A STABILITY RESULT FOR NONLINEAR NEUMANN PROBLEMS UNDER BOUNDARY VARIATIONS

GIANNI DAL MASO, FRANÇOIS EBOBISSE, AND MARCELLO PONSIGLIONE

Abstract. In this paper we study, in dimension two, the stability of the solutions of some nonlinear elliptic equations with Neumann boundary conditions, under perturbations of the domains in the Hausdorff complementary topology. More precisely, for every bounded open subset Ω of ℝ^{2}, we consider the problem

− div a(x, ∇u_{Ω}) + b(x, u_{Ω}) = 0 in Ω,
a(x, ∇u_{Ω}) · ν = 0  on ∂Ω,

where a: ℝ^{2} ×ℝ^{2} → ℝ^{2} and b: ℝ^{2} ×ℝ → ℝ are two Carathéodory functions which satisfy the standard monotonicity and growth conditions of order p, with 1 < p ≤ 2.

Let Ω_{n} be a uniformly bounded sequence of open sets in ℝ^{2}, whose complements Ω_{n}^{c} have a uniformly bounded number of connected components. We prove that, if Ω_{n}^{c} → Ω^{c} in the Hausdorff metric and |Ω_{n}| → |Ω|, then u_{Ω_{n}} → u_{Ω} and ∇u_{Ω_{n}} → ∇u_{Ω} strongly in L^{p}. The proof is obtained by showing the Mosco convergence of the Sobolev spaces W^{1,p}(Ω_{n}) to the Sobolev space W^{1,p}(Ω).

Key words: Boundary value problems, nonlinear elliptic equations, capacity, Hausdorff metric, monotone operators, Mosco convergence

2000 Mathematics Subject Classification: 35J65, 31A15, 47H05, 49Q10, 49J45.

1. Introduction

In this paper we consider nonlinear elliptic equations with Neumann boundary conditions of the form

− div a(x, ∇u) + b(x, u) = 0 in Ω,
a(x, ∇u) · ν = 0  on ∂Ω,

(1.1)

where Ω is a bounded open subset of ℝ^{2} and a: ℝ^{2} × ℝ^{2} → ℝ^{2} and b: ℝ^{2} × ℝ → ℝ are two Carathéodory functions which satisfy suitable monotonicity, coerciveness, and growth conditions (see (2.2)-(2.4) below). Our purpose is to study the continuity of the mapping Ω ↦ u_{Ω} which associates to every Ω the corresponding solution u_{Ω}. The notion of convergence we consider on the sets Ω is given by the Hausdorff complementary topology, which is induced by the Hausdorff distance of the complements of the sets Ω (see Section 2.3).

Many examples (see [13], [23], [14], [10], [11]) show that, if we consider a uniformly bounded sequence Ω_{h} of open subsets of ℝ^{2} which converges to an open set Ω in the Hausdorff complementary topology, it may happen that u_{Ω_{h}} does not converge to u_{Ω}. Therefore, in this paper we investigate additional conditions on Ω_{h} which imply that u_{Ω_{h}} converges to u_{Ω} for any choice of the functions a and b. These stability results are useful in the proof of the existence of solutions of some shape optimization problems. More recently, similar stability results have been applied in [12] to study some models in fracture mechanics.

In the special case of the linear problems

\begin{cases}− Δu + u = f & in Ω, \\ \frac{∂u}{∂ν} = 0 & on ∂Ω, \end{cases} (1.2)

the stability with respect to Ω was first studied by Chenais [9] under the assumption that the domains Ω_{h} satisfy a uniform cone condition, which allows to use extension operators with uniformly bounded norms. This condition excludes a large class of domains, like for instance domains with cracks, for which there is an increasing interest in view of the applications to fracture mechanics.

The stability of (1.2) in nonsmooth domains is studied in [5], [6], [7] under variuos assumptions on Ω_{h}. In [8] the problem is studied under the hypothesis that Ω_{h} converges to Ω in the Hausdorff complementary topology, assuming also a uniform bound for the lenghts of the boundaries H^{1}(∂Ω_{h}) and for the number of connected components of ∂Ω_{h}. This result has been recently improved in [2], where the bound on H^{1}(∂Ω_{h}) is replaced by the weaker assumption of convergence of the two-dimensional measures of the domains, i.e., |Ω_{h}| → |Ω|, which is also necessary for the stability of (1.2).

In the present paper we study the stability of the nonlinear problems (1.1) with respect to the Hausdorff complementary topology, assuming that |Ω_{h}| → |Ω| and that the number of connected components of the complements Ω_{h}^{c} is uniformly bounded. To obtain this result we reduce the problem to the convergence in the sense of Mosco of the Sobolev spaces W^{1,p}(Ω_{h}) to the Sobolev space W^{1,p}(Ω), where the exponent p is related in the usual way to the growth condition of the functions a and b (see Section 2.6).

The proof of this property for 1 < p ≤ 2 is obtained in two steps. First, under the same assumptions on Ω_{h} and Ω, we prove the continuity of the map Ω ↦ ∇u_{Ω} for the solutions u_{Ω} of following nonlinear Neumann problems

− div a(x, ∇u) = 0 in Ω,
a(x, ∇u) · ν = 0  on ∂Ω.

(1.3)

This result is obtained by using the fact that the rotation by π/2 of the vectorfield a(x, ∇u) (extended to 0 on the complemenent Ω^{c}) is the gradient of a function v_{Ω} which is constant on each connected component of Ω^{c} (Proposition 3.6). This function plays the role of the conjugate of u_{Ω} used in [3] and [12] in the linear case. Another important ingredient in the proof is a result on the stability of nonlinear Dirichlet problems proved in [4], which allows to show that, if each function v_{Ω_{h}} is constant on each connected component of Ω_{h}^{c}, then their weak limit is constant on each connected component of Ω^{c} (Lemmas 3.3 and 3.5).

The second step in the proof of the convergence of the Sobolev spaces W^{1,p}(Ω_{h}) is the approximation of locally constant functions in Ω by functions belonging to W^{1,p}(Ω_{h}), which relies on a result obtained in [2].

The main difference with respect to the linear case studied in [2] and [3] is that for the nonlinear problems (1.3) we can not use the method of conformal mappings.

The hypothesis p ≤ 2 is used both in the first and in the second step. In the case p > 2 the stability result for (1.1) and (1.3) is not true under our hypotheses, as shown in Remarks 3.7 and 4.6. The general form of the limit problem will be studied in [16].

In the last part of the paper we consider the case of unbounded open sets and the case of mixed boundary value problems, with a Dirichlet condition on a fixed part of the boundary.

Contents

1. Introduction 1 2. Notation and preliminaries 3 2.1. Deny-Lions spaces 3 2.2. Capacity 4 2.3. Convergence of sets 4 2.4. The Neumann problems 5 2.5. Stability of Neumann problems 6 2.6. Mosco convergence 6 3. Mosco convergence of Deny-Lions spaces 11 4. Mosco convergence of Sobolev spaces 15 5. The case of unbounded domains 21 6. Problems with Dirichlet boundary conditions 22 References 25

2. Notation and preliminaries

Throughout the paper p and q are real numbers, with 1 < p ≤ 2 ≤ q < +∞ and p^{−1}+q^{−1} = 1. The scalar product of two vectors ξ, ζ ∈ ℝ^{2} is denoted by ξ · ζ, and the norm of ξ by |ξ|. For any E, F ⊂ ℝ^{2}, E△F := (E \ F) ∪ (F \ E) is the symmetric difference of E and F, and |E| is the Lebesgue (outer) measure of E.

2.1. Deny-Lions spaces. Given an open subset Ω of ℝ^{2}, the Deny-Lions space is defined by

L^{1,p}(Ω) := {u ∈ L_{loc}^{p}(Ω) : ∇u ∈ L^{p}(Ω, ℝ^{2})}.

It is well-known that L^{1,p}(Ω) = W^{1,p}(Ω) whenever Ω is bounded and has a Lipschitz boundary. It is also known that the set {∇u : u ∈ L^{1,p}(Ω)} is a closed subspace of L^{p}(Ω, ℝ^{2}). For further properties of the spaces L^{1,p}(Ω) the reader is referred to [15] and [21].

In many problems it is useful to consider the following equivalence relation in L^{1,p}(Ω):

(2.1)  v_{1} ∼ v_{2}  if and only if  ∇v_{1} = ∇v_{2} a.e. in Ω.

The corresponding quotient space is denoted by L^{1,p}(Ω)/_{∼}.

2.2. Capacity. Let 1 < r < ∞. For every subset E of ℝ^{2}, the (1, r)-capacity of E in ℝ^{2}, denoted by C_{r}(E), is defined as the infimum of ∫_{ℝ^{2}}(|∇u|^{r} + |u|^{r}) dx over the set of all functions u ∈ W^{1,r}(ℝ^{2}) such that u ≥ 1 a.e. in a neighborhood of E. If r > 2, then C_{r}(E) > 0 for every nonempty set E. On the contrary, if r = 2 there are nonempty sets E with C_{r}(E) = 0 (for instance, C_{r}({x}) = 0 for every x ∈ ℝ^{2}).

We say that a property P(x) holds C_{r}-quasi everywhere (abbreviated as C_{r}-q.e.) in a set E if it holds for all x ∈ E except a subset N of E with C_{r}(N) = 0. We recall that the expression almost everywhere (abbreviated as a.e.) refers, as usual, to the Lebesgue measure.

A function u: E → \overline{ℝ} is said to be quasi-continuous if for every ε there exists A_{ε} ⊂ ℝ^{2}, with C_{r}(A_{ε}) < ε, such that the restriction of u to E \ A_{ε} is continuous. If r > 2 every quasicontinuous function is continuous, while for r = 2 there are quasi-continuous functions that are not continuous. It is well known that, for any open subset Ω of ℝ^{2}, every function u ∈ W^{1,r}(Ω) has a quasi-continuous representative \overline{u}: Ω → ℝ, which satisfies

lim_{ρ→0^{+}} \frac{1}{|B(x, ρ)|} ∫_{B(x,ρ)} |u(y) − \overline{u}(x)| dy = 0 for q.e. x ∈ Ω,

where B(x, ρ) is the open ball with centre x and radius ρ. We recall that if u_{h} converges strongly to u in W^{1,r}(Ω), then a subsequence of \overline{u}_{h} converges to \overline{u} pointwise C_{r}-q.e. on Ω. For these and other properties on quasi-continuous representatives the reader is referred to [17], [19], [21], [25].

To simplify the notation, we always identify each function u ∈ W^{1,r}(Ω) with its quasicontinuous representative \overline{u}.

2.3. Convergence of sets. We recall here the notion of convergence in the sense of Kuratowski. We say that a sequence (C_{h}) of closed subsets of ℝ^{2} converges to a closed set C in the sense of Kuratowski if the following two properties hold: (K_{1}) for every x ∈ C, there exists a sequence x_{h} ∈ C_{h} such that x_{h} → x; (K_{2}) if (h_{k}) is a sequence of indices converging to ∞, (x_{k}) is a sequence such that x_{k} ∈ C_{h_{k}} for

every k, and x_{k} converges to some x ∈ ℝ^{2}, then x ∈ C. Let us recall also that the Hausdorff distance between two nonempty closed subsets C_{1} and

C_{2} of ℝ^{2} is defined by

d_{H}(C_{1}, C_{2}) := max { sup_{x∈C_{1}} dist(x, C_{2}) , sup_{x∈C_{2}} dist(x, C_{1})} .

We say that a sequence (C_{h}) of nonempty closed subsets of ℝ^{2} converges to a nonempty closed subset C in the Hausdorff metric if d_{H}(C_{h} , C) converges to 0.

A sequence of subsets of ℝ^{2} is said to be uniformly bounded if there exists a bounded subset of ℝ^{2} which contains all sets of the sequence.

The convergence in the Hausdorff metric implies the convergence in the sense of Kuratowski, while in general the converse is false. However, if (C_{h}) is a uniformly bounded sequence of nonempty closed sets in ℝ^{2}, then (C_{h}) converges to a closed set C in the Hausdorff metric if and only if (C_{h}) converges to C in the sense of Kuratowski.

We say that a sequence (Ω_{h}) of open subsets of ℝ^{2} converges to an open set Ω in the Hausdorff complementary topology, if d_{H}(Ω_{h}^{c} , Ω^{c}) converges to 0, where Ω_{h}^{c} and Ω^{c} are the complements of Ω_{h} and Ω in ℝ^{2}, It is well-known (see, e.g., [18, Blaschke’s Selection Theorem]) that every uniformly bounded sequence of nonempty closed sets is compact with respect to the Hausdorff convergence. This implies that every uniformly bounded sequence of open sets is compact with respect to the Hausdorff complementary topology.

Moreover, a uniformly bounded sequence of open sets (Ω_{h}) converges to an open set Ω in the Hausdorff complementary topology, if and only if the sequence (Ω_{h}^{c}) converges to Ω^{c} in the sense of Kuratowski.

2.4. The Neumann problems. Let a: ℝ^{2}×ℝ^{2} → ℝ^{2} and b: ℝ^{2}×ℝ → ℝ be two Carathéodory functions that satisfy the following assumptions: there exist 0 < c_{1} ≤ c_{2}, α ∈ L^{q}(ℝ^{2}), and β ∈ L^{1}(ℝ^{2}) such that, for almost every x ∈ ℝ^{2} and for every ξ, ξ_{1}, ξ_{2} ∈ ℝ^{2} with ξ_{1} ≠ ξ_{2}

(a(x, ξ_{1}) − a(x, ξ_{2})) · (ξ_{1} − ξ_{2}) > 0; (2.2)

|a(x, ξ)| ≤ α(x) + c_{2}|ξ|^{p−1}; (2.3)

a(x, ξ) · ξ ≥ −β(x) + c_{1}|ξ|^{p}. (2.4)

We assume that b satisfies the same inequalities for every ξ, ξ_{1}, ξ_{2} ∈ ℝ.

