ASYMPTOTIC BEHAVIOUR AND CORRECTORS

FOR LINEAR DIRICHLET PROBLEMS

WITH SIMULTANEOUSLY VARYING

OPERATORS AND DOMAINS

Gianni DAL MASO (^{∗})

François MURAT (^{∗∗})

Abstract

We consider a sequence of Dirichlet problems in varying domains (or, more generally, of relaxed Dirichlet problems involving measures in M_{0}^{+}(Ω)) for second order linear elliptic operators in divergence form with varying matrices of coefficients. When the matrices H -converge to a matrix A^{0} , we prove that there exist a subsequence and a measure μ^{0} in M_{0}^{+}(Ω) such that the limit problem is the relaxed Dirichlet problem corresponding to A^{0} and μ^{0} . We also prove a corrector result which provides an explicit approximation of the solutions in the H^{1} -norm, and which is obtained by multiplying the corrector for the H -converging matrices by some special test function which depends both on the varying matrices and on the varying domains.

(^{∗}) SISSA, via Beirut 4, 34014 Trieste, Italy

     e-mail: dalmaso@sissa.it

(^{∗∗}) Laboratoire Jacques-Louis Lions, Boîte courrier 187, Université Pierre et Marie Curie, 75252 Paris cedex 05, France

     e-mail: murat@ann.jussieu.fr

Ref. S.I.S.S.A. 40/02/M (May 2002)

1. Introduction

In this paper we consider a sequence of linear Dirichlet problems

(1.1)  \{ \begin{matrix}u^{ε} ∈ H_{0}^{1}(Ω^{ε}) , \\ −div(A^{ε}Du^{ε}) = f in D′(Ω^{ε}) , \end{matrix}

where the matrices A^{ε} and the domains Ω^{ε} both depend on the parameter ε. We assume that the open sets Ω^{ε} are all contained in a fixed bounded open subset Ω of R^{n}, and that the matrices A^{ε}, defined on Ω with measurable coefficients, are coercive and bounded, uniformly with respect to ε. Our goal is to study the behaviour of the solutions u^{ε} as ε tends to zero.

In the special case Ω^{ε} = Ω it is known (see Section 3) that there exist a subsequence, still denoted by (A^{ε}), and a matrix A^{0} , called the H -limit of (A^{ε}), such that for every f ∈ H^{−1}(Ω) the solutions v^{ε} of the problems

\{ \begin{matrix}v^{ε} ∈ H_{0}^{1}(Ω) , \\ −div(A^{ε}Dv^{ε}) = f in D′(Ω) , \end{matrix}

converge weakly in H_{0}^{1}(Ω) to the solution v^{0} of

\{ \begin{matrix}v^{0} ∈ H_{0}^{1}(Ω) , \\ −div(A^{0}Dv^{0}) = f in D′(Ω) , \end{matrix}

and satisfy also

A^{ε}Dv^{ε} ⇀ A^{0}Dv^{0} weakly in L^{2}(Ω, R^{n}) .

Without making any further hypothesis on the open sets Ω^{ε}, we prove in the present paper that there exists a subsequence, still denoted by (Ω^{ε}), such that for every f ∈ H^{−1}(Ω) the solutions u^{ε} of (1.1) converge to the solution u^{0} of the problem

(1.2)  \{ \begin{matrix}u^{0} ∈ H_{0}^{1}(Ω) ∩ L^{2}(Ω, μ^{0}) , \\ ∫_{Ω} A^{0}Du^{0}Dy dx + ∫_{Ω} u^{0}y dμ^{0} = ⟨f, y⟩  ∀y ∈ H_{0}^{1}(Ω) ∩ L^{2}(Ω, μ^{0}) , \end{matrix}

where μ^{0} belongs to M_{0}^{+}(Ω), a class of a nonnegative Borel measures which vanish on all sets of capacity zero, but can take the value +∞ on some subsets of Ω (see Section 2).

Problems like (1.2) are called relaxed Dirichlet problems (see Section 4) and have been extensively studied to describe the limits of the solutions of (1.1) when the matrices A^{ε} do not depend on ε. On the other hand, problems (1.1) can be written as relaxed Dirichlet problems (see Remark 4.1) by considering the measures μ^{ε} defined by

(1.3)  μ^{ε}(B) = \{ \begin{matrix}0, & if cap(B \ Ω^{ε}) = 0, \\ +∞, & otherwise. \end{matrix}

Actually in the paper we consider not only the case of Dirichlet problems (1.1), which correspond to the measures μ^{ε} defined by (1.3), but more in general we study the case of a sequence of relaxed Dirichlet problems with arbitrary μ^{ε} ∈ M_{0}^{+}(Ω).

In the limit problem (1.2) the measure μ^{0} does not depend on f, but, as shown in Section 6, it depends both on the sequence of sets (Ω^{ε}) and on the sequence of matrices (A^{ε}) (and not only on its H -limit A^{0} ). Nevertheless the sequence (Ω^{ε}) has a stronger influence than the sequence (A^{ε}): indeed the limit measures corresponding to the same sequence (Ω^{ε}) but to different sequences (A_{i}^{ε}) are equivalent (see Theorem 8.1).

In Section 5 we give a fairly general and flexible method to construct the limit measure μ^{0} using suitable test functions ω^{ε} associated to Ω^{ε} and A^{ε}. We then pass to the limit in the sequence of problems (1.1) by a duality argument and obtain (1.2).

In Section 7 we continue the study of the behaviour of the solutions u^{ε} of (1.1) by giving a corrector result. By this we mean the following: when the solution u^{0} of the limit problem (1.2) can be written as

(1.4)  u^{0} = ψ ω^{0} ,

where ω^{0} is the limit of the above test functions ω^{ε} and ψ is sufficiently smooth (actually in H^{2}(Ω) ∩ W^{1,∞}(Ω)), we prove that

(1.5)  u^{ε} = (ψ + ∑_{j=1}^{n} D_{j}ψ z_{j}^{ε}) ω^{ε} + r^{ε}  with r^{ε} → 0 strongly in H_{0}^{1}(Ω) ,

where the functions z_{j}^{ε} depend only on the matrices A^{ε}. This provides an approximation of u^{ε} in the norm of H_{0}^{1}(Ω) by means of functions that are constructed explicitly.

When (1.4) is not satisfied with a smooth ψ, a similar but more technical result holds (see Theorem 7.2). We also prove a local version of this corrector result.

Moreover, we prove (global and local) convergence and corrector results when also the right hand side of (1.1) depends on ε and converges strongly in a convenient sense (see Section 10).

Let us finally note that results similar to those presented in this paper have been recently obtained by Calvo Jurado and Casado Diaz in [6] for a class of nonlinear monotone elliptic equations.

Contents

1. Introduction

2. Preliminaries on capacity and measures

3. H -convergence

4. Relaxed Dirichlet problems

5. A global convergence result

6. An example

7. Global and local corrector results

8. A comparison theorem

9. Proofs of the corrector results

10. Problems with more general data

2. Preliminaries on capacity and measures

In this section we first introduce a few notation. Then we recall some known results on measures, capacity, and fine properties of Sobolev functions.

Notation

Throughout the paper Ω is a bounded open subset of R^{n}, n ≥ 1. The space D′(Ω) of distributions in Ω is the dual of the space C_{c}^{∞}(Ω). The space W_{0}^{1,p}(Ω), 1 ≤ p < +∞, is the closure of C_{c}^{∞}(Ω) in the Sobolev space W^{1,p}(Ω), while W^{−1,q}(Ω), 1 ≤ q < +∞, is the space of all distributions of the form f = f_{0} + ∑_{j} D_{j}f_{j}, with f_{0}, f_{1}, …, f_{n} ∈ L^{q}(Ω) (if 1/p + 1/q = 1, then W^{−1,q}(Ω) is the dual of W_{0}^{1,p}(Ω)). In the Hilbert case p = q = 2 these spaces are denoted by H_{0}^{1}(Ω), H^{1}(Ω), and H^{−1}(Ω), respectively. The norm in H_{0}^{1}(Ω) is defined by

∥u∥_{H_{0}^{1}(Ω)} = ( ∫_{Ω} |Du|^{2}dx)^{\frac{1}{2}} ,

while the duality pairing between H^{−1}(Ω) and H_{0}^{1}(Ω) is denoted by ⟨·, ·⟩. We shall sometimes use also the Sobolev space H^{2}(Ω) = W^{2,2}(Ω).

The adjoint of a matrix A is denoted by \overline{A}. Since complex numbers are not used in this paper, the bar never denotes complex conjugation. If w is an object related to the matrix A, then \overline{w} denotes the corresponding object related to the adjoint \overline{A}.

Throughout the paper ε varies in a stricly decreasing sequence of positive real numbers which converges to 0. When we write ε > 0, we consider only the elements of this sequence, while when we write ε ≥ 0 we also consider its limit ε = 0.

Capacity and measures

For every subset E of Ω the capacity of E in Ω, denoted by cap(E), is defined as the infimum of ∫_{Ω} |Du|^{2} dx over the set of all functions u ∈ H_{0}^{1}(Ω) such that u ≥ 1 a.e. in a neighbourhood of E. We say that a property P(x) holds quasi everywhere (abbreviated as q.e.) in a set E if it holds for all x ∈ E except for a subset N of E with cap(N) = 0. The expression almost everywhere (abbreviated as a.e.) refers, as usual, to the analogous property for the Lebesgue measure.

A function u: Ω → R is said to be quasi continuous if for every ε > 0 there exists a set E ⊆ Ω, with cap(E) < ε, such that the restriction of u to Ω \ E is continuous. A subset U of Ω is said to be quasi open if for every ε > 0 there exists an open set V ⊆ Ω, with cap(V △ U) < ε, where △ denotes the symmetric difference.

Every u ∈ H^{1}(Ω) has a quasi continuous representative, which is uniquely defined up to a set of capacity zero. In the sequel we shall always identify u with its quasi continuous representative, so that the pointwise values of a function u ∈ H^{1}(Ω) are defined quasi everywhere in Ω. If u ∈ H^{1}(Ω), then

(2.1)  u ≥ 0 a.e. in Ω ⟺ u ≥ 0 q.e. in Ω .

If a sequence (u_{j}) converges to u strongly in H_{0}^{1}(Ω), then a subsequence of (u_{j}) converges to u q.e. in Ω. For all these properties concerning quasi continuous representatives of Sobolev functions we refer to [17], Section 4.8, [20], Section 7.2.4, [18], Section 4, or [27], Chapter 3.

The characteristic function 1_{E} of a set E ⊆ Ω is defined by 1_{E}(x) = 1 if x ∈ E and 1_{E}(x) = 0 if x ∈ Ω \ E. The following lemma (see [9], Lemma 1.5, or [11], Lemma 1.1) concerns the pointwise approximation of the characteristic function of a quasi open set.

Lemma 2.1. For every quasi open set U of Ω there exists an increasing sequence (z_{k}) of nonnegative functions of H_{0}^{1}(Ω) converging to 1_{U} pointwise q.e. in Ω.

By a nonnegative Borel measure on Ω we mean a countably additive set function defined on the Borel subsets of Ω with values in [0, +∞]. By a nonnegative Radon measure on Ω we mean a nonnegative Borel measure which is finite on every compact subset of Ω. Every nonnegative Borel measure μ on Ω can be extended to a Borel regular outer measure on Ω by setting for every subset E of Ω

μ(E) = inf{μ(B) : B Borel, E ⊆ B ⊆ Ω} .

If μ is a nonnegative Borel measure on Ω, we shall use L^{r}(Ω, μ), 1 ≤ r ≤ +∞, to denote the usual Lebesgue space with respect to the measure μ. We adopt the standard notation L^{r}(Ω) when μ is the Lebesgue measure.

We will consider the cone M_{0}^{+}(Ω) of all nonnegative Borel measures μ on Ω such that

(a) μ(B) = 0 for every Borel set B ⊆ Ω with cap(B) = 0,

(b) μ(B) = inf{μ(U) : U quasi open , B ⊆ U} for every Borel set B ⊆ Ω. If E ⊆ Ω and cap(E) = 0, then E is contained in a Borel set B ⊆ Ω with cap(B) = 0. Therefore E is μ-measurable by (a). Property (b) is a weak regularity property of the measure μ. It is always satisfied if μ is a nonnegative Radon measure. Since any quasi open set differs from a Borel set by a set of capacity zero, every quasi open set is μ-measurable for every nonnegative Borel measure μ which satisfies (a).

Let us explicitly observe that the notation is not fixed in the literature and that in other works (see, e.g., [14]) M_{0}(Ω) denotes the set of nonnegative Borel measures which only satisfy (a), while the set that we call M_{0}^{+}(Ω) in the present paper is sometimes denoted by M_{0}^{∗}(Ω) (see, e.g., [10]).

For every quasi open set U ⊆ Ω we define the Borel measure μ_{U} by

(2.2)  μ_{U}(B) = \{ \begin{matrix}0, & if cap(B \ U) = 0, \\ +∞, & otherwise. \end{matrix}

Roughly speaking, μ_{U} is identically zero on U and identically +∞ on Ω \ U. It is easy to see that this measure belongs to the class M_{0}^{+}(Ω). Indeed, property (a) follows immediately from the definition, and it is enough to verify (b) only for every Borel set with μ_{U}(B) < +∞; in this case cap(B\U) = 0, and this implies that V = U ∪B is quasi open (since U is quasi open), contains B, and μ_{U}(V) = 0 (since cap(V \U) = cap(B\U) = 0), so that (b) is satisfied. The measures μ_{U} will be used to transform a sequence of Dirichlet problems on varying domains into a sequence of relaxed Dirichlet problems on a fixed domain (see Remark 4.1 and the proof of Corollary 5.5).

If μ ∈ M_{0}^{+}(Ω), then the space H^{1}(Ω)∩L^{2}(Ω, μ) is well defined, since every function u in H^{1}(Ω) is defined μ-almost everywhere and is μ-measurable in Ω (recall that u is quasi continuous, so that {u > t} is quasi open for every t ∈ R). It is easy to see that H^{1}(Ω) ∩ L^{2}(Ω, μ) is a Hilbert space for the scalar product

(2.3)  (u, v)_{H^{1}(Ω)∩L^{2}(Ω,μ)} = ∫_{Ω} Du Dv dx + ∫_{Ω} u v dx + ∫_{Ω} u v dμ

(see [5], Proposition 2.1).

The space of all (signed) Radon measures on Ω will be denoted by M(Ω), while M_{b}(Ω) will be the space of all μ ∈ M(Ω) with |μ|(Ω) < +∞, where |μ| denotes the total variation of μ. A subset A of M(Ω) is bounded if for every compact set K ⊆ Ω we have

sup_{μ∈A} |μ|(K) < +∞ .

Every Radon measure on Ω will be identified with an element of D′(Ω) in the usual way. Therefore μ belongs to M(Ω) ∩ W^{−1,q}(Ω) if and only if there exist f_{0}, f_{1}, …, f_{n} ∈ L^{q}(Ω) such that

∫_{Ω} φ dμ = ∫_{Ω} f_{0}φ dx − ∑_{j=1}^{n} ∫_{Ω} f_{j}D_{j}φ dx ∀φ ∈ C_{c}^{∞}(Ω) .

Note that, by the Riesz theorem, every nonnegative element of W^{−1,p}(Ω) is a nonnegative Radon measure on Ω.

The cone of all nonnegative elements of H^{−1}(Ω) will be denoted by H^{−1}(Ω)^{+} . It is well known that every element of H^{−1}(Ω)^{+} is a nonnegative Radon measure which belongs also to M_{0}^{+}(Ω). In other words we have the inclusion H^{−1}(Ω)^{+} ⊆ M(Ω) ∩ M_{0}^{+}(Ω).

3. H -convergence

In this section we recall the definition of H -convergence and the corresponding corrector result. Moreover we prove a fairly general convergence theorem for right hand sides which do not converge strongly in H^{−1}(Ω).

Throughout the paper we fix two constants α and β such that

0 < α ≤ β < +∞ .

We define M_{α}^{β}(Ω) as the set of all matrices A in L^{∞}(Ω, R^{n×n}) such that

(3.1)  A(x) ≥ αI ,  (A(x))^{−1} ≥ β^{−1}I ,  for a.e. x ∈ Ω .

In (3.1) I is the identity matrix in R^{n×n}, and the inequalities are in the sense of the quadratic forms defined byA(x)ξ ξ for ξ ∈ R^{n}. Note that (3.1) implies that

(3.2)  |A(x)| ≤ β  for a.e. x ∈ Ω ,

and that necessarily α ≤ β.

Definition of H -convergence

A sequence (A^{ε}) of matrices in M_{α}^{β}(Ω) H -converges to a matrix A^{0} in M_{α}^{β}(Ω) if for every f ∈ H^{−1}(Ω) the sequence (u^{ε}) of the solutions to the problems

(3.3)  \{ \begin{matrix}u^{ε} ∈ H_{0}^{1}(Ω) , \\ −div(A^{ε}Du^{ε}) = f in D′(Ω) , \end{matrix}

satisfies

u^{ε} ⇀ u^{0} weakly in H_{0}^{1}(Ω) ,
A^{ε}Du^{ε} ⇀ A^{0}Du^{0} weakly in L^{2}(Ω, R^{n}) ,

where u^{0} is the solution to the problem

(3.4)  \{ \begin{matrix}u^{0} ∈ H_{0}^{1}(Ω) , \\ −div(A^{0}Du^{0}) = f in D′(Ω) . \end{matrix}

Every sequence of matrices in M_{α}^{β}(Ω) has a subsequence which H -converges to a matrix in M_{α}^{β}(Ω) (see [25] and [23]).

Denoting the adjoint of A^{ε} by \overline{A}^{ε}, it is easy to prove that the sequence (\overline{A}^{ε}) H - converges to \overline{A}^{0} when the sequence (A^{ε}) H -converges to A^{0} .

If U is an open set contained in Ω, we can consider also the notion of H -convergence in U, replacing Ω by U in the definition. It is not difficult to prove that (A^{ε}) H - converges to A^{0} in U, for every open set U ⊆ Ω, if (A^{ε}) H -converges to A^{0} in Ω.

Corrector result

Besides the compactness result mentioned above, one of the main theorems is the corrector result (see [23], and [1], [24] in the periodic case). Let (e_{1}, e_{2}, …, e_{n}) be the canonical basis of R^{n}. For j = 1, 2, …, n there exists a sequence (z_{j}^{ε}) in H^{1}(Ω) such that

(3.5)  z_{j}^{ε} ⇀ 0  weakly in H^{1}(Ω) ,
(3.6)  A^{ε}(Dz_{j}^{ε} + e_{j}) ⇀ A^{0}e_{j}  weakly in L^{2}(Ω, R^{n}) ,
(3.7)  −div(A^{ε}(Dz_{j}^{ε} + e_{j})) → −div(A^{0}e_{j})  strongly in H^{−1}(Ω) .

Throughout the paper we will also assume that

(3.8)  z_{j}^{ε} → 0  strongly in L^{∞}(Ω) ,
(3.9)  z_{j}^{ε} ⇀ 0  weakly in W^{1,p}(Ω) for some p > 2 ;

using De Giorgi’s and Meyers’ regularity theorems, such a sequence can be constructed, for instance, by solving the problems

z_{j}^{ε} ∈ H_{0}^{1}(Ω′) ,
−div(A^{ε}(Dz_{j}^{ε} + e_{j})) = −div(A^{0}e_{j}) in D′(Ω′) ,

where Ω′ is a bounded open set with Ω ⊂⊂ Ω′, and A^{ε} is extended by αI on Ω′ \ Ω. (The use of Ω′ is needed here only to obtain a global W^{1,p}(Ω) bound for z_{j}^{ε} in the case where ∂Ω is not smooth.)

