A stability result for nonlinear Neumann . . .

In this paper we prove that if 
k is a sequence of Reifenberg-flat
domains in RN that converges to for the complementary Haus-
dorff distance and if in addition the sequence 
k has a �uniform
size of holes�, then the solutions uk of a Neumann problem of the
form
(0.1)
(
-div a(x,ruk) + b(x, uk) = 0 in 
k
a(x,ruk) ·  = 0 on @
k
converge to the solution u of the same Neumann problem in 
.
The result is obtained by proving the Mosco convergence of
some Sobolev spaces, that follows from the extension property
of Reifenberg-flat domains.


Introduction
In this paper we study the stability of solutions for the following nonlinear Neumann problem where Ω is a bounded subset of R N , a : R N × R N → R N and b : R N × R → R are two Carathéodory functions satisfying suitable monotonicity, coerciveness and growth conditions (see (1.2)-(1.3)below).More precisely, we are interested in the following question.Let Ω k be a sequence of open sets in R N that converges to Ω for the Hausdorff distance.Let u k be the sequence of solutions for the problem (0.2) in Ω k and let u be the solution associated to Ω.Is it true that u k converges to u ?If the answer is positive we say that the problem (0.2) is stable along the sequence Ω k (see Definition 3).
This question was studied by many authors in the last decade, principally in dimension 2, for smooth and non smooth domains.The case of non smooth domain, and more specifically domains with cracks, appears typically in applications from fracture mechanics.In addition, the problem of stability is linked to the notions of Mosco convergence or Gamma convergence, which are powerful tools to study the semicontinuity of functionals in shape optimization problems or for instance the Mumford-Shah functional in image processing.
Recently, G. Dal Maso, F. Ebobisse and M. Ponsiglione [6] proved that in dimension 2, if Ω k tends to Ω for the complementary Hausdorff distance in R 2 , |Ω k | tends to |Ω| and if the number of the connected components of the complements Ω c k are uniformly bounded, then the problem (0.2) is stable.This stability property extends the corresponding results of A. Chambolle and F. Doveri [5] and also D. Bucur and N. Varchon [4].In particular, the authors showed that the stability is equivalent to the convergence of some Banach spaces in the sense of Mosco.
According to the authors knowledge, only very few results have been proven in higher dimensions.In [12], A. Giacomini proves a stability result in R N for fractured locally Lipschitz domains where an approach involving the Mosco convergence is also used.
A famous example called the "Neumann Sieve" ( [10], [19], [21]), shows that in many cases the stability is not true.The general idea, for instance in dimension 2, is that if one considers a sequence of 1-dimensional sets E k in B(0, 1) with more and more holes of size 1 k but with always same global length, it could happen that E k would converge to a segment while the problem (0.2) is not stable along the sequence B(0, 1)\E k .One can notice that this example is very close to the one that makes the Hausdorff measure K → H 1 (K) not lower semicontinuous with respect to the Hausdorff distance.
In this paper, we partially extend the results of [6] and [12].We prove that if Ω k converges to Ω for the complementary Hausdorff distance in R N and if {Ω k } is a sequence of Reifenberg flat domains with "uniform size of holes", then problem (0.2) is stable along the sequence {Ω k }.The main point is to prove that the limit in the sense of Mosco, of the space of restrictions of functions in W 1,p (Ω k ) is the space of restrictions of functions in W 1,p (Ω).
The notion of "uniform size of holes" (Definition 8) is here to avoid the "Neumann Sieve" problem as described above.Indeed, we want to allow crack domains but according to our definition of uniform size of holes, the "tips" of the cracks can never become arbitrary closer to each other at the limit.
Our theorem partially extends the result of [6] in any dimension, and it also can be considered as an extension of [12] since the regularity of our sets is weaker than Lipschitz.Indeed, Reifenberg flat sets are ones that are locally well-approximated by hyperplanes, at every scale.This allows for instance some "Hölderian spirals" or fractal boundaries (see [9]).
Reifenberg flat sets are naturally used in the study of boundary regularity and the regularity of free boundaries coming from a minimization problem (see for instance [7], [8], [15], [18] and references therein).As we shall see in the sequel, Reifenberg flat domains are moreover very well-adapted in the construction of good extensions for functions in the Sobolev space, which is the key ingredient for proving the Mosco convergence.This sort of technics were also used in [14], and probably appeared for the first time in [13] while Peter Jones was seeking some geometrical conditions to define a class of extension domains.
In the second part of the paper, we prove a decay estimate for the solution to nonlinear elliptic operators of divergence form at the boundary.At first glance we prove the decay when the boundary is flat.This is done by establishing a differential inequality and using some estimates on the smallest eigenvalue in the half-sphere.Then we use the stability to prove that this estimate is still true when the boundary is a Reifenberg flat set close enough to a hyperplane.This decay result gives a nice example of how the stability property can be used.It can be considered as a partial extension of [14] for nonlinear minimizers.In addition, such a decay estimate is useful for studying minimizers of fully nonlinear convex functionals.We anticipate that a similar argument is needed in our study for the regularity of minimizers of the Mumford-Shah functional with a nonlinear term of energy.This fact along with a slightly different approach will be the content of a forthcoming paper [16].
If Suppose that Ω k converges to Ω for the complementary Hausdorff distance, and let u k be a weak solution of the problem (1.5) Definition 3. We say that problem (0.2) is stable along the sequence Ω k , if the following holds: Let u k ∈ W 1,p (Ω k ) be a sequence of solutions of the problem (1.5) in Ω k .Then We seek conditions on Ω k to make the problem (0.2) stable.dist(y, P (x, r)) ≤ δ. ( Our definition of Reifenberg-flat domains is not exactly the same that could be found in the literature, essentially from the fact that we didn't take a bilateral definition of the distance in (1.6) in order to allow cracks (when the domain lies in each side of its boundary).
Using the terminology of [14], our Reifenberg-flat domains should be called "weak Reifenbergflat domains".Observe that our definition allows the fact that ∂Ω could have a infinite number of connected components.Moreover in our definition, Ω is not supposed to be connected.
The topological disc theorem of Reifenberg [20] says that, under some additional separation conditions and if δ is small enough, then the boundary of a Reifenberg-flat domain is locally the bi-hölderian image of a N − 1 dimensional unit disc.In addition, this is optimal since a Reifenberg-flat domain can admit some Hölder spiral.It is worth mentioning that a Reifenberg-flat domain could have a fractal "snowflake-like" boundary as it is shown in [9].
As a consequence, the regularity of our domains is weaker than the one considered in [12] which are essentially locally Lipschitz domains.