For every open set Ω ⊂ ℝ^{2}, we consider the following nonlinear Neumann boundary value problems, where ν denotes the outward unit normal to ∂Ω:

− div a(x, ∇u) + b(x, u) = 0 in Ω,
a(x, ∇u) · ν = 0  on ∂Ω,

(2.5)

and

− div a(x, ∇v) = 0 in Ω ,
a(x, ∇v) · ν = 0  on ∂Ω .

(2.6)

A function u is a solution of (2.5) if

\begin{cases}u ∈ W^{1,p}(Ω), \\ ∫_{Ω}[a(x, ∇u) · ∇z + b(x, u)z]dx = 0 & ∀z ∈ W^{1,p}(Ω). \end{cases} (2.7)

while v is a solution of (2.6) if

\begin{cases}v ∈ L^{1,p}(Ω), \\ ∫_{Ω} a(x, ∇v) · ∇z dx = 0 & ∀z ∈ L^{1,p}(Ω). \end{cases} (2.8)

By well-known existence results for nonlinear elliptic equations with strictly monotone operators (see, e.g., Lions [20]), one can easily see that (2.7) has a unique solution in W^{1,p}(Ω). Similarly one can prove that (2.8) has a solution, and that if v_{1} and v_{2} are solutions of (2.8), then ∇v_{1} = ∇v_{2} a.e. in Ω. Note that problem (2.8) can be formulated in the quotient space L^{1,p}(Ω)/_{∼}, where a uniqueness result holds.

2.5. Stability of Neumann problems. In order to study the stability of (2.5) and (2.6) with respect to the variations of the open set Ω, we should be able to compare two solutions defined in two different domains. For any subset E of ℝ^{2}, the characteristic function 1_{E} of E is defined by 1_{E}(x) := 1 for x ∈ E and 1_{E}(x) := 0 for x ∈ E^{c}. For every u ∈ L^{1,p}(Ω), the functions u1_{Ω} and ∇u1_{Ω} are the extensions of the functions u and ∇u which vanish in Ω^{c}. By means of these extensions, W^{1,p}(Ω) will be identified with the closed linear subspace X_{Ω} of L^{p}(ℝ^{2})×L^{p}(ℝ^{2}, ℝ^{2}) defined by

X_{Ω} := {(u1_{Ω}, ∇u1_{Ω}) : u ∈ W^{1,p}(Ω)}, (2.9)

while the quotient space L^{1,p}(Ω)/_{∼} will be identified with the closed linear subspace Y_{Ω} of L^{p}(ℝ^{2}, ℝ^{2}) defined by

Y_{Ω} := {∇u1_{Ω} : u ∈ L^{1,p}(Ω)}. (2.10)

Let Ω be an open subset of ℝ^{2} and let (Ω_{h}) be a sequence of open subsets of ℝ^{2}. Given a pair of Carathéodory functions a: ℝ^{2} × ℝ^{2} → ℝ^{2} and b: ℝ^{2} × ℝ → ℝ satisfying (2.2)–(2.4), let u be the solution to problem (2.5) in Ω and, for every h, let u_{h} be the solution to problem (2.5) in Ω_{h}.

Definition 2.1. We say that Ω is stable for the Neumann problems (2.5) along the sequence (Ω_{h}) if for every pair of functions a, b satisfying (2.2)–(2.4) the sequence (u_{h}1_{Ω_{h}}) converges to u1_{Ω} strongly in L^{p}(ℝ^{2}) and the sequence (∇u_{h}1_{Ω_{h}}) converges to ∇u1_{Ω} strongly in L^{p}(ℝ^{2}, ℝ^{2}).

Similarly, let v be a solution to problem (2.6) in Ω and, for every h, let v_{h} be a solution to problem (2.6) in Ω_{h}.

Definition 2.2. We say that Ω is stable for the Neumann problems (2.6) along the sequence (Ω_{h}) if for every function a satisfying (2.2)–(2.4) the sequence (∇v_{h}1_{Ω_{h}}) converges to ∇v1_{Ω} strongly in L^{p}(ℝ^{2}, ℝ^{2}).

2.6. Mosco convergence. We shall prove that the notion of stability introduced in the previous definitions is equivalent to a notion of convergence for subspaces of a Banach space introduced by Mosco in [22].

Let Ω_{h} and Ω be open subsets of ℝ^{2}, and let X_{Ω_{h}} and X_{Ω} be the corresponding subspaces defined by (2.9). We recall that X_{Ω_{h}} converges to X_{Ω} in the sense of Mosco (see [22, Definition 1.1]) if the following two properties hold: (M_{1}) for every u ∈ W^{1,p}(Ω), there exists a sequence u_{h} ∈ W^{1,p}(Ω_{h}) such that u_{h}1_{Ω_{h}} converges

to u1_{Ω} strongly in L^{p}(ℝ^{2}) and ∇u_{h}1_{Ω_{h}} converges to ∇u1_{Ω} strongly in L^{p}(ℝ^{2}, ℝ^{2}); (M_{2}) if (h_{k}) is a sequence of indices converging to ∞, (u_{k}) is a sequence such that u_{k} ∈ W^{1,p}(Ω_{h_{k}})

for every k, and u_{k}1_{Ω_{h_{k}}} converges weakly in L^{p}(ℝ^{2}) to a function ϕ, while ∇u_{k}1_{Ω_{h_{k}}} converges weakly in L^{p}(ℝ^{2}, ℝ^{2}) to a function ψ, then there exists u ∈ W^{1,p}(Ω) such that ϕ = u1_{Ω} and ψ = ∇u1_{Ω} a.e. in ℝ^{2}.

Analogously, the convergence in the sense of Mosco of the spaces Y_{Ω_{h}} to Y_{Ω} defined by (2.10) is obtained by using only the convergence of the extensions of gradients, that is:

(M′_{1}) for every u ∈ L^{1,p}(Ω), there exists a sequence u_{h} ∈ L^{1,p}(Ω_{h}) such that ∇u_{h}1_{Ω_{h}} converges

strongly to ∇u1_{Ω} in L^{p}(ℝ^{2}, ℝ^{2}); (M′_{2}) if (h_{k}) is a sequence of indices converging to ∞, (u_{k}) is a sequence such that u_{k} ∈ L^{1,p}(Ω_{h_{k}})

for every k, and ∇u_{k}1_{Ω_{h_{k}}} converges weakly in L^{p}(ℝ^{2}, ℝ^{2}) to a function ψ, then there exists

u ∈ L^{1,p}(Ω) such that ψ = ∇u1_{Ω} a.e. in ℝ^{2}.

Theorem 2.3. Let Ω_{h} and Ω be open subsets of ℝ^{2}, and let X_{Ω_{h}} and X_{Ω} be the corresponding subspaces defined by (2.9). Then Ω is stable for the Neumann problems (2.5) along the sequence (Ω_{h}) if and only if X_{Ω_{h}} converges to X_{Ω} in the sense of Mosco.

Proof. Assume that Ω is stable for the Neumann problems (2.5) along the sequence (Ω_{h}). We want to prove that X_{Ω_{h}} converges to X_{Ω} in the sense of Mosco by using only the stability of the solutions corresponding to functions a and b of the special form

a(x, ξ) := a_{0}(x) + a_{1}(ξ), b(x, t) := b_{0}(x) + b_{1}(t), (2.11)

with

a_{1}(ξ) := |ξ|^{p−2}ξ,  b_{1}(t) := |t|^{p−2}t,  a_{0} ∈ L^{q}(ℝ^{2}, ℝ^{2}),  b_{0} ∈ L^{q}(ℝ^{2}). (2.12)

Let us prove (M_{1}). Given u ∈ W^{1,p}(Ω), let a_{0} := −|∇u|^{p−2}∇u1_{Ω} and b_{0} := −|u|^{p−2}u1_{Ω}. Then u is the solution of (2.7) in Ω with a and b given by (2.11). Let u_{h} ∈ W^{1,p}(Ω_{h}) be the solution of (2.7) in Ω_{h} with the same a and b. By Definition 2.1 the sequence (u_{h}1_{Ω_{h}}) converges to u1_{Ω} strongly in L^{p}(ℝ^{2}) and (∇u_{h}1_{Ω_{h}}) converges to ∇u1_{Ω} strongly in L^{p}(ℝ^{2}, ℝ^{2}). This proves (M_{1}).

Let us prove (M_{2}). Let (h_{k}) be a sequence of indices converging to ∞ and let (u_{k}) be a sequence, with u_{k} ∈ W^{1,p}(Ω_{h_{k}}) for every k, such that (u_{k}1_{Ω_{h_{k}}}) converges weakly in L^{p}(ℝ^{2}) to a function ϕ, while (∇u_{k}1_{Ω_{h_{k}}}) converges weakly in L^{p}(ℝ^{2}, ℝ^{2}) to a function ψ. Let a_{0} := −a_{1}(ψ) = −|ψ|^{p−2}ψ and b_{0} := −b_{1}(ϕ) = −|ϕ|^{p−2}ϕ, let a and b be defined by (2.11), and let u^{∗} and u_{h_{k}}^{∗} be the solutions of problems (2.7) in Ω and Ω_{h_{k}} respectively. By the stability assumption the sequence (u_{h_{k}}^{∗}1_{Ω_{h_{k}}}) converges to u^{∗}1_{Ω} strongly in L^{p}(ℝ^{2}) and (∇u_{h_{k}}^{∗}1_{Ω_{h_{k}}}) converges to ∇u^{∗}1_{Ω} strongly in L^{p}(ℝ^{2}, ℝ^{2}). This implies that a(x, ∇u_{h_{k}}^{∗}1_{Ω_{h_{k}}}) converges to a(x, ∇u^{∗}1_{Ω}) strongly in L^{q}(ℝ^{2}, ℝ^{2}) and b(x, u_{h_{k}}^{∗}1_{Ω_{h_{k}}}) converges to b(x, u^{∗}1_{Ω}) strongly in L^{q}(ℝ^{2}). Therefore

lim_{k→∞} ∫_{Ω_{h_{k}}} [a(x, ∇u_{h_{k}}^{∗}) · (∇u_{h_{k}} − ∇u_{h_{k}}^{∗}) + b(x, u_{h_{k}}^{∗})(u_{h_{k}} − u_{h_{k}}^{∗})]dx =

= ∫_{ℝ^{2}}[a(x, ∇u^{∗}1_{Ω}) · (ψ − ∇u^{∗}1_{Ω}) + b(x, u^{∗}1_{Ω})(ϕ − u^{∗}1_{Ω})]dx. (2.13)

By (2.7) the left hand side of (2.13) is zero. Therefore, using (2.11) and (2.12) we obtain

∫_{ℝ^{2}}[(a_{1}(∇u^{∗}1_{Ω}) − a_{1}(ψ)) · (ψ − ∇u^{∗}1_{Ω}) + (b_{1}(u^{∗}1_{Ω}) − b_{1}(ϕ))(ϕ − u^{∗}1_{Ω})]dx = 0.

Using the strict monotonicity of a_{1} and b_{1} we obtain that ψ = ∇u^{∗}1_{Ω} and ϕ = u^{∗}1_{Ω} a.e. in ℝ^{2}.

Conversely, assume now that X_{Ω_{h}} converges to X_{Ω} in the sense of Mosco and let us prove the stability. Let a and b be two Carathéodory functions satisfying (2.2)–(2.4) and let u_{h} and u be the solutions to problems (2.5) in Ω_{h} and Ω. The weak convergence in L^{p}(ℝ^{2}) × L^{p}(ℝ^{2}, ℝ^{2}) of (u_{h}1_{Ω_{h}}, ∇u_{h}1_{Ω_{h}}) to (u1_{Ω}, ∇u1_{Ω}) is a particular case of [22, Theorem A]. For the reader’s convenience, we give here the simple proof.

Using z := u_{h} as test function in (2.7) for Ω_{h}, from (2.4) we obtain that ∥u_{h}∥_{W^{1,p}(Ω_{h})} is bounded. Passing to a subsequence, we have that u_{h}1_{Ω_{h}} converges weakly in L^{p}(ℝ^{2}) to a function ϕ, while ∇u_{h}1_{Ω_{h}} converges weakly in L^{p}(ℝ^{2}, ℝ^{2}) to a function ψ. By (M_{2}) there exists u^{∗} ∈ W^{1,p}(Ω) such that ϕ = u^{∗}1_{Ω} and ψ = ∇u^{∗}1_{Ω} a.e. in ℝ^{2}. By monotonicity for every v ∈ W^{1,p}(Ω) we have

∫_{ℝ^{2}}[a(x, ∇v1_{Ω}) · (∇v1_{Ω} − ∇u_{h}1_{Ω_{h}}) + b(x, v1_{Ω})(v1_{Ω} − u_{h}1_{Ω_{h}})]dx ≥ (2.14)

≥ ∫_{ℝ^{2}}[a(x, ∇u_{h}1_{Ω_{h}}) · (∇v1_{Ω} − ∇u_{h}1_{Ω_{h}}) + b(x, u_{h}1_{Ω_{h}})(v1_{Ω} − u_{h}1_{Ω_{h}})]dx.

