla u \nabla v}{{(1 + b (x) | u |)}^{2}} + \int_{Ω} u v = \int_{Ω} f v, \\ \forall v \in W_{0}^{1, 2} (Ω), \end{matrix}$ (6) since we have $\int_{Ω} \frac{M (x) \nabla u \nabla u}{{(1 + b (x) | u |)}^{4}} ⩽ \int_{Ω} \frac{M (x) \nabla u \nabla u}{{(1 + b (x) | u |)}^{2}} < \infty,$ which implies that $\frac{M (x) \nabla u}{{(1 + b (x) | u |)}^{2}} \in {(L^{2} (Ω))}^{N} .$ We point out that the existence in the Sobolev space $W_{0}^{1, 1} (Ω)$ (instead of $BV (Ω)$ ) is unusual in the framework of elliptic problems.

In this paper we study the stability of the solutions $u_{ε} \in W_{0}^{1, 1} (Ω)$ of (1) with respect to the G-convergence of the differential operators. The study of the stability of the solutions of (4) can be found in [12].

2. G-convergence

Definition 2 (Spagnolo–Murat–Tartar).

Let ${M_{ε} (x)}$ be a sequence of matrices which satisfies (2), for almost every $x \in Ω$ and for every ξ in $R^{N}$ . The sequence ${M_{ε} (x)}$ is said to G-converges to a bounded, elliptic matrix $M_{0} (x)$ if for every g in $W^{- 1, 2} (Ω)$ the sequence ${w_{ε}}$ of the unique solutions $w_{ε} \in W_{0}^{1, 2} (Ω), - div (M_{ε} (x) \nabla w_{ε}) = g in Ω,$ (7) satisfies $\begin{matrix} w_{ε} ⇀ w_{0} & weakly in W_{0}^{1, 2} (Ω), \\ M_{ε} (x) \nabla w_{ε} ⇀ M_{0} (x) \nabla w_{0} & weakly in {(L^{2} (Ω))}^{N}, \end{matrix}$ (8) where $w_{0}$ is the unique solution of $w_{0} \in W_{0}^{1, 2} (Ω), - div (M_{0} (x) \nabla w_{0}) = g in Ω .$ (9)

This notion of G-convergence was introduced by Spagnolo (in [13]) in the symmetric case. He proved the following compactness theorem: any sequence of symmetric, uniformly coercive and uniformly bounded matrices $M_{ε} (x)$ admits a subsequence which G-converges to a matrix $M (x)$ of the same type.

A relationship between G-convergence of differential operators and Γ(weak- $W_{0}^{1, 2} (Ω)$ )-convergence can be found in [7].

The above general definition (non-symmetric matrices, convergence (8)) with the name of G-convergence is due to Murat–Tartar ([10]).

For every $f \in L^{2} (Ω)$ , we consider the boundary value problem $\{\begin{matrix} - div (\frac{M_{ε} (x) \nabla u_{ε}}{{(1 + b (x) | u_{ε} |)}^{2}}) + u_{ε} = f & in Ω, \\ u_{ε} = 0 & on \partial Ω, \end{matrix}$ (10) in the sense that it satisfies $\int_{Ω} \frac{M_{ε} (x) \nabla u_{ε}}{{(1 + b (x) | u_{ε} |)}^{2}} \nabla v + \int_{Ω} u v = \int_{Ω} f v \forall v \in W_{0}^{1, 2} (Ω)$ and $\int_{Ω} \frac{| \nabla u_{ε} |^{2}}{{(1 + | u_{ε} |)}^{2}} < \infty .$ We repeat that there exists a solution $u_{ε} \in W_{0}^{1, 1} (Ω) \cap L^{2} (Ω)$ . Our G-convergence result is contained in the following theorem.

Theorem 3.