Now we consider a topological assumption in order to avoid the problem of "Neumann
Sieves" (see [19]).For a Reifenberg-flat domain Ω and for any ball B(x, r) centered at ∂Ω and with radius r ≤ r 0 , let us define the sets D + (x, r) and D − (x, r) by the following way.
Let P (x, r) be the hyperplane given by the definition of Reifenberg flatness of Ω. Denote by z ± (x, r) two points of B(x, r) that lie at distance 3r/4 from P (x, r) and whose orthogonal projections on P (x, r) are equal to x.Then we set D ± (x, r) := B(z ± , r/4) as in the following picture.
P (x, r) Let Ω be a (δ, r 0 )-Reifenberg flat domain.We say that Ω has a uniform size of holes with constant C 0 ≥ 1/2, if for every ball B(x, r) centered at ∂Ω with radius r ≤ 1 2 r 0 and such that D + (x, r) and D − (x, r) lie in the same connected component of B(x, r) ∩ Ω, there exists a ball B(y, s) centered on P (x, 2r) ∩ B(x, 2r) with radius s > 1 C 0 r such that B(y, s) ∩ ∂Ω = ∅.By convention if Ω is such that D ± (x, r) always lie in different connected components we say that Ω has a uniform size of holes with C 0 = 1/3.

Remark 6.
If Ω has a uniform size of holes with constant C 0 ≥ 1/2 then Ω has a uniform size of holes with any constant C > C 0 .Therefore, we will usually take C 0 as being the minimal constant satisfying the property of Definition 5 which can never be less than 1/2 for obvious geometrical reasons.Moreover when we mean that a sequence Ω k has a uniform size of holes with same constant C 0 , we say that all the minimal constants associated to Ω n are bounded by a same constant C 0 (see also the examples given below).
The rest of the section is devoted to the proof of the following result.
0 , having a uniform size of holes with same constant C 0 , and such that Ω k converges to Ω for the complementary Hausdorff distance.
Then the Neumann problem (0.2) is stable along the sequence Ω k .
Before passing to the details of the proof of Theorem 7, we give two examples for domains described in the discussion above.