By (M_{1}) there exists a sequence v_{h} ∈ W^{1,p}(Ω_{h}) such that v_{h}1_{Ω_{h}} converges to v1_{Ω} strongly in L^{p}(ℝ^{2}) and ∇v_{h}1_{Ω_{h}} converges to ∇v1_{Ω} strongly in L^{p}(ℝ^{2}, ℝ^{2}). As v_{h} − u_{h} ∈ W^{1,p}(Ω_{h}), by (2.7) we have

∫_{ℝ^{2}}[a(x, ∇u_{h}1_{Ω_{h}}) · (∇v1_{Ω} − ∇u_{h}1_{Ω_{h}}) + b(x, u_{h}1_{Ω_{h}})(v1_{Ω} − u_{h}1_{Ω_{h}})]dx = (2.15)

= ∫_{ℝ^{2}}[a(x, ∇u_{h}1_{Ω_{h}}) · (∇v1_{Ω} − ∇v_{h}1_{Ω_{h}}) + b(x, u_{h}1_{Ω_{h}})(v1_{Ω} − v_{h}1_{Ω_{h}})]dx.

Since a(x, ∇u_{h}1_{Ω_{h}}) is bounded in L^{q}(ℝ^{2}, ℝ^{2}) and b(x, u_{h}1_{Ω_{h}}) is bounded in L^{q}(ℝ^{2}), passing to the limit in (2.14) and (2.15) we obtain

∫_{Ω}[a(x, ∇v) · (∇v − ∇u^{∗}) + b(x, v)(v − u^{∗})]dx ≥ (2.16)

≥ lim_{h→∞} ∫_{ℝ^{2}}[a(x, ∇u_{h}1_{Ω_{h}}) · (∇v1_{Ω} − ∇v_{h}1_{Ω_{h}}) + b(x, u_{h}1_{Ω_{h}})(v1_{Ω} − v_{h}1_{Ω_{h}})]dx = 0.

Then we take v = u^{∗} ± εz in (2.16), with z ∈ W^{1,p}(Ω) and ε > 0. Dividing by ε, and passing to the limit as ε tends to 0, we obtain that u^{∗} satisfies (2.7) in Ω. This proves that u^{∗} = u. Therefore u_{h}1_{Ω_{h}} converges to u1_{Ω} weakly in L^{p}(ℝ^{2}) and ∇u_{h}1_{Ω_{h}} converges to ∇u1_{Ω} weakly in L^{p}(ℝ^{2}, ℝ^{2}).

Taking v := u in (2.16) we obtain that

∫_{ℝ^{2}}(a(x, ∇u_{h}1_{Ω_{h}}) − a(x, ∇u1_{Ω})) · (∇u_{h}1_{Ω_{h}} − ∇u1_{Ω})dx +
+ ∫_{ℝ^{2}}(b(x, u_{h}1_{Ω_{h}}) − b(x, u1_{Ω}))(u_{h}1_{Ω_{h}} − u1_{Ω})dx

tends to 0 as h → ∞. Using the monotonicity of a and b we conclude that each integral tends to 0. The strong convergence of (u_{h}1_{Ω_{h}}, ∇u_{h}1_{Ω_{h}}) is now a consequence of the following lemma.

Lemma 2.4. Let (ψ_{h}) be a sequence in L^{p}(ℝ^{2}, ℝ^{2}) converging weakly in L^{p}(ℝ^{2}, ℝ^{2}) to a function ψ. Assume that

lim_{h→∞} ∫_{ℝ^{2}}(a(x, ψ_{h}) − a(x, ψ)) · (ψ_{h} − ψ)dx = 0. (2.17)

Then ψ_{h} converges to ψ strongly in L^{p}(ℝ^{2}, ℝ^{2}).

Proof. Various forms of this lemma have been used in the study of Leray-Lions operators (see, e.g., [1, Lemma 5]). For the sake of completeness, we give here the short proof of the present version.

Let g_{h} := (a(x, ψ_{h}) − a(x, ψ)) · (ψ_{h} − ψ). By monotonicity we have g_{h} ≥ 0 a.e. in ℝ^{2}, thus (2.17) implies that g_{h} converges to 0 strongly in L^{1}(ℝ^{2}). Passing to a subsequence, we may assume that g_{h} converges to 0 a.e. in ℝ^{2}. Using the Cauchy inequality, from (2.3) and (2.4) we obtain for every ε > 0

c_{1}|ψ_{h}|^{p} ≤ β + a(x, ψ_{h}) · ψ_{h} = β + g_{h} + a(x, ψ_{h}) · ψ + a(x, ψ) · (ψ_{h} − ψ) ≤

≤ g_{h} + β + α|ψ| + |a(x, ψ)||ψ| + c_{2}(\frac{ε^{q}}{q} + \frac{ε^{p}}{p} )|ψ_{h}|^{p} + c_{2}(\frac{|ψ|^{p}}{pε^{p}} + \frac{|a(x, ψ)|^{q}}{qε^{q}} ).

Choosing ε small enough, we obtain that there exist a constant c_{3} > 0 and a function γ ∈ L^{1}(ℝ^{2}) such that

c_{3}|ψ_{h}|^{p} ≤ g_{h} + γ. (2.18)

Let us fix a point x ∈ ℝ^{2} where γ(x) < +∞ and where g_{h}(x) tends to 0. By (2.18) the sequence ψ_{h}(x) is bounded in ℝ^{2}, thus a subsequence (depending on x) converges to a vector ξ ∈ ℝ^{2}. By the definition of g_{h}(x) and by the continuity of a(x, ·) we get (a(x, ξ)−a(x, ψ(x)))·(ξ−ψ(x)) = 0, which implies ξ = ψ(x) by (2.2). Therefore the whole sequence ψ_{h}(x) converges to ψ(x). Since this is true for a.e. x ∈ ℝ^{2}, the strong convergence in L^{p}(ℝ^{2}, ℝ^{2}) follows from (2.18) by the dominated convergence theorem.

Remark 2.5. Let us observe that, if (Ω_{h}) is a uniformly bounded sequence of open subsets of ℝ^{2} such that X_{Ω_{h}} converges to X_{Ω} in the sense of Mosco, then |Ω_{h}△Ω| converges to 0.

Let Σ ⊂ ℝ^{2} be a bounded closed set such that Ω_{h} ⊂ Σ for every h. From property (M_{1}) it follows that Ω ⊂ Σ. Indeed, if Ω \ Σ ≠ Ø, let B ⊂ Ω \ Σ be an open ball and let φ ∈ C_{c}^{∞}(B) with φ ≠ 0. By property (M_{1}), there exists u_{h} ∈ W^{1,p}(Ω_{h}) such that u_{h}1_{Ω_{h}} → φ strongly in L^{p}(ℝ^{2}). So,

0 < ∫_{B} |φ|^{p} dx = lim_{h→∞} ∫_{B} |u_{h}1_{Ω_{h}}|^{p} dx = 0,

which is absurd.

Now, let u := 1_{Ω}. Since u ∈ W^{1,p}(Ω), by property (M_{1}) there exists u_{h} ∈ W^{1,p}(Ω_{h}) such that u_{h}1_{Ω_{h}} → 1_{Ω} strongly in L^{p}(ℝ^{2}). As |u_{h}1_{Ω_{h}} − 1_{Ω}|^{p} = 1 a.e. on Ω \ Ω_{h}, we have

(2.19)  lim_{h→∞} |Ω \ Ω_{h}| ≤ lim_{h→∞} ∫_{ℝ^{2}} |u_{h}1_{Ω_{h}} − 1_{Ω}|^{p} dx = 0.

On the other hand, up to a subsequence, 1_{Ω_{h}} converges weakly in L^{p}(ℝ^{2}, ℝ^{2}) to some ϕ. Hence from property (M_{2}), there exists v ∈ W^{1,p}(Ω) such that ϕ = v1_{Ω} a.e. in ℝ^{2}. So we have that

lim_{h→∞} |Ω_{h} \ Ω| = lim_{h→∞} ∫_{ℝ^{2}} 1_{Ω_{h}}1_{Σ\Ω} dx = ∫_{ℝ^{2}} v1_{Ω}1_{Σ\Ω} dx = 0,

which together with (2.19) gives |Ω_{h}△Ω| → 0.

Note that, if the open sets Ω_{h} are not uniformly bounded, it is possible that X_{Ω_{h}} converges to X_{Ω} in the sense of Mosco while |Ω_{h}△Ω| does not converge to 0. Consider, for example, the sequence of open sets Ω_{h} := B(0, 1) ∪ ((B(0, h + h^{−1}) \ \overline{B(0, h)}) and Ω := B(0, 1). We have that |Ω_{h}△Ω| = 2π + h^{−2}π → 2π.

Let us verify that X_{Ω_{h}} converges to X_{Ω} in the sense of Mosco. For every u ∈ W^{1,p}(Ω), property (M_{1}) is satisfied by the sequence u_{h} := u1_{Ω}. For property (M_{2}), let (h_{k}) be a sequence of indices converging to ∞; let (u_{k}) be a sequence, with u_{k} ∈ W^{1,p}(Ω_{h_{k}}) for every k, such that u_{k}1_{Ω_{h_{k}}} converges weakly in L^{p}(ℝ^{2}) to a function ϕ, while ∇u_{k}1_{Ω_{h_{k}}} converges weakly in L^{p}(ℝ^{2}, ℝ^{2}) to a function ψ. We set u := ϕ|_{Ω}. As u_{k} ⇀ u weakly in L^{p}(Ω) and ∇u_{k} ⇀ ψ|_{Ω} weakly in L^{p}(Ω, ℝ^{2}), we have u ∈ W^{1,p}(Ω) and ∇u = ψ|_{Ω} a.e. on Ω. Now, for every ball D ⊂ Ω^{c} we have D ⊂ Ω_{h}^{c} for h large enough, hence

∫_{D} ϕ dx = lim_{k→∞} ∫_{D} u_{k}1_{Ω_{h_{k}}} dx = 0.

So, ϕ = 0 a.e. in Ω^{c} and similarly also ψ = 0 a.e. in Ω^{c}. Hence, ϕ = u1_{Ω} and ψ = ∇u1_{Ω} a.e. in ℝ^{2}.

Note that, in this case, Ω_{h} converges to Ω in the Hausdorff complementary topology, since d_{H}(Ω_{h}^{c}, Ω^{c}) = h^{−1} → 0. By adding a small strip whose width tends to zero one can obtain an example with connected sets.

The following theorem can be proved as Theorem 2.3.

Theorem 2.6. Let Ω_{h} and Ω be open subsets of ℝ^{2}, and let Y_{Ω_{h}} and Y_{Ω} be the corresponding subspaces defined by (2.10). Then Ω is stable for the Neumann problems (2.6) along the sequence (Ω_{h}) if and only if Y_{Ω_{h}} converges to Y_{Ω} in the sense of Mosco.

Remark 2.7. If (Ω_{h}) is a uniformly bounded sequence of open subsets of ℝ^{2} such that Y_{Ω_{h}} converges to Y_{Ω} in the sense of Mosco, then |Ω_{h}△Ω| converges to 0. Let Σ ⊂ ℝ^{2} be a bounded set such that Ω_{h} ⊂ Σ for every h. Arguing as in Remark 2.5, we get that Ω ⊂ Σ.

Now, let u(x) := ξ·x with ξ ∈ ℝ^{2} and |ξ| = 1. Since u ∈ L^{1,p}(Ω), by property (M′_{1}) there exists u_{h} ∈ L^{1,p}(Ω_{h}) such that ∇u_{h}1_{Ω_{h}} → ∇u1_{Ω} strongly in L^{p}(ℝ^{2}, ℝ^{2}). As |∇u_{h}1_{Ω_{h}} − ∇u1_{Ω}|^{p} = 1 a.e. on Ω \ Ω_{h}, we have

(2.20)  lim_{h→∞} |Ω \ Ω_{h}| ≤ lim_{h→∞} ∫_{ℝ^{2}} |∇u_{h}1_{Ω_{h}} − ∇u1_{Ω}|^{p} dx = 0.

On the other hand, we consider the functions v_{h} ∈ L^{1,p}(Ω_{h}) defined by v_{h}(x) := ξ · x. Up to a subsequence, ∇v_{h}1_{Ω_{h}} = ξ1_{Ω_{h}} converges weakly in L^{p}(ℝ^{2}, ℝ^{2}) to some function ψ. By property (M′_{2}), there exists a function v ∈ L^{1,p}(Ω) such that ψ = ∇v1_{Ω} a.e. in ℝ^{2}. So, it follows that

ξ lim_{h→∞} |Ω_{h} \ Ω| = lim_{h→∞} ∫_{ℝ^{2}} ∇v_{h}1_{Ω_{h}}1_{Σ\Ω} dx = ∫_{ℝ^{2}} ∇v1_{Ω}1_{Σ\Ω} dx = 0,

which together with (2.20) gives |Ω△Ω_{h}| → 0.

Remark 2.8. Theorems 2.3 and 2.6 can be applied in the following easy case. If (Ω_{h}) is increasing and Ω is the union of the sequence, then it is easy to see that X_{Ω_{h}} converges to X_{Ω} and Y_{Ω_{h}} converges to Y_{Ω} in the sense of Mosco. Therefore every open set is stable for the Neumann problems (2.5) and (2.6) along any increasing sequence converging to it.

3. Mosco convergence of Deny-Lions spaces

In this section we study the Mosco convergence of the subspaces Y_{Ω} introduced in (2.10) and corresponding to the Deny-Lions spaces L^{1,p}(Ω). By Theorem 2.6, this is equivalent to the stability for the Neumann problems (2.6).