Let f ∈ H^{−1}(Ω), let (u^{ε}) be the sequence of the solutions to (3.3), and let u^{0} be the solution to (3.4). Given δ > 0, let ψ_{δ} be a function in C_{c}^{∞}(Ω) which satisfies

(3.10)  β ∫_{Ω} |Du^{0} − Dψ_{δ}|^{2}dx < δ ,

and let v_{δ}^{ε} be defined by

(3.11)  v_{δ}^{ε} = ψ_{δ} + ∑_{j=1}^{n} D_{j}ψ_{δ} z_{j}^{ε} .

Then (see [23])

(3.12)  lim sup_{ε→0} α ∫_{Ω} |Du^{ε} − Dv_{δ}^{ε}|^{2}dx < δ .

If u^{0} belongs to C_{c}^{∞}(Ω), we can take ψ_{δ} = u^{0} in (3.10) for every δ > 0, so that

(3.13)  v_{δ}^{ε} = v^{ε} = u^{0} + ∑_{j=1}^{n} D_{j}u^{0}z_{j}^{ε} ,

and (3.12) implies that

(3.14)  Du^{ε} − Dv^{ε} → 0  strongly in L^{2}(Ω, R^{n}) ,

which means that Du^{ε} is equivalent to Dv^{ε} (and also to Du^{0} + ∑_{j} D_{j}u^{0} Dz_{j}^{ε}, using (3.5)), as far as convergences in L^{2}(Ω, R^{n}) are concerned.

In the general case where u^{0} only belongs to H_{0}^{1}(Ω), we obtain from (3.12) that

Du^{ε} = Dψ_{δ} + ∑_{j=1}^{n} D_{j}ψ_{δ} Dz_{j}^{ε} + R_{δ}^{ε} ,  with  lim sup_{ε→0} ∥R_{δ}^{ε}∥_{L^{2}(Ω,R^{n})}^{2} < \frac{δ}{α} .

This is a corrector result: indeed it allows one to replace Du^{ε} by an explicit expression, up to a remainder R_{δ}^{ε} which is small in L^{2}(Ω, R^{n}) for δ small, uniformly in ε; similar corrector results have been obtained also in the case of local solutions. Applications can be found, e.g., in [2] and [8].

A convergence result

We conclude this section with the following convergence result, which is implicitly used in various works (see, e.g., [3]). Observe that there is no boundary condition on the solutions u^{ε} and that the right hand sides f^{ε} do not converge strongly in H^{−1}(Ω).

Theorem 3.1. Let (A^{ε}) be a sequence of matrices in M_{α}^{β}(Ω) which H -converges to a matrix A^{0} in M_{α}^{β}(Ω), and let (u^{ε}) be a sequence in H^{1}(Ω) such that

(3.15)

u^{ε} ⇀ u^{0} weakly in H^{1}(Ω) ,
−div(A^{ε}Du^{ε}) = f^{ε} in D′(Ω) for every ε ≥ 0 .

Assume that f^{ε} = g^{ε} + μ^{ε} + ν^{ε} for every ε > 0, where

(3.16)

(g^{ε}) is relatively compact in W_{loc}^{−1,p}(Ω) for some p > 1 ,
(μ^{ε}) is bounded in M(Ω) ,
ν^{ε} ≥ 0 in D′(Ω) .

Then

(3.17)

f^{ε} ⇀ f^{0} weakly in H^{−1}(Ω) and strongly in W_{loc}^{−1,q}(Ω) for every q < 2 ,
A^{ε}Du^{ε} ⇀ A^{0}Du^{0} weakly in L^{2}(Ω, R^{n}) .

In the present paper, this theorem will be used with μ^{ε} = 0 and (g^{ε}) relatively compact (or even constant) in H^{−1}(Ω).

Proof. Let K be any compact set of R^{n} with K ⊆ Ω, and let φ ∈ C_{c}^{∞}(Ω) with φ ≥ 0 on Ω and φ = 1 on K. We have

(3.18)  0 ≤ ∫_{K} dν^{ε} ≤ ∫_{Ω} φ dν^{ε} = ∫_{Ω} A^{ε}Du^{ε}Dφ dx − ⟨g^{ε}, φ⟩ − ∫_{Ω} φ dμ^{ε} .

Because of (3.2), (3.15), and (3.16), the right hand side of (3.18) is bounded independently of ε. This implies that

(3.19)  (ν^{ε}) is bounded in M(Ω) .

For every bounded open set U of R^{n}, the embedding W_{0}^{1,r}(U) ⊆ C_{0}^{0}(U) is compact for every r > n. This implies that the embedding M_{b}(U) ⊆ W^{−1,s}(U) is compact for every s < n/(n − 1), and therefore the embedding M(Ω) ⊆ W_{loc}^{−1,s}(Ω) is compact for every s < n/(n − 1) . Therefore (3.16) and (3.19) imply that (μ^{ε} + ν^{ε}) is relatively compact in W_{loc}^{−1,s}(Ω) , which implies that (f^{ε}) is relatively compact in W_{loc}^{−1,t}(Ω) for some t > 1. On the other hand, we deduce from (3.15) and (3.2) that (f^{ε}) is bounded in H^{−1}(Ω). By interpolation, (f^{ε}) is relatively compact in W_{loc}^{−1,q}(Ω) for every q < 2.

Let now \overline{v}^{0} be an arbitrary function in C_{c}^{∞}(Ω), and, for every ε > 0, let \overline{v}^{ε} be the solution to the problem

(3.20)  \{ \begin{matrix}\overline{v}^{ε} ∈ H_{0}^{1}(Ω) , \\ −div(\overline{A}^{ε}D\overline{v}^{ε}) = −div(\overline{A}^{0}D\overline{v}^{0}) in D′(Ω) . \end{matrix}

Recall that the sequence (\overline{A}^{ε}) H -converges to \overline{A}^{0} , so that

(3.21)

\overline{v}^{ε} ⇀ \overline{v}^{0} weakly in H_{0}^{1}(Ω) ,
\overline{A}^{ε}D\overline{v}^{ε} ⇀ \overline{A}^{0}D\overline{v}^{0} weakly in L^{2}(Ω, R^{n}) ,
\overline{v}^{ε} ⇀ \overline{v}^{0} weakly in W_{loc}^{1,p}(Ω) for some p > 2 ,

where in the last assertion we have used Meyers’ regularity result (see [21]).

Let φ ∈ C_{c}^{∞}(Ω). Using \overline{v}^{ε}φ as test function in (3.15), and u^{ε}φ as test function in (3.20), we have

(3.22)  \begin{matrix}⟨f^{ε}, \overline{v}^{ε}φ⟩ = ∫_{Ω} A^{ε}Du^{ε}D\overline{v}^{ε}φ dx + ∫_{Ω} A^{ε}Du^{ε}Dφ \overline{v}^{ε}dx = \\ = ⟨−div(\overline{A}^{0}D\overline{v}^{0}), u^{ε}φ⟩ − ∫_{Ω} \overline{A}^{ε}D\overline{v}^{ε}Dφ u^{ε} dx + ∫_{Ω} A^{ε}Du^{ε}Dφ \overline{v}^{ε}dx . \end{matrix}

Passing to a subsequence, we may assume that

(3.23)

A^{ε}Du^{ε} ⇀ σ weakly in L^{2}(Ω, R^{n}) ,

f^{ε} ⇀ f weakly in H^{−1}(Ω) and strongly in W_{loc}^{−1,q}(Ω) for every q < 2 , for some σ ∈ L^{2}(Ω, R^{n}) and f ∈ H^{−1}(Ω). It is now easy to pass to the limit in the left and right hand sides of (3.22) by using (3.21), (3.23), and Rellich’s compactness theorem. One obtains

(3.24)

⟨f,\overline{v}^{0}φ⟩ =
= ⟨−div(\overline{A}^{0}D\overline{v}^{0}), u^{0}φ⟩ − ∫_{Ω} \overline{A}^{0}D\overline{v}^{0}Dφ u^{0}dx + ∫_{Ω} σ Dφ \overline{v}^{0}dx
= ∫_{Ω} \overline{A}^{0}D\overline{v}^{0}Du^{0}φ dx + ∫_{Ω} σ Dφ \overline{v}^{0}dx
= ∫_{Ω} A^{0}Du^{0}D\overline{v}^{0}φ dx + ∫_{Ω} σ Dφ \overline{v}^{0}dx .

Since

(3.25)  −div(σ) = f  in D′(Ω) ,

one deduces from (3.24) that

(3.26)  ∫_{Ω} σ D\overline{v}^{0}φ dx = ∫_{Ω} A^{0}Du^{0}D\overline{v}^{0}φ dx ,

for every φ ∈ C_{c}^{∞}(Ω) and every \overline{v}^{0} ∈ C_{c}^{∞}(Ω). Since, for every point x ∈ Ω, the vector D\overline{v}^{0}(x) can be chosen to coincide with any prescribed vector of R^{n}, (3.26) implies that

σ = A^{0}Du^{0} a.e. in Ω ,

which, together with (3.25), gives f = f^{0} . The uniqueness of the limits in (3.23) implies that the whole sequences converge, and this completes the proof of (3.17).

4. Relaxed Dirichlet problems

In this section we recall the definition, introduced in [13] and [14], of relaxed Dirichlet problems associated with measures μ ∈ M_{0}^{+}(Ω), and prove that, under some conditions on the data, the measure μ can be reconstructed from the solution of the corresponding relaxed Dirichlet problem.

Relaxed Dirichlet problems

Given A ∈ M_{α}^{β}(Ω), μ ∈ M_{0}^{+}(Ω), and f ∈ H^{−1}(Ω), we call relaxed Dirichlet problem the problem of finding u such that

(4.1)  \{ \begin{matrix}u ∈ H_{0}^{1}(Ω) ∩ L^{2}(Ω, μ) , \\ ∫_{Ω} A Du Dy dx + ∫_{Ω} u y dμ = ⟨f, y⟩  ∀y ∈ H_{0}^{1}(Ω) ∩ L^{2}(Ω, μ) . \end{matrix}

By a straightforward application of the Lax-Milgram lemma problem (4.1) has a unique solution u (see [14], Theorem 2.4) and u satisfies the estimate

(4.2)  α ∫_{Ω} |Du|^{2}dx + ∫_{Ω} |u|^{2}dμ ≤ \frac{1}{α}∥f∥_{H^{−1}(Ω)}^{2} .

A connection between classical Dirichlet problems on open subsets of Ω and relaxed Dirichlet problems of the form (4.1) is given by the following remark.

Remark 4.1. Using Theorem 4.5 of [18] it is easy to check that, if U ⊆ Ω is open and μ_{U} is the measure introduced in (2.2), then u ∈ H_{0}^{1}(Ω) ∩ L^{2}(Ω, μ_{U}) if and only if the restriction of u to U belongs to H_{0}^{1}(U) and u = 0 q.e. in Ω \ U. Therefore when μ = μ_{U} problem (4.1) reduces to the following boundary value problem on U:

(4.3)  \{ \begin{matrix}u ∈ H_{0}^{1}(U) , \\ −div(A Du) = f in D′(U) , \end{matrix}

in the sense that u is the solution of (4.1) if and only if its restriction to U is the solution of (4.3) and u = 0 q.e. in Ω \ U.

The name “relaxed Dirichlet problem” is motivated by the fact that the limit of the solutions to Dirichlet problems on varying domains Ω^{ε} always satisfies a relaxed Dirichlet problem (see, e.g., [14] and [11], and also Corollary 5.5 below). Moreover, the results proved in [14] and [12] ensure that every relaxed Dirichlet problem on Ω can be approximated in a convenient sense by classical Dirichlet problems on a suitable sequence of open sets (Ω^{ε}) included in Ω.

Reconstructing the measure μ

We now want to reconstruct the measure μ from one particular solution of the relaxed Dirichlet problem (4.1). In view of the applications we consider also solutions of the equation in (4.1) which do not necessarily satisfy the homogeneous Dirichlet boundary condition on ∂Ω, but we study only the case where the solution and the right hand side are nonnegative. Let us fix

(4.4)  A ∈ M_{α}^{β}(Ω) ,  μ ∈ M_{0}^{+}(Ω) ,  λ ∈ H^{−1}(Ω)^{+} ,

and a solution ω to the problem

(4.5)  \{ \begin{matrix}ω ∈ H^{1}(Ω) ∩ L^{2}(Ω, μ) , \\ ∫_{Ω} A Dω Dy dx + ∫_{Ω} ω y dμ = ∫_{Ω} y dλ  ∀y ∈ H_{0}^{1}(Ω) ∩ L^{2}(Ω, μ) , \end{matrix}

which satisfies

(4.6)  ω ≥ 0 q.e. in Ω .

Remark 4.2. From the Lax-Milgram lemma, there exists a solution of (4.5) which belongs to H_{0}^{1}(Ω); by the comparison principle (Theorem 2.10 in [13]) this solution satisfies (4.6), so that the set of such functions ω is not empty.

The following proposition (proved in [13], Proposition 2.6) will be frequently used throughout the paper.

Proposition 4.3. Assume (4.4), (4.5), and (4.6). Then there exists ν ∈ H^{−1}(Ω)^{+} such that

(4.7)  −div(A Dω) + ν = λ in D′(Ω) .

For technical reasons, the reconstruction of the measure μ from ω requires the following assumption: for every quasi open set U in Ω we have

(4.8)  cap(U ∩ {ω = 0}) > 0 ⟹ λ(U) > 0 .

Remark 4.4. Condition (4.8) is satisfied in the following (extreme) cases:

(a) ω > 0 q.e. in Ω;

(b) λ(U) > 0 for every quasi open set U ⊆ Ω with cap(U) > 0. Note that (b) is always satisfied if λ(U) = ∫_{U} f dx with f ∈ L_{loc}^{1}(Ω) and f > 0 a.e. in Ω, since, by Lemma 2.1 and (2.1), every quasi open set with positive capacity has positive Lebesgue measure.

Proposition 4.5. Assume (4.4), (4.5), (4.6), and (4.8). Then

(4.9)  u ∈ H_{0}^{1}(Ω) ∩ L^{2}(Ω, μ) ⟹ u = 0 q.e. in {ω = 0} .

Moreover for every Borel set B ⊆ Ω

(4.10)  cap(B ∩ {ω = 0}) > 0 ⟹ μ(B) = +∞ .

Proof. The proof is along the lines of Lemma 3.2 of [11], with some important variants, due to the fact that now λ is not the Lebesgue measure.

To prove (4.9) it is enough to consider a function u ∈ H_{0}^{1}(Ω) ∩ L^{2}(Ω, μ) such that 0 ≤ u ≤ 1 q.e. in Ω. For every k ∈ N let u_{k} be the solution of the relaxed Dirichlet problem

(4.11)  \{ \begin{matrix}u_{k} ∈ H_{0}^{1}(Ω) ∩ L^{2}(Ω, μ) , \\ ∫_{Ω} A Du_{k}Dy dx + ∫_{Ω} u_{k}y dμ + k ∫_{Ω} u_{k}y dλ = k ∫_{Ω} u y dλ \\ ∀y ∈ H_{0}^{1}(Ω) ∩ L^{2}(Ω, μ) . \end{matrix}

By the comparison principle (see [13], Proposition 2.10) we have 0 ≤ u_{k} ≤ k ω q.e. in Ω, hence u_{k} = 0 q.e. in {ω = 0}.

Taking y = u_{k} − u as test function in (4.11), from (3.1) we obtain, by using Cauchy inequality,

\begin{matrix}α ∫_{Ω} |Du_{k}|^{2}dx + ∫_{Ω} |u_{k}|^{2}dμ + 2k ∫_{Ω} |u_{k} − u|^{2}dλ ≤ \\ ≤ \frac{1}{α} ∫_{Ω} |A Du|^{2}dx + ∫_{Ω} |u|^{2}dμ . \end{matrix}

It follows that (u_{k}) is bounded in H_{0}^{1}(Ω) and converges to u strongly in L^{2}(Ω, λ). Therefore a subsequence, still denoted by (u_{k}), converges weakly in H_{0}^{1}(Ω) to some function v in H_{0}^{1}(Ω) such that v = u λ-a.e. in Ω. Since u_{k} = 0 q.e. in {ω = 0}, and since suitable convex combinations of (u_{k}) converge to v strongly in H_{0}^{1}(Ω), we conclude that v = 0 q.e. in {ω = 0}. Let V = {v ≠ u}. Then V is quasi open and λ(V) = 0. It follows from (4.8) that cap(V ∩ {ω = 0}) = 0. As u = v in Ω \ V and v = 0 q.e. in {ω = 0}, this implies that u = 0 q.e. in {ω = 0}.

Let is prove (4.10). Let U be a quasi open subset of Ω such that μ(U) < +∞. By Lemma 2.1 there exists an increasing sequence (z_{k}) in H_{0}^{1}(Ω) converging to 1_{U} pointwise q.e. in Ω and such that 0 ≤ z_{k} ≤ 1_{U} q.e. in Ω for every k ∈ N. As μ(U) < +∞, each function z_{k} belongs to L^{2}(Ω, μ), hence z_{k} = 0 q.e. on {ω = 0} by the previous step. This implies that 1_{U} = 0 q.e. on {ω = 0}, hence cap(U ∩ {ω = 0}) = 0.

Let us consider a Borel set B with cap(B ∩ {ω = 0}) > 0. For every quasi open set U containing B we have cap(U ∩ {ω = 0}) > 0, hence μ(U) = +∞ by the previous step of the proof. Then the regularity property (b) in the definition of M_{0}^{+}(Ω) implies that μ(B) = +∞.

Proposition 4.6. Assume (4.4), (4.5), (4.6), and (4.8), and let ν be the measure of H^{−1}(Ω)^{+} defined in (4.7). Then for every Borel set B ⊆ Ω we have

(4.12)  μ(B) = \{ \begin{matrix}∫_{B} \frac{dν}{ω} , & if cap(B ∩ \{ω = 0\}) = 0, \\ +∞ , & if cap(B ∩ \{ω = 0\}) > 0, \end{matrix}

and

(4.13)  ν(B ∩ {ω > 0}) = ∫_{B} ω dμ .

In particular, this implies that ν = ωμ on {ω > 0}.

Proof. The proof is along the lines of Lemma 3.3 and Proposition 3.4 of [11]. For every η > 0 let ν_{η} be the Borel measure defined by

(4.14)  ν_{η}(B) = ∫_{B∩{ω>η}} ω dμ .

As ω ∈ L^{2}(Ω, μ), we have ν_{η}(Ω) ≤ \frac{1}{η} ∫_{Ω} ω^{2}dμ < +∞. Let us prove that

(4.15)  ν_{η}(B) = ν(B ∩ {ω > η}) ,

for every Borel set B ⊆ Ω. Since ν_{η} is a Radon measure, it is enough to prove that ν_{η}(U) = ν(U ∩ {ω > η}) for every open set U ⊆ Ω. Let us fix an open set U, and let U_{η} = U ∩ {ω > η}. As U_{η} is quasi open, by Lemma 2.1 there exists an increasing sequence (z_{k}) of nonnegative functions of H_{0}^{1}(Ω) converging to 1_{U_{η}} pointwise q.e. in Ω. Since μ(U_{η}) < +∞, the functions z_{k} belong to L^{2}(Ω, μ). Using z_{k} as test function in (4.5) and (4.7) we obtain

∫_{Ω} z_{k}dν = ∫_{Ω} ω z_{k}dμ .

Taking the limit as k tends to ∞ we get ν(U ∩ {ω > η}) = ν_{η}(U_{η}) = ν_{η}(U), which proves (4.15). When η tends to 0, we obtain (4.13) from (4.14) and (4.15) (recall that ω ≥ 0 q.e. in Ω).