Examples
We would like to emphasize the fact that according to Definition 5, in Theorem 7 we do not consider "sieve domains" with holes in the boundary that are becoming smaller when Ω k tends to Ω, but only "crack domains" which contain some holes but with a fixed size bigger than C −1 0 r 0 and controlled shape.To illustrate this, let us show two basic examples in dimension 2.

A counterexample
Let us consider the classical 2 dimensional Sieve Domain defined as follow.For a fixed ε set and Ω n := B((1/2, 0), 1)\Γ n . -- Since the boundary of Ω n is "smooth", for every δ ∈ (0, 1), one can easily obtain a radius r δ ∈ (0, 1  10 ) such that all the Ω n are (δ, r δ )-Reifenberg flat domains.Now for n big enough, we claim that the minimal constant C n 0 of uniform size of holes in Ω n is bounded from below by C2 n .Indeed, we denote x 0 := (1/2, 0) ∈ R 2 and consider the ball B(x 0 , r δ /2).
The corresponding domains D ± (x 0 , r δ /2) lie in the same connected components of Ω and P (x, r δ /2) is the first axis.Now since r δ < 1  10 , any ball B(y, s) centered on P (x 0 , r δ )∩B(x 0 , r δ ) and such that B(y, s) ∩ ∂Ω n = ∅ must have a radius s < ε2 −(n+1) .This means that thus the sequence Ω n cannot have a uniform size of hole with same constant C 0 .

An example with fractured domains
Further, we give another example where the domains Ω n have now a uniform size of holes with same constant.Of course we could also take some "non-fractured" domains for which the constant is fixed to 1 3 by convention, but let us consider the following more instructive crack situation.Let S be the segment [−1, 1] × {0} ⊂ R 2 and Ω := B(0, 2)\S.Now let It is not difficult to see that the sequence Ω n := B(0, 2)\Γ n is a sequence of (δ, r 0 )-Reifenberg-flat domains converging to Ω\S for the complementary Hausdorff distance.Indeed, they are Reifenberg-flat with a good choice of δ and r 0 depending only on the Lipschitz constant L. Moreover we have that d Finaly, Ω n and Ω have all a uniform size of holes with constant C 0 ≤ 10.This is easy to prove for Ω because the only way for a ball B(x, r) to be centered on ∂Ω and having the property that B(x, r) ∩ Ω is connected is to be centered on S with a radius r > dist(x, E(S)) where E(S) are the two endpoints of S. In this case it is clear that B(x, 2r) contains a ball of radius s > 10 −1 r centered on the first axis that does not meet ∂Ω.Now one can see this also for Ω n by exactly the same argument, replacing the endpoints of S by the endpoints of Γ n , using also that Γ n is a graph.Of course in our example, Lipschitz graphs was assumed for convenience and one could try to weaken the regularity assumption on Γ n taking for instance a sequence of connected Reifenberg-flat sets but the proof become more technical in this case.

Whitney Extension
The main ingredient for proving the Mosco convergence will be the extension Lemma contained in this section.For every function u ∈ L 1 loc (B(x, r)) we denote by m ± (u) the average of u on D ± (x, r) (1.7) We begin by the following fact.
for any function u ∈ W 1,1 (B(x, 3r) ∩ Ω), and where C is depending on dimension N and constant C 0 .
Proof.The proof is an easy consequence of the classical Poincaré inequality.Indeed, the definition of Uniform size of holes implies that there exist a ball B(x, s) centered on P (x, 2r)∩ B(x, 2r) such that B(x, s)∩∂Ω = ∅, and with s > 1 C 0 r.Now the proposition follows from the following fact : since ∂Ω is at distance less than 2δr < 2.10 −3 C −1 0 r from P (x, 2r) in B(x, 2r), one can define a Lipschitz domain A contained in B(x, 3r), that contains D + (x, r)∪D − (x, r)∪ B(x, s 2 ), and such that the Poincaré constant in A is less than C(C 0 , N )r, where C(C 0 , N ) is only depending on C 0 and N and is independent from all the possible positions of B(x, s).
Next we give the Extension lemma.
Proof.Let τ ∈ (0, r 0 ) be fixed and let B i := B(x i , τ ) be a family of balls of radius τ , centered at x i ∈ ∂Ω 1 , and maximal for the property that 1 10 B i ∩ 1 10 B j = ∅ for all i, j ∈ I with i = j.Notice that by this way, We want to construct a partition of unity associated to {B i } i∈I .For all i, define a function Indeed, such a function ϕ 0 can be obtained by setting where l is a Lipschitz function equal to 0 in [0, 8], equal to 1 in [10, +∞) and l (x) ≤ 10.