Theorem 3.1. Let (Ω_{h}) be a uniformly bounded sequence of open subsets of ℝ^{2} that converges to an open set Ω in the Hausdorff complementary topology. Assume that |Ω_{h}| converges to |Ω| and that Ω_{h}^{c} has a uniformly bounded number of connected components. Then Ω is stable for the Neumann problems (2.6) along the sequence (Ω_{h}).

To prove Theorem 3.1 we use the following lemmas.

Lemma 3.2. Let (Ω_{h}) be a uniformly bounded sequence of open subsets of ℝ^{2} which converges to an open set Ω in the Hausdorff complementary topology. Assume that |Ω_{h}| → |Ω|. Then 1_{Ω_{h}} → 1_{Ω} in measure, i.e., |Ω_{h}△Ω| → 0. Moreover, if φ_{h} ⇀ φ weakly in L^{r}(ℝ^{2}) for some 1 < r < +∞, and φ_{h} = 0 a.e. in Ω_{h}^{c}, then φ = 0 a.e. in Ω^{c}.

Proof. From the convergence of Ω_{h} to Ω in the Hausdorff complementary topology we have that 1_{Ω\Ω_{h}} → 0 pointwise, hence |Ω \ Ω_{h}| → 0 by the dominated convergence theorem. Since |Ω_{h}| − |Ω| = |Ω_{h} \ Ω| − |Ω \ Ω_{h}|, and the left hand side tends to 0 by hypothesis, we conclude that |Ω_{h} \ Ω| → 0 too.

Now, let ψ ∈ L^{∞}(ℝ^{2}). As 1_{Ω_{h}} → 1_{Ω} strongly in L^{s}(ℝ^{2}) for every 1 ≤ s < +∞, we have

∫_{Ω^{c}} φψ dx = lim_{h→∞} ∫_{Ω^{c}} φ_{h}ψ dx = lim_{h→∞} ∫_{Ω^{c}} 1_{Ω_{h}}φ_{h}ψ dx = ∫_{Ω^{c}} 1_{Ω}φψ dx = 0,

which implies φ = 0 a.e. in Ω^{c}.

Lemma 3.3. Let (Ω_{h}) be a uniformly bounded sequence of open subsets of ℝ^{2}, converging to an open set Ω in the Hausdorff complementary topology. Assume that the sets Ω_{h}^{c} have a uniformly bounded number of connected components. Let (v_{h}) be a sequence in W^{1,q}(ℝ^{2}) converging weakly in W^{1,q}(ℝ^{2}) to a function v and with v_{h} = 0 C_{q}-q.e. on Ω_{h}^{c}. Then v = 0 C_{q}-q.e. on Ω^{c}.

Proof. Let Δ_{q} be the q-Laplacian, defined by Δ_{q}u := div (|∇u|^{q−2}∇u). Let w_{h} and w be the solutions of the problems

(3.1)  w_{h} ∈ W^{1,q}(ℝ^{2}),  Δ_{q}w_{h} = Δ_{q}v in Ω_{h},  w_{h} = 0 C_{q}-q.e. in Ω_{h}^{c},
w ∈ W^{1,q}(ℝ^{2}),  Δ_{q}w = Δ_{q}v in Ω,  w = 0 C_{q}-q.e. in Ω^{c}.

Using a result on the stability of Dirichlet problems proved by Bucur and Trebeschi in [4] (see also Šverák [24] for the case q = 2), we obtain that w_{h} converges to w strongly in W^{1,q}(ℝ^{2}). Taking v_{h} − w_{h} as test function in (3.1), which is possible since v_{h} − w_{h} ∈ W_{0}^{1,q}(Ω_{h}) (see, e.g., [19, Theorem 4.5]), we obtain

⟨Δ_{q}w_{h}, v_{h} − w_{h}⟩ = ⟨Δ_{q}v, v_{h} − w_{h}⟩, (3.2) where ⟨·, ·⟩ is the duality pairing between W^{−1,p}(ℝ^{2}) and W^{1,q}(ℝ^{2}). Passing to the limit in (3.2) we obtain

⟨Δ_{q}w, v − w⟩ = ⟨Δ_{q}v, v − w⟩,

which implies v = w by the strict monotonicity of −Δ_{q}. Since, by definition, w = 0 C_{q}-q.e. in Ω^{c}, we conclude that v = 0 C_{q}-q.e. in Ω^{c}.

Lemma 3.4. Let v ∈ W^{1,q}(ℝ^{2}) and let C_{1} and C_{2} be two connected closed subsets of ℝ^{2} with C_{1} ∩ C_{2} ≠ Ø. If v is constant C_{q}-q.e. in C_{1} and in C_{2}, then v is constant C_{q}-q.e. in C_{1} ∪ C_{2}.

Proof. For q = 2 we the reader is referred to Proposition 2.5 in [12], while for q > 2 the result follows from the Sobolev embedding theorem, which yields the continuity of v.

Lemma 3.5. Let (Ω_{h}) be a uniformly bounded sequence of open subsets of ℝ^{2} which converges to an open set Ω in the Hausdorff complementary topology, and let (v_{h}) be a sequence in W^{1,q}(ℝ^{2}), which converges to a function v weakly in W^{1,q}(ℝ^{2}). Assume that Ω_{h}^{c} has a uniformly bounded number of connected components and that every function v_{h} is constant C_{q}-q.e. in each connected component of Ω_{h}^{c}. Then v is constant C_{q}-q.e. in each connected component of Ω^{c}.

Proof. Let C_{h}^{1}, …, C_{h}^{n_{h}} be the connected components of Ω_{h}^{c}. Passing to a subsequence, we can assume that n_{h} does not depend on h, and that the sets C_{h}^{i} converge in the Hausdorff metric to some connected sets C^{i} as h → ∞. Let us prove that v is constant q.e. in each C^{i}.

This is trivial if C^{i} contains only a single point. If C^{i} has more than one point, there exists r > 0 such that diam(C_{h}^{i}) > 2r for h large enough. Let us prove that the constant values c_{h}^{i} taken by v_{h} on C_{h}^{i} are bounded uniformly with respect to h. To this aim let us consider a point x_{h} ∈ C_{h}^{i}. Since diam(C_{h}^{i}) > 2r, we have C_{h}^{i} \ B(x_{h}, r) ≠ Ø, and by connectedness

(3.3)  C_{h}^{i} ∩ ∂B(x_{h}, ρ) ≠ Ø  for every 0 < ρ < r.

As v_{h} = c_{h}^{i} C_{q}-q.e. on C_{h}^{i}, by using polar coordinates we deduce from (3.3) the Poincaré inequality

∫_{B(x_{h},r)} |v_{h} − c_{h}^{i}|^{q} dx ≤ Mr^{q} ∫_{B(x_{h},r)} |∇v_{h}|^{q} dx,

where the constant M is independent of h, i, and r. Since v_{h} is bounded in W^{1,q}(ℝ^{2}), it follows that c_{h}^{i} is bounded, and so it converges (up to a subsequence) to some constant c^{i}.

To prove that v = c^{i} C_{q}-q.e. on C^{i}, we fix two open balls B_{1} and B_{2} with B_{1} ⊂⊂ B_{2}, and a cut-off function φ ∈ C_{c}^{∞}(B_{2}) with φ = 1 in B_{1}. Then we have that φ(v_{h} − c_{h}^{i}) = 0 C_{q}-q.e. on (B_{2} \ C_{h}^{i})^{c}. By Lemma 3.3 we get φ(v − c^{i}) = 0 C_{q}-q.e. in (B_{2} \ C^{i})^{c}, hence v = c^{i} C_{q}-q.e on B_{1} ∩ C^{i}. As B_{1} is arbitrary, we obtain v = c^{i} C_{q}-q.e. on C^{i}. If C^{i} ∩ C^{j} ≠ Ø, by Lemma 3.4 we have that v is constant C_{q}-q.e. on C^{i} ∪ C^{j}. As Ω^{c} is the union of the sets C^{i}, we conclude that v is constant C_{q}-q.e. on each connected component of Ω^{c}.

Lemma 3.6. Let Ω be a bounded open subset of ℝ^{2} and let u be a solution of problem (2.6). Let R be the rotation on ℝ^{2} defined by R(y_{1}, y_{2}) := (−y_{2}, y_{1}). Then there exists a unique function v ∈ W^{1,q}(ℝ^{2}) such that ∇v = Ra(x, ∇u)1_{Ω} a.e. in ℝ^{2}. Moreover v is constant C_{q}-q.e. on each connected component of Ω^{c}.

Proof. We consider the vector field Φ ∈ L^{q}(ℝ^{2}, ℝ^{2}) defined by Φ := a(x, ∇u)1_{Ω}. By (2.8) we have div Φ = 0 in D′(ℝ^{2}), hence rot(RΦ) = 0 in D′(ℝ^{2}). Since Ω is bounded, there exists a potential v ∈ W^{1,q}(ℝ^{2}) such that ∇v = RΦ a.e. in ℝ^{2} and v = 0 a.e. in the interior of the unbounded connected component of Ω^{c}.

Given a connected component C of Ω^{c}, it remains to prove that v is constant C_{q}-q.e. on C. For every ε > 0 let C_{ε} := {x ∈ ℝ^{2} : dist(x, C) < ε}, and let u_{ε} be a solution of problem (2.6) in Ω_{ε} := Ω \ \overline{C}_{ε}. Let v_{ε} be the unique function in W^{1,q}(ℝ^{2}) such that ∇v_{ε} = Ra(x, ∇u_{ε})1_{Ω_{ε}} a.e. in ℝ^{2}. By Remark 2.8, ∇u_{ε} converges to ∇u strongly in L^{p}(ℝ^{2}, ℝ^{2}) and so v_{ε} converges to v strongly in W^{1,q}(ℝ^{2}). By construction ∇v_{ε} = 0 in C_{ε}. As C_{ε} is a connected open set containing C, we have that v_{ε} is constant C_{q}-q.e. on C. Since a subsequence of v_{ε} converges to v C_{q}-q.e. on ℝ^{2}, we conclude that v is constant C_{q}-q.e. on C.

Proof of Theorem 3.1. Let a: ℝ^{2} × ℝ^{2} → ℝ^{2} be a Carathéodory function satisfying (2.2)–(2.4) and let u_{h} and u be solutions to problems (2.6) in Ω_{h} and Ω. Taking u_{h} as test function in (2.8) in Ω_{h} and using (2.4) we obtain that ∇u_{h}1_{Ω_{h}} is bounded in L^{p}(ℝ^{2}, ℝ^{2}). By (2.3) we obtain also that a(x, ∇u_{h})1_{Ω_{h}} is bounded in L^{q}(ℝ^{2}). Passing to a subsequence, we may assume that ∇u_{h}1_{Ω_{h}} ⇀ Ψ weakly in L^{p}(ℝ^{2}, ℝ^{2}) and a(x, ∇u_{h})1_{Ω_{h}} ⇀ Φ weakly in L^{q}(ℝ^{2}, ℝ^{2}). By (2.8) we have div(a(x, ∇u_{h})1_{Ω_{h}}) = 0 in D′(ℝ^{2}), hence div Φ = 0 in D′(ℝ^{2}).

If Ω′ ⊂⊂ Ω, by the Hausdorff complementary convergence we have Ω′ ⊂⊂ Ω_{h} for h large enough. Since the set of gradients of functions of L^{1,p}(Ω′) is closed in L^{p}(Ω′, ℝ^{2}), the vectorfield Ψ is the gradient of a function of L^{1,p}(Ω′). As Ω′ is arbitrary, we can construct u^{∗} ∈ L^{1,p}(Ω) such that Ψ = ∇u^{∗} a.e. in Ω. On the other hand, by Lemma 3.2 we have Ψ = 0 a.e. in Ω^{c}, hence Ψ = ∇u^{∗}1_{Ω} a.e. in ℝ^{2}.

Let us prove that Φ = a(x, ∇u^{∗})1_{Ω} a.e. in ℝ^{2}. By Lemma 3.2 it is enough to prove the equality in every open ball B ⊂⊂ Ω. Note that by the Hausdorff complementary convergence we have B ⊂⊂ Ω_{h} for h large enough. By adding suitable constants, we may assume that the mean values of u_{h} and u^{∗} on B are zero. Thus the Poincaré inequality and the Rellich theorem imply that u_{h} converges to u^{∗} strongly in L^{p}(B).

Let z ∈ W^{1,p}(B) and let φ ∈ C_{c}^{∞}(B) with φ ≥ 0. For h large enough we have B ⊂⊂ Ω_{h}, thus by monotonicity we have

∫_{B}(a(x, ∇z) − a(x, ∇u_{h})) · (∇z − ∇u_{h})φdx ≥ 0. (3.4)

By (2.8) we have also

∫_{B} a(x, ∇u_{h}) · ∇((z − u_{h}) φ)dx = 0,

which, together with (3.4), gives

∫_{B} a(x, ∇z) · ∇((z − u_{h})φ)dx − ∫_{B}(a(x, ∇z) − a(x, ∇u_{h})) · ∇φ (z − u_{h})dx ≥ 0. (3.5)

We can pass to the limit in each term of (3.5) and we get

∫_{B} a(x, ∇z) · ∇((z − u^{∗})φ)dx − ∫_{B}(a(x, ∇z) − Φ) · ∇φ (z − u^{∗})dx ≥ 0. (3.6)

As div Φ = 0 in D′(B), we have

∫_{B} Φ · ∇((z − u^{∗})φ)dx = 0. (3.7)

From (3.6) and (3.7) we obtain

∫_{B}(a(x, ∇z) − Φ) · (∇z − ∇u^{∗}) φdx ≥ 0.