From (4.13) we have

μ(B ∩ \{ω > η\}) = ∫_{B∩\{ω>η\}} \frac{dν}{ω} ,

for every Borel set B ⊆ Ω and every η > 0. Taking the limit as η tends to 0 we obtain

(4.16)  μ(B) = ∫_{B} \frac{dν}{ω} ,

for every Borel set B ⊆ {ω > 0}. Since μ vanishes on all sets with capacity zero, (4.16) holds also when cap(B ∩ {ω = 0}) = 0. Finally, if cap(B ∩ {ω = 0}) > 0, then μ(B) = +∞ by Proposition 4.5.

Density and uniqueness results In the next proposition we assume, in addition, that

(4.17)  ω ∈ L^{∞}(Ω) .

The following density result will be crucial in Sections 7 and 9. The proof is along the lines of Proposition 5.5 of [15], with one important variant, due to the fact that now the solutions u_{k} of the penalized problem (4.11) may not converge to u weakly in H_{0}^{1}(Ω) (see the proof of Proposition 4.5).

Proposition 4.7. Assume (4.4), (4.5), (4.6), (4.8), and (4.17). Then the set {ω φ : φ ∈ C_{c}^{∞}(Ω)} is dense in H_{0}^{1}(Ω) ∩ L^{2}(Ω, μ).

Proof. For every u ∈ H_{0}^{1}(Ω) ∩ L^{2}(Ω, μ) we have to construct a sequence (φ_{k}) in C_{c}^{∞}(Ω) such that (ω φ_{k}) converges to u both in H_{0}^{1}(Ω) and in L^{2}(Ω, μ). Clearly it is enough to consider the case u ≥ 0 q.e. in Ω. For every j ∈ N let v_{j} = u∧(j ω). Since ω ≥ 0 q.e. in Ω and u = 0 q.e. in {ω = 0} by Proposition 4.5, the sequence (v_{j}) is nondecreasing and converges to u q.e. in Ω. By Lemma 1.6 of [9] there exists a sequence (u_{j}) in H_{0}^{1}(Ω), converging to u strongly in H_{0}^{1}(Ω), such that 0 ≤ u_{j} ≤ v_{j} ≤ u q.e. in Ω for every j ∈ N. By the dominated convergence theorem it turns out that (u_{j}) converges to u in L^{2}(Ω, μ) too.

We are thus reconduced to the case where u ∈ H_{0}^{1}(Ω) is such that 0 ≤ u ≤ c ω q.e. in Ω for some constant c > 0. Since {(u − c ε)^{+} > 0} ⊆ {ω > ε}, and (u − c ε)^{+} converges to u in H_{0}^{1}(Ω)∩L^{2}(Ω, μ) as ε tends to 0, we may also assume that there exists ε > 0 such that {u > 0} ⊆ {ω > ε}. Then u/ω = u/(ω ∨ ε). Since ω ∈ H^{1}(Ω)∩L^{∞}(Ω), we have u ∈ H_{0}^{1}(Ω) ∩ L^{∞}(Ω), and thus u/ω ∈ H_{0}^{1}(Ω) ∩ L^{∞}(Ω). Therefore there exists a sequence (φ_{k}) in C_{c}^{∞}(Ω), bounded in L^{∞}(Ω), which converges to z = u/ω strongly in H_{0}^{1}(Ω) and q.e. in Ω, hence μ-a.e. in Ω. Since ω ∈ H^{1}(Ω) ∩ L^{∞}(Ω), the sequence (ω φ_{k}) converges to ω z = u strongly in H_{0}^{1}(Ω). As ω ∈ L^{2}(Ω, μ) and (φ_{k}) is bounded in L^{∞}(Ω, μ) and converges to z = u/ω μ-a.e. in Ω, by the dominated convergence theorem the sequence (ω φ_{k}) converges to ω z = u strongly in L^{2}(Ω, μ).

The following uniqueness result will be crucial in Theorems 5.1 and 5.4. The proof is along the lines of Lemma 3.5 of [11], with one important variant, due to the fact that now the condition ∫_{Ω} u^{2}dλ = 0 does not imply that u = 0 q.e. in Ω.

Proposition 4.8. Assume (4.4), (4.5), (4.6), (4.8), and (4.17). Let u be a solution of the problem

(4.18) \{ \begin{matrix}u ∈ H_{0}^{1}(Ω) ∩ L^{∞}(Ω) , \\ ∫_{Ω} A Dφ Du ω dx − ∫_{Ω} A Dω Dφ u dx + ∫_{Ω} u φ dλ = 0  ∀φ ∈ C_{c}^{∞}(Ω) . \end{matrix}

Then u = 0 q.e. in Ω.

Proof. Since ω ∈ H_{0}^{1}(Ω) ∩ L^{∞}(Ω), it is easy to see that the equation in (4.18) is satisfied also for φ ∈ H_{0}^{1}(Ω) ∩ L^{∞}(Ω). Using φ = u as test function in this equation we obtain

(4.19)  ∫_{Ω} A Du Du ω dx − \frac{1}{2} ∫_{Ω} A Dω D(u^{2}) dx + ∫_{Ω} u^{2}dλ = 0 .

Using y = u^{2} as test function in (4.5), from (4.19) we get

∫_{Ω} A Du Du ω dx + \frac{1}{2} ∫_{Ω} ω u^{2}dμ + \frac{1}{2} ∫_{Ω} u^{2}dλ = 0 .

This implies

Du = 0 a.e. in {ω > 0} , (4.20) u = 0 λ-a.e. in Ω . (4.21)

Let U = {u ≠ 0}. Then U is quasi open and λ(U) = 0 by (4.21). Therefore (4.8) implies that u = 0 q.e. in {ω = 0}, and consequently Du = 0 a.e. in {ω = 0}. By (4.20) we conclude that Du = 0 a.e. in Ω. Since u ∈ H_{0}^{1}(Ω), this yields u = 0 q.e. in Ω.

5. A global convergence result

For every ε ≥ 0 we consider a matrix A^{ε} in M_{α}^{β}(Ω) and a measure μ^{ε} in M_{0}^{+}(Ω), that will remain fixed throughout the rest of the paper. We assume that

(5.1)  A^{ε} H-converges to A^{0} .

In this section we use a duality argument to prove that, under suitable hypotheses on (μ^{ε}) (which are always satisfied by a subsequence), the solutions u^{ε} of the relaxed Dirichlet problems (4.1) for A = A^{ε} and μ = μ^{ε} converge to the solution u^{0} of the relaxed Dirichlet problem for A = A^{0} and μ = μ^{0} .

Definition of special test fuctions

For every ε ≥ 0 we define the functions w^{ε} and \overline{w}^{ε} as the unique solutions to the problems

\{ \begin{matrix}w^{ε} ∈ H_{0}^{1}(Ω) ∩ L^{2}(Ω, μ^{ε}) , \\ ∫_{Ω} A^{ε}Dw^{ε}Dy dx + ∫_{Ω} w^{ε}y dμ^{ε} = ∫_{Ω} y dx ∀y ∈ H_{0}^{1}(Ω) ∩ L^{2}(Ω, μ^{ε}) , \end{matrix} (5.2)

\{ \begin{matrix}\overline{w}^{ε} ∈ H_{0}^{1}(Ω) ∩ L^{2}(Ω, μ^{ε}) , \\ ∫_{Ω} \overline{A}^{ε}D\overline{w}^{ε}Dy dx + ∫_{Ω} \overline{w}^{ε}y dμ^{ε} = ∫_{Ω} y dx ∀y ∈ H_{0}^{1}(Ω) ∩ L^{2}(Ω, μ^{ε}) . \end{matrix} (5.3)

By the comparison principle (Theorem 2.10 of [13]) we have

(5.4)  w^{ε} ≥ 0 and \overline{w}^{ε} ≥ 0 q.e. in Ω .

Moreover, by the maximum principle, we have also

(5.5)  sup_{ε≥0} ∥w^{ε}∥_{L^{∞}(Ω)} < +∞  and  sup_{ε≥0} ∥\overline{w}^{ε}∥_{L^{∞}(Ω)} < +∞

(see [11], Section 3). By Proposition 4.3 there exists two measures ν^{ε} and \overline{ν}^{ε} in H^{−1}(Ω)^{+} such that

(5.6)  −div(A^{ε}Dw^{ε}) + ν^{ε} = 1 , −div(\overline{A}^{ε}D\overline{w}^{ε}) + \overline{ν}^{ε} = 1 in D′(Ω) .

Finally, from (4.2) we obtain

(5.7)  sup_{ε≥0} ∫_{Ω} |Dw^{ε}|^{2}dx < +∞ ,  sup_{ε≥0} ∫_{Ω} |D\overline{w}^{ε}|^{2}dx < +∞ ,
(5.8)  sup_{ε≥0} ∫_{Ω} |w^{ε}|^{2}dμ^{ε} < +∞ ,  sup_{ε≥0} ∫_{Ω} |\overline{w}^{ε}|^{2}dμ^{ε} < +∞ .

The main convergence result

Given, for every ε ≥ 0, f^{ε} and \overline{f}^{ε} in H^{−1}(Ω), we consider the solutions u^{ε} and \overline{u}^{ε} to the following problems

\{ \begin{matrix}u^{ε} ∈ H_{0}^{1}(Ω) ∩ L^{2}(Ω, μ^{ε}) , \\ ∫_{Ω} A^{ε}Du^{ε}Dy dx + ∫_{Ω} u^{ε}y dμ^{ε} = ⟨f^{ε}, y⟩ ∀y ∈ H_{0}^{1}(Ω) ∩ L^{2}(Ω, μ^{ε}) , \end{matrix} (5.9)

\{ \begin{matrix}\overline{u}^{ε} ∈ H_{0}^{1}(Ω) ∩ L^{2}(Ω, μ^{ε}) , \\ ∫_{Ω} \overline{A}^{ε}D\overline{u}^{ε}Dy dx + ∫_{Ω} \overline{u}^{ε}y dμ^{ε} = ⟨\overline{f}^{ε}, y⟩ ∀y ∈ H_{0}^{1}(Ω) ∩ L^{2}(Ω, μ^{ε}) . \end{matrix} (5.10)

Theorem 5.1. Assume (5.1) and let w^{ε} and \overline{w}^{ε} be the solutions of (5.2) and (5.3). The following conditions are equivalent:

(a) w^{ε} ⇀ w^{0} weakly in H_{0}^{1}(Ω);

(b) \overline{w}^{ε} ⇀ \overline{w}^{0} weakly in H_{0}^{1}(Ω);

(c) for every (f^{ε}) and (u^{ε}) satisfying (5.9), if f^{ε} → f^{0} strongly in H^{−1}(Ω), then u^{ε} ⇀ u^{0} weakly in H_{0}^{1}(Ω);

(d) for every (\overline{f}^{ε}) and (\overline{u}^{ε}) satisfying (5.10), if \overline{f}^{ε} → \overline{f}^{0} strongly in H^{−1}(Ω), then \overline{u}^{ε} ⇀ \overline{u}^{0} weakly in H_{0}^{1}(Ω).

Proof. (a) ⇒ (d). Assume (a). By (4.2) it is enough to prove (d) when \overline{f}^{ε} = \overline{f}^{0} = \overline{f} ∈ L^{∞}(Ω). Since the equation is linear, it suffices to consider the case 0 ≤ \overline{f} ≤ 1 a.e. in Ω, so that

0 ≤ \overline{u}^{ε} ≤ \overline{w}^{ε} q.e. in Ω by the comparison principle (Theorem 2.10 of [13]).

By (4.2) the sequence (\overline{u}^{ε}) is bounded in H_{0}^{1}(Ω) and by (5.5) it is bounded in L^{∞}(Ω). Extracting a subsequence, we may assume that

(5.11)  \overline{u}^{ε} ⇀ \overline{u}  weakly in H_{0}^{1}(Ω) ,

for some function \overline{u} ∈ H_{0}^{1}(Ω) ∩ L^{∞}(Ω). We want to show that \overline{u} = \overline{u}^{0} . Since the limit does not depend on the subsequence, this will prove that the whole sequence (\overline{u}^{ε}) converges to \overline{u}^{0} .

By Proposition 4.3 we have

(5.12)  −div(\overline{A}^{ε}D\overline{u}^{ε}) + \overline{γ}^{ε} = \overline{f} in D′(Ω) ,

for some \overline{γ}^{ε} ∈ H^{−1}(Ω)^{+} . By Theorem 3.1, from (5.6) and (5.12) we deduce that

(5.13)

A^{ε}Dw^{ε} ⇀ A^{0}Dw^{0} weakly in L^{2}(Ω, R^{n}) ,
\overline{A}^{ε}D\overline{u}^{ε} ⇀ \overline{A}^{0}D\overline{u} weakly in L^{2}(Ω, R^{n}) .

Let φ ∈ C_{c}^{∞}(Ω). Using y = w^{ε}φ as test function in (5.10) and y = \overline{u}^{ε}φ as test function in (5.2), by difference we obtain

(5.14)  \begin{matrix}∫_{Ω} \overline{A}^{ε}D\overline{u}^{ε}Dφ w^{ε}dx − ∫_{Ω} A^{ε}Dw^{ε}Dφ \overline{u}^{ε}dx = \\ = ∫_{Ω} \overline{f} w^{ε}φ dx − ∫_{Ω} \overline{u}^{ε}φ dx , \end{matrix}

for every ε ≥ 0. Since (w^{ε}) converges to w^{0} strongly in L^{2}(Ω) by (a) and (\overline{u}^{ε}) converges to \overline{u} strongly in L^{2}(Ω) by (5.11), using (5.13) we can pass to the limit in each term of (5.14) and we obtain

(5.15)  \begin{matrix}∫_{Ω} \overline{A}^{0}D\overline{u} Dφ w^{0}dx − ∫_{Ω} A^{0}Dw^{0}Dφ \overline{u} dx = \\ = ∫_{Ω} \overline{f} w^{0}φ dx − ∫_{Ω} \overline{u} φ dx . \end{matrix}

Since (5.14), with ε = 0, and (5.15) hold for every φ ∈ C_{c}^{∞}(Ω), the difference u = \overline{u}^{0} −\overline{u} belongs to H_{0}^{1}(Ω) ∩ L^{∞}(Ω) and satisfies (4.18) with A = A^{0} , ω = w^{0} , and λ = 1. This implies \overline{u} = \overline{u}^{0} q.e. in Ω by Proposition 4.8.

(d) ⇒ (b). It is enough to take \overline{f}^{ε} = \overline{f}^{0} = 1 in condition (d).

(b) ⇒ (c). Since (\overline{A}^{ε}) H -converges to \overline{A}^{0} , we can replace A^{ε} by \overline{A}^{ε} and \overline{f}^{ε} by f^{ε} in the proof of the implication (a) ⇒ (d).

(c) ⇒ (a). It is enough to take f^{ε} = f^{0} = 1 in condition (c).

A compactness result

We now prove that the conditions of Theorem 5.1 are always satisfied by a subsequence.

Theorem 5.2. Assume (5.1). For every sequence (μ^{ε})_{ε>0} in M_{0}^{+}(Ω) there exist a subsequence, still denoted by (μ^{ε}), and a measure μ^{0} in M_{0}^{+}(Ω), such that the equivalent conditions (a)–(d) of Theorem 5.1 are satisfied.

Proof. By (5.7) the sequence (w^{ε}) is bounded in H_{0}^{1}(Ω). Passing to a subsequence, we may assume that (w^{ε}) converges weakly in H_{0}^{1}(Ω) to some function w ∈ H_{0}^{1}(Ω). By (5.4) we have w ≥ 0 q.e. in Ω. Now we want to construct a measure μ^{0} ∈ M_{0}^{+}(Ω) such that w coincides with the solution w^{0} of (5.2) for ε = 0.

By (5.6) and Theorem 3.1 the sequence (A^{ε}Dw^{ε}) converges to A^{0}Dw weakly in L^{2}(Ω, R^{n}). Therefore (ν^{ε}) converges to ν weakly in H^{−1}(Ω), where ν ∈ H^{−1}(Ω)^{+} is defined by

(5.16)  −div(A^{0}Dw) + ν = 1 in D′(Ω) .

Let us define the measure μ^{0} by

(5.17)  μ^{0}(B) = \{ \begin{matrix}∫_{B} \frac{dν}{w} & if cap(B ∩ \{w = 0\}) = 0, \\ +∞ & if cap(B ∩ \{w = 0\}) > 0. \end{matrix}

Using (5.16), from Proposition 3.4 of [11] we obtain that μ^{0} ∈ M_{0}^{+}(Ω) and that w coincides with the unique solution w^{0} to problem (5.2) for ε = 0. This shows that condition (a) of Theorem 5.1 is satisfied.

More general test functions

We introduce now a more general family of test functions (ω^{ε}). While it is very difficult to compute explicitly the functions w^{ε} defined by (5.2), in some interesting situations it will be very easy to construct explicitly the new family (ω^{ε}), from which one can determine immediately the limit measure μ^{0} .

For every ε ≥ 0 let λ^{ε} ∈ H^{−1}(Ω)^{+} and let ω^{ε} be a solution of the problem

(5.18) \{ \begin{matrix}ω^{ε} ∈ H^{1}(Ω) ∩ L^{2}(Ω, μ^{ε}) , \\ ∫_{Ω} A^{ε}Dω^{ε}Dy dx + ∫_{Ω} ω^{ε}y dμ^{ε} = ∫_{Ω} y dλ^{ε}  ∀y ∈ H_{0}^{1}(Ω) ∩ L^{2}(Ω, μ^{ε}) . \end{matrix}

We assume that (5.19) λ^{ε} ∈ H^{−1}(Ω)^{+} for every ε ≥ 0 , (5.20) λ^{ε} → λ^{0} strongly in H^{−1}(Ω) , (5.21) ω^{ε} ≥ 0 q.e. in Ω for every ε ≥ 0 ,

(5.22)  ω^{ε} ⇀ ω^{0}  weakly in H^{1}(Ω) .

Moreover we assume that for every quasi open set U in Ω we have

(5.23)  cap(U ∩ {ω^{0} = 0}) > 0 ⟹ λ^{0}(U) > 0 ,

and that

(5.24)  ω^{0} ∈ L^{∞}(Ω) .

Remark 5.3. If condition (a) of Theorem 5.1 is satisfied, then the functions w^{ε}, ε ≥ 0, defined by (5.2) satisfy conditions (5.18)–(5.24) with λ^{ε} = 1 for every ε ≥ 0 (see Remark 4.4). Other sequences (ω^{ε})_{ε≥0} and (λ^{ε})_{ε≥0} satisfying (5.18)–(5.24), with ω^{0} = 1, are constructed in [7] when μ^{0} ∈ H^{−1}(Ω)^{+} .

If conditions (5.18)–(5.24) are satisfied in Ω, then they are satisfied in every open set U ⊆ Ω.

Theorem 5.4. Assume that (5.1) holds and that (ω^{ε})_{ε≥0} and (λ^{ε})_{ε≥0} satisfy (5.18)– (5.24). Then the equivalent conditions (a)–(d) of Theorem 5.1 are fulfilled.

Proof. We will prove that condition (b) holds. By (5.7) the sequence (\overline{w}^{ε}) is bounded in H_{0}^{1}(Ω) and by (5.5) it is bounded in L^{∞}(Ω). Extracting a subsequence, we may assume that

(5.25)  \overline{w}^{ε} ⇀ \overline{w}  weakly in H_{0}^{1}(Ω) ,

for some function \overline{w} ∈ H_{0}^{1}(Ω) ∩ L^{∞}(Ω). We will show that \overline{w} = \overline{w}^{0} . Since the limit does not depend on the subsequence, this will prove that the whole sequence (\overline{w}^{ε}) converges to \overline{w}^{0} .