Finally, define
thus we now have a partition of unity in Ω 1 ∪ i∈I B i .To define a function ṽ Let P i := P (x i , 10τ ) be the hyperplane associated to Ω 1 given by definition 4. Recall For each ball B i , we denote by D ± (x i , 10τ ) the two balls defined just before Definition 8 associated to Ω 1 .We need to consider three cases.
• Cracktip case : D + (x i , 10τ ) and D − (x i , 10τ ) lie both in the same connected component of B(x, 10τ ) ∩ Ω 2 .Let I c be the set of indices corresponding to the balls in this situation and for every i ∈ I c define m i := m + (v) (as in (1.7)).
• Boundary case 1: D + (x i , 10τ ) and D − (x i , 10τ ) lie in different connected components of B(x, 10τ ) ∩ Ω 2 .Let I b 1 be the set of indices corresponding to the balls in this situation.For every i ∈ I b 1 , let A + i and A − i be the two connected components of 10B i \∂Ω 2 that contain respectively D + (x i , 10τ ) and D − (x i , 10τ ) and define m ± i := m ± (v).• Boundary case 2: One of D ± (x i , 10τ ) lies in Ω 2 while the other one lies in Ω c 2 .Let I b 2 be the set of indices corresponding to the balls in this situation.For every i ∈ I b 2 , let m i be equal to the one of m ± (v) corresponding to ball D ± (x i , 10τ ) that lies in Ω 2 .Finally as for I b 1 , let A + i and A − i be the two connected components of 10B i \∂Ω 2 that contain respectively D + (x i , 10τ ) and D − (x i , 10τ ).
We are now ready to define ṽ.It could be that Ω 2 has some tiny connected components hidden in some 10B i \(A + i ∪ A − i ) for i ∈ I b 1 .In those components, let us define ṽ to be equal to 0. Then, anywhere else, i.e. for all The function ṽ is now well defined for every x ∈ Ω 2 .We claim that ṽ ∈ W 1,p (Ω 2 ) and that (1.9) and (1.10) are satisfied.Let us first show (1.9).Using Hölder inequality and extending v by 0 outside of Ω 1 , we have for all i ∈ I b 2 ∪ I c , and by the same way for i ∈ I b 1 we also have In addition, since B i are in a bounded cover (with a universal constant C), the sums in (1.11) are locally finite.Thus, since θ i (x) ≤ 1 10B i (x) and using (1.12) and (1.13), and by the same way thus taking the L p norm in (1.11) we deduce that (1.9) holds.Let us now prove (1.10), which will also imply that ṽ ∈ W 1,p (Ω 2 ).By definition, ṽ = v in Ω 2 \ i∈I 10B i , thus all we have to prove is that Let x ∈ A ∩ i∈I 10B i and let I x be the finite set of indices i ∈ I such that x ∈ B i .
All balls B i for i ∈ I x are contained in B(x i 0 , 20τ ) for any i 0 ∈ I x .Let such an i 0 be fixed.
Let us define two convex domains D + x and D − x such that D ± x is the convex hull of all the D ± (x i , 10τ ) for i ∈ I x .Since (∂Ω 1 ∪ ∂Ω 2 ) ∩ B(x i 0 , 20τ ) is contained in {y; d(y, P 0 ) ≤ 2τ 10 } for some hyperplane P 0 , we have that D ± x do not meet (∂Ω 1 ∪ ∂Ω 2 ) ∩ B(x i 0 , 20τ ).Suppose firstly that if there exists an i ∈ I Since ∇θ 0 + i∈I ∇θ i = 0 we have Now by the Poincaré inequality, (1.15) If i ∈ I c , we have the same inequality with m i instead of m + i .On the other hand, since D + x is convex, and after a mollification of v if necessary, Therefore we have proved, Now suppose that if there exists an i ∈ I x ∩ (I b 1 ∪ I b 2 ), then x lies in A − i .Then we can proceed as above, except for inequality (1.15) with i ∈ I c since by convention in this case we set m i := m + (v).However, since the boundary of ∂Ω 1 has a uniform size of holes, we can estimate the difference |m + (v) − m − (v)| by (1.8).Indeed, we know that D ± Ω 1 (x i , 10τ ) lie in the same connected component of B(x i , 10τ ) ∩ Ω 2 .This means that there is a ball B(y i , s) ), lie also in the same connected components of B(x i , 20τ ) ∩ Ω 2 .Then Proposition 8, together with an other application of the Poincaré inequality to estimate the difference between the average on D ± Ω 2 (x i , 20τ ) and the average on D ± Ω 1 (x i , 10τ ), gives where C is now depending on C 0 .
In conclusion, extending ∇v by 0 out of Ω 1 we have obtained, for all (1.17) Following, we will denote i x instead of i 0 .Then we have On the other hand, |∇v| p dx thus (1.10) follows and the proof is complete.