As φ is arbitrary, we get (a(x, ∇z) − Φ) · (∇z − ∇u^{∗}) ≥ 0 a.e. in B. In particular, taking z(x) := u^{∗}(x) ± εξ · x, with ξ ∈ ℝ^{2} and ε > 0, we obtain ±(a(x, ∇u^{∗} ± εξ) − Φ) · ξ ≥ 0 a.e. in B. As ε tends to zero we get (a(x, ∇u^{∗}) − Φ) · ξ = 0 a.e. in B, which implies that a(x, ∇u^{∗}) = Φ a.e. in B by the arbitrariness of ξ.

Let us prove now that

∫_{Ω} a(x, ∇u^{∗}) · ∇z dx = 0 ∀z ∈ L^{1,p}(Ω). (3.8)

By Lemma 3.6 for every h there exists v_{h} ∈ W^{1,q}(ℝ^{2}) such that ∇v_{h} = Ra(x, ∇u_{h})1_{Ω_{h}} a.e. in ℝ^{2}. Moreover v_{h} is constant C_{q}-q.e. on each connected component of Ω_{h}^{c}. As a(x, ∇u_{h})1_{Ω_{h}} converges to a(x, ∇u^{∗})1_{Ω} weakly in L^{q}(ℝ^{2}, ℝ^{2}), there exists a function v ∈ W^{1,q}(ℝ^{2}) such that v_{h} ⇀ v weakly in W^{1,q}(ℝ^{2}) and ∇v = Ra(x, ∇u^{∗})1_{Ω} a.e. in ℝ^{2}. So, we have to prove that

∫_{Ω} R∇v · ∇z dx = 0 ∀z ∈ L^{1,p}(Ω).

From the Lemma 3.5 it follows that v is constant C_{q}-q.e. on the connected components of Ω^{c}. By [19, Theorem 4.5] we can approximate v strongly in W^{1,q}(ℝ^{2}) by a sequence of functions v_{h} ∈ C_{c}^{∞}(ℝ^{2}) that are constant in suitable neighborhoods U_{h}^{i} of each connected component C^{i} of Ω^{c}. Let z ∈ L^{1,p}(Ω) and z_{h} ∈ W_{0}^{1,p}(Ω) such that z_{h} = z in Ω \ ⋃_{i} U_{h}^{i}. Then, we have

∫_{Ω} R∇v_{h} · ∇z dx = ∫_{Ω} R∇v_{h} · ∇z_{h} dx = 0, (3.9)

where the last equality follows from the fact that the vector field R∇v_{h} is divergence free. Then, passing to the limit in (3.9) for h → ∞, we get

∫_{Ω} a(x, ∇u^{∗}) · ∇z dx = − ∫_{Ω} R∇v · ∇zdx = 0.

So u^{∗} is a solution of (2.8) in Ω, hence ∇u^{∗} = ∇u a.e. in Ω by uniqueness of the gradients. This implies that ∇u_{h}1_{Ω_{h}} converges to ∇u1_{Ω} weakly in L^{p}(ℝ^{2}, ℝ^{2}) and a(x, ∇u_{h})1_{Ω_{h}} converges to a(x, ∇u)1_{Ω} weakly in L^{q}(ℝ^{2}, ℝ^{2}). Since |Ω_{h}△Ω| tends to 0 by Lemma 3.2, from the identity a(x, ∇u_{h}1_{Ω_{h}}) = a(x, ∇u_{h})1_{Ω_{h}} + a(x, 0)1_{Ω_{h}^{c}} we conclude also that a(x, ∇u_{h}1_{Ω_{h}}) converges to a(x, ∇u1_{Ω}) weakly in L^{q}(ℝ^{2}, ℝ^{2}).

To prove the strong convergence, we consider the integral

I_{h} := ∫_{ℝ^{2}}(a(x, ∇u_{h}1_{Ω_{h}}) − a(x, ∇u1_{Ω})) · (∇u_{h}1_{Ω_{h}} − ∇u1_{Ω})dx.

Since by (2.8)

∫_{Ω_{h}} a(x, ∇u_{h}) · ∇u_{h} dx = 0  and  ∫_{Ω} a(x, ∇u) · ∇u dx = 0,

we have

I_{h} = − ∫_{Ω} a(x, ∇u_{h}1_{Ω_{h}}) · ∇u dx − ∫_{Ω_{h}} a(x, ∇u1_{Ω}) · ∇u_{h}1_{Ω_{h}} dx.

Therefore

lim_{h→∞} I_{h} = −2 ∫_{Ω} a(x, ∇u) · ∇u dx = 0, (3.10)

where the last equality can be deduced from (2.8). The strong convergence in L^{p}(ℝ^{2}, ℝ^{2}) of ∇u_{h}1_{Ω_{h}} to ∇u1_{Ω} follows now from (3.10) and from Lemma 2.4.

Remark 3.7. As we said in the introduction, in the case p > 2 the stability result for problem (2.6) is not true under our hypotheses. Indeed, let us consider

S := [1, 3] × {0}, S_{h} := ([1, 2 − 1/h] ∪ [2 + 1/h, 3]) × {0},
Ω := B(0, 3) \ (\overline{B(0, 1)} ∪ S) and Ω_{h} := B(0, 3) \ (\overline{B(0, 1)} ∪ S_{h}).

Let φ ∈ C_{c}^{∞}(0, ∞) be such that φ(ρ) = ρ^{−p} for 1 ≤ ρ ≤ 3. We set

a(x, ξ) := |ξ|^{p−2}ξ − φ(|x|)Rx,

where R is the rotation by π/2 defined by Rx := (−x_{2}, x_{1}). Let u_{h} and u be solutions of problems (2.6) in Ω and Ω_{h}, with ∫_{Ω_{h}} u_{h} dx = ∫_{Ω} u dx = 0. For every x ∈ Ω, let 0 < θ(x) < 2π be the angle between x and the positive x_{1}-axis. As ∇θ(x) = Rx/|x|^{2}, we have that u = θ − π in Ω.

If the open set Ω were stable for problem (2.6) along the sequence (Ω_{h}), then ∇u_{h} would converge strongly to ∇u in L^{p}(Ω, ℝ^{2}). By the Poincaré inequality we would have that u_{h} converges strongly to u in W^{1,p}(Ω).

For every v ∈ W^{1,p}(Ω), let v^{+} and v^{−} be the upper and lower traces of v on S, defined by

(3.11) v^{+}(x) := lim_{\begin{matrix}y→x \\ y_{2}>0 \end{matrix}} v(y) and v^{−}(x) := lim_{\begin{matrix}y→x \\ y_{2}<0 \end{matrix}} v(y).

From the convergence of u_{h} to u in W^{1,p}(Ω), we obtain that u_{h}^{+} → u^{+} and u_{h}^{−} → u^{−} uniformly on S (recall that p > 2 here). Since u_{h}^{+}(2, 0) = u_{h}^{−}(2, 0) by the continuity of u_{h}, we obtain u^{+}(2, 0) = u^{−}(2, 0), which contradicts the fact that u^{+}(2, 0) = −π and u^{−}(2, 0) = π, being u = θ − π.

4. Mosco convergence of Sobolev spaces

In this section we study the convergence in the sense of Mosco of the subspaces X_{Ω} introduced in (2.9) and corresponding to the Sobolev spaces W^{1,p}(Ω). The convergence of X_{Ω_{h}} to X_{Ω} will be obtained from the convergence of Y_{Ω_{h}} to Y_{Ω} and from the following approximation theorem for functions which are locally constant on the limit open set Ω.

Theorem 4.1. Let (Ω_{h}) be a uniformly bounded sequence of open subsets of ℝ^{2} which converges to an open set Ω in the Hausdorff complementary topology. Assume that |Ω_{h}| converges to |Ω| and that Ω_{h}^{c} is connected for every h. Then for every u ∈ W^{1,p}(Ω) with ∇u = 0 a.e. in Ω, there exists a sequence u_{h} ∈ W^{1,p}(Ω_{h}) such that u_{h}1_{Ω_{h}} converges to u1_{Ω} strongly in L^{p}(ℝ^{2}) and ∇u_{h}1_{Ω_{h}} converges to 0 strongly in L^{p}(ℝ^{2}, ℝ^{2}).

The proof of this theorem is postponed. We are now in a position to state the main result of the paper.

Theorem 4.2. Let (Ω_{h}) be a uniformly bounded sequence of open subsets of ℝ^{2} which converges to an open set Ω in the Hausdorff complementary topology, with |Ω_{h}| converging to |Ω|. Assume that Ω_{h}^{c} has a uniformly bounded number of connected components. Then X_{Ω_{h}} converges to X_{Ω} in the sense of Mosco.

To prove Theorem 4.2 we need the following localization lemma.

Lemma 4.3. Let (Ω_{h}) be a uniformly bounded sequence of open subsets of ℝ^{2}, and let Ω be a bounded open subset of ℝ^{2}. Assume that for every x ∈ ℝ^{2} there exists ε > 0 such that the sequence X_{B(x,ε)∩Ω_{h}} converges to X_{B_{ε}(x)∩Ω} in the sense of Mosco. Then X_{Ω_{h}} converges to X_{Ω} in the sense of Mosco.

Proof. Condition (M_{2}) is easy, and condition (M_{1}) can be obtained by using a partition of unity.

Proof of Theorem 4.2. Step 1. Assume that Ω_{h}^{c} is connected for every h. Let us prove (M_{2}). Let (h_{k}) be a sequence of indices converging to ∞, (u_{k}) be a sequence, with u_{k} ∈ W^{1,p}(Ω_{h_{k}}) for every k, such that u_{k}1_{Ω_{h_{k}}} converges weakly in L^{p}(ℝ^{2}) to a function ϕ, while ∇u_{k}1_{Ω_{h_{k}}} converges weakly in L^{p}(ℝ^{2}, ℝ^{2}) to a function ψ. From Lemma 3.2 it follows that ϕ and ψ vanish a.e. in Ω^{c}.

Let Ω′ ⊂⊂ Ω be an open set. By the Hausdorff complementary convergence we have Ω′ ⊂⊂ Ω_{h} for h large enough. So, u_{k}|_{Ω′} converges weakly to ϕ|_{Ω′} in L^{p}(Ω′) and ∇u_{k}|_{Ω′} converges weakly to ψ|_{Ω′} in L^{p}(Ω′, ℝ^{2}). Hence ϕ|_{Ω′} ∈ W^{1,p}(Ω′) and ψ|_{Ω′} = ∇ϕ|_{Ω′} in Ω′. From the arbitrariness of Ω′, it follows that the function u := ϕ|_{Ω} belongs to W^{1,p}(Ω), ϕ = u1_{Ω} and ψ = ∇u1_{Ω} a.e. in ℝ^{2}.

Now let us prove (M_{1}). Let u ∈ W^{1,p}(Ω). We write Ω := ⋃_{i=1}^{m} Ω_{i}, where 1 ≤ m ≤ ∞ and (Ω_{i}) is the family of connected components of Ω. Since the set of functions u satisfying (M_{1}) is a closed linear subspace of W^{1,p}(Ω), by a density argument it is enough to prove (M_{1}) when u belongs to L^{∞}(Ω) and vanishes on all connected components of Ω except one. By renumbering the sequence (Ω_{i}), we may assume that u vanishes on Ω_{i} for every i ≥ 2.

From Theorem 3.1 on the convergence of Y_{Ω_{h}} to Y_{Ω} in the sense of Mosco, there exists a sequence z_{h} ∈ L^{1,p}(Ω_{h}) such that ∇z_{h}1_{Ω_{h}} converges strongly to ∇u1_{Ω} in L^{p}(ℝ^{2}, ℝ^{2}). Let us fix a nonempty open set A_{0} ⊂⊂ Ω_{1}. We can assume that ∫_{A_{0}} z_{h}dx = ∫_{A_{0}} udx. For every smooth connected open set A, with A_{0} ⊂⊂ A ⊂⊂ Ω_{1}, by the Poincaré inequality we have that z_{h}|_{A} → u|_{A} strongly in W^{1,p}(A).

We consider now w_{h} := (−∥u∥_{∞}) ∨ z_{h} ∧ ∥u∥_{∞}. We have that w_{h}|_{A} → u|_{A} strongly in W^{1,p}(A) for every open set A ⊂⊂ Ω_{1}. Moreover, for every open set E ⊂⊂ Ω the function w_{h}|_{E} belongs to W^{1,p}(E) for h large enough. As ∥w_{h}∥_{∞} ≤ ∥u∥_{∞} and |∇w_{h}|_{E}| ≤ |∇z_{h}|_{E}|, the sequence (w_{h}|_{E}) is bounded in W^{1,p}(E). By the Rellich theorem, there exists w ∈ W^{1,p}(Ω) such that w_{h}|_{E} converges to w|_{E} strongly in L^{p}(E) for every open set E ⊂⊂ Ω with smooth boundary. As w_{h}|_{A} → u|_{A} strongly in W^{1,p}(A) for every open set A ⊂⊂ Ω_{1}, we have that w = u a.e. in Ω_{1}.

For every open set E ⊂⊂ Ω \ Ω_{1}, since |∇w_{h}| ≤ |∇z_{h}| and ∇z_{h}|_{E} → ∇u|_{E} = 0 strongly in L^{p}(E, ℝ^{2}), we have ∇w_{h}|_{E} → 0 strongly in L^{p}(E, ℝ^{2}). Therefore we get ∇w = 0 = ∇u a.e. in Ω \ Ω_{1}, which together with the result obtained in Ω_{1} implies that ∇w_{h}|_{E} converges to ∇u|_{E} strongly in L^{p}(E, ℝ^{2}) for every E ⊂⊂ Ω. In particular, we obtain that the function u − w is locally constant in Ω.