By Proposition 4.3 and Theorem 3.1 we have

(5.26)

A^{ε}Dω^{ε} ⇀ A^{0}Dω^{0} weakly in L^{2}(Ω, R^{n}) ,
\overline{A}^{ε}D\overline{w}^{ε} ⇀ \overline{A}^{0}D\overline{w}^{0} weakly in L^{2}(Ω, R^{n}) .

Let φ ∈ C_{c}^{∞}(Ω). Using y = ω^{ε}φ as test function in (5.3) and y = \overline{w}^{ε}φ as test function in (5.18), by difference we obtain

(5.27)  \begin{matrix}∫_{Ω} \overline{A}^{ε}D\overline{w}^{ε}Dφ ω^{ε}dx − ∫_{Ω} A^{ε}Dω^{ε}Dφ \overline{w}^{ε}dx = \\ = ∫_{Ω} ω^{ε}φ dx − ∫_{Ω} \overline{w}^{ε}φ dλ^{ε} , \end{matrix}

for every ε ≥ 0. Since (ω^{ε}) converges to ω^{0} strongly in L_{loc}^{2}(Ω) by (5.22) and (\overline{w}^{ε}) converges to \overline{w} strongly in L^{2}(Ω) by (5.25), using (5.26) we can pass to the limit in each term of (5.27) and we obtain

(5.28)  \begin{matrix}∫_{Ω} \overline{A}^{0}D\overline{w} Dφ ω^{0}dx − ∫_{Ω} A^{0}Dω^{0}Dφ \overline{w} dx = \\ = ∫_{Ω} ω^{0}φ dx − ∫_{Ω} \overline{w} φ dλ^{0} . \end{matrix}

Since (5.27), with ε = 0, and (5.28) hold for every φ ∈ C_{c}^{∞}(Ω), the difference \overline{w}^{0} − \overline{w} belongs to H_{0}^{1}(Ω) ∩ L^{∞}(Ω) and satisfies (4.18) with A = A^{0} and λ = λ^{0} . This implies \overline{w} = \overline{w}^{0} q.e. in Ω by Proposition 4.8.

Dirichlet problems on varying domains

We conclude this section by considering the particular case of classical Dirichlet problems on varying domains. Let (Ω^{ε})_{ε>0} be a sequence of open sets, with Ω^{ε} ⊆ Ω, and let μ^{0} be a measure in M_{0}^{+}(Ω). For every ε > 0 let w^{ε} and \overline{w}^{ε} be the unique solutions to the problems

\{ \begin{matrix}w^{ε} ∈ H_{0}^{1}(Ω^{ε}) , \\ −div(A^{ε}Dw^{ε}) = 1 in D′(Ω^{ε}) , \end{matrix} (5.29)

\{ \begin{matrix}\overline{w}^{ε} ∈ H_{0}^{1}(Ω^{ε}) , \\ −div(\overline{A}^{ε}D\overline{w}^{ε}) = 1 in D′(Ω^{ε}) , \end{matrix} (5.30)

and let w^{0} and \overline{w}^{0} be the solutions of (5.2) and (5.3) with ε = 0.

Given f^{ε} and \overline{f}^{ε} in H^{−1}(Ω), for ε > 0, we consider the solutions u^{ε} and \overline{u}^{ε} to the following problems

\{ \begin{matrix}u^{ε} ∈ H_{0}^{1}(Ω^{ε}) , \\ −div(A^{ε}Du^{ε}) = f^{ε} in D′(Ω^{ε}) , \end{matrix} (5.31)

\{ \begin{matrix}\overline{u}^{ε} ∈ H_{0}^{1}(Ω^{ε}) , \\ −div(\overline{A}^{ε}D\overline{u}^{ε}) = \overline{f}^{ε} in D′(Ω^{ε}) . \end{matrix} (5.32)

Given f^{0} and \overline{f}^{0} in H^{−1}(Ω), let u^{0} and \overline{u}^{0} be the solutions of (5.9) and (5.10) with ε = 0. All functions in H_{0}^{1}(Ω^{ε}) are considered as functions in H_{0}^{1}(Ω) which are equal to 0 q.e. in Ω \ Ω^{ε}. (Observe that u^{ε}, \overline{u}^{ε}, f^{ε}, and \overline{f}^{ε} are defined in the whole of Ω, for ε ≥ 0.)

Corollary 5.5. Assume (5.1) and let w^{ε} and \overline{w}^{ε} be the solutions of (5.29) and (5.30) for ε > 0, and of (5.2) and (5.3) for ε = 0. The following conditions are equivalent:

(a) w^{ε} ⇀ w^{0} weakly in H_{0}^{1}(Ω);

(b) \overline{w}^{ε} ⇀ \overline{w}^{0} weakly in H_{0}^{1}(Ω);

(c) for every (f^{ε}) and (u^{ε}) satisfying (5.31) for ε > 0 and (5.9) for ε = 0, if f^{ε} → f^{0} strongly in H^{−1}(Ω), then u^{ε} ⇀ u^{0} weakly in H_{0}^{1}(Ω);

(d) for every (\overline{f}^{ε}) and (\overline{u}^{ε}) satisfying (5.32) for ε > 0 and (5.10) for ε = 0, if \overline{f}^{ε} → \overline{f}^{0} strongly in H^{−1}(Ω), then \overline{u}^{ε} ⇀ \overline{u}^{0} weakly in H_{0}^{1}(Ω).

Proof. For every ε > 0 let μ_{Ω^{ε}} be the measures introduced in (2.2) with U = Ω^{ε}. By Remark 4.1 the functions w^{ε} and \overline{w}^{ε} defined in (5.29) and (5.30) coincide with the solutions of (5.2) and (5.3) with μ^{ε} = μ_{Ω^{ε}}. For the same reason the functions u^{ε} and \overline{u}^{ε} defined in (5.31) and (5.32) coincide with the solutions of (5.9) and (5.10) with μ^{ε} = μ_{Ω^{ε}}. The conclusion follows now from Theorem 5.1.

Remark 5.6. Let (λ^{ε}) be a sequence in H^{−1}(Ω)^{+} and, for every ε > 0, let ω^{ε} be a function in H^{1}(Ω) such that ω^{ε} = 0 q.e. in Ω \ Ω^{ε} and

−div(A^{ε}Dω^{ε}) = λ^{ε} in D′(Ω^{ε}) .

Let λ^{0} ∈ H^{−1}(Ω)^{+} and let ω^{0} be a solution of (5.18) with ε = 0. If conditions (5.19)– (5.24) are satisfied, then the equivalent conditions (a)–(d) of Corollary 5.5 are satisfied. To prove this fact, it is enough to use Remark 4.1 and Theorem 5.4.

6. An example

In this section we apply Corollary 5.5 and Remark 5.6 to a model problem that has not yet been considered in the literature. The purpose of this example is to show that the measure μ^{0} which appears in the limit problem depends not only on the sequence (Ω^{ε}) and on A^{0} , but also on the sequence (A^{ε}).

To simplify the exposition, we assume n ≥ 3 (the case n = 2 requires obvious modifications, as in [7]). Let us fix an exponent γ with

(6.1)  1 < γ < \frac{n}{n − 2} .

For every ε > 0 and i ∈ Z^{n} we consider the point x_{i}^{ε} = εi, the open ball B_{i}^{ε} with centre x_{i}^{ε} and radius ε^{γ}, and the concentric closed ball C_{i}^{ε} with radius ε^{\frac{n}{n−2}} . By (6.1) we have C_{i}^{ε} ⊆ B_{i}^{ε} for 0 < ε < 1, and the sets (B_{i}^{ε})_{i∈Z^{n}} are pairwise disjoint for 0 < ε < 2^{\frac{1}{1−γ}} . Given a bounded open set Ω ⊆ R^{n} we define

B^{ε} = Ω ∩ ⋃_{i∈Z^{n}} B_{i}^{ε} , C^{ε} = Ω ∩ ⋃_{i∈Z^{n}} C_{i}^{ε} .

Let us fix two constants a, b ∈ [α, β] and let us define the matrices A^{ε}, for ε ≥ 0, by

(6.2)  A^{ε}(x) = \{ \begin{matrix}a I & for x ∈ Ω \ B^{ε}, \\ b I & for x ∈ B^{ε}, \end{matrix}

where we set B^{0} = Ø, so that A^{0}(x) = a I for every x ∈ Ω.

Since (A^{ε}) converges in measure to A^{0} by (6.1), it is easy to prove that (A^{ε}) H - converges to A^{0} .

Finally, let Ω^{ε} = Ω \ C^{ε} for every ε > 0.

We will determine μ^{0} ∈ M_{0}^{+}(Ω) such that the equivalent conditions (a)–(d) of Corollary 5.5 are satisfied. Using Remark 5.6 we will construct, for ε ≥ 0, a measure λ^{ε} in H^{−1}(Ω)^{+} and, for ε > 0, a function ω^{ε} in H^{1}(Ω) such that ω^{ε} = 0 q.e. in C^{ε} and

(6.3)  −div(A^{ε}Dω^{ε}) = λ^{ε} in D′(Ω^{ε}) .

Then we will prove that conditions (5.19)–(5.24) are satisfied, where ω^{0} is a solution of (5.18) with ε = 0.

For every ε > 0 and i ∈ Z^{n} let ω_{i}^{ε} ∈ H^{1}(B_{i}^{ε} \ C_{i}^{ε}) be the solution of the equation Δω_{i}^{ε} = 0 on B_{i}^{ε} \ C_{i}^{ε} which satisfies the boundary conditions ω_{i}^{ε} = 0 on ∂C_{i}^{ε} and ω_{i}^{ε} = 1 on ∂B_{i}^{ε}. By explicit computation we find that

(6.4)  ω_{i}^{ε}(x) = c^{ε} − c^{ε}ε^{n}|x − x_{i}^{ε}|^{2−n} for x ∈ B_{i}^{ε} \ C_{i}^{ε} ,

where

(6.5)  c^{ε} = \frac{1}{1 − ε^{n−γ(n−2)}} ⟶ 1

by (6.1). For 0 < ε < 2^{\frac{1}{1−γ}} we define ω^{ε} as the function which is equal to ω_{i}^{ε} on (B_{i}^{ε} \ C_{i}^{ε}) ∩ Ω, and is extended by 0 on C^{ε} and by 1 on Ω \ B^{ε}.

By direct computation we find that

∫_{B_{i}^{ε}\C_{i}^{ε}} |Dω_{i}^{ε}|^{2}dx = (n − 2) S_{n−1} c^{ε}ε^{n} ,

where S_{n−1} is the (n − 1)-dimensional measure of the boundary of the unit ball in R^{n}. This yields

(6.6)  ∫_{Ω} |Dω^{ε}|^{2}dx ≤ (n − 2) S_{n−1} c^{ε}N^{ε}ε^{n} ,

where N^{ε} is the number of indices i ∈ Z^{n} such that the distance from x_{i}^{ε} to Ω is less than ε. Since

(6.7)  lim_{ε→0} N^{ε}ε^{n} = meas(\overline{Ω}) < +∞ ,

from (6.5) and (6.6) we deduce that (ω^{ε}) is bounded in H^{1}(Ω). As (ω^{ε}) converges to ω^{0} = 1 in measure, we conclude that (ω^{ε}) converges to ω^{0} weakly in H^{1}(Ω), i.e., condition (5.22) is fulfilled.

Let σ^{ε} denote the (n − 1)-dimensional measure on Ω ∩ ∂B^{ε} and let λ^{ε} be the measure defined by

λ^{ε} = b (n − 2) c^{ε}ε^{n−γ(n−1)}σ^{ε} .

Since, by (6.4),

\frac{∂ω_{i}^{ε}}{∂ν} = (n − 2) c^{ε}ε^{n−γ(n−1)} on ∂B_{i}^{ε} ,

we obtain that −b Δω^{ε} = λ^{ε} in D′(Ω^{ε}). As Dω^{ε} = 0 a.e. in Ω^{ε} \ B^{ε}, we have A^{ε}Dω^{ε} = b Dω^{ε} a.e. in Ω^{ε} by (6.2), and we conclude that (6.3) holds.

From the properties of σ^{ε} and from (6.5) it follows that λ^{ε} ∈ H^{−1}(Ω)^{+} and that

(6.8)  lim_{ε→0} ∫_{Ω} φ dλ^{ε} = b (n − 2) S_{n−1} ∫_{Ω} φ dx ,

for every φ ∈ C_{c}^{∞}(Ω).

We now define μ^{0} = λ^{0} = b (n − 2) S_{n−1} . Then condition (5.19) is satisfied and ω^{0} = 1 is a solution to problem (5.18) for ε = 0. Therefore it remains to prove that (λ^{ε}) converges to λ^{0} strongly in H^{−1}(Ω).

To this aim for every ε > 0 and i ∈ Z^{n} we consider the functions v_{i}^{ε} defined by

v_{i}^{ε}(x) = \{ \begin{matrix}b c^{ε}ε^{n}|x − x_{i}^{ε}|^{2−n} & if x ∈ D_{i}^{ε} \ B_{i}^{ε}, \\ b c^{ε}ε^{n−γ(n−2)} & if x ∈ B_{i}^{ε}, \end{matrix}

where D_{i}^{ε} is the open ball with centre x_{i}^{ε} and radius ε/2. By computing the normal derivatives of v_{i}^{ε} on both sides of ∂B_{i}^{ε} we obtain that

(6.9)  −Δv_{i}^{ε} = λ^{ε} on D_{i}^{ε} ,

for 0 < ε < 2^{\frac{1}{1−γ}} .

Let E_{i}^{ε} be the open ball with centre x_{i}^{ε} and radius ε/4. We take a cut-off function φ_{i}^{ε} ∈ C_{c}^{∞}(D_{i}^{ε}) such that φ_{i}^{ε} = 1 on E_{i}^{ε}, and 0 ≤ φ_{i}^{ε} ≤ 1, |Dφ_{i}^{ε}| ≤ c/ε, and |Δφ_{i}^{ε}| ≤ c/ε^{2} on D_{i}^{ε}, where c is a suitable constant independent of ε and i.

Finally, we define v^{ε} ∈ H^{1}(Ω) by

v^{ε} = ∑_{i∈Z^{n}} φ_{i}^{ε}v_{i}^{ε} .

By (6.9) we have

(6.10)  −Δv^{ε} = λ^{ε} + g^{ε} ,

where

g^{ε} = −2 ∑_{i∈Z^{n}} Dφ_{i}^{ε}Dv_{i}^{ε} − ∑_{i∈Z^{n}} Δφ_{i}^{ε}v_{i}^{ε} .

From the definition of v_{i}^{ε} and from the estimates for Dφ_{i}^{ε} and Δφ_{i}^{ε} we obtain that the sequence (g^{ε}) is bounded in L^{∞}(Ω). Therefore, passing to a subsequence, we may assume that

(6.11)  g^{ε} ⇀ g  weakly in L^{2}(Ω) and strongly in H^{−1}(Ω) .
Moreover we have, for 0 < ε < 4^{\frac{1}{1−γ}} ,

\begin{aligned}∫_{Ω} |Dv^{ε}|^{2}dx & ≤ 2 ∑_{i∈Z^{n}} \{ ∫_{D_{i}^{ε}\B_{i}^{ε}} |Dv_{i}^{ε}|^{2}dx + \frac{1}{ε^{2}} ∫_{D_{i}^{ε}\E_{i}^{ε}} |v_{i}^{ε}|^{2}dx\} ≤ \\ & ≤ M N^{ε}ε^{n}(ε^{n−γ(n−2)} + ε^{2}) , \end{aligned}

for a suitable constant M independent of ε. Taking (6.1) and (6.7) into account, we conclude that (Dv^{ε}) converges to 0 strongly in L^{2}(Ω, R^{n}), hence (Δv^{ε}) converges to 0 strongly in H^{−1}(Ω). By (6.10) and (6.11) we obtain that (λ^{ε}) converges to −g strongly in H^{−1}(Ω), and by (6.8) we have −g = b (n − 2) S_{n−1} = λ^{0} . Since the limit does not depend on the subsequence, we conclude that (λ^{ε}) converges to λ^{0} strongly in H^{−1}(Ω).

Therefore, by Remark 5.6, if (f^{ε})converges to f^{0} strongly in H^{−1}(Ω), then the solutions u^{ε} of the classical Dirichlet problems

\{ \begin{matrix}u^{ε} ∈ H_{0}^{1}(Ω^{ε}) , \\ −div(A^{ε}Du^{ε}) = f^{ε} in D′(Ω^{ε}) , \end{matrix}

extended by 0 on Ω \ Ω^{ε}, converge weakly in H_{0}^{1}(Ω) to the solution u^{0} of the problem

\{ \begin{matrix}u^{0} ∈ H_{0}^{1}(Ω) , \\ −div(A^{0}Du^{0}) + μ^{0}u^{0} = f^{0} in D′(Ω) , \end{matrix}

where μ^{0} = b (n − 2) S_{n−1} .

Note that, if we change the constant b in the definition of A^{ε} (see (6.2)), the H - limit A^{0} does not change, but the measure μ^{0} changes. This shows that μ^{0} depends on the whole sequence (A^{ε}), and not only on A^{0} .

7. Global and local corrector results

In this section we prove a corrector result for the solutions of problems (5.9) in the special case f^{ε} = f^{0} = f, with f ∈ L^{∞}(Ω). In Section 10 we shall consider the case where (f^{ε}) converges to f^{0} strongly in H^{−1}(Ω), together with the case of more general data.

Assume that (ω^{ε})_{ε≥0} and (λ^{ε})_{ε≥0} satisfy (5.18)–(5.24). In order to obtain the corrector result we assume, in addition, that

sup_{ε≥0} ∥ω^{ε}∥_{L^{∞}(Ω)} < +∞ , (7.1)

sup_{ε≥0} ∫_{Ω} |ω^{ε}|^{2}dμ^{ε} < +∞ . (7.2)

Remark 7.1. If conditions (5.18)–(5.24), (7.1), and (7.2) are satisfied in Ω, then they are satisfied in every open set U ⊆ Ω.

The functions w^{ε} introduced in (5.2) satisfy conditions (7.1) and (7.2), as stated in (5.5) and (5.8).

Global corrector result

For j = 1, 2, …, n let us fix a sequence (z_{j}^{ε}) in H^{1}(Ω) satisfying (3.5)–(3.9). Let u^{0} be the solution of (5.9) with ε = 0 and f^{0} = f ∈ L^{∞}(Ω). Let us fix δ > 0 and ψ_{δ} ∈ H^{2}(Ω) ∩ W^{1,∞}(Ω) such that

(7.3)  β ∫_{Ω} |Du^{0} − D(ψ_{δ}ω^{0})|^{2}dx + ∫_{Ω} |u^{0} − ψ_{δ}ω^{0}|^{2}dμ^{0} < δ .

Such a ψ_{δ} exists since the set {ω^{0}φ : φ ∈ C_{c}^{∞}(Ω)} is dense in H_{0}^{1}(Ω) ∩ L^{2}(Ω, μ^{0}) by Proposition 4.7.

For every ε > 0 let v_{δ}^{ε} be the function defined by

(7.4)  v_{δ}^{ε} = (ψ_{δ} + ∑_{j=1}^{n} D_{j}ψ_{δ}z_{j}^{ε}) ω^{ε} .

By (3.5), (3.8), (5.22), and (7.1) we have

(7.5)  v_{δ}^{ε} ⇀ ψ_{δ}ω^{0}  weakly in H^{1}(Ω) and weakly^{∗} in L^{∞}(Ω) .