Taking a subsequence of ε such that ${M_{ε} (x)}$ H-converges to a matrix function $M_{0}$ , we have that for every $f \in L^{2} (Ω)$ , the function $u_{ε}$ is bounded in $L^{2} (Ω)$ and for every subsequence of $u_{ε}$ , still denoted by $u_{ε}$ , which converges weakly in $L^{2} (Ω)$ to some function $u_{0}$ , we have $\begin{array}{rclr} u_{ε} ⇀ u_{0} in W_{0}^{1, 1} (Ω), u_{ε} \to u_{0} in L^{2} (Ω), & (11) \\ \frac{M_{ε} (x) \nabla u_{ε}}{{(1 + b (x) | u_{ε} |)}^{2}} ⇀ \frac{M_{0} (x) \nabla u_{0}}{{(1 + b (x) | u_{0} |)}^{2}} in {(L^{2} (Ω))}^{N}, & (12) \end{array}$ and $u_{0}$ satisfies $\{\begin{matrix} - div (\frac{M_{0} (x) \nabla u_{0}}{{(1 + b (x) | u_{0} |)}^{2}}) + u_{0} = f & in Ω, \\ u_{0} = 0 & on \partial Ω . \end{matrix}$ (13)

Proof.

In the proof, we also reconsider some technical tools of [4] and we adapt them to the present framework. Step 1.

The existence of $u_{ε}$ in proved in [4].

Step 2.

We prove some a priori estimates on the sequence ${u_{ε}}$ . Let $k ⩾ 0$ , $i > 0$ , and let $ψ_{i, k} (s)$ be the function defined by $ψ_{i, k} (s) = \{\begin{matrix} 0 & if 0 ⩽ s ⩽ k, \\ i (s - k) & if k < s ⩽ k + \frac{1}{i}, \\ 1 & if s > k + \frac{1}{i}, \\ ψ_{i, k} (s) = - ψ_{i, k} (- s) & if s < 0 . \end{matrix}$ Note that $lim_{i \to + \infty} ψ_{i, k} (s) = \{\begin{matrix} 1 & if s > k, \\ 0 & if | s | ⩽ k, \\ - 1 & if s < - k . \end{matrix}$ We choose $| u_{ε} | ψ_{i, k} (u_{ε})$ as a test function in (1), and we obtain $\begin{array}{rcl} \int_{Ω} \frac{M_{ε} (x) | \nabla u_{ε} |^{2}}{{(1 + b (x) | u_{ε} |)}^{2}} | ψ_{i, k} (u_{ε}) | + \int_{Ω} \frac{M_{ε} (x) | \nabla u_{ε} |^{2}}{{(1 + b (x) | u_{ε} |)}^{2}} ψ_{i, k}^{'} (u_{ε}) | u_{ε} | + \int_{Ω} u_{ε} | u_{ε} | ψ_{i, k} (u_{ε}) \\ = \int_{Ω} f | u_{ε} | ψ_{i, k} (u_{ε}) . \end{array}$ Since $ψ_{i, k}^{'} (s) ⩾ 0$ , we can drop the second term; using (2), we have $α \int_{Ω} \frac{| \nabla u_{ε} |^{2}}{{(1 + b (x) | u_{ε} |)}^{2}} | ψ_{i, k} (u_{ε}) | + \int_{Ω} u_{ε} | u_{ε} | ψ_{i, k} (u_{ε}) ⩽ \int_{Ω} | f | | u_{ε} | | ψ_{i, k} (u_{ε}) | .$ Letting i tend to infinity, we thus obtain, by Fatou’s lemma (on the left-hand side) and by Lebesgue’s theorem (on the right-hand side, recall that $u_{ε}$ belongs to $L^{2} (Ω)$ ), $α \int_{{| u_{ε} | ⩾ k}} \frac{| \nabla u_{ε} |^{2}}{{(1 + b (x) | u_{ε} |)}^{2}} + \int_{{| u_{ε} | ⩾ k}} | u_{ε} |^{2} ⩽ \int_{{| u_{ε} | ⩾ k}} | f | | u_{ε} | .$ (14) Dropping the nonnegative first term in (14) and using Hölder’s inequality on the right-hand side, we obtain $\int_{{| u_{ε} | ⩾ k}} | u_{ε} |^{2} ⩽ {[\int_{{| u_{ε} | ⩾ k}} | f |^{2}]}^{\frac{1}{2}} {[\int_{{| u_{ε} | ⩾ k}} | u_{ε} |^{2}]}^{\frac{1}{2}} .$ Simplifying equal terms we thus have $\int_{{| u_{ε} | ⩾ k}} | u_{ε} |^{2} ⩽ \int_{{| u_{ε} | ⩾ k}} | f |^{2} .$ (15) For $k = 0$ , the inequality (15) gives $\int_{Ω} | u_{ε} |^{2} ⩽ \int_{Ω} | f |^{2},$ (16) so that ${u_{ε}}$ is bounded in $L^{2} (Ω)$ . This fact implies in particular that $lim_{k \to + \infty} meas ({| u_{ε} | ⩾ k}) = 0, uniformly with respect to ε .$ (17)