We claim that ∇w_{h}1_{Ω_{h}} converges to ∇w1_{Ω} strongly in L^{p}(ℝ^{2}, ℝ^{2}). Indeed for every E ⊂⊂ Ω we have ∥∇w_{h}1_{Ω_{h}} − ∇u1_{Ω}∥_{L^{p}(ℝ^{2},ℝ^{2})} ≤ ∥∇w_{h}1_{E} − ∇u1_{E}∥_{L^{p}(E,ℝ^{2})} + ∥∇z_{h}1_{Ω_{h}}1_{E^{c}}∥_{L^{p}(ℝ^{2},ℝ^{2})} + ∥∇u1_{Ω\E}∥_{L^{p}(ℝ^{2},ℝ^{2})}. Hence

lim sup_{h→∞} ∥∇w_{h}1_{Ω_{h}} − ∇u1_{Ω}∥_{L^{p}(ℝ^{2},ℝ^{2})} ≤ 2∥∇u1_{Ω\E}∥_{L^{p}(ℝ^{2},ℝ^{2})},

and by letting E ↗ Ω we prove the claim. In a similar way, we obtain also that w_{h}1_{Ω_{h}} converges to w1_{Ω} strongly in L^{p}(ℝ^{2}).

Since u − w is locally constant in Ω, from Theorem 4.1 there exists v_{h} ∈ W^{1,p}(Ω_{h}) such that v_{h}1_{Ω_{h}} → (u − w)1_{Ω} strongly in L^{p}(ℝ^{2}) and ∇v_{h}1_{Ω_{h}} → 0 strongly in L^{p}(ℝ^{2}, ℝ^{2}). Therefore w_{h} + v_{h} ∈ W^{1,p}(Ω_{h}), (w_{h} + v_{h})1_{Ω_{h}} → u1_{Ω} strongly in L^{p}(ℝ^{2}), and ∇(w_{h} + v_{h})1_{Ω_{h}} → ∇u1_{Ω} strongly in L^{p}(ℝ^{2}, ℝ^{2}), which give property (M_{1}).

Step 2. We now remove the hypothesis that Ω_{h}^{c} is connected. Let C_{h}^{1}, …, C_{h}^{n_{h}} be the connected components of Ω_{h}^{c}. Passing to a subsequence we can assume that n_{h} does not depend on h and that the sets C_{h}^{i} converge in the Hausdorff metric to some connected sets C^{i} as h → ∞. Let C^{i_{1}}, …, C^{i_{d}} be those C^{i} having at least two points. We set

Ω^{∗} := (⋃_{j=1}^{d} C^{i_{j}})^{c} and Ω_{h}^{∗} := (⋃_{j=1}^{d} C_{h}^{i_{j}})^{c}.

We have that Ω ⊂ Ω^{∗}, Ω_{h} ⊂ Ω_{h}^{∗}, and, by construction, Ω_{h}^{∗} converges in the Hausdorff complementary topology to Ω^{∗} and |Ω_{h}^{∗}| → |Ω^{∗}| (because |C_{h}^{i}| → 0 if i ≠ i_{1}, …, i_{d}). There exists some η > 0 such that diam C^{i_{j}} > η for every j, hence diam C_{h}^{i_{j}} > η for h large enough. Let us observe that, for every x ∈ ℝ^{2}, the sequence B(x, η/2) ∩ Ω_{h}^{∗} converges to B(x, η/2) ∩ Ω^{∗} in the Hausdorff complementary topology and also |B(x, η/2) ∩ Ω_{h}^{∗}| → |B(x, η/2) ∩ Ω^{∗}| (by Lemma 3.2). As diam C_{h}^{i_{j}} > η, it is easy to see that (B(x, η/2) ∩ Ω_{h}^{∗})^{c} is connected for h large enough. So, from Step 1 we obtain the Mosco convergence of X_{B(x,η/2)∩Ω_{h}^{∗}} to X_{B(x,η/2)∩Ω^{∗}}. Now, using Lemma 4.3 we get the Mosco convergence of X_{Ω_{h}^{∗}} to X_{Ω^{∗}}.

As Ω_{h}^{∗} \ Ω_{h} is the union of a uniformly bounded number of sets with diameter tending to 0, using the fact that 1 < p ≤ 2, we deduce that C_{p}(Ω_{h}^{∗} \ Ω_{h}) → 0. Let us show that this implies that X_{Ω_{h}} converges to X_{Ω} in the sense of Mosco.

For property (M_{1}), let u ∈ W^{1,p}(Ω). Since the set Ω^{∗}\Ω is finite, we have that C_{p}(Ω^{∗}\Ω) = 0. Hence, u ∈ W^{1,p}(Ω^{∗}). So, there exists u_{h}^{∗} ∈ W^{1,p}(Ω_{h}^{∗}) such that u_{h}^{∗}1_{Ω_{h}^{∗}} converges to u1_{Ω^{∗}} strongly in L^{p}(ℝ^{2}) and ∇u_{h}^{∗}1_{Ω_{h}^{∗}} converges to ∇u1_{Ω^{∗}} strongly in L^{p}(ℝ^{2}, ℝ^{2}). Setting u_{h} = u_{h}^{∗}|_{Ω_{h}}, we obtain that u_{h}1_{Ω_{h}} converges to u1_{Ω} strongly in L^{p}(ℝ^{2}) and ∇u_{h}1_{Ω_{h}} converges to ∇u1_{Ω} strongly in L^{p}(ℝ^{2}, ℝ^{2}), and so property (M_{1}) holds.

Let us prove property (M_{2}). Let (h_{k}) be a sequence of indices converging to ∞, (u_{k}) be a sequence, with u_{k} ∈ W^{1,p}(Ω_{h_{k}}) for every k, such that u_{k}1_{Ω_{h_{k}}} converges weakly in L^{p}(ℝ^{2}) to a function ϕ, while ∇u_{k}1_{Ω_{h_{k}}} converges weakly in L^{p}(ℝ^{2}, ℝ^{2}) to a function ψ. From Lemma 3.2 it follows that ϕ and ψ vanish a.e. in Ω^{c}.

As C_{p}(Ω_{h_{k}}^{∗} \ Ω_{h_{k}}) → 0, there exists a sequence φ_{k} ∈ W^{1,p}(ℝ^{2}) converging strongly to 0 in W^{1,p}(ℝ^{2}) such that φ_{k} = 1 a.e. in a neighborhood of Ω_{h_{k}}^{∗} \ Ω_{h_{k}}.

We set

u_{k}^{∗} := u_{k}(1 − φ_{k}).

Note that u_{k}^{∗} ∈ W^{1,p}(Ω_{h_{k}}^{∗}) and that u_{k}^{∗}1_{Ω_{h_{k}}^{∗}} converges weakly in L^{p}(ℝ^{2}) to ϕ, while ∇u_{k}^{∗}1_{Ω_{h_{k}}^{∗}} converges weakly in L^{p}(ℝ^{2}, ℝ^{2}) to ψ. So, from the Mosco convergence of X_{Ω_{h}^{∗}} to X_{Ω^{∗}}, it follows that there exists u^{∗} ∈ W^{1,p}(Ω^{∗}) such that ϕ = u^{∗}1_{Ω^{∗}} and ψ = ∇u^{∗}1_{Ω^{∗}} a.e. in ℝ^{2}. By setting u = u^{∗}|_{Ω}, we get that ϕ = u1_{Ω} and ψ = ∇u1_{Ω} a.e. in ℝ^{2} and the proof of (M_{2}) is complete.

The rest of this section is devoted to the proof of Theorem 4.1. To this aim we will need some preliminary results.

Lemma 4.4. Let (Ω_{h}) be a uniformly bounded sequence of open subsets of ℝ^{2}, which converges to an open set Ω in the Hausdorff complementary topology, with |Ω_{h}| converging to |Ω|. Assume that Ω_{h} = Ω_{h}^{1} ∪ Ω_{h}^{2} with Ω_{h}^{i} open and Ω_{h}^{1} ∩ Ω_{h}^{2} = Ø. Assume also that (Ω_{h}^{i}) converges to an open set Ω^{i}, i = 1, 2, in the Hausdorff complementary topology. Then

(i) Ω^{1} ∩ Ω^{2} = Ø, (ii) Ω^{1} ∪ Ω^{2} = Ω, (iii) |Ω^{i}| = lim_{h} |Ω_{h}^{i}|, i = 1, 2. In particular, if Ω_{h}^{0} is union of connected components of Ω_{h} and converges to an open set Ω^{0} in the Hausdorff complementary topology, then Ω^{0} is union of connected components of Ω and |Ω_{h}^{0}| converges to |Ω^{0}|.

Proof. (i) and (ii) are easy consequences of the convergence in the Hausdorff complementary topology, while (iii) follows from Lemma 3.2. As Ω_{h}^{0} is union of connected components of Ω_{h}, then the set Ω′_{h} := Ω_{h} \ Ω_{h}^{0} is open in the relative topology of Ω_{h}. Up to a subsequence, Ω′_{h} converges to an open set Ω′ in the Hausdorff complementary topology. From (i) and (ii) we have that Ω^{0} ∩ Ω′ = Ø, and Ω^{0} ∪ Ω′ = Ω; hence Ω^{0} is union of connected components of Ω. The last assertion follows from (iii).

The following lemma, proved by Bucur and Varchon in [2], will also be used in the proof of Theorem 4.1.

Lemma 4.5. Let Ω be a bounded open set in ℝ^{2} and let a and b be two points in two different connected components Ω_{a} and Ω_{b} of Ω, whose distance from Ω^{c} is greater than 10δ for some δ > 0. Let U be an open subset of ℝ^{2} such that U^{c} is connected and d_{H}(U^{c}, Ω^{c}) < δ. Then there exists x ∈ Ω^{c} such that the closed square Q(x, 9δ), with centre x and side length 9δ, intersects any curve contained in U and joining the points a and b.

Proof of Theorem 4.1. By a density argument, it is sufficient to prove that for every connected component Ω^{0} of Ω there exists a sequence u_{h} ∈ W^{1,p}(Ω_{h}) such that u_{h}1_{Ω_{h}} converges strongly to 1_{Ω^{0}} in L^{p}(ℝ^{2}) and ∇u_{h}1_{Ω_{h}} converges strongly to 0 in L^{p}(ℝ^{2}, ℝ^{2}). Let a^{0} ∈ Ω^{0} and let Ω_{h}^{0} be the connected component of Ω_{h} which contains a^{0} (which is defined for h large enough). Up to a subsequence, Ω_{h}^{0} converges in the Hausdorff complementary topology to some open set E. From Lemma 4.4 it follows that

E = ⋃_{i=0}^{m} Ω^{i},

where 0 ≤ m ≤ ∞ and (Ω^{i}) is a family of connected components of Ω (including Ω^{0}), and

|Ω_{h}^{0}| → |E|. (4.1)

Let 0 < ε < |Ω^{0}| be fixed. There exist a finite integer n_{ε} ≥ 1 and an open set Ω_{ε} such that

E = ⋃_{i=0}^{n_{ε}} Ω^{i} ∪ Ω_{ε}, (4.2)

where |Ω_{ε}| ≤ ε and Ω^{i} ∩ Ω_{ε} = Ø for every i ≤ n_{ε}.

We fix now a point a^{i} in each set Ω^{i}. Let δ > 0 be such that dist(a^{i}, Ω^{c}) > 10δ for every i ≤ n_{ε}. From Lemma 4.5, for h big enough there exist some points (x_{h}^{δ,i})_{i=1}^{n_{ε}}, uniformly bounded in Ω^{c}, such that for every i ≤ n_{ε} the square Q(x_{h}^{δ,i}, 9δ) intersects any curve contained in Ω_{h}^{0} and joining the points a^{0} and a^{i}. Up to a subsequence, we have that x_{h}^{δ,i} → x^{δ,i} as h → ∞, for some x^{δ,i} ∈ Ω^{c}. Once again up to a subsequence, we have that x^{δ,i} → x^{i} as δ → 0, for some x^{i} ∈ Ω^{c}. Let

K^{δ,ε} := ⋃_{i=1}^{n_{ε}} \overline{B(x^{i}, 10δ)}

and, for i = 0, …, n_{ε}, let Ω_{h}^{δ,ε,i} be the connected component of Ω_{h}^{0} \ K^{δ,ε} containing a^{i}. As K^{δ,ε} ⊃ Q(x_{h}^{δ,i}, 9δ) for δ small enough and h large enough, we have

Ω_{h}^{δ,ε,0} ≠ Ω_{h}^{δ,ε,i} for i ≠ 0. (4.3)

Let φ^{δ,ε} be the C_{p}-capacitary potential of K^{δ,ε}, i.e., the solution of the minimum problem

min {∫_{ℝ^{2}}[|∇φ|^{p} + |φ|^{p}] dx : φ ∈ W^{1,p}(ℝ^{2}), φ = 1 C_{p}-a.e. on K^{δ,ε}} .