Moreover we have

Dv_{δ}^{ε} = (ψ_{δ} + ∑_{j=1}^{n} D_{j}ψ_{δ}z_{j}^{ε})Dω^{ε} + ∑_{j=1}^{n} D_{j}ψ_{δ}(e_{j} + Dz_{j}^{ε}) ω^{ε} + ∑_{j=1}^{n} DD_{j}ψ_{δ}z_{j}^{ε}ω^{ε} .

The last sum in the right hand side converges to 0 strongly in L^{2}(Ω, R^{n}) by (3.8) and (7.1), while (D_{j}ψ_{δ}z_{j}^{ε}Dω^{ε}) converges to 0 strongly in L^{2}(Ω, R^{n}) by (3.8) and (5.22). Therefore

(7.6)  Dv_{δ}^{ε} = ψ_{δ}Dω^{ε} + ∑_{j=1}^{n} D_{j}ψ_{δ}(e_{j} + Dz_{j}^{ε}) ω^{ε} + H_{δ}^{ε} ,

where (H_{δ}^{ε}) converges to 0 strongly in L^{2}(Ω, R^{n}) as ε tends to 0.

Since ψ_{δ} ∈ W^{1,∞}(Ω) and (z_{j}^{ε}) is bounded in L^{∞}(Ω), from (7.2) we deduce that

(7.7)  sup_{ε>0} ∫_{Ω} |v_{δ}^{ε}|^{2}dμ^{ε} < +∞ .

Theorem 7.2. Assume (5.1), (5.18)–(5.24), (7.1), and (7.2). Let δ > 0 and let ψ_{δ} be a function in H^{2}(Ω) ∩ W^{1,∞}(Ω) which satisfies (7.3). Assume that the functions v_{δ}^{ε} defined by (7.4) belong to H_{0}^{1}(Ω). Then for every f ∈ L^{∞}(Ω) the solutions u^{ε} of problems (5.9) with f^{ε} = f satisfy the estimate

(7.8)  lim sup_{ε→0} {α ∫_{Ω} |Du^{ε} − Dv_{δ}^{ε}|^{2}dx + ∫_{Ω} |u^{ε} − v_{δ}^{ε}|^{2}dμ^{ε}} < δ .

Remark 7.3. In the special case u^{0} = ψ ω^{0} , for some ψ ∈ H^{2}(Ω) ∩ W^{1,∞}(Ω), we can take ψ_{δ} = ψ for every δ > 0 in (7.3), so that

(7.9)  v_{δ}^{ε} = v^{ε} = (ψ + ∑_{j=1}^{n} D_{j}ψz_{j}^{ε}) ω^{ε} .

Therefore (7.8) implies

(7.10)  lim_{ε→0} {α ∫_{Ω} |Du^{ε} − Dv^{ε}|^{2}dx + ∫_{Ω} |u^{ε} − v^{ε}|^{2}dμ^{ε}} = 0 ,

which is a corrector result.

When the measures μ^{ε} are fixed and equal to 0 (so that we can choose ω^{ε} = ω^{0} = 1 and ψ = u^{0} ), formulas (7.9) and (7.10) provide the classical corrector result for H - converging operators stated in (3.14) (see [23] and, in the periodic case, [1] and [24]). When the matrices A^{ε} are fixed and equal to some matrix A^{0} (so that we can choose z_{j}^{ε} = 0), formulas (7.9) and (7.10) with ω^{ε} = w^{ε} defined by (5.2) provide the corrector result of [11] and [15]; with a different choice of ω^{ε}, which leads to ω^{0} = 1, the same formulas give also the corrector result of [7] in the periodic case. When both A^{ε} and μ^{ε} depend on ε, but ω^{0} = 1, so that we have ψ = u^{0} , the combination of H -converging operators and varying domains results in the multiplication of the corresponding correctors.

In the general case, ψ_{δ} and v_{δ}^{ε} depend on δ and we obtain from (7.8) that

Du^{ε} = Dv_{δ}^{ε} + R_{δ}^{ε}  with  lim sup_{ε→0} ∥R_{δ}^{ε}∥_{L^{2}(Ω,R^{n})}^{2} < \frac{δ}{α} ,

which is still a corrector result, but in a more technical form.

Local convergence and corrector results We consider now the case where the functions u^{ε} are solutions of the problems

(7.11) \{ \begin{matrix}u^{ε} ∈ H^{1}(Ω) ∩ L^{2}(Ω, μ^{ε}) , \\ ∫_{Ω} A^{ε}Du^{ε}Dy dx + ∫_{Ω} u^{ε}y dμ^{ε} = ∫_{Ω} f y dx  ∀y ∈ H_{0}^{1}(Ω) ∩ L^{2}(Ω, μ^{ε}) , \end{matrix}

but are not required to satisfy the boundary condition u^{ε} = 0 on ∂Ω. We still consider the case of data f ∈ L^{∞}(Ω). More general data will be studied in Section 10.

The following theorem is a local version of the convergence result given in Theorem 5.4. It will be proved in Section 9.

Theorem 7.4. Assume (5.1), (5.18)–(5.24), (7.1), and (7.2). Let f ∈ L^{∞}(Ω) and, for every ε > 0, let u^{ε} be a solution of (7.11). Assume that

(7.12)  u^{ε} ⇀ u^{0}  weakly in H^{1}(Ω) ,

for some function u^{0} ∈ H^{1}(Ω), and that

sup_{ε>0} ∥u^{ε}∥_{L^{∞}(Ω)} < +∞ , (7.13)

sup_{ε>0} ∫_{Ω} |u^{ε}|^{2}dμ^{ε} < +∞ . (7.14)

Then u^{0} is a solution of (7.11) for ε = 0.

The following lemma, which will be proved in Section 9, shows that (under the other assumptions of Theorem 7.4) conditions (7.13) and (7.14) are always satisfied in every open set U ⊂⊂ Ω, and also in Ω if every u^{ε} belongs to H_{0}^{1}(Ω).

Lemma 7.5. Assume (5.1), (5.18)–(5.24), (7.1), and (7.2). Let f ∈ L^{∞}(Ω) and, for every ε > 0, let u^{ε} be a solution of (7.11). Assume that (7.12) holds for some function u^{0} ∈ H^{1}(Ω). Then we have

sup_{ε>0} ∥u^{ε}∥_{L^{∞}(U)} < +∞ , (7.15)

sup_{ε>0} ∫_{U} |u^{ε}|^{2}dμ^{ε} < +∞ , (7.16)

for every open set U ⊂⊂ Ω. If, in addition, u^{ε} ∈ H_{0}^{1}(Ω) for every ε > 0, then (7.15) and (7.16) also hold for U = Ω.

In the next corollary H_{c}^{1}(Ω) denotes the space of all functions u ∈ H^{1}(Ω) with compact support in Ω. The first assertion of the corollary follows immediately from Theorem 7.4 and Lemma 7.5, while the last assertion is easily obtained by approximating any nonnegative function y ∈ H_{0}^{1}(Ω) ∩ L^{2}(Ω, μ^{0}) by the sequence (φ_{j} ∧ y), where φ_{j} ∈ C_{c}^{∞}(Ω) converges to y in H_{0}^{1}(Ω).

Corollary 7.6. Under the assumptions of Lemma 7.5, u^{0} is a solution to the problem

(7.17) \{ \begin{matrix}u^{0} ∈ H^{1}(Ω) ∩ L_{loc}^{2}(Ω, μ^{0}) , \\ ∫_{Ω} A^{0}Du^{0}Dy dx + ∫_{Ω} u^{0}y dμ^{0} = ∫_{Ω} f y dx  ∀y ∈ H_{c}^{1}(Ω) ∩ L^{2}(Ω, μ^{0}) . \end{matrix}

If, in addition, u^{0} ∈ L^{2}(Ω, μ^{0}), then the last line of (7.17) holds for every y ∈ H_{0}^{1}(Ω) ∩ L^{2}(Ω, μ^{0}).

Let us fix an open set U ⊂⊂ Ω and a function ζ ∈ C_{c}^{∞}(Ω) such that ζ = 1 in U. Given u^{0} ∈ H_{loc}^{1}(Ω) ∩ L_{loc}^{2}(Ω, μ^{0}), by Proposition 4.7 we can approximate the function ζ u^{0} in H_{0}^{1}(Ω) ∩ L^{2}(Ω, μ^{0}) by functions of the form ψ ω^{0} with ψ ∈ C_{c}^{∞}(Ω). Therefore for every δ > 0 there exists ψ_{δ} ∈ H^{2}(U) ∩ W^{1,∞}(U) such that

(7.18)  β ∫_{U} |Du^{0} − D(ψ_{δ}ω^{0})|^{2}dx + ∫_{U} |u^{0} − ψ_{δ}ω^{0}|^{2}dμ^{0} < δ .

The following theorem is a local version of the corrector result given in Theorem 7.2.

Theorem 7.7. Under the hypotheses of Lemma 7.5, let U be an open set with U ⊂⊂ Ω, let δ > 0, let ψ_{δ} be a function in H^{2}(U) ∩ W^{1,∞}(U) which satisfies (7.18), and let v_{δ}^{ε} be the functions defined in U by (7.4). Then

(7.19)  lim sup_{ε→0} {α ∫_{V} |Du^{ε} − Dv_{δ}^{ε}|^{2}dx + ∫_{V} |u^{ε} − v_{δ}^{ε}|^{2}dμ^{ε}} < δ ,

for every open set V ⊂⊂ U.

Theorems 7.2 and 7.7 can be deduced from the following theorem, which will be proved in Section 9. Indeed, by Theorem 5.4 and Lemma 7.5, the assumptions of Theorem 7.2 imply all assumptions of Theorem 7.4, so that (7.8) follows from (3.1), (3.2), and (7.21) with φ = 1. Similarly, the assumptions of Theorem 7.7 imply, by Lemma 7.5, that all assumptions of Theorem 7.4 are satisfied in every open set U ⊂⊂ Ω, so that we can apply Theorem 7.8 with Ω replaced by U and with φ ∈ C_{c}^{∞}(U) such that φ = 1 in V and φ ≥ 0 in U \ V .

Theorem 7.8. Under the hypotheses of Theorem 7.4, let ψ be a function in H^{2}(Ω) ∩ W^{1,∞}(Ω), and let v^{ε} be defined by

(7.20)  v^{ε} = (ψ + ∑_{j=1}^{n} D_{j}ψz_{j}^{ε}) ω^{ε} .

Then for every φ ∈ C_{c}^{∞}(Ω) we have

(7.21)  \begin{matrix}lim_{ε→0} & \{ ∫_{Ω} A^{ε}D(u^{ε} − v^{ε})D(u^{ε} − v^{ε}) φ dx + ∫_{Ω} |u^{ε} − v^{ε}|^{2}φ dμ^{ε}\} = \\ = ∫_{Ω} A^{0}D(u^{0} − ψ ω^{0})D(u^{0} − ψ ω^{0}) φ dx + ∫_{Ω} |u^{0} − ψ ω^{0}|^{2}φ dμ^{0} . \end{matrix}

If the functions u^{ε} and v^{ε} belong to H_{0}^{1}(Ω) ∩ L^{2}(Ω, μ^{ε}) for every ε > 0, then (7.21) also holds with φ = 1.

8. A comparison theorem

In this section we state and prove a comparison result for the limit measures μ_{1}^{0} and μ_{2}^{0} corresponding to different sequences of H -convergent matrices A_{1}^{ε} and A_{2}^{ε}. This result has its own interest and will be crucial in the proof of the corrector results stated in the previous section.

For every ε ≥ 0 let A_{1}^{ε} and A_{2}^{ε} be two matrices in M_{α}^{β}(Ω). We assume that

(8.1)  A_{i}^{ε} H-converges to A_{i}^{0} for i = 1, 2.

For every ε > 0 let μ^{ε} be a measure in M_{0}^{+}(Ω), and let μ_{1}^{0} and μ_{2}^{0} be two measures in M_{0}^{+}(Ω). For i = 1, 2 and ε > 0 let w_{i}^{ε} be the solutions of the problems

(8.2) \{ \begin{matrix}w_{i}^{ε} ∈ H_{0}^{1}(Ω) ∩ L^{2}(Ω, μ^{ε}) , \\ ∫_{Ω} A_{i}^{ε}Dw_{i}^{ε}Dy dx + ∫_{Ω} w_{i}^{ε}y dμ^{ε} = ∫_{Ω} y dx  ∀y ∈ H_{0}^{1}(Ω) ∩ L^{2}(Ω, μ^{ε}) , \end{matrix}

and let w_{i}^{0} be the solutions of the problems

(8.3) \{ \begin{matrix}w_{i}^{0} ∈ H_{0}^{1}(Ω) ∩ L^{2}(Ω, μ_{i}^{0}) , \\ ∫_{Ω} A_{i}^{0}Dw_{i}^{0}Dy dx + ∫_{Ω} w_{i}^{0}y dμ_{i}^{0} = ∫_{Ω} y dx  ∀y ∈ H_{0}^{1}(Ω) ∩ L^{2}(Ω, μ_{i}^{0}) . \end{matrix}

We assume that

(8.4)  w_{i}^{ε} ⇀ w_{i}^{0}  weakly in H_{0}^{1}(Ω) .

Note that, by Theorem 5.2, these hypotheses are always satisfied by a subsequence. In this section we shall prove the following comparison theorem.

Theorem 8.1. Assume (8.1) and (8.4). Then

\frac{α^{2}}{β^{2}} μ_{2}^{0} ≤ μ_{1}^{0} ≤ \frac{β^{2}}{α^{2}} μ_{2}^{0} in Ω , (8.5)

cap({w_{1}^{0} > 0} △ {w_{2}^{0} > 0}) = 0 . (8.6)

In particular we have L^{2}(Ω, μ_{1}^{0}) = L^{2}(Ω, μ_{2}^{0}).

In order to prove Theorem 8.1, for ε ≥ 0 and i = 1, 2 we consider the measures ν_{i}^{ε} ∈ H^{−1}(Ω)^{+} defined by

(8.7)  −div(A_{i}^{ε}Dw_{i}^{ε}) + ν_{i}^{ε} = 1  in D′(Ω)

(see Proposition 4.3). By Proposition 4.6 we have

(8.8)  ν_{i}^{0} = w_{i}^{0}μ_{i}^{0} on {w_{i}^{0} > 0} .

By Theorem 3.1 we have

(8.9)  A_{i}^{ε}Dw_{i}^{ε} ⇀ A_{i}^{0}Dw_{i}^{0}  weakly in L^{2}(Ω, R^{n}) .

Therefore

(8.10)  ν_{i}^{ε} ⇀ ν_{i}^{0}  weakly in H^{−1}(Ω) .

As ν_{i}^{ε} ≥ 0, by Theorem 1 of [22] we have

(8.11)  ψ ν_{i}^{ε} → ψ ν_{i}^{0}  strongly in W^{−1,q}(Ω) ,

for every ψ ∈ C_{c}^{∞}(Ω) and for every q < 2.

Let ζ_{i}^{ε} be the solution of the problem

(8.12)  \{ \begin{matrix}ζ_{i}^{ε} ∈ H_{0}^{1}(Ω) , \\ −div(A_{i}^{ε}Dζ_{i}^{ε}) = −div(A_{i}^{0}Dw_{i}^{0}) in D′(Ω) . \end{matrix}

By the definition of H -convergence we have

(8.13)  ζ_{i}^{ε} ⇀ w_{i}^{0}  weakly in H_{0}^{1}(Ω) ,
(8.14)  A_{i}^{ε}Dζ_{i}^{ε} ⇀ A_{i}^{0}Dw_{i}^{0}  weakly in L^{2}(Ω, R^{n}) .

Lemma 8.2. For every φ ∈ C_{c}^{∞}(Ω) and i = 1, 2 we have

lim_{ε→0} { ∫_{Ω} A_{1}^{ε}D(w_{1}^{ε} − ζ_{1}^{ε})D(w_{2}^{ε} − ζ_{2}^{ε})φ dx + ∫_{Ω} w_{1}^{ε}w_{2}^{ε}φ dμ^{ε}} = ∫_{Ω} w_{2}^{0}φ dν_{1}^{0} , (8.15)

lim_{ε→0} { ∫_{Ω} A_{i}^{ε}D(w_{i}^{ε} − ζ_{i}^{ε})D(w_{i}^{ε} − ζ_{i}^{ε})φ dx + ∫_{Ω} |w_{i}^{ε}|^{2}φ dμ^{ε}} = ∫_{Ω} w_{i}^{0}φ dν_{i}^{0} . (8.16)

Proof. Let us first prove (8.15). For every ε > 0 we write

(8.17)  ∫_{Ω} A_{1}^{ε}D(w_{1}^{ε} − ζ_{1}^{ε})D(w_{2}^{ε} − ζ_{2}^{ε})φ dx + ∫_{Ω} w_{1}^{ε}w_{2}^{ε}φ dμ^{ε} = I^{ε} + II^{ε} + III^{ε} ,

where

\begin{aligned}I^{ε} & = ∫_{Ω} A_{1}^{ε}Dw_{1}^{ε}Dw_{2}^{ε}φ dx + ∫_{Ω} w_{1}^{ε}w_{2}^{ε}φ dμ^{ε} , \\ II^{ε} & = − ∫_{Ω} A_{1}^{ε}Dw_{1}^{ε}Dζ_{2}^{ε}φ dx , \\ III^{ε} & = − ∫_{Ω} A_{1}^{ε}Dζ_{1}^{ε}D(w_{2}^{ε} − ζ_{2}^{ε})φ dx . \end{aligned}

Using y = w_{2}^{ε}φ as test function in (8.2) we get

I^{ε} = ∫_{Ω} w_{2}^{ε}φ dx − ∫_{Ω} A_{1}^{ε}Dw_{1}^{ε}Dφ w_{2}^{ε} dx .

Since (w_{2}^{ε}) converges to w_{2}^{0} strongly in L^{2}(Ω) by (8.4) and since (A_{1}^{ε}Dw_{1}^{ε}) converges to A_{1}^{0}Dw_{1}^{0} weakly in L^{2}(Ω, R^{n}) by (8.9), we have

(8.18)  \begin{matrix}lim_{ε→0} I^{ε} = ∫_{Ω} w_{2}^{0}φ dx − ∫_{Ω} A_{1}^{0}Dw_{1}^{0}Dφw_{2}^{0} dx = \\ = ∫_{Ω} A_{1}^{0}Dw_{1}^{0}Dw_{2}^{0}φ dx + ∫_{Ω} w_{2}^{0}φ dν_{1}^{0} , \end{matrix}

where in the last equality we used (8.7) for ε = 0. Note that we can not use w_{2}^{0}φ as test function in (8.3) for i = 1 because we do not know yet that w_{2}^{0}φ ∈ L^{2}(Ω, μ_{1}^{0}).

From (8.7) we obtain

(8.19)  \begin{matrix}II^{ε} = − ∫_{Ω} A_{1}^{ε}Dw_{1}^{ε}D(ζ_{2}^{ε}φ) dx + ∫_{Ω} A_{1}^{ε}Dw_{1}^{ε}Dφ ζ_{2}^{ε} dx = \\ = ⟨ν_{1}^{ε}, ζ_{2}^{ε}φ⟩ − ∫_{Ω} ζ_{2}^{ε}φ dx + ∫_{Ω} A_{1}^{ε}Dw_{1}^{ε}Dφ ζ_{2}^{ε} dx . \end{matrix}

Since (ζ_{2}^{ε}) converges to w_{2}^{0} strongly in L^{2}(Ω) by (8.13) and (A_{1}^{ε}Dw_{1}^{ε}) converges to A_{1}^{0}Dw_{1}^{0} weakly in L^{2}(Ω, R^{n}) by (8.9), we have

(8.20)  \begin{matrix}lim_{ε→0} \{ − ∫_{Ω} ζ_{2}^{ε}φ dx + ∫_{Ω} A_{1}^{ε}Dw_{1}^{ε}Dφ ζ_{2}^{ε} dx\} = \\ = − ∫_{Ω} w_{2}^{0}φ dx + ∫_{Ω} A_{1}^{0}Dw_{1}^{0}Dφ w_{2}^{0} dx . \end{matrix}

We will prove in Lemma 8.3 that

(8.21)  lim_{ε→0} ⟨ν_{1}^{ε}, ζ_{2}^{ε}φ⟩ = ⟨ν_{1}^{0}, w_{2}^{0}φ⟩ .