From (14), written for $k = 0$ , dropping the nonnegative second term and using that $b (x) ⩽ B$ , we have $α \int_{Ω} \frac{| \nabla u_{ε} |^{2}}{{(1 + B | u_{ε} |)}^{2}} ⩽ \int_{Ω} | f | | u_{ε} | .$ Hölder’s inequality on the right-hand side then gives $α \int_{Ω} \frac{| \nabla u_{ε} |^{2}}{{(1 + B | u_{ε} |)}^{2}} ⩽ {[\int_{Ω} | f |^{2}]}^{\frac{1}{2}} {[\int_{Ω} | u_{ε} |^{2}]}^{\frac{1}{2}},$ so that, by (16), we infer that $α \int_{Ω} \frac{| \nabla u_{ε} |^{2}}{{(1 + B | u_{ε} |)}^{2}} ⩽ \int_{Ω} | f |^{2} .$ (18)

Step 3.

We prove that, up to subsequences, the sequence ${u_{ε}}$ strongly converges in $L^{2} (Ω)$ to some function u.

From (18) we deduce that $v_{ε} = log (1 + B | u_{ε} |) sgn (u_{ε})$ is bounded in $W_{0}^{1, 2} (Ω)$ . Therefore, up to subsequences, it converges to some function v weakly in $W_{0}^{1, 2} (Ω)$ , strongly in $L^{2} (Ω)$ , and almost everywhere in Ω. If we define $u = \frac{e^{| v |} - 1}{B} sgn (v)$ , then $u_{ε}$ converges almost everywhere to u in Ω. Let now E be a measurable subset of Ω; then $\begin{array}{rcl} \int_{E} | u_{ε} |^{2} & ⩽ & \int_{E \cap {| u_{ε} | ⩾ k}} | u_{ε} |^{2} + \int_{E \cap {| u_{ε} | < k}} | u_{ε} |^{2} \\ ⩽ & \int_{{| u_{ε} | ⩾ k}} | f |^{2} + k^{2} meas (E) \forall k > 0, \end{array}$ where we have used (15) in the last passage. Thanks to (17), we may choose k large enough so that the first integral is small, uniformly with respect to ε; once k is chosen, we may choose the measure of E small enough such that the second term is small. Thus, the sequence ${u_{ε}^{2}}$ is equi-integrable and so, by Vitali’s theorem, $u_{ε}$ strongly converges to u in $L^{2} (Ω)$ .

Step 4.

We prove that, up to subsequences, the sequence ${u_{ε}}$ weakly converges to u in $W_{0}^{1, 1} (Ω)$ .