Mosco-convergence
For every open set Ω ⊂ R N we define the closed linear subspace by X Ω be the corresponding subspaces of L p (R N ) × L p (R N , R N ) defined by (1.18).We say that X Ω k converges to X Ω in the sense of Mosco if the following two properties hold: (M 1) for every u ∈ W 1,p (Ω), there exists a sequence for every k, and The Mosco convergence is a great tool to study stability for Neumann problems.In particular, we have the following result coming from [6] (Theorem 2.3.), which is stated in R 2 in [6] but can be extended in R N with the same proof.
Theorem 11. [6] Let Ω k and Ω be open subsets of R N .Then Ω is stable for the problems (0.2) along the sequence Ω k , if and only if X Ω k converges to X Ω in the sense of Mosco.
According to Theorem 11, Theorem 7 will be a consequence of the following result.Assume that Ω k converges to Ω for the complementary Hausdorff distance.Then X Ω k converges to X Ω in the sense of Mosco.
Proof.Let u ∈ W 1,p (Ω) and assume without loss of generality that Set W (t) := {x; d(x, ∂Ω) ≤ t} and let ũk be the extension function given in Lemma 9 for which tends to zero when k tends to +∞ because ∂Ω is closed and |∂Ω| = 0.For the gradients, a similar argument can be done, using (1.10).That is, which tends to 0, thus (M 1) is proved.
Let us now prove (M 2).Let ϕ ∈ C ∞ (R N ) be compactly supported in Ω.Then we know by the weak convergence that On the other hand, since Ω h k converges to Ω for the complementary Hausdorff distance, for k large enough 1 Ω h k is equal to 1 everywhere on the support of φ.Thus (1.19) shows that u k converges to φ in D (Ω) and ∇u k converges to ψ in D (Ω).By uniqueness of the limit in in Ω and therefore φ| Ω ∈ W 1,p (Ω).To conclude, all we have to show is that ϕ = ψ = 0 in Ω c .
To see this, we use a similar argument as above by defining a function ϕ compactly supported in Ω c .By the weak convergence, and because Ω k converges to Ω for the complementary Hausdorff distance, we deduce that Ω φϕ dx = 0.This holds for any function ϕ compactly supported in Ω c .Since φ ∈ L p (R N ) we conclude that φ = 0 in Ω c .In a similar way we obtain that ψ = 0 in Ω c and since the Lebesgue measure of ∂Ω is zero, the proof is complete.
2 Stability for problems with mixed boundary conditions Following the proof of Theorem 6.3 in [6] and using the Mosco convergence (Theorem 12), one can show a similar stability result for the problem with mixed boundary conditions.
The argument is based on taking some judicious extensions as in [6].Let us recall here the definitions and statements since it will be used in the sequel.
Definition 13.We say that the problem (2.1) is stable along the sequence (K k , g k ), if the following holds: Let u k ∈ W 1,p (A\K) be a sequence of solutions of the problem (2.1) corre- For all g ∈ W 1,p (A) we denote As in (1.18) we define the closed linear subspace X g K (A) of L p (A) × L p (A, R N ) by Definition 14 (Mosco-convergence).We say that X g k K k (A) converges to X g K (A) in the sense of Mosco if the following two properties hold: (M 1 ) for every u ∈ W 1,p g (A\K, ∂ D A\K), there exists a sequence g k (A\K k ) for every k, and u k 1 K c k converges weakly in L p (A) to a function φ, while ∇u k 1 K c k converges weakly in L p (A, R N ) to a function ψ, then there exists u ∈ W 1,p g (A\K, ∂ D A\K) such that φ = u1 K c and ψ = ∇u1 K c a.e. in R N .
Remark 15.As it is pointed out in [6], by adopting the same proof as in Theorem 11, the Mosco convergence of X g k K k (A) to X g K (A) is equivalent to the stability of the mixed problem (2.1) along the sequence (K k , g k ).
Finally we obtain the corresponding result in the case of the mixed boundary conditions.be a relatively open subset of ∂A and let C 0 be a positive constant.Let g h be a sequence in W 1,p (A) converging strongly to a function g ∈ W 1,p (A), and let K h be a sequence of compact subsets of Ā converging to a set K in the Hausdorff metric.Assume in addition that for every h, A\K h is a (δ, r 0 )−Reifenberg flat domain with δ < 10 −3 C −1 0 and having a uniform size of holes with constant C 0 .Then X g h K h (A) converges to X g K (A) in the sense of Mosco.
Proof.The proof is essentially the same as the proof of Theorem 6.3 in [6], consisting of an idea due to Chambolle.The only difference is that the sets where Σ is an open ball in R N such that Ā ⊂ Σ, are now satisfying the assumptions of Theorem 12 and thus the Mosco convergence in that setting.We refer the reader to [6], Theorem 6.3 for the details in that approach.