From (8.19), (8.20), and (8.21) it follows that

(8.22)  \begin{matrix}lim_{ε→0} II^{ε} = ∫_{Ω} w_{2}^{0}φ dν_{1}^{0} − ∫_{Ω} w_{2}^{0}φ dx + ∫_{Ω} A_{1}^{0}Dw_{1}^{0}Dφ w_{2}^{0} dx = \\ = − ∫_{Ω} A_{1}^{0}Dw_{1}^{0}Dw_{2}^{0}φ dx , \end{matrix}

where the last equality is obtained by using w_{2}^{0}φ as test function in (8.7) for ε = 0. From (8.12) it follows that

III^{ε} = − ∫_{Ω} A_{1}^{ε}Dζ_{1}^{ε}D((w_{2}^{ε} − ζ_{2}^{ε}) φ) dx + ∫_{Ω} A_{1}^{ε}Dζ_{1}^{ε}Dφ (w_{2}^{ε} − ζ_{2}^{ε}) dx =
= − ∫_{Ω} A_{1}^{0}Dw_{1}^{0}D((w_{2}^{ε} − ζ_{2}^{ε}) φ) dx + ∫_{Ω} A_{1}^{ε}Dζ_{1}^{ε}Dφ (w_{2}^{ε} − ζ_{2}^{ε}) dx .

Since (w_{2}^{ε} − ζ_{2}^{ε}) converges to 0 weakly in H_{0}^{1}(Ω) and strongly in L^{2}(Ω) by (8.4) and (8.13), while (A_{1}^{ε}Dζ_{1}^{ε}) converges to A_{1}^{0}Dζ_{1}^{0} weakly in L^{2}(Ω, R^{n}) by (8.14), we have

(8.23)  lim_{ε→0} III^{ε} = 0 .

Equality (8.15) now follows from (8.17), (8.18), (8.22), and (8.23).

Let us prove now (8.16) for a given i = 1, 2. To this aim for every ε > 0 we define Â_{1}^{ε} = Â_{2}^{ε} = A_{i}^{ε} and μ̂^{ε} = μ^{ε}, so that ŵ_{1}^{ε} = ŵ_{2}^{ε} = w_{i}^{ε}, ζ̂_{1}^{ε} = ζ̂_{2}^{ε} = ζ_{i}^{ε}, and ν̂_{1}^{0} = ν̂_{2}^{0} = ν_{i}^{0} . Applying (8.15) in this new setting gives (8.16).

Lemma 8.3. For every φ ∈ C_{c}^{∞}(Ω) we have

(8.24)  lim_{ε→0} ⟨ν_{1}^{ε}, ζ_{2}^{ε}φ⟩ = ⟨ν_{1}^{0}, w_{2}^{0}φ⟩ .

Proof. Given δ > 0, let ζ^{0} ∈ C_{c}^{∞}(Ω) be a function such that

(8.25)  ∥ζ^{0} − w_{2}^{0}∥_{H_{0}^{1}(Ω)} < δ ,

and let ζ^{ε} be the solution of the problem

(8.26)  \{ \begin{matrix}ζ^{ε} ∈ H_{0}^{1}(Ω) , \\ −div(A_{2}^{ε}Dζ^{ε}) = −div(A_{2}^{0}Dζ^{0}) in D′(Ω) . \end{matrix}

Using ζ^{ε} − ζ_{2}^{ε} as test function in (8.12) and (8.26), from (3.1) and (3.2) we obtain

(8.27)  ∥ζ^{ε} − ζ_{2}^{ε}∥_{H_{0}^{1}(Ω)} ≤ \frac{β}{α}∥ζ^{0} − w_{2}^{0}∥_{H_{0}^{1}(Ω)} .

As (ν_{1}^{ε}) is bounded in H^{−1}(Ω), form (8.25) and (8.27) we obtain that there exists a constant M, independent of δ, such that

(8.28)  \begin{matrix}|⟨ν_{1}^{ε}, ζ_{2}^{ε}φ⟩ − ⟨ν_{1}^{0}, w_{2}^{0}φ⟩| ≤ \\ ≤ |⟨ν_{1}^{ε}, (ζ_{2}^{ε} − ζ^{ε})φ⟩| + |⟨ν_{1}^{ε}, ζ^{ε}φ⟩ − ⟨ν_{1}^{0}, ζ^{0}φ⟩| + |⟨ν_{1}^{0}, (ζ^{0} − w_{2}^{0})φ⟩| ≤ \\ ≤ Mδ + |⟨ν_{1}^{ε}, ζ^{ε}φ⟩ − ⟨ν_{1}^{0}, ζ^{0}φ⟩| . \end{matrix}

By Meyers’ estimate, there exists p > 2 such that (ζ^{ε}φ) is bounded in W_{0}^{1,p}(Ω). As (ζ^{ε}) converges to ζ^{0} weakly in H_{0}^{1}(Ω) by the definition of H -convergence, we conclude that (ζ^{ε}φ) converges to ζ^{0}φ weakly in W_{0}^{1,p}(Ω). Since by (8.11) the sequence (ψν_{1}^{ε}) converges to ψν_{1}^{0} strongly in W^{−1,q}(Ω) for 1/p + 1/q = 1 and for every ψ ∈ C_{c}^{∞}(Ω), we obtain that

lim_{ε→0} ⟨ν_{1}^{ε}, ζ^{ε}φ⟩ = lim_{ε→0} ⟨ψ ν_{1}^{ε}, ζ^{ε}φ⟩ = ⟨ψ ν_{1}^{0}, ζ^{0}φ⟩ = ⟨ν_{1}^{0}, ζ^{0}φ⟩ ,

where ψ is any function in C_{c}^{∞}(Ω) which is equal to 1 in a neighbourhood of supp(φ). Therefore by (8.28)

lim sup_{ε→0} |⟨ν_{1}^{ε}, ζ_{2}^{ε}φ⟩ − ⟨ν_{1}^{0}, w_{2}^{0}φ⟩| ≤ Mδ .

As δ > 0 is arbitrary, we obtain (8.24).

38 G. DAL MASO and F. MURAT

Lemma 8.4. For every φ ∈ C_{c}^{∞}(Ω), with φ ≥ 0 in Ω, and for every t > 0 we have

(8.29)  ∫_{Ω} w_{2}^{0}φ dν_{1}^{0} ≤ \frac{β}{α}\{\frac{t}{2} ∫_{Ω} w_{1}^{0}φ dν_{1}^{0} + \frac{1}{2t} ∫_{Ω} w_{2}^{0}φ dν_{2}^{0}\} .

Proof. By (3.1) and (3.2) we have the estimates

∫_{Ω} A_{1}^{ε}D(w_{1}^{ε} − ζ_{1}^{ε})D(w_{2}^{ε} − ζ_{2}^{ε})φ dx + ∫_{Ω} w_{1}^{ε}w_{2}^{ε}φ dμ^{ε} ≤ ≤ β ∫_{Ω} |D(w_{1}^{ε} − ζ_{1}^{ε})||D(w_{2}^{ε} − ζ_{2}^{ε})|φ dx + ∫_{Ω} w_{1}^{ε}w_{2}^{ε}φ dμ^{ε} ≤

≤ \frac{t}{2}\{β ∫_{Ω} |D(w_{1}^{ε} − ζ_{1}^{ε})|^{2}φ dx + ∫_{Ω} |w_{1}^{ε}|^{2}φ dμ^{ε}\} +

+ \frac{1}{2t}\{β ∫_{Ω} |D(w_{2}^{ε} − ζ_{2}^{ε})|^{2}φ dx + ∫_{Ω} |w_{2}^{ε}|^{2}φ dμ^{ε}\} ≤

≤ \frac{β}{α} \frac{t}{2}\{ ∫_{Ω} A_{1}^{ε}D(w_{1}^{ε} − ζ_{1}^{ε})D(w_{1}^{ε} − ζ_{1}^{ε})φ dx + ∫_{Ω} |w_{1}^{ε}|^{2}φ dμ^{ε}\} +

+ \frac{β}{α} \frac{1}{2t}\{ ∫_{Ω} A_{2}^{ε}D(w_{2}^{ε} − ζ_{2}^{ε})D(w_{2}^{ε} − ζ_{2}^{ε})φ dx + ∫_{Ω} |w_{2}^{ε}|^{2}φ dμ^{ε}\} .

Inequality (8.29) is obtained by applying Lemma 8.2. Lemma 8.5. The following inequality holds:

(8.30)  w_{2}^{0}ν_{1}^{0} ≤ \frac{β^{2}}{α^{2}} w_{1}^{0}ν_{2}^{0} in Ω .

Proof. Let ν = ν_{1}^{0} + ν_{2}^{0} . From Lemma 8.4 it follows that for every t > 0

(8.31) w_{2}^{0} \frac{dν_{1}^{0}}{dν} ≤ \frac{β}{α}\{\frac{t}{2}w_{1}^{0} \frac{dν_{1}^{0}}{dν} + \frac{1}{2t}w_{2}^{0} \frac{dν_{2}^{0}}{dν} \} ν-a.e. in Ω .

If we minimize with respect to t we obtain

w_{2}^{0} \frac{dν_{1}^{0}}{dν} ≤ \frac{β^{2}}{α^{2}} w_{1}^{0} \frac{dν_{2}^{0}}{dν} ν-a.e. in Ω ,

which implies (8.30).

Proof of Theorem 8.1. We prove only the second inequality in (8.5) and

(8.32)  cap({w_{2}^{0} > 0} \ {w_{1}^{0} > 0}) = 0 .

The other inequality and the equality cap({w_{1}^{0} > 0} \ {w_{2}^{0} > 0}) = 0 are proved by exchanging the roles of A_{1}^{ε} and A_{2}^{ε}.

By (8.8) we have ν_{2}^{0} = w_{2}^{0}μ_{2}^{0} on {w_{2}^{0} > 0}, so that (8.30) gives

(8.33)  ν_{1}^{0} ≤ \frac{β^{2}}{α^{2}} w_{1}^{0}μ_{2}^{0}  on \{w_{2}^{0} > 0\} .

If y ∈ H_{0}^{1}(Ω)∩L^{2}(Ω, μ_{2}^{0}), then y = 0 q.e. on {w_{2}^{0} = 0} (see Proposition 4.5). From (8.7) and (8.33) it follows that

(8.34)  ∫_{Ω} A_{1}^{0}Dw_{1}^{0}Dy dx + \frac{β^{2}}{α^{2}} ∫_{Ω} w_{1}^{0}y dμ_{2}^{0} ≥ ∫_{Ω} y dx

for every y ∈ H_{0}^{1}(Ω) ∩ L^{2}(Ω, μ_{2}^{0}) with y ≥ 0 q.e. in Ω.

Let w be the solution of the problem

(8.35) \{ \begin{matrix}w ∈ H_{0}^{1}(Ω) ∩ L^{2}(Ω, μ_{2}^{0}) , \\ ∫_{Ω} A_{1}^{0}DwDy dx + \frac{β^{2}}{α^{2}} ∫_{Ω} wy dμ_{2}^{0} = ∫_{Ω} y dx  ∀y ∈ H_{0}^{1}(Ω) ∩ L^{2}(Ω, μ_{2}^{0}) . \end{matrix}

As 0 ≤ (w − w_{1}^{0})^{+} ≤ w q.e. in Ω, the function y = (w − w_{1}^{0})^{+} can be taken as test function in (8.35) and (8.34). By difference we obtain

∫_{Ω} A_{1}^{0}D(w − w_{1}^{0})D(w − w_{1}^{0})^{+}dx + \frac{β^{2}}{α^{2}} ∫_{Ω}(w − w_{1}^{0})(w − w_{1}^{0})^{+}dμ_{2}^{0} ≤ 0 ,

which implies (w − w_{1}^{0})^{+} = 0 a.e. in Ω, and hence w ≤ w_{1}^{0} q.e. in Ω by (2.1). Therefore

(8.36)  cap({w > 0} ∩ {w_{1}^{0} = 0}) = 0 .

Let us prove that

(8.37)  cap({w_{2}^{0} > 0} ∩ {w = 0}) = 0 .

It is enough to show that

(8.38)  cap({w_{2}^{0} > δ} ∩ {w = 0}) = 0

for every δ > 0. If (8.38) is not satisfied, by Proposition 4.5 we have \frac{β^{2}}{α^{2}} μ_{2}^{0}(\{w_{2}^{0} > δ\}) = +∞, which contradicts the fact that w_{2}^{0} ∈ L^{2}(Ω, μ_{2}^{0}). This proves (8.37).

As w ≥ 0 and w_{1}^{0} ≥ 0 q.e. on Ω by the comparison principle (Theorem 2.10 of [13]), from (8.36) and (8.37) it follows that

(8.39)  cap({w_{2}^{0} > 0} ∩ {w_{1}^{0} = 0}) = cap({w_{2}^{0} > 0} \ {w_{1}^{0} > 0}) = 0 ,

which proves (8.32). Since ν_{1}^{0} = w_{1}^{0}μ_{1}^{0} on {w_{1}^{0} > 0} by (8.8), it follows from (8.39) that ν_{1}^{0} = w_{1}^{0}μ_{1}^{0} on {w_{2}^{0} > 0}, so that (8.33) yields

w_{1}^{0}μ_{1}^{0} ≤ \frac{β^{2}}{α^{2}} w_{1}^{0}μ_{2}^{0} on \{w_{2}^{0} > 0\} .

As w_{1}^{0} > 0 q.e. on {w_{2}^{0} > 0}, we conclude that

(8.40)  μ_{1}^{0} ≤ \frac{β^{2}}{α^{2}} μ_{2}^{0}  on \{w_{2}^{0} > 0\} .

Let us finally prove that

(8.41)  μ_{1}^{0} ≤ \frac{β^{2}}{α^{2}} μ_{2}^{0}  on \{w_{2}^{0} = 0\} .

Let B be a Borel set contained in {w_{2}^{0} = 0}. If cap(B) = 0, then μ_{1}^{0}(B) = μ_{2}^{0}(B) = 0, because μ_{1}^{0} and μ_{2}^{0} belong to M_{0}^{+}(Ω). If cap(B) > 0, then μ_{2}^{0}(B) = +∞ by Proposition 4.5. In both cases we have μ_{1}^{0}(B) ≤ \frac{β^{2}}{α^{2}} μ_{2}^{0}(B), hence (8.41) is proved.

Inequality (8.5) now follows from (8.40) and (8.41).

9. Proofs of the corrector results

In this section we prove Lemma 7.5 and Theorems 7.4 and 7.8, which give immediately all results of Section 7 (see the comments before the statement of Theorem 7.8).

We begin by the following theorem, which is proved by using the comparison result of Section 8.

Theorem 9.1. For every ε > 0, let y^{ε} ∈ H^{1}(Ω) ∩ L^{2}(Ω, μ^{ε}). Assume that

(9.1)  y^{ε} ⇀ y^{0}  weakly in H^{1}(Ω) ,

sup_{ε>0} ∫_{Ω} |y^{ε}|^{2}dμ^{ε} < +∞ . (9.2)

Then y^{0} ∈ L^{2}(Ω, μ^{0}).

Proof. We use the notion of γ -convergence, introduced in [14] and further developed in [10], which concerns the convergence of minima and minimizers of the functionals J_{f}^{ε} defined on H_{0}^{1}(Ω) ∩ L^{2}(Ω, μ^{ε}) by

J_{f}^{ε}(y) = α ∫_{Ω} |Dy|^{2}dx + ∫_{Ω} |y|^{2}dμ^{ε} − 2⟨f, y⟩ ,

for any given f ∈ H^{−1}(Ω). Note that the minimizer of J_{f}^{ε} is the unique solution to problem (4.1) with A = αI and μ = μ^{ε}. By Theorem 4.14 of [14] there exists a subsequence, still denoted by (μ^{ε}), which γ -converges (with respect to the operator −αΔ) to a measure μ̂^{0} ∈ M_{0}^{+}(Ω) (the regularity property (b) of μ̂^{0} is obtained by using Theorem 3.10 of [10]). By Lemma 5.5 of [10] we have

(9.3) α ∫_{Ω} |Dy^{0}|^{2}dx + ∫_{Ω} |y^{0}|^{2}dμ̂^{0} ≤ lim inf_{ε→0} {α ∫_{Ω} |Dy^{ε}|^{2}dx + ∫_{Ω} |y^{ε}|^{2}dμ^{ε}} < +∞ .

Let ŵ^{ε} be the unique solution to the problem

(9.4)  \{ \begin{matrix}ŵ^{ε} ∈ H_{0}^{1}(Ω) ∩ L^{2}(Ω, μ^{ε}) , \\ α ∫_{Ω} Dŵ^{ε}Dy dx + ∫_{Ω} ŵ^{ε}y dμ^{ε} = ∫_{Ω} y dx  ∀y ∈ H_{0}^{1}(Ω) ∩ L^{2}(Ω, μ^{ε}) . \end{matrix}

By Proposition 4.10 of [14] the sequence (ŵ^{ε}) converges weakly in H_{0}^{1}(Ω) to the solution ŵ^{0} of the problem

(9.5)  \{ \begin{matrix}ŵ^{0} ∈ H_{0}^{1}(Ω) ∩ L^{2}(Ω, μ̂^{0}) , \\ α ∫_{Ω} Dŵ^{0}Dy dx + ∫_{Ω} ŵ^{0}y dμ̂^{0} = ∫_{Ω} y dx  ∀y ∈ H_{0}^{1}(Ω) ∩ L^{2}(Ω, μ̂^{0}) . \end{matrix}

If we apply Theorem 8.1 with A_{1}^{ε} = A^{ε} and A_{2}^{ε} = αI, we obtain

(9.6)  μ^{0} ≤ \frac{β^{2}}{α^{2}} μ̂^{0} ,

so that (9.3) implies that y^{0} ∈ L^{2}(Ω, μ^{0}).

Lemma 9.2. Under the hypotheses of Theorem 7.4, we have

(9.7)  A^{ε}Du^{ε} ⇀ A^{0}Du^{0}  weakly in L^{2}(Ω, R^{n}) .

Moreover there exists σ^{0} ∈ M(Ω) ∩ H^{−1}(Ω), with |σ^{0}| ∈ M(Ω) ∩ H^{−1}(Ω), such that

(9.8)  −div(A^{0}Du^{0}) + σ^{0} = f  in D′(Ω) .

Proof. Since the positive and the negative parts (u^{ε})^{+} and (u^{ε})^{−} of u^{ε} belong to H_{0}^{1}(Ω) ∩ L^{2}(Ω, μ^{ε}), by Theorem 2.4 of [13] for every ε > 0 we can consider the solutions uε_{⊕} and uε_{⊖} to the problems

\{ \begin{matrix}uε_{⊕} − (u^{ε})^{+} ∈ H_{0}^{1}(Ω) ∩ L^{2}(Ω, μ^{ε}) , \\ ∫_{Ω} A^{ε}Duε_{⊕}Dy dx + ∫_{Ω} uε_{⊕}y dμ^{ε} = ∫_{Ω} f^{+}y dx ∀y ∈ H_{0}^{1}(Ω) ∩ L^{2}(Ω, μ^{ε}) , \end{matrix} (9.9)

\{ \begin{matrix}uε_{⊖} − (u^{ε})^{−} ∈ H_{0}^{1}(Ω) ∩ L^{2}(Ω, μ^{ε}) , \\ ∫_{Ω} A^{ε}Duε_{⊖}Dy dx + ∫_{Ω} uε_{⊖}y dμ^{ε} = ∫_{Ω} f^{−}y dx ∀y ∈ H_{0}^{1}(Ω) ∩ L^{2}(Ω, μ^{ε}) . \end{matrix} (9.10)

By linearity we have

(9.11)  u^{ε} = uε_{⊕} − uε_{⊖} q.e. in Ω .