Let again E be a measurable subset of Ω, and let i be in ${1, \dots, N}$ . Then $\begin{array}{rcl} \int_{E} | \partial_{i} u_{ε} | & ⩽ & \int_{E} | \nabla u_{ε} | = \int_{E} \frac{| \nabla u_{ε} |}{1 + B | u_{ε} |} (1 + B | u_{ε} |) \\ ⩽ & {[\int_{E} \frac{| \nabla u_{ε} |^{2}}{{(1 + B | u_{ε} |)}^{2}}]}^{\frac{1}{2}} {[\int_{E} {(1 + B | u_{ε} |)}^{2}]}^{\frac{1}{2}} \\ ⩽ & {[\frac{1}{α} \int_{Ω} | f |^{2}]}^{\frac{1}{2}} {[2 meas (E) + 2 B^{2} \int_{E} | u_{ε} |^{2}]}^{\frac{1}{2}}, \end{array}$ where we have used (18) in the last passage. Since the sequence ${u_{ε}}$ is compact in $L^{2} (Ω)$ , we have that the sequence ${\partial_{i} u_{ε}}$ is equi-integrable. Thus, by Dunford–Pettis theorem, and up to subsequences, there exists $Y_{i}$ in $L^{1} (Ω)$ such that $\partial_{i} u_{ε}$ weakly converges to $Y_{i}$ in $L^{1} (Ω)$ . Since $\partial_{i} u_{ε}$ is the distributional derivative of $u_{ε}$ , we have, for every n in $N$ , $\int_{Ω} \partial_{i} u_{ε} φ = - \int_{Ω} u_{ε} \partial_{i} φ \forall φ \in C_{0}^{\infty} (Ω) .$ We now pass to the limit in the above identities, using that $\partial_{i} u_{ε}$ weakly converges to $Y_{i}$ in $L^{1} (Ω)$ , and that $u_{ε}$ strongly converges to u in $L^{2} (Ω)$ ; we obtain $\int_{Ω} Y_{i} φ = - \int_{Ω} u_{ε} \partial_{i} φ \forall φ \in C_{0}^{\infty} (Ω),$ which implies that $Y_{i} = \partial_{i} u$ , and this result is true for every i. Since $Y_{i}$ belongs to $L^{1} (Ω)$ for every i, u belongs to $W_{0}^{1, 1} (Ω)$ , as desired.

Note now that, since $s \mapsto log (1 + B s)$ is Lipschitz continuous on $R^{+}$ , and u belongs to $W_{0}^{1, 1} (Ω)$ , by the chain rule we have $\nabla [log (1 + B | u_{ε} |) sgn (u)] = \frac{\nabla u_{ε}}{1 + B | u_{ε} |}, almost everywhere in Ω .$ Hence, from the weak convergence of $v_{ε}$ to v in $W_{0}^{1, 2} (Ω)$ we deduce that $lim_{ε \to 0} \frac{\nabla u_{ε}}{1 + B | u_{ε} |} = \frac{\nabla u}{1 + B | u |} weakly in {(L^{2} (Ω))}^{N} .$ (19)

Step 5.

We now pass to the limit in problem (10). For this purpose, we consider the function $z_{ε} = (1 - \frac{1}{1 + | u_{ε} |}) sgn (u_{ε}),$ which, by using (10), satisfies the problem $\{\begin{matrix} - div ({(\frac{1 + | u_{ε} |}{1 + b (x) | u_{ε} |})}^{2} M_{ε} (x) \nabla z_{ε}) + u_{ε} = f & in Ω, \\ z_{ε} = 0 & on \partial Ω . \end{matrix}$ (20) Moreover, taking into account $\int_{Ω} | \nabla z_{ε} |^{2} = \int_{Ω} \frac{| \nabla u_{ε} |^{2}}{{(1 + | u_{ε} |)}^{4}} ⩽ \int_{Ω} \frac{| \nabla u_{ε} |^{2}}{{(1 + | u_{ε} |)}^{2}},$ we have that the sequence ${z_{ε}}$ is bounded in $W_{0}^{1, 2} (Ω)$ , which combined with $z_{ε}$ converging a.e. in Ω to $z = (1 - \frac{1}{1 + | u |}) sgn (u),$ implies that $z_{ε}$ converges weakly in $W_{0}^{1, 2} (Ω)$ to z. Using then that $f - u_{ε}$ converges to $f - u$ strongly in $L^{2} (Ω)$ and hence in $H^{- 1} (Ω)$ , we can use Lemma 4 to pass to the limit in (20) to deduce ${(\frac{1 + | u_{ε} |}{1 + b (x) | u_{ε} |})}^{2} M_{ε} (x) \nabla z_{ε} ⇀ {(\frac{1 + | u |}{1 + b (x) | u |})}^{2} M_{0} (x) \nabla z in {(L^{2} (Ω))}^{N},$ and that z satisfies $\{\begin{matrix} - div ({(\frac{1 + | u |}{1 + b (x) | u |})}^{2} M (x) \nabla z) + u = f in Ω, \\ z \in H_{0}^{1} (Ω) . \end{matrix}$ Now we recall the uniqueness of the solution of (13) proved in [5]. Then using the definitions of $z_{ε}$ and z, this proves that $u_{0} = u$ satisfies (12) and (13).