A decay estimate and its stability
In this section we prove a boundary estimate on the gradient of solutions for a non-linear Neumann problem similar to (0.2).In the first part we will prove the estimate when the boundary is flat, and then using the stability of problem (2.1) we will extend the result for Reifenberg flat domains, whose boundaries are close enough to a hyperplane for the Hausdorff distance.
Let Ω be an open subset of R N and let I be a relatively open and Lipschitz subset of ∂Ω.
For every g ∈ W 1,2 (Ω) we denote by W 1,2 g (Ω) the functions of W 1,2 (Ω) that are equal to g on I (in terms of trace).We focus on minimizers u ∈ W 1,2 g (Ω) for the following functional where for some positive constants c, C. We define f (p) := F (|p| 2 ), which is a convex function due to (3.2).We also assume hereby that f is uniformly elliptic that is, there exists positive constants λ, Λ such that for all ξ ∈ R N .If we combine the ellipticity condition with (3.2) we get that f satisfies Let Ω be an open subset of R N and u ∈ W 1,2 g (Ω) be a minimizer for the functional F. Then u is a weak solution for the problem Proof.This is a standard variational argument.Let u be such a minimizer and let ϕ ∈ C ∞ ( Ω), such that ϕ = 0 on I.Then, comparing u with u + tϕ, we obtain that Then, using that dividing by t and letting t goes to 0 we obtain that By a similar argument dividing by −t we obtain the reverse inequality and the lemma follows.Let us recall this result from [11] that can be also found in [1] p. 347.
for all balls B(x 0 , r) ⊂ Ω where C 1 depends only on N , λ, and Λ.
Following, we want to prove the same sort of estimate but for a local minimizer in Ω \ K where K is a compact set such that Ω\K is a Reifenberg flat domain and for balls centered on ∂K ∩ ∂Ω.We begin with the case when K is a half-space.
Now by solving the corresponding eigenvalue problem as in [2], we find that for w ∈ H g (B + 1 ) be a minimizer of the functional F. Then u is locally Lipschitz in B + 1 and for all r < 1 and for all 0 < a < 1 for a positive exponent β, where C 1 depends only on N , the Lipschitz constant of u, λ and Λ.
Proof.Let r 0 and C 0 be fixed, β is an exponent strictly greater than the one in (3.13), and