Using y = uε_{⊕} − (u^{ε})^{+} as test function in (9.9), and then (3.1) and (3.2), as well as Poincaré’s and Young’s inequalities, we obtain

(9.12)  sup_{ε>0} ∥uε_{⊕}∥_{H^{1}(Ω)} < +∞ .

Passing to a subsequence, we can assume that (uε_{⊕}) converges weakly in H^{1}(Ω) to some function u0_{⊕}. Since uε_{⊕} ≥ 0 q.e. in Ω by the comparison principle (Theorem 2.10 of [13]), by Proposition 4.3 there exists σε_{⊕} ∈ H^{−1}(Ω)^{+} such that

(9.13)  −div(A^{ε}Duε_{⊕}) + σε_{⊕} = f^{+} in D′(Ω) .

From Theorem 3.1 we obtain that

(9.14)  A^{ε}Duε_{⊕} ⇀ A^{0}Du0_{⊕}  weakly in L^{2}(Ω, R^{n}) ,

and we deduce from (9.13) that there exists σ0_{⊕} ∈ H^{−1}(Ω)^{+} such that

(9.15)  −div(A^{0}Du0_{⊕}) + σ0_{⊕} = f^{+}  in D′(Ω) .

Properties (9.7) and (9.8) now follow from (9.14) and (9.15), from the analogous results for uε_{⊖}, and from (9.11).

Proof of Lemma 7.5. By (9.11) we have u^{ε} = uε_{⊕} − uε_{⊖} q.e. in Ω, where uε_{⊕} and uε_{⊖} are the solutions of (9.9) and (9.10). Let vε_{⊕} be the solution to the problem

\{ \begin{matrix}vε_{⊕} − (u^{ε})^{+} ∈ H_{0}^{1}(Ω) , \\ −div(A^{ε}Dvε_{⊕}) = f^{+} in D′(Ω) . \end{matrix}

By the comparison principle (Theorem 2.10 of [13]) we have 0 ≤ uε_{⊕} ≤ vε_{⊕} q.e. in Ω.

As (u^{ε})^{+} is bounded in H^{1}(Ω), the sequence (vε_{⊕}) is bounded in H^{1}(Ω) too. On the other hand the classical local L^{∞} estimate for solutions of elliptic equations (see, e.g., [26]) asserts that for every open set U ⊂⊂ Ω

(9.16)  ∥vε_{⊕}∥_{L^{∞}(U)} ≤ C_{U}∥vε_{⊕}∥_{L^{2}(Ω)} ,

therefore (vε_{⊕}), and hence (uε_{⊕}), is bounded in L^{∞}(U). If u^{ε} ∈ H_{0}^{1}(Ω), we have also vε_{⊕} ∈ H_{0}^{1}(Ω), and the global L^{∞} estimate in Ω implies that (vε_{⊕}), and hence (uε_{⊕}), is bounded in L^{∞}(Ω). A similar argument holds for (uε_{⊖}), so that (u^{ε}) is bounded in L^{∞}(U) (and also in L^{∞}(Ω) if u^{ε} ∈ H_{0}^{1}(Ω)) and (7.15) is proved.

Let φ be a function in C_{c}^{∞}(Ω) such that φ = 1 in U. Using y = u^{ε}φ^{2} as test function in (7.11), and then (3.1), (3.2), and the boundedness of (u^{ε}) in H^{1}(Ω), we easily obtain (7.16). If u^{ε} ∈ H_{0}^{1}(Ω), we simply use y = u^{ε} as test function in (7.11).

The proof of Theorems 7.4 and 7.8 will be divided in three lemmas. For every ε > 0 let y^{ε} be a function of H^{1}(Ω) ∩ L^{2}(Ω, μ^{ε}) such that

(9.17)  y^{ε} ⇀ y^{0}  weakly in H^{1}(Ω) ,

for some function y^{0} in H^{1}(Ω). Assume that

sup_{ε>0} ∥y^{ε}∥_{L^{∞}(Ω)} < +∞ , (9.18)

sup_{ε>0} ∫_{Ω} |y^{ε}|^{2}dμ^{ε} < +∞ . (9.19)

Lemma 9.3. Under the hypotheses of Theorem 7.8, let y^{ε}, ε ≥ 0, be functions in H^{1}(Ω) which satisfy (9.17), (9.18), and (9.19). Then y^{0} belongs to L^{2}(Ω, μ^{0}) and for every φ ∈ C_{c}^{∞}(Ω) we have

(9.20)  \begin{aligned}lim_{ε→0} & ∫_{Ω} A^{ε}Dv^{ε}Dy^{ε}φ dx + ∫_{Ω} v^{ε}y^{ε}φ dμ^{ε} = \\ & = ∫_{Ω} A^{0}D(ψ ω^{0})Dy^{0}φ dx + ∫_{Ω} y^{0}ψ ω^{0}φ dμ^{0} . \end{aligned}

If, in addition, y^{ε} ∈ H_{0}^{1}(Ω) for every ε > 0, then (9.20) also holds with φ = 1.

Proof. We prove (9.20) only in the case φ ∈ C_{c}^{∞}(Ω), since, under the additional hypothesis y^{ε} ∈ H_{0}^{1}(Ω), the proof with φ = 1 is similar. In this proof (η^{ε}) will denote a sequence of real numbers converging to 0 as ε tends to 0, whose value can change from line to line.

Theorem 9.1, (9.17), and (9.19) imply that y^{0} ∈ L^{2}(Ω, μ^{0}).

By (7.6) for every φ ∈ C_{c}^{∞}(Ω) we have

\begin{aligned}_{∫Ω} & A^{ε}Dv^{ε}Dy^{ε}φ dx = \\ & = ∑_{j=1}^{n} ∫_{Ω} D_{j}ψA^{ε}(e_{j} + Dz_{j}^{ε})Dy^{ε}ω^{ε}φ dx + ∫_{Ω} ψA^{ε}Dω^{ε}Dy^{ε}φ dx + η^{ε} . \end{aligned}

By (5.22) and (7.1) the sequence (ω^{ε}) converges to ω^{0} strongly in L^{r}(Ω) for every 1 ≤ r < +∞. Since, by (3.9), (z_{j}^{ε}) converges to 0 weakly in in W^{1,p}(Ω) for some p > 2, we conclude that

∫_{Ω} D_{j}ψA^{ε}(e_{j} + Dz_{j}^{ε})Dy^{ε}ω^{ε}φ dx = ∫_{Ω} D_{j}ψA^{ε}(e_{j} + Dz_{j}^{ε})Dy^{ε}ω^{0}φ dx + η^{ε} .

Therefore

(9.21)  ∫_{Ω} A^{ε}Dv^{ε}Dy^{ε}φ dx + ∫_{Ω} v^{ε}y^{ε}φ dμ^{ε} = I^{ε} + II^{ε} + III^{ε} + η^{ε} ,

where

\begin{aligned}I^{ε} & = ∑_{j=1}^{n} ∫_{Ω} D_{j}ψA^{ε}(e_{j} + Dz_{j}^{ε})Dy^{ε}ω^{0}φ dx , \\ II^{ε} & = ∫_{Ω} ψA^{ε}Dω^{ε}Dy^{ε}φ dx , \\ III^{ε} & = ∫_{Ω} v^{ε}y^{ε}φ dμ^{ε} . \end{aligned}

We now pass to the limit in I^{ε}, II^{ε}, and III^{ε}. For what concerns I^{ε}, we write

I^{ε} = ∫_{Ω} D_{j}ψA^{ε}(e_{j} + Dz_{j}^{ε})Dy^{ε}ω^{0}φ dx =

= ⟨−div(A^{ε}(e_{j} + Dz_{j}^{ε})), D_{j}ψ y^{ε}ω^{0}φ⟩ − ∫_{Ω} A^{ε}(e_{j} + Dz_{j}^{ε})DD_{j}ψ y^{ε}ω^{0}φ dx −

− ∫_{Ω} A^{ε}(e_{j} + Dz_{j}^{ε})D_{j}ψ y^{ε}Dω^{0}φ dx − ∫_{Ω} A^{ε}(e_{j} + Dz_{j}^{ε})D_{j}ψ y^{ε}ω^{0}Dφ dx .

Properties (3.6) and (3.7) of z_{j}^{ε}, together with properties (9.17) and (9.18) of y^{ε}, imply that we can pass to the limit in each term of the right hand side of the previous formula, so that

(9.22)  I^{ε} = ∑_{j=1}^{n} ∫_{Ω} D_{j}ψA^{0}e_{j}Dy^{0}ω^{0}φ dx + η^{ε} = ∫_{Ω} A^{0}DψDy^{0}ω^{0}φ dx + η^{ε} .

As for II^{ε}, we write

(9.23) II^{ε} = ∫_{Ω} ψA^{ε}Dω^{ε}Dy^{ε}φ dx = ∫_{Ω} A^{ε}Dω^{ε}D(y^{ε}ψ φ) dx − ∫_{Ω} A^{ε}Dω^{ε}y^{ε}D(ψ φ) dx .

As ω^{ε} satisfies (5.18), we have

∫_{Ω} A^{ε}Dω^{ε}D(y^{ε}ψ φ) dx + ∫_{Ω} ω^{ε}y^{ε}ψ φ dμ^{ε} = ∫_{Ω} y^{ε}ψ φ dλ^{ε} ,

and by (5.20) and (9.17) we conclude that

(9.24)  ∫_{Ω} A^{ε}Dω^{ε}D(y^{ε}ψ φ) dx = ∫_{Ω} y^{0}ψ φ dλ^{0} − ∫_{Ω} ω^{ε}y^{ε}ψ φ dμ^{ε} + η^{ε} .

By Proposition 4.3 and Theorem 3.1 (A^{ε}Dω^{ε}) converges to A^{0}Dω^{0} weakly in L^{2}(Ω, R^{n}), while by (9.17) (y^{ε}) converges to y^{0} strongly in L_{loc}^{2}(Ω). Therefore

(9.25)  − ∫_{Ω} A^{ε}Dω^{ε}y^{ε}D(ψ φ) dx = − ∫_{Ω} A^{0}Dω^{0}y^{0}D(ψ φ) dx + η^{ε} .

From (9.23), (9.24), and (9.25) we obtain that

(9.26)  \begin{aligned}II^{ε} & = ∫_{Ω} y^{0}ψ φ dλ^{0} − ∫_{Ω} ω^{ε}y^{ε}ψ φ dμ^{ε} − ∫_{Ω} A^{0}Dω^{0}y^{0}D(ψ φ) dx + η^{ε} = \\ & = ∫_{Ω} A^{0}Dω^{0}Dy^{0}ψ φ dx + ∫_{Ω} ω^{0}y^{0}ψ φ dμ^{0} − ∫_{Ω} ω^{ε}y^{ε}ψ φ dμ^{ε} + η^{ε} , \end{aligned}

where the last equality follows from (5.18) for ε = 0, since y^{0} ∈ L^{2}(Ω, μ^{0}). Finally, we write III^{ε} as

III^{ε} = ∫_{Ω} v^{ε}y^{ε}φ dμ^{ε} = ∫_{Ω} ψ ω^{ε}y^{ε}φ dμ^{ε} + ∑_{j=1}^{n} ∫_{Ω} D_{j}ψ z_{j}^{ε}ω^{ε}y^{ε}φ dμ^{ε} .

Since, by (3.8), (z_{j}^{ε}) converges to 0 uniformly, while, by (7.2) and (9.19), the norms of ω^{ε} and y^{ε} in L^{2}(Ω, μ^{ε}) remain bounded, we conclude that

(9.27)  III^{ε} = ∫_{Ω} ψ ω^{ε}y^{ε}φ dμ^{ε} + η^{ε} .

From (9.21), (9.22), (9.26), and (9.27) we obtain (9.20).

Lemma 9.4. Under the hypotheses of Theorem 7.8, for every φ ∈ C_{c}^{∞}(Ω) we have

(9.28)

lim_{ε→0}{ ∫_{Ω} A^{ε}D(u^{ε} − v^{ε})D(u^{ε} − v^{ε}) φ dx + ∫_{Ω} |u^{ε} − v^{ε}|^{2}φ dμ^{ε}} =
= ∫_{Ω} A^{0}D(u^{0} − ψ ω^{0})D(u^{0} − ψ ω^{0}) φ dx +
+ ∫_{Ω}(u^{0} − ψ ω^{0}) φ dσ^{0} − ∫_{Ω}(u^{0} − ψ ω^{0})ψ ω^{0}φ dμ^{0} ,

where σ^{0} is defined by (9.8). If the functions u^{ε} and v^{ε} belong to H_{0}^{1}(Ω) for every ε > 0, then (9.28) also holds with φ = 1.

Proof. We prove the lemma only in the case φ ∈ C_{c}^{∞}(Ω), since, under the additional hypothesis u^{ε}, v^{ε} ∈ H_{0}^{1}(Ω), the proof with φ = 1 is similar.

Let y^{ε} = u^{ε} − v^{ε} and let y^{0} = u^{0} − ψω^{0} . Then properties (9.17), (9.18), and (9.19) are satisfied by the definition (7.20) of v^{ε} and by (3.5), (3.8), (5.22), (7.1), (7.2), (7.12), (7.13), and (7.14). Using y = y^{ε}φ as test function in (7.11) we get

(9.29)  \begin{aligned}_{∫Ω} & A^{ε}Du^{ε}Dy^{ε}φ dx + ∫_{Ω} u^{ε}y^{ε}φ dμ^{ε} = \\ & = ∫_{Ω} fy^{ε}φ dx − ∫_{Ω} A^{ε}Du^{ε}Dφ y^{ε} dx . \end{aligned}

Using (9.7) and (9.8) we obtain

(9.30)

lim_{ε→0}{ ∫_{Ω} A^{ε}Du^{ε}Dy^{ε}φ dx + ∫_{Ω} u^{ε}y^{ε}φ dμ^{ε}} =
= ∫_{Ω} fy^{0}φ dx − ∫_{Ω} A^{0}Du^{0}Dφ y^{0} dx =
= ∫_{Ω} A^{0}Du^{0}Dy^{0}φ dx + ∫_{Ω} y^{0}φ dσ^{0} .

From (9.30) and (9.20) we deduce (9.28).

Lemma 9.5. Under the hypotheses of Theorem 7.8, for every y ∈ H_{0}^{1}(Ω) ∩ L^{2}(Ω, μ^{0}) we have

(9.31)  ∫_{Ω} y dσ^{0} = ∫_{Ω} y u^{0} dμ^{0} ,

where σ^{0} is defined by (9.8).

Proof. First of all we recall that u^{0} ∈ H^{1}(Ω) ∩ L^{2}(Ω, μ^{0}) by Theorem 9.1. Let us fix φ ∈ C_{c}^{∞}(Ω) with φ ≥ 0 in Ω. By Lemma 9.4 we have

(9.32)  \begin{aligned}_{∫Ω} & A^{0}D(u^{0} − ψ ω^{0})D(u^{0} − ψ ω^{0}) φ dx + \\ & + ∫_{Ω}(u^{0} − ψ ω^{0}) φ dσ^{0} − ∫_{Ω}(u^{0} − ψ ω^{0}) ψ ω^{0}φ dμ^{0} ≥ 0 , \end{aligned}

for every ψ ∈ H^{2}(Ω) ∩ W^{1,∞}(Ω). By Proposition 4.7 the set {ψ ω^{0} : ψ ∈ C_{c}^{∞}(Ω)} is dense in H_{0}^{1}(Ω) ∩ L^{2}(Ω, μ^{0}). Therefore (9.32) implies that

(9.33) ∫_{Ω} A^{0}D(u^{0} − z)D(u^{0} − z) φ dx + ∫_{Ω}(u^{0} − z) φ dσ^{0} − ∫_{Ω}(u^{0} − z) z φ dμ^{0} ≥ 0 ,

for every z ∈ H_{0}^{1}(Ω) ∩ L^{2}(Ω, μ^{0}).

We now use Minty’s trick, and we take in (9.33) z = u^{0}ζ + ty, with t ∈ R, y ∈ H_{0}^{1}(Ω) ∩ L^{2}(Ω, μ^{0}), and ζ ∈ C_{c}^{∞}(Ω) with ζ = 1 on supp φ. Dividing by t and passing to the limit as t tends to 0 we obtain

∫_{Ω} y φ dσ^{0} = ∫_{Ω} y u^{0}φ dμ^{0} .

Since u^{0} ∈ L^{2}(Ω, μ^{0}), we obtain (9.31) by approximating 1 by a sequence (φ_{k}) of functions in C_{c}^{∞}(Ω).

Proof of Theorem 7.4. In view of Theorem 9.1 the function u^{0} belongs to H^{1}(Ω) ∩ L^{2}(Ω, μ^{0}). From (9.8) we have

∫_{Ω} A^{0}Du^{0}Dy dx + ∫_{Ω} y dσ^{0} = ∫_{Ω} fy dx ∀y ∈ H_{0}^{1}(Ω) ∩ L^{2}(Ω, μ^{0}) .

By Lemma 9.5 this implies (7.11) for ε = 0.

Proof of Theorem 7.8. Since u^{0} belongs to H^{1}(Ω) ∩ L^{2}(Ω, μ^{0}) by Theorem 9.1, it is enough to apply Lemmas 9.4 and 9.5.

10. Problems with more general data

In this section we state and prove global and local convergence and corrector results for relaxed Dirichlet problems of the form (5.9) and (7.11), when the right hand sides f^{ε} and f are replaced by more general linear functionals L^{ε}, and when the strong convergence of (f^{ε}) in H^{−1}(Ω) is replaced by the strong convergence of (L^{ε}) “along the sequence” of spaces (H_{0}^{1}(Ω) ∩ L^{2}(Ω, μ^{ε}))′.

Strong convergence of the data

For every ε ≥ 0 we consider an element of the dual space (H_{0}^{1}(Ω) ∩ L^{2}(Ω, μ^{ε}))′, i.e., a linear functional L^{ε}: H_{0}^{1}(Ω) ∩ L^{2}(Ω, μ^{ε}) → R such that

|L^{ε}(y)| ≤ C^{ε}\{α ∫_{Ω} |Dy|^{2}dx + ∫_{Ω} |y|^{2}dμ^{ε}\}^{\frac{1}{2}} ∀y ∈ H_{0}^{1}(Ω) ∩ L^{2}(Ω, μ^{ε}) ,

for a suitable constant C^{ε} < +∞ (the constant α is introduced in this formula for future convenience). It is easy to prove that each functional L^{ε} can be represented in the form

(10.1)  L^{ε}(y) = ⟨f^{ε}, y⟩ + ∫_{Ω} g^{ε}y dμ^{ε} ,

where f^{ε} ∈ H^{−1}(Ω) and g^{ε} ∈ L^{2}(Ω, μ^{ε}).