□

Lemma 4.

Assume that ${M_{ε} (x)}$ H-converges to a matrix function $M_{0}$ , then, for every sequence ${q_{ε} (x)}$ in $L^{\infty} (Ω)$ , such that there exist $ρ, γ > 0$ with $ρ ⩽ q_{ε} ⩽ γ, a.e. in Ω$ and such that $q_{ε} \to q_{0}, a.e. in Ω,$ for some $q_{0}$ , we have that ${q_{ε} (x) M_{ε} (x)}$ H-converges to $q_{0} (x) M_{0} (x)$ .

Proof.

We need to prove that for every $g \in H^{- 1} (Ω)$ , the solution $w_{ε}$ of $w_{ε} \in W_{0}^{1, 2} (Ω), - div (q_{ε} (x) M_{ε} (x) \nabla w_{ε}) = g in Ω,$ (21) satisfies $w_{ε} ⇀ w_{0} in W_{0}^{1, 2} (Ω), q_{ε} (x) M_{ε} (x) \nabla w_{ε} ⇀ q_{0} (x) M_{0} (x) \nabla w_{0} in {(L^{2} (Ω))}^{N},$ with $w_{0}$ the unique solution of $w_{0} \in W_{0}^{1, 2} (Ω), - div (q_{0} (x) M_{0} (x) \nabla w_{0}) = g in Ω .$ (22)

We follow the ideas in the classical proof of the G-convergence compactness method ([10]). Since $w_{ε}$ is bounded in $W_{0}^{1, 2} (Ω)$ , we know that up to a subsequence, there exist some functions $w \in W_{0}^{1, 2} (Ω)$ and $σ \in {(L^{2} (Ω))}^{N}$ , such that, for a subsequence $w_{ε} ⇀ w in W_{0}^{1, 2} (Ω), q_{ε} (x) M_{ε} (x) \nabla w_{ε} ⇀ σ in {(L^{2} (Ω))}^{N} .$ (23) The problem is to characterize w and σ. For an open subset ω strictly contained in Ω we consider $ψ \in C_{c}^{\infty} (Ω)$ , with $ψ = 1$ in ω, and we introduce $z_{ε}^{i}$ , $i \in {1, \dots, N}$ , solution of $z_{ε}^{i} \in W_{0}^{1, 2} (Ω), - div (M_{ε} (x) \nabla z_{ε}^{i}) = - div (M_{0} (x) \nabla (ψ x_{i})) in Ω .$ (24) Then, by definition of G-convergence, we have that $z_{ε}^{i}$ satisfies $z_{ε}^{i} ⇀ ψ x_{i} in W_{0}^{1, 2} (Ω), M_{ε} \nabla z_{ε}^{i} ⇀ M_{0} \nabla (ψ x_{i}) in {(L^{2} (Ω))}^{N},$ (25) where, as a consequence of Meyer’s theorem ([9]), it is also known that ${| \nabla z_{ε}^{i} |}^{2} is equi-integrable in Ω .$ (26) Now, for $λ = (λ_{1}, \dots, λ_{N}) \in R^{N}$ , and $φ \in C_{0}^{\infty} (ω)$ let us pass to the limit in $0 ⩽ \int_{Ω} q_{ε} (x) M_{ε} (x) \nabla (w_{ε} - \sum_{i = 1}^{N} λ_{i} z_{ε}^{i}) \nabla (w_{ε} - \sum_{i = 1}^{N} λ_{i} z_{ε}^{i}) φ d x .$ (27) For this purpose let us use the div–curl lemma ([11,14]), which establishes that if ${ξ_{ε}}$ , ${η_{ε}}$ converge weakly in ${(L^{2} (Ω))}^{N}$ to $ξ_{0}$ and $η_{0}$ respectively and are such that the divergence of $ξ_{ε}$ and the curl of $η_{ε}$ are compact in $W^{- 1, 2} (Ω)$ and ${(W^{- 1, 2} (Ω))}^{N \times N}$ respectively, then ${ξ_{ε} η_{ε}}$ converges in the distribution sense to $ξ η$ .