W 1 ,
p (Ω) := the Sobolev space.C 1 c (Ω) := space of C 1 (Ω) functions with compact support.ν := the outward normal vector.|A| := the Lebesgue measure of the Borel set A. d H := the Hausdorff distance (defined in (1.4)).A c := the complement of the set A. A B := A\B ∪ B\A.B(x, r):= the ball with center at x and radius r.C := a positive constant, that could vary from line to line, and that only could depend on dimension.1 Stability for the nonlinear Neumann problem Let Ω be an open bounded subset of R N .In this section we prove the stability for problem (0.2) where a : R N × R N → R N and b : R N × R → R are two Carathéodory functions satisfying the following assumptions : there exist 0

Definition 4 .
A (δ, r 0 )-Reifenberg-flat domain Ω ⊂ R N is an open bounded set such that for each x ∈ ∂Ω and for any r ≤ r 0 there exist a hyperplane P (x, r) containing x such that 1 r sup y∈∂Ω∩B(x,r)

Proposition 8 .
Let C 0 ≥ 1/2 and let Ω be a (δ, r 0 )−Reifenberg flat domain with δ < 10 −3 C −1 0 and having uniform size of holes with constant C 0 .Then for every ball B(x, r) centered at ∂Ω with r ≤ 1 4 r 0 such that D + (x, r) and D − (x, r) lie in the same connected component of B(x, r) ∩ Ω, we have that |m

. 18 )
Definition 10 (Mosco-convergence).Let Ω k and Ω be open subsets of R N and let X Ω k and
Let A ⊂ R N be a bounded open set with Lipschitz boundary ∂A, and let ∂ D A be a relatively open subset of ∂A.For every compact set K ⊂ Ā, for every g ∈ W 1,p (A), and for every pair of function a and b satisfying the properties (1.2)-(1.3)we consider the solutions u of the mixed problem

Theorem 16 .
Let A be a bounded open subset of R N with Lipschitz boundary ∂A, let ∂ D A

1
is the constant in(3.13).Suppose Theorem 22 is false.Then for every ε k > 0 there exists a compact set K k of B(0, 1) such that B(0, 1)\K k is a Reifenberg flat Domain with maximum radius r 0 , with uniform size of holes with constant C 0 andd H (K k , B− ) ≤ ε k .In addition there exists a sequence of solutions u k ∈ W 1,2 g (A\K k ) of the problem (3.4) withI := ∂B + 1 \K k and Ω := B(0, 1)\K k and a sequence of radii a k ∈ (a 0 , 1) such that Ba\K k |∇u k | 2 dx > C 1 a N −1+β k B\K k |∇u k | 2 dx.(3.15)By extracting a subsequence (not relabeled), we may assume that a k converges to a certain a ≤ 1.Now we claim that lim infk→+∞ B\K k |∇u k | 2 dx ≥ Eλ −1 > 0Indeed, let us consider the compact set K 0 := B(0, 1)∩{x N ≤ 3ε 0 } and let v ∈ W 1,2 g (B(0, 1)\K 0 ) be the solution of the problem(3.4).Since for every k, u k is a competitor for v we have thatE := B(0,1)\K 0 F (|∇v|) 2 ≤ B(0,1)\K 0 F (|∇u k | 2 ) ≤ B(0,1)\K k F (|∇u k | 2 ) ≤ Λ B(0,1)\K k |∇u k | 2which proves the claim.Then, by the stability of Problem (2.1) we know that ũk 1 K c converges strongly in L 2 (A) to a function u, and ∇ũ k 1 K c converges strongly in L 2 (A) to ∇u1 B + .Moreover, u is a solution of the problem (3.4) in B + with I = ∂B + and boundary values on I equal to g.Thus by passing to the limit in(3.15)we obtain a contradiction according to Lemma 21.
A and B are two nonempty closed subsets of R N , we define the Hausdorff distance Let {Ω k } k∈N and Ω be some nonempty open subsets of R N .We say that Ω k