We set

u_{h}^{ε,δ} := \begin{cases}1 & in Ω_{h}^{δ,ε,0}, \\ φ^{δ,ε} & in Ω_{h} \ Ω_{h}^{δ,ε,0}. \end{cases} (4.4)

As Ω_{h} ∩ ∂Ω_{h}^{δ,ε,0} ⊂ K^{δ,ε}, we have that u_{h}^{ε,δ} ∈ W^{1,p}(Ω_{h}). We observe that

∥u_{h}^{δ,ε}1_{Ω_{h}} − 1_{Ω_{h}^{δ,ε,0}}∥_{L^{p}(ℝ^{2})} + ∥∇u_{h}^{δ,ε}1_{Ω_{h}}∥_{L^{p}(ℝ^{2},ℝ^{2})} ≤ 2C_{p}(K^{δ,ε})^{\frac{1}{p}} .

As C_{p}(K^{δ,ε}) ≤ n_{ε}C_{p}(B(0, 10δ)) and p ≤ 2, we conclude that

(4.5)  lim sup_{δ→0} lim sup_{h→∞}[∥u_{h}^{δ,ε,0}1_{Ω_{h}} − 1_{Ω_{h}^{δ,ε}}∥_{L^{p}(ℝ^{2})} + ∥∇u_{h}^{δ,ε}1_{Ω_{h}}∥_{L^{p}(ℝ^{2},ℝ^{2})}] = 0.

We claim that

(4.6)  lim sup_{δ→0} lim sup_{h→∞} ∥u_{h}^{ε,δ,0}1_{Ω_{h}} − 1_{Ω^{0}}∥_{L^{p}(ℝ^{2})}^{p} ≤ ε,

and

(4.7)  lim sup_{δ→0} lim sup_{h→∞} ∥∇u_{h}^{ε,δ}1_{Ω_{h}}∥_{L^{p}(ℝ^{2},ℝ^{2})} = 0,

from which the proof of the theorem is achieved by the arbitrariness of ε.

It is easy to see that (4.7) follows from (4.5), while (4.6) is a consequence of (4.5) and of the following inequality

(4.8)  lim sup_{δ→0} lim sup_{h→∞} |Ω_{h}^{δ,ε,0}△Ω^{0}| ≤ ε.

So, let us prove (4.8). For every i = 0, … n_{ε}, up to a subsequence, Ω_{h}^{δ,ε,i} converges in the Hausdorff complementary topology, when h → ∞, to some open set Ω^{δ,ε,i} ⊂ E. We observe that Ω_{h}^{0} \ K^{δ,ε} converges to E \ K^{δ,ε} in the Hausdorff complementary topology when h → ∞. Let E^{δ,ε,i} be the connected component of E \ K^{δ,ε} which contains a^{i}. It is easy to see that

E^{δ,ε,i} ⊂ Ω^{δ,ε,i}. (4.9)

Note that, as δ ↘ 0, K^{δ,ε} converges decreasingly to the set {x_{1}, …, x_{n_{ε}}}, E^{δ,ε,i} converges increasingly to Ω^{i}, and Ω^{δ,ε,i} converges increasingly to some open set Ω^{ε,i} ⊂ E. From (4.9), it follows that Ω^{i} ⊂ Ω^{ε,i}. From (4.1) and from Lemma 3.2 it follows that

|Ω_{h}^{0} \ K^{δ,ε}| → |E \ K^{δ,ε}|. (4.10)

By Lemma 4.4 applied to Ω_{h}^{1} := Ω_{h}^{δ,ε,0} and Ω_{h}^{2} := (Ω_{h}^{0}\K^{δ,ε})\Ω_{h}^{δ,ε,0}, we have that Ω^{δ,ε,0}∩Ω^{δ,ε,i} = Ø for every i ≠ 0, from which it follows that Ω^{ε,0} ∩ Ω^{ε,i} = Ø and hence Ω^{ε,0} ∩ Ω^{i} = Ø for every 1 ≤ i ≤ n_{ε}. Therefore, there exists an open set Ω′_{ε}, contained in the set Ω_{ε} introduced in (4.2), such that

Ω^{ε,0} = Ω^{0} ∪ Ω′_{ε}.

From (4.10) and from Lemmas 4.4 and 3.2, it follows that

|Ω_{h}^{δ,ε,0}△Ω^{δ,ε,0}| → 0. (4.11)

As Ω^{δ,ε,0} ⊂ Ω^{ε,0} = Ω^{0} ∪ Ω′_{ε}, it follows that

(4.12) |Ω_{h}^{δ,ε,0} \ Ω^{0}| ≤ |Ω_{h}^{δ,ε,0} \ Ω^{δ,ε,0}| + |Ω^{δ,ε,0} \ Ω^{0}| ≤ |Ω_{h}^{δ,ε,0} \ Ω^{δ,ε,0}| + |Ω′_{ε}|,

and

(4.13) |Ω^{0} \ Ω_{h}^{δ,ε,0}| ≤ |Ω^{0} \ Ω^{δ,ε,0}| + |Ω^{δ,ε,0} \ Ω_{h}^{δ,ε,0}| ≤ |Ω^{ε,0} \ Ω^{δ,ε,0}| + |Ω^{δ,ε,0} \ Ω_{h}^{δ,ε,0}|.

As Ω^{δ,ε,0} converges increasingly to Ω^{ε,0} as δ → 0^{+}, we have |Ω^{ε,0} \ Ω^{δ,ε,0}| → 0 as δ → 0. By (4.11), passing to the limit in (4.12) and (4.13) first as h → ∞ and then as δ → 0^{+} we obtain

lim sup_{δ→0} lim sup_{h→∞} |Ω_{h}^{δ,ε,0} \ Ω^{0}| ≤ |Ω_{ε}| + lim sup_{h→∞} |Ω_{h}^{δ,ε,0} \ Ω^{δ,ε,0}| ≤ ε

and

lim sup_{δ→0} lim sup_{h→∞} |Ω^{0} \ Ω_{h}^{δ,ε,0}| = 0,

which give inequality (4.8).

Remark 4.6. In the case p > 2 the stability result for problem (1.1) is not true under our hypotheses. Indeed, let us consider

S := [−1, 1] × {0}, S_{h} := ([−1, −1/h] ∪ [1/h, 1]) × {0},
Ω := (−1, 1)^{2} \ S, and Ω_{h} := (−1, 1)^{2} \ S_{h}.

We set

a(x, ξ) := |ξ|^{p−2}ξ and b(x, η) := |η|^{p−2}η − x_{2},

where x = (x_{1}, x_{2}). Let u_{h} and u be solutions of problems (1.1) in Ω_{h} and Ω respectively. By the symmetry of Ω, the solution u will depend only on x_{2}. Therefore, for every x = (x_{1}, x_{2}) ∈ Ω

u(x) = \begin{cases}w(x_{2}) & if x_{2} ∈ (0, 1), \\ −w(−x_{2}) & if x_{2} ∈ (−1, 0) \end{cases}

where w is the solution of the one-dimensional problem

\begin{cases}−(|w′|^{p−2}w′)′ + |w|^{p−2}w = t & in (0, 1), \\ w′(0) = w′(1) = 0, \end{cases} (4.14)

which turns out to be of class C^{1}([0, 1]). For every v ∈ W^{1,p}(Ω), let v^{+} and v^{−} be the upper and lower traces of v on S, defined as in (3.11).

If the open set Ω were stable for problem (1.1) along the sequence (Ω_{h}), then u_{h} would converge strongly to u in W^{1,p}(Ω). Hence we would have that u_{h}^{+} → u^{+} and u_{h}^{−} → u^{−} uniformly on S (recall that p > 2 here). Since u_{h}^{+}(0, 0) = u_{h}^{−}(0, 0) by the continuity of u_{h}, we would obtain u^{+}(0, 0) = u^{−}(0, 0), which implies that w(0) = 0. Let us prove that this is false. Indeed, by the maximum principle we have that w(t) ≥ 0 for every t ∈ [0, 1]. Since w′(0) = 0 and p > 2, we have that w^{p−1}(t) − t < 0 in a small neighborhood I of 0 in [0, 1]. So, from equation (4.14) the function |w′|^{p−2}w′ is decreasing in I and hence w′(t) < 0 for every t ∈ I. If w(0) were equal to 0, we would obtain w(t) < 0 for every t ∈ I, which contradicts the fact that w(t) ≥ 0 for every t ∈ [0, 1]. This proves that w(0) > 0, and hence Ω is not stable for problem (1.1) along the sequence (Ω_{h}) for p > 2.

5. The case of unbounded domains

We now extend the results of the previous sections to the case of unbounded domains.

Theorem 5.1. Let (Ω_{h}) be a sequence of open subsets of ℝ^{2} such that (Ω_{h}^{c}) converges to Ω^{c} in the sense of Kuratowski for some open subset Ω. Assume that, for every R > 0, |Ω_{h} ∩ B(0, R)| converges to |Ω ∩ B(0, R)| and that the number of connected components of (Ω_{h} ∩ B(0, R))^{c} is uniformly bounded with respect to h. Then the sequence of subspaces X_{Ω_{h}} (resp. Y_{Ω_{h}}) converges to X_{Ω} (resp. Y_{Ω}).

Proof. We prove only the Mosco convergence of X_{Ω_{h}} to X_{Ω}, since the convergence of Y_{Ω_{h}} to Y_{Ω} can be proved in the same way. First of all note that, from the convergence of Ω_{h}^{c} to Ω^{c} in the sense of Kuratowski, it follows that the sequence Ω_{h} ∩ B(0, R) converges to Ω ∩ B(0, R) in the Hausdorff complementary topology. Moreover, by the assumptions of the theorem, we can apply Theorem 4.2 to the sequence Ω_{h} ∩ B(0, R). So, we get that X_{Ω_{h}∩B(0,R)} converges to X_{Ω∩B(0,R)} in the sense of Mosco.

Now let us prove (M_{2}) for X_{Ω_{h}} and X_{Ω}. Let (h_{k}) be a sequence of indices converging to ∞, (u_{k}) be a sequence, with u_{k} ∈ W^{1,p}(Ω_{h_{k}}) for every k, such that u_{k}1_{Ω_{h_{k}}} converges weakly in L^{p}(ℝ^{2}) to a function ϕ, while ∇u_{k}1_{Ω_{h_{k}}} converges weakly in L^{p}(ℝ^{2}, ℝ^{2}) to a function ψ. It follows that u_{k}1_{Ω_{h_{k}} ∩B(0,R)} converges to ϕ1_{B(0,R)} weakly in L^{p}(ℝ^{2}), while ∇u_{k}1_{Ω_{h_{k}} ∩B(0,R)} converges to ψ1_{B(0,R)} weakly in L^{p}(ℝ^{2}, ℝ^{2}). So, by property (M_{2}) relative to the Mosco convergence of X_{Ω_{h}∩B(0,R)} to X_{Ω∩B(0,R)}, there exists a function u_{R} ∈ W^{1,p}(Ω ∩ B(0, R)) such that ϕ1_{B(0,R)} = u_{R}1_{Ω∩B(0,R)} and ψ1_{B(0,R)} = ∇u_{R}1_{Ω∩B(0,R)} a.e. in ℝ^{2}. Since R is arbitrary, it is easy to construct u ∈ W^{1,p}(Ω) such that ϕ = u1_{Ω} and ψ = ∇u1_{Ω} a.e. in ℝ^{2}.

Let us prove property (M_{1}). Let u ∈ W^{1,p}(Ω) and let ε > 0. There exists R_{ε} > 0 such that

∫_{Ω\B(0,R_{ε})}[|u|^{p} + |∇u|^{p}] dx ≤ ε.

By property (M_{1}) relative to the Mosco convergence of X_{Ω_{h}∩B(0,R_{ε}+1)} to X_{Ω∩B(0,R_{ε}+1)}, there exists a sequence w_{h}^{ε} ∈ W^{1,p}(Ω_{h} ∩ B(0, R_{ε} + 1)) such that w_{h}^{ε}1_{Ω_{h}∩B(0,R_{ε}+1)} converges strongly to u1_{Ω∩B(0,R_{ε}+1)} in L^{p}(ℝ^{2}) and ∇w_{h}^{ε}1_{Ω_{h}∩B(0,R_{ε}+1)} converges strongly to ∇u1_{Ω∩B(0,R_{ε}+1)} in L^{p}(ℝ^{2}, ℝ^{2}). Let φ_{ε} ∈ C_{c}^{1}(B(0, R_{ε} + 1)) such that 0 ≤ φ_{ε} ≤ 1, φ_{ε} = 1 in B(0, R_{ε}), and ∥∇φ_{ε}∥_{∞} ≤ C. Now we set u_{h}^{ε} := φ_{ε}w_{h}^{ε}. By construction u_{h}^{ε} ∈ W^{1,p}(Ω_{h}), u_{h}^{ε}1_{Ω_{h}} → φ_{ε}u1_{Ω} strongly in L^{p}(ℝ^{2}), and ∇u_{h}^{ε}1_{Ω_{h}} → φ_{ε}∇u1_{Ω} + u∇φ_{ε}1_{Ω} strongly in L^{p}(ℝ^{2}, ℝ^{2}). On the other hand

lim sup_{h→∞} ∫_{ℝ^{2}} |u_{h}^{ε}1_{Ω_{h}} − u1_{Ω}|^{p} + |∇u_{h}^{ε}1_{Ω_{h}} − ∇u1_{Ω}|^{p} dx ≤ 2^{p−1}(C^{p} + 1)ε,

φ_{ε}u1_{Ω} → u1_{Ω} strongly in L^{p}(ℝ^{2}), and φ_{ε}∇u1_{Ω} + u∇φ_{ε}1_{Ω} → ∇u1_{Ω} strongly in L^{p}(ℝ^{2}, ℝ^{2}). Therefore, we can construct a sequence u_{h} ∈ W^{1,p}(Ω_{h}) which satisfies (M_{1}) by a standar argument on double sequences.