Thus, using that $w_{ε}$ satisfies (21), the convergences (23) and (25), and $ψ = 1$ in the support of φ, we get $\int_{Ω} q_{ε} (x) M_{ε} (x) \nabla w_{ε} \nabla (w_{ε} - \sum_{i = 1}^{N} λ_{i} z_{ε}^{i}) φ d x \to \int_{Ω} σ (\nabla w - λ) φ d x .$ Analogously, the div–curl lemma and $z_{ε}^{i}$ solution of (24) show that $M_{ε} (x) \nabla (\sum_{i = 1}^{N} λ_{i} z_{ε}^{i}) \nabla (w_{ε} - \sum_{i = 1}^{N} λ_{i} z_{ε}^{i}) φ ⇀ M_{0} (x) λ (\nabla w - λ) φ$ in the distribution sense. From (26), this convergence holds in fact weakly in $L^{1} (Ω)$ and then using that $q_{ε}$ is bounded in $L^{\infty} (Ω)$ and converges a.e. in Ω, a well-known consequence of Egorov’s theorem proves $\begin{array}{rcl} \int_{Ω} q_{ε} (x) M_{ε} (x) \nabla (\sum_{i = 1}^{N} λ_{i} z_{ε}^{i}) \nabla (w_{ε} - \sum_{i = 1}^{N} λ_{i} z_{ε}^{i}) φ d x \\ \to \int_{Ω} q_{0} (x) M_{0} (x) λ (\nabla w - λ) φ d x . \end{array}$ Therefore, passing to the limit in (27), we have proved $0 ⩽ \int_{Ω} (σ - q_{0} (x) M_{0} (x) λ) (\nabla w - λ) φ d x,$ which by the arbitrariness of ω and φ shows $0 ⩽ (σ - q_{0} (x) M_{0} (x) λ) (\nabla w - λ) \forall λ \in R^{N}, a.e. in Ω .$ Taking $λ = \nabla w - t μ$ with $t > 0$ and $μ \in R^{N}$ , the above inequality reads as $0 ⩽ (σ - q_{0} (x) M_{0} (x) (\nabla w - t μ)) t μ \forall μ \in R^{N}, \forall t > 0, a.e. in Ω .$ Dividing by t and letting then t converging to 0, we get $0 ⩽ (σ - q_{0} (x) M_{0} (x) \nabla w) μ \forall μ \in R^{N}, a.e. in Ω .$ This proves that σ, defined by (23), agrees with $q_{0} (x) M_{0} \nabla w$ and then passing to the limit in (21) we get that w agrees with the solution $w_{0}$ of (22). □

Footnotes

Acknowledgements

This work has been partially supported by the projects MTM 2011-24457 of the “Ministerio de Ciencia e Innovación” of Spain.

This paper contains the unpublished part of the results presented by the first author in a talk at “Topics in Elliptic and Parabolic PDE’s, celebrating Guido Trombetti’s 65th birthday” (Napoli, September 2014).

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