In this section we assume that

(10.2)  L^{ε} → L^{0}  strongly along the sequence (H_{0}^{1}(Ω) ∩ L^{2}(Ω, μ^{ε}))′ ,

in the sense that

(10.3)  lim_{ε′→0} L^{ε′}(y^{ε′}) = L^{0}(y^{0}) ,

for every subsequence ε′ of ε (see Notation in Section 2) and every sequence (y^{ε′}) which satisfies

y^{ε′} ∈ H_{0}^{1}(Ω) ∩ L^{2}(Ω, μ^{ε′}) ∀ε′ > 0 , (10.4)

(10.5)  y^{ε′} ⇀ y^{0}  weakly in H_{0}^{1}(Ω) ,

sup_{ε′>0} ∫_{Ω} |y^{ε′}|^{2}dμ^{ε′} < +∞ . (10.6)

Note that y^{0} ∈ L^{2}(Ω, μ^{0}) by Theorem 9.1. Since (10.3) holds true for every sequence (y^{ε′}) which satisfies (10.4), (10.5), and (10.6), it is easy to prove by contradiction that there exists a constant C < +∞ such that for every ε > 0

(10.7)  |L^{ε}(y)| ≤ C\{α ∫_{Ω} |Dy|^{2}dx + ∫_{Ω} |y|^{2}dμ^{ε}\}^{\frac{1}{2}}  ∀y ∈ H_{0}^{1}(Ω) ∩ L^{2}(Ω, μ^{ε}) .

When L^{ε} is represented as in (10.1) with g^{ε} = 0, it is easy to see that (10.2) is satisfied if (f^{ε}) converges to f^{0} strongly in H^{−1}(Ω) (this condition is also necessary if all measures μ^{ε} are zero). The case where the functions g^{ε} are not identically zero is of course more difficult to handle, since the measures μ^{ε} vary, and the corresponding spaces L^{2}(Ω, μ^{ε}) may be different for different values of ε. This leads in a natural way to definition (10.2), where we used the word “strongly” since the test functions y^{ε′} in (10.3) are only assumed to be uniformly bounded in the corresponding spaces H_{0}^{1}(Ω) ∩ L^{2}(Ω, μ^{ε}).

In this definition the presence in (10.3) of subsequences ε′ (and not just of the whole sequence ε) is due, among other reasons, to the fact that we want that the convergence of (L^{ε}) implies the convergence of any subsequence.

Global convergence and corrector results

By the Lax-Milgram lemma for every ε ≥ 0 there exists a unique solution u^{ε} to the problem

(10.8)  \{ \begin{matrix}u^{ε} ∈ H_{0}^{1}(Ω) ∩ L^{2}(Ω, μ^{ε}) , \\ ∫_{Ω} A^{ε}Du^{ε}Dy dx + ∫_{Ω} u^{ε}y dμ^{ε} = L^{ε}(y)  ∀y ∈ H_{0}^{1}(Ω) ∩ L^{2}(Ω, μ^{ε}) . \end{matrix}

The following theorem is a generalization of Theorem 5.4.

Theorem 10.1. Assume (5.1), (5.18)–(5.24), and (10.2). For every ε ≥ 0, let u^{ε} be the unique solution to problem (10.8). Then (u^{ε}) converges to u^{0} weakly in H_{0}^{1}(Ω).

Proof. By (3.1), (3.2), and (10.7), using y = u^{ε} as test function in (10.8) we obtain the estimate

(10.9)  α ∫_{Ω} |Du^{ε}|^{2}dx + ∫_{Ω} |u^{ε}|^{2}dμ^{ε} ≤ C^{2} .

Extracting a subsequence, we may assume that

(10.10)  u^{ε} ⇀ u  weakly in H_{0}^{1}(Ω) ,

for some function u ∈ H_{0}^{1}(Ω). By Theorem 9.1 we have u ∈ L^{2}(Ω, μ^{0}). We will prove that u = u^{0} . Since the limit does not depend on the subsequence, this will prove that the whole sequence (u^{ε}) converges to u^{0} .

If y ∈ H_{0}^{1}(Ω) ∩ L^{2}(Ω, μ^{0}) satisfies ∫_{Ω} fy dx = 0 for every f ∈ L^{∞}(Ω), then y = 0 a.e. in Ω. By the Hahn-Banach theorem, this implies that L^{∞}(Ω) is dense in the dual space of H_{0}^{1}(Ω) ∩ L^{2}(Ω, μ^{0}). Therefore, given η > 0, there exists f_{η} ∈ L^{∞}(Ω) such that

(10.11) |L^{0}(y)−∫_{Ω} f_{η}y dx| ≤ η\{α ∫_{Ω} |Dy|^{2}dx+∫_{Ω} |y|^{2}dμ^{0}\}^{\frac{1}{2}} ∀y ∈ H_{0}^{1}(Ω)∩L^{2}(Ω, μ^{0}) .

For every ε ≥ 0 let u_{η}^{ε} be the unique solution to problem (5.9) with f^{ε} = f_{η}. By Theorem 5.4 we have

(10.12) u_{η}^{ε} ⇀ u_{η}^{0} weakly in H_{0}^{1}(Ω) ,

and taking y = u_{η}^{ε} as test function in (5.9), with f^{ε} = f_{η}, we obtain

(10.13)  sup_{ε>0} ∫_{Ω} |u_{η}^{ε}|^{2}dμ^{ε} < +∞ .

Using y = u^{ε} − u_{η}^{ε} as test function in (10.8) and (5.9), with f^{ε} = f_{η}, we obtain by difference

(10.14) α ∫_{Ω} |D(u^{ε} − u_{η}^{ε})|^{2}dx + ∫_{Ω} |u^{ε} − u_{η}^{ε}|^{2}dμ^{ε} ≤ L^{ε}(u^{ε} − u_{η}^{ε}) − ∫_{Ω} f_{η}(u^{ε} − u_{η}^{ε}) dx ,

for every ε ≥ 0. By (10.3), (10.9), (10.10), (10.12), and (10.13) we have

(10.15) lim_{ε→0} {L^{ε}(u^{ε} − u_{η}^{ε}) + ∫_{Ω} f_{η}(u^{ε} − u_{η}^{ε}) dx} = L^{0}(u − u_{η}^{0}) − ∫_{Ω} f_{η}(u − u_{η}^{0}) dx .

Let μ̂^{0} be the measure defined in the proof of Theorem 9.1. By (9.3) we have

(10.16)

α ∫_{Ω} |D(u − u_{η}^{0})|^{2}dx + ∫_{Ω} |u − u_{η}^{0}|^{2}dμ̂^{0} ≤

≤ lim inf_{ε→0} {α ∫_{Ω} |D(u^{ε} − u_{η}^{ε})|^{2}dx + ∫_{Ω} |u^{ε} − u_{η}^{ε}|^{2}dμ^{ε}} .

From (9.6), (10.11), (10.14), (10.15), and (10.16) we obtain that

(10.17)  α ∫_{Ω} |D(u − u_{η}^{0})|^{2}dx + ∫_{Ω} |u − u_{η}^{0}|^{2}dμ^{0} ≤ \frac{β^{4}}{α^{4}} η^{2} .

Using (10.14) for ε = 0, we obtain from (10.11)

(10.18)  α ∫_{Ω} |D(u^{0} − u_{η}^{0})|^{2}dx + ∫_{Ω} |u^{0} − u_{η}^{0}|^{2}dμ^{0} ≤ η^{2} .

From (10.17) and (10.18) we get

α ∫_{Ω} |D(u − u^{0})|^{2}dx ≤ 4\frac{β^{4}}{α^{4}} η^{2} .

Since η > 0 is arbitrary, we conclude that u = u^{0} .

The next theorem is a generalization of Theorem 7.2.

Theorem 10.2. Assume (5.1), (5.18)–(5.24), (7.1), (7.2), and (10.2). Let δ > 0 and let ψ_{δ} be a function in H^{2}(Ω) ∩ W^{1,∞}(Ω) which satisfies (7.3). Assume that the functions v_{δ}^{ε} defined by (7.4) belong to H_{0}^{1}(Ω). Then we have

(10.19)  lim sup_{ε→0} {α ∫_{Ω} |Du^{ε} − Dv_{δ}^{ε}|^{2}dx + ∫_{Ω} |u^{ε} − v_{δ}^{ε}|^{2}dμ^{ε}} < δ .

Proof. Let us fix δ′ < δ such that

(10.20)  β ∫_{Ω} |Du^{0} − D(ψ_{δ}ω^{0})|^{2}dx + ∫_{Ω} |u^{0} − ψ_{δ}ω^{0}|^{2}dμ^{0} < δ′ .

For η > 0, let f_{η}, u_{η}^{ε}, and u_{η}^{0} be as in the proof of Theorem 10.1. Using (10.20) and (10.18), we fix η > 0 small enough such that

\sqrt{δ′} + \frac{β}{α}η < \sqrt{δ} , (10.21)

β ∫_{Ω} |Du_{η}^{0} − D(ψ_{δ}ω^{0})|^{2}dx + ∫_{Ω} |u_{η}^{0} − ψ_{δ}ω^{0}|^{2}dμ^{0} < δ′ . (10.22)

Therefore we can apply Theorem 7.2 with f = f_{η} and we obtain

(10.23)  lim sup_{ε→0} {α ∫_{Ω} |Du_{η}^{ε} − Dv_{δ}^{ε}|^{2}dx + ∫_{Ω} |u_{η}^{ε} − v_{δ}^{ε}|^{2}dμ^{ε}} < δ′ .

As (u^{ε}) converges to u^{0} weakly in H_{0}^{1}(Ω) by Theorem 10.1, using (10.11), (10.14), (10.15), and (10.17) we deduce that

(10.24)  lim sup_{ε→0} \{α ∫_{Ω} |Du^{ε} − Du_{η}^{ε}|^{2}dx + ∫_{Ω} |u^{ε} − u_{η}^{ε}|^{2}dμ^{ε}\} ≤ \frac{β^{2}}{α^{2}} η^{2} .

From (10.21), (10.23), and (10.24) we obtain (10.19).

Local convergence and corrector results We consider now the case where the functions u^{ε} are solutions to the problems

(10.25) \{ \begin{matrix}u^{ε} ∈ H^{1}(Ω) ∩ L^{2}(Ω, μ^{ε}) , \\ ∫_{Ω} A^{ε}Du^{ε}Dy dx + ∫_{Ω} u^{ε}y dμ^{ε} = L^{ε}(y)  ∀y ∈ H_{0}^{1}(Ω) ∩ L^{2}(Ω, μ^{ε}) , \end{matrix}

but are not required to satisfy the boundary condition u^{ε} = 0 on ∂Ω.

The next theorem is a generalization of Corollary 7.6.

Theorem 10.3. Assume (5.1), (5.18)–(5.24), (7.1), (7.2), and (10.2). For every ε > 0, let u^{ε} be a solution to problem (10.25). Assume that

(10.26)  u^{ε} ⇀ u^{0}  weakly in H^{1}(Ω) ,

for some function u^{0} ∈ H^{1}(Ω). Then u^{0} is a solution to the problem

(10.27) \{ \begin{matrix}u^{0} ∈ H^{1}(Ω) ∩ L_{loc}^{2}(Ω, μ^{0}) , \\ ∫_{Ω} A^{0}Du^{0}Dy dx + ∫_{Ω} u^{0}y dμ^{0} = L^{0}(y)  ∀y ∈ H_{c}^{1}(Ω) ∩ L^{2}(Ω, μ^{0}) , \end{matrix}

where H_{c}^{1}(Ω) denotes the space of all functions u ∈ H^{1}(Ω) with compact support in Ω. If, in addition, u^{0} ∈ L^{2}(Ω, μ^{0}), then the last line in (10.27) holds for every y ∈ H_{0}^{1}(Ω) ∩ L^{2}(Ω, μ^{0}).

Proof. As we have seen in the proof of Theorem 10.1, for every η > 0 there exists f_{η} ∈ L^{∞}(Ω) which satisfies (10.11). Let us fix an open set U ⊂⊂ Ω and let φ ∈ C_{c}^{∞}(Ω) such that φ = 1 on U. Using y = u^{ε}φ^{2} as test function in (10.25), and then (3.1), (3.2), (10.7), and (10.26) we obtain

(10.28)  sup_{ε>0} ∫_{U} |u^{ε}|^{2}dμ^{ε} < +∞ ,

which implies that u^{0} ∈ L^{2}(U, μ^{0}) by Theorem 9.1. For every ε ≥ 0 let u_{η}^{ε} be the unique solution to the problem

(10.29) \{ \begin{matrix}u_{η}^{ε} − u^{ε} ∈ H_{0}^{1}(U) ∩ L^{2}(U, μ^{ε}) , \\ ∫_{U} A^{ε}Du_{η}^{ε}Dy dx + ∫_{U} u_{η}^{ε}y dμ^{ε} = ∫_{U} f_{η}y dx  ∀y ∈ H_{0}^{1}(U) ∩ L^{2}(U, μ^{ε}) . \end{matrix}

Taking y = u_{η}^{ε} − u^{ε} as test function in (10.29) and (10.25), we obtain by difference

(10.30) α ∫_{U} |D(u^{ε} − u_{η}^{ε})|^{2}dx + ∫_{U} |u^{ε} − u_{η}^{ε}|^{2}dμ^{ε} ≤ L^{ε}(u^{ε} − u_{η}^{ε}) − ∫_{U} f_{η}(u^{ε} − u_{η}^{ε}) dx ,

for every ε ≥ 0. By (10.7) this implies that (u^{ε} − u_{η}^{ε}) is bounded in H_{0}^{1}(U) and that the integrals ∫_{U} |u^{ε} − u_{η}^{ε}|^{2}dμ^{ε} are bounded. Using (10.26) and (10.28), we conclude that (u_{η}^{ε}) is bounded in H^{1}(U) and

(10.31)  sup_{ε>0} ∫_{U} |u_{η}^{ε}|^{2}dμ^{ε} < +∞ .

Extracting a subsequence, we may assume that

(10.32)  u_{η}^{ε} ⇀ u  weakly in H^{1}(U) ,

for some function u ∈ H^{1}(U) with u − u^{0} ∈ H_{0}^{1}(U). By (10.31) and by Theorem 9.1 the function u belongs to L^{2}(U, μ^{0}). Using both assertions of Corollary 7.6, u is a solution to the problem

\{ \begin{matrix}u ∈ H^{1}(U) ∩ L^{2}(U, μ^{0}) , \\ ∫_{U} A^{0}Du Dy dx + ∫_{U} u y dμ^{0} = ∫_{U} f_{η}y dx  ∀y ∈ H_{0}^{1}(U) ∩ L^{2}(U, μ^{0}) . \end{matrix}

Since u − u^{0} ∈ H_{0}^{1}(U) ∩ L^{2}(U, μ^{0}), by uniqueness, we have u = u_{η}^{0} .

By (10.2), (10.26), (10.28), (10.31), and (10.32) we have

(10.33) lim_{ε→0} {L^{ε}(u^{ε} − u_{η}^{ε}) − ∫_{U} f_{η}(u^{ε} − u_{η}^{ε}) dx} = L^{0}(u^{0} − u_{η}^{0}) − ∫_{U} f_{η}(u^{0} − u_{η}^{0}) dx .

Let μ̂^{0} be the measure defined in the proof of Theorem 9.1. By (9.3) we have

(10.34)  \begin{aligned}α & ∫_{U} |D(u^{0} − u_{η}^{0})|^{2}dx + ∫_{U} |u^{0} − u_{η}^{0}|^{2}dμ̂^{0} ≤ \\ & ≤ lim inf_{ε→0} \{α ∫_{U} |D(u^{ε} − u_{η}^{ε})|^{2}dx + ∫_{U} |u^{ε} − u_{η}^{ε}|^{2}dμ^{ε}\} . \end{aligned}

From (9.6), (10.11), (10.30), (10.33), and (10.34) we obtain that

(10.35)  α ∫_{U} |D(u^{0} − u_{η}^{0})|^{2}dx + ∫_{U} |u^{0} − u_{η}^{0}|^{2}dμ^{0} ≤ \frac{β^{4}}{α^{4}} η^{2} .

Since, by (10.11), f_{η} converges to L^{0} in the dual space of H_{0}^{1}(Ω) ∩ L^{2}(Ω, μ^{0}) as η tends to 0, the solution u_{η}^{0} of (10.29) for ε = 0 converges in H_{0}^{1}(U) ∩ L^{2}(U, μ^{0}), as η tends to 0, to the solution v^{0} of the problem

(10.36) \{ \begin{matrix}v^{0} − u^{0} ∈ H_{0}^{1}(U) ∩ L^{2}(U, μ^{0}) , \\ ∫_{U} A^{0}Dv^{0}Dy dx + ∫_{U} v^{0}y dμ^{0} = L^{0}(y)  ∀y ∈ H_{0}^{1}(U) ∩ L^{2}(U, μ^{0}) . \end{matrix}

On the other hand, by (10.35), (u_{η}^{0}) converges to u^{0} in H_{0}^{1}(U) ∩ L^{2}(U, μ^{0}) as η tends to 0. We conclude that u^{0} = v^{0} and is the solution of (10.36). Since this holds for every open set U ⊂⊂ Ω, this implies that u^{0} is a solution of (10.27).

The final statement of the theorem can be proved as explained before Corollary 7.6.

The next theorem is a generalization of Theorem 7.7

Theorem 10.4. Assume (5.1), (5.18)–(5.24), (7.1), (7.2), and (10.2). For every ε > 0, let u^{ε} be a solution to problem (10.25). Assume that

u^{ε} ⇀ u^{0} weakly in H^{1}(Ω) ,

for some function u^{0} ∈ H^{1}(Ω). Let U be an open set with U ⊂⊂ Ω, let δ > 0, let ψ_{δ} be a function in H^{2}(U) ∩ W^{1,∞}(U) which satisfies (7.18), and let v_{δ}^{ε} be the functions defined in U by (7.4). Then

(10.37)  lim sup_{ε→0} {α ∫_{V} |Du^{ε} − Dv_{δ}^{ε}|^{2}dx + ∫_{V} |u^{ε} − v_{δ}^{ε}|^{2}dμ^{ε}} < δ ,

for every open set V ⊂⊂ U.

Proof. Let us fix δ′ < δ such that

(10.38)  β ∫_{U} |Du^{0} − D(ψ_{δ}ω^{0})|^{2}dx + ∫_{U} |u^{0} − ψ_{δ}ω^{0}|^{2}dμ^{0} < δ′ .

For η > 0, let f_{η}, u_{η}^{ε}, and u_{η}^{0} be as in the proof of Theorem 10.3. Since (u_{η}^{0}) converges to u^{0} in H_{0}^{1}(U) ∩ L^{2}(U, μ^{0}), we fix η small enough such that

\sqrt{δ′} + \frac{β}{α}η < \sqrt{δ} (10.39)

β ∫_{U} |Du_{η}^{0} − D(ψ_{δ}ω^{0})|^{2}dx + ∫_{U} |u_{η}^{0} − ψ_{δ}ω^{0}|^{2}dμ^{0} < δ′ . (10.40)

Therefore we can apply Theorem 7.7 with f = f_{η} and we obtain

(10.41)  lim sup_{ε→0} {α ∫_{V} |Du_{η}^{ε} − Dv_{δ}^{ε}|^{2}dx + ∫_{V} |u_{η}^{ε} − v_{δ}^{ε}|^{2}dμ^{ε}} < δ′ ,

for every open set V ⊂⊂ U. Using (10.11), (10.30), (10.33), and (10.35) we deduce that

(10.42)  lim sup_{ε→0} \{α ∫_{U} |Du^{ε} − Du_{η}^{ε}|^{2}dx + ∫_{U} |u^{ε} − u_{η}^{ε}|^{2}dμ^{ε}\} ≤ \frac{β^{2}}{α^{2}} η^{2} .

From (10.39), (10.41), and (10.42) we obtain (10.37).

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