6. Problems with Dirichlet boundary conditions

In this section we study the Mosco convergence of Sobolev and Deny-Lions spaces with prescribed Dirichlet conditions on part of the boundary.

Let A ⊂ ℝ^{2} be a bounded open set with Lipschitz boundary ∂A, and let ∂_{D}A be a relatively open subset of ∂A with a finite number of connected components. For every compact set K ⊂ \overline{A}, for every g ∈ W^{1,p}(A), and for every pair of function a and b satisfying the properties (2.2)–(2.4), we consider the solutions u and v of the mixed problems

− div a(x, ∇u) + b(x, u) = 0 in A \ K,
u = g  on ∂_{D}A \ K,
a(x, ∇u) · ν = 0  on ∂(A \ K) \ (∂_{D}A \ K),

(6.1) and

− div a(x, ∇v) = 0 in A \ K,
v = g  on ∂_{D}A \ K,
a(x, ∇v) · ν = 0  on ∂(A \ K) \ (∂_{D}A \ K).

(6.2)

Let (K_{h}) be a sequence of compact subsets of \overline{A}, let (g_{h}) be a sequence in W^{1,p}(A), and let (u_{h}) be the sequence of the solutions of problems (6.1) corresponding to K_{h} and g_{h}.

Definition 6.1. We say that the pair (K, g) is stable for the mixed problems (6.1) along the sequence (K_{h}, g_{h}) if for every pair of functions a, b satisfying (2.2)–(2.4) the sequence (u_{h}1_{K_{h}^{c}}) converges to u1_{K^{c}} strongly in L^{p}(A) and the sequence (∇u_{h}1_{K_{h}^{c}}) converges to ∇u1_{K^{c}} strongly in L^{p}(A, ℝ^{2}).

The stability for problems (6.2) is defined in a similar way by using only the convergence of the gradients (as in Definition 2.2).

The stabilty for problems (6.2) has been recently studied in [12] in the case a(x, ξ) = ξ. In this section we will study the stability in the general case by using again the notion of Mosco convergence.

We set

W_{g}^{1,p}(A \ K, ∂_{D}A \ K) := {u ∈ W^{1,p}(A \ K) : u = g on ∂_{D}A \ K},

and

L_{g}^{1,p}(A \ K, ∂_{D}A \ K) := {u ∈ L^{1,p}(A \ K) : u = g on ∂_{D}A \ K},

where the equality u = g on ∂_{D}A \ K is intended in the usual sense of traces.

As in Section 2.5 the space W_{g}^{1,p}(A \ K, ∂_{D}A \ K) will be identified with the closed linear subspace X_{K}^{g}(A) of L^{p}(A) × L^{p}(A, ℝ^{2}) defined by

X_{K}^{g}(A) := {(u1_{K^{c}}, ∇u1_{K^{c}}) : u ∈ W_{g}^{1,p}(A \ K, ∂_{D}A \ K)}. (6.3)

For problem (6.2), we consider in L_{g}^{1,p}(A \ K, ∂_{D}A \ K) the equivalence relation ∼ defined in (2.1). Note that in this case v_{1} ∼ v_{2} if and only if v_{1} = v_{2} a.e. in those connected components of A\K whose boundary intersects ∂_{D}A\K and ∇v_{1} = ∇v_{2} a.e. in the other connected components of A\K. The corresponding quotient space, denoted by L_{g}^{1,p}(A\K, ∂_{D}A\K)/_{∼}, will be identified with the closed linear subspace Y_{K}^{g}(A) of L^{p}(A, ℝ^{2}) defined by

Y_{K}^{g}(A) := {∇u1_{K^{c}} : u ∈ L_{g}^{1,p}(A \ K, ∂_{D}A \ K)}. (6.4)

Let K_{h}, K be compact subsets of \overline{A} and let g_{h}, g ∈ W^{1,p}(A). Let X_{K_{h}}^{g_{h}}(A) and X_{K}^{g}(A) be the corresponding subspaces defined by (6.3). We recall that X_{K_{h}}^{g_{h}}(A) converges to X_{K}^{g}(A) in the sense of Mosco if the following two properties hold: (M′′_{1}) for every u ∈ W_{g}^{1,p}(A \ K, ∂_{D}A \ K), there exists a sequence u_{h} ∈ W_{g_{h}}^{1,p}(A \ K_{h}, ∂_{D}A \ K_{h})

such that u_{h}1_{K_{h}^{c}} converges strongly to u1_{K^{c}} in L^{p}(A) and ∇u_{h}1_{K_{h}^{c}} converges strongly to

∇u1_{K^{c}} in L^{p}(A, ℝ^{2});

(M′′_{2}) if (h_{k}) is a sequence of indices converging to ∞, (u_{k}) is a sequence, with u_{k} ∈ W_{g_{h_{k}}}^{1,p} (A \ K_{h_{k}}, ∂_{D}A \ K_{h_{k}}) for every k, such that u_{k}1_{K_{h_{k}}^{c}} converges weakly in L^{p}(A) to a function ϕ, while ∇u_{k}1_{K_{h_{k}}^{c}} converges weakly in L^{p}(A, ℝ^{2}) to a function ψ, then there exists u ∈

W_{g}^{1,p}(A \ K, ∂_{D}A \ K) such that ϕ = u1_{K^{c}} and ψ = ∇u1_{K^{c}} a.e. in A.

Analogously, the convergence of Y_{K_{h}}^{g_{h}}(A) to Y_{K}^{g}(A) in the sense of Mosco can be characterized by using only the convergence of the extensions of the gradients.

Remark 6.2. As in Section 2.6 we can prove that the Mosco convergence of X_{K_{h}}^{g_{h}}(A) to X_{K}^{g}(A) (resp. of Y_{K_{h}}^{g_{h}}(A) to Y_{K}^{g}(A)) is equivalent to the stability of (K, g) for the mixed problems (6.1) (resp. (6.2) along the sequence (K_{h}, g_{h})).

The following theorem is the main result of this section.

Theorem 6.3. Let A be a bounded open subset of ℝ^{2} with Lipschitz boundary ∂A and let ∂_{D}A be a relatively open subset of ∂A with a finite number of connected components. Let (g_{h}) be a sequence in W^{1,p}(A) converging strongly to a function g in W^{1,p}(A), and let (K_{h}) be a sequence of compact subsets of \overline{A} converging to a set K in the Hausdorff metric. Assume that |K_{h}| converges to |K| and that the sets K_{h} have a uniformly bounded number of connected components. Then X_{K_{h}}^{g_{h}}(A) converges to X_{K}^{g}(A) (resp. Y_{K_{h}}^{g_{h}}(A) converges to Y_{K}^{g}(A)) in the sense of Mosco.

Proof. The main idea of this proof is due to Chambolle. Let us first prove the Mosco convergence of X_{K_{h}}^{g_{h}}(A) to X_{K}^{g}(A). Let Σ be an open ball in ℝ^{2} such that \overline{A} ⊂ Σ. Let g̃_{h}, g̃ ∈ W^{1,p}(Σ) be extensions of g_{h} and g to Σ such that g̃_{h} converges to g̃ strongly in W^{1,p}(Σ). We set

Ω_{h} := Σ \ (K_{h} ∪ (∂A \ ∂_{D}A)) and Ω := Σ \ (K ∪ (∂A \ ∂_{D}A)).

Note that Ω_{h} and Ω satisfy the assumptions of Theorem 4.2. Let us prove property (M2′′). Let (h_{k}) be a sequence of indices that tends to ∞, and u_{k} ∈ W_{g_{h_{k}}}^{1,p} (A \ K_{h_{k}}, ∂_{D}A \ K_{h_{k}}) such that u_{k}1_{K_{h_{k}}^{c}} converges weakly to ϕ in L^{p}(A) and ∇u_{k}1_{K_{h_{k}}^{c}} converges weakly to ψ in L^{p}(A, ℝ^{2}). Let ũ_{k} be the extension of u_{k} defined by

ũ_{k} := \begin{cases}u_{k}1_{K_{h_{k}}^{c}} & in A, \\ g̃_{h_{k}} & in Σ \ A, \end{cases}

and let ϕ̃ and ψ̃ be defined by

ϕ̃:= \begin{cases}ϕ & in A, \\ g̃ & in Σ \ A, \end{cases}  and  ψ̃ := \begin{cases}ψ & in A, \\ ∇g̃ & in Σ \ A. \end{cases}

As u_{k} = g_{h_{k}} on ∂_{D}A \ K_{h_{k}}, we have ũ_{k} ∈ W^{1,p}(Ω_{h_{k}}). Since ũ_{k}1_{Ω_{h_{k}}} converges to ϕ̃1_{Σ} weakly in L^{p}(ℝ^{2}) and ∇ũ_{k}1_{Ω_{h_{k}}} converges to ψ̃1_{Σ} weakly in L^{p}(ℝ^{2}, ℝ^{2}), by Theorem 4.2 we conclude that there exists ũ ∈ W^{1,p}(Ω) such that ϕ̃1_{Σ} = ũ1_{Ω} and ψ̃1_{Σ} = ∇ũ1_{Ω}. Let u be the restriction of ũ to A \ K. Then u ∈ W^{1,p}(A \ K) and we have that ϕ = u1_{K^{c}} and ψ = ∇u1_{K^{c}} a.e. in A. As ũ ∈ W^{1,p}(Ω), the traces of ũ on both sides of ∂_{D}A \ K coincide. Since ũ = g̃ a.e. in Σ \ A, we conclude that u = g̃ = g in the sense of traces on ∂_{D}A\K. Therefore u ∈ W_{g}^{1,p}(A\K, ∂_{D}A\K).

Now we prove property (M′′_{1}). Let u ∈ W_{g}^{1,p}(A \ K, ∂_{D}A \ K). Let ũ be the extension of u defined by

ũ = \begin{cases}u1_{K^{c}} & a.e. in A, \\ g̃ & a.e. in Σ \ A. \end{cases}

As u = g on ∂_{D}A \ K, we have that ũ ∈ W^{1,p}(Ω). By Theorem 4.2 there exists a sequence ũ_{h} ∈ W^{1,p}(Ω_{h}) such that ũ_{h}1_{Ω_{h}} converges to ũ1_{Ω} strongly in L^{p}(ℝ^{2}) and ∇ũ_{h}1_{Ω_{h}} converges to ∇ũ1_{Ω} strongly in L^{p}(ℝ^{2}, ℝ^{2}). We consider the function

ϕ_{h} := (ũ_{h} − g̃_{h})|_{Σ\\overline{A}}.

By construction, ϕ_{h} → 0 strongly in W^{1,p}(Σ\\overline{A}). Therefore there exists a sequence v_{h} ∈ W^{1,p}(Σ), converging to 0 strongly in W^{1,p}(Σ), such that v_{h}|_{Σ\\overline{A}} = ϕ_{h} a.e. in Σ \ \overline{A}. We set

u_{h} := (ũ_{h} − v_{h})|_{A\K_{h}}.

By construction, we have that u_{h} ∈ W^{1,p}(A\K_{h}) and u_{h} = g_{h} in the sense of traces on ∂_{D}A\K_{h}. Moreover, we have that u_{h}1_{K_{h}^{c}} converges to u1_{K^{c}} strongly in L^{p}(A) and ∇u_{h}1_{K_{h}^{c}} converges to ∇u1_{K^{c}} strongly in L^{p}(A, ℝ^{2}).

Now let us prove that Y_{K_{h}}^{g_{h}}(A) converges to Y_{K}^{g}(A) in the sense of Mosco. Property (M′′_{2}) is obtained arguing as in [12, Lemma 4.1]. So, let us prove (M′′_{1}). Let u ∈ L_{g}^{1,p}(A \ K, ∂_{D}A \ K). We set for every k ∈ ℕ

u^{k} := (g − k) ∨ u ∧ (g + k).

Then u^{k} ∈ W_{g}^{1,p}(A \ K, ∂_{D}A \ K) and ∇u^{k}1_{K^{c}} → ∇u1_{K^{c}} strongly in L^{p}(A, ℝ^{2}). From property (M′′_{1}) proved above for the Mosco convergence of X_{K_{h}}^{g_{h}}(A) to X_{K}^{g}(A), for every k there exists u_{h}^{k} ∈ W_{g_{h}}^{1,p}(A \ K_{h}, ∂_{D}A \ K_{h}) such that ∇u_{h}^{k}1_{K_{h}^{c}} → ∇u^{k}1_{K^{c}} in L^{p}(A, ℝ^{2}). Hence, by a standard argument on double sequences, we obtain a sequence of indices k_{h} converging to ∞ such that, setting u_{h} := u_{h}^{k_{h}}, we get ∇u_{h}1_{K_{h}^{c}} → ∇u1_{K^{c}} in L^{p}(A, ℝ^{2}).

Acknowledgements

This work is part of the European Research Training Network “Homogenization and Multiple Scales” under contract HPRN-2000-0010, and of the Research Project “Calculus of Variations” supported by SISSA and by the Italian Ministry of Education, University, and Research.

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(Gianni Dal Maso) S.I.S.S.A., Via Beirut 2-4, 34014, Trieste, Italy

   E-mail address: dalmaso@sissa.it

(François Ebobisse) S.I.S.S.A., Via Beirut 2-4, 34014, Trieste, Italy

   E-mail address: ebobisse@sissa.it

(Marcello Ponsiglione) S.I.S.S.A., Via Beirut 2-4, 34014, Trieste, Italy

   E-mail address: ponsigli@sissa.it