New York Journal of Mathematics

New York J. Math. 8 (2002) 189–213.

Some Geometric Properties for a Class of Non-Lipschitz Domains

Mohammed Barkatou

Abstract. In this paper, we introduce a class C, of domains of ℝ^{N}, N ≥ 2, which satisfy a geometric property of the inward normal (such domains are not Lipschitz, in general). We begin by giving various results concerning this property, and we show the stability of the solution of the Dirichlet problem when the domain varies in C.

Contents

1. Introduction 189 2. The geometric normal property 190 3. The geometric normal property and the convergence of domains 202 4. Stability of the solution of Dirichlet problem 207 References 212

1. Introduction

Most of the work concerning the stability of the solution of a boundary value problem with the Laplacian operator was done by V. Keldyš in the 40’s and is presented in his original paper [12], see also [9], [14], [10]. In [6], the author considered the class of all open domains satisfying the restricted cone property with a given height and angle of the cone, say ε (each open domain is said to satisfy the ε-cone property) which are stable in the sense of Keldyš (see Theorem 4.1). For other kind of constraints, see for example [4], [5], [16]. In this paper, we introduce a very simple constraint involving a geometric property of the inward normal vector. The domains we consider must satisfy the following condition: for almost every point of the boundary, the inward normal (if it exists) intersects a fixed compact convex set C. We shall call this property C-GNP. Such property is satisfied by the solution of the quadrature surface free boundary problem, see for example [1], [2], [8] and [13].

189^{ISSN 1076-9803/02}

Received April 1, 2002.

Mathematics Subject Classification. 35J05, 51A05 and 52A20.

Key words and phrases. convergence of domains, normal cone, Sobolev capacity, stability of the Dirichlet problem, Steiner symmetrization, Wiener criterion.

The aim of this paper is, first, to show some important results concerning the domains which satisfy C-GNP. We will prove that the boundary of a domain Ω which satisfies C-GNP has a uniform cone property outside C. Moreover, even though cusps can be formed at the points of ∂Ω ∩ ∂C (one can consider, in two dimensions, the convex C = [−1, 1] × {0} and the domain Ω = B(−1, 1) ∪ B(1, 1)), it is shown that these cusps are not sharper than (i.e., contain) a canonical cusp (which is obtained by revolving the cusp between two touching circles of large radius around its axis). In particular, this implies that every point of ∂Ω is regular for the Dirichlet problem, as one can easily verify the Wiener criterion. We also obtain a characterization of C-GNP where we don’t need the normal and give an example of a domain which satisfies C-GNP while its Steiner symmetrization doesn’t have C-GNP.

Next, in Section 3, we will prove that if Ω_{n} is a sequence of open subsets included in a fixed ball D and satisfying C-GNP, then there exists an open subset Ω ⊂ D and a subsequence (still denoted by Ω_{n}) such that:

1. Ω_{n} converges to Ω in the Hausdorff sense,

2. Ω_{n} converges to Ω in the compact sense: every compact subset of Ω is included in Ω_{n} for n large enough and every compact subset of \overline{Ω}^{c} is included in \overline{Ω}_{n}^{c} for n large enough (\overline{Ω}^{c} is the exterior of Ω),

3. Ω_{n} converges to Ω in the sense of characteristic functions, and

4. Ω satisfies C-GNP.

In the last section, we study the behaviour of the solution of the Dirichlet problem on Ω_{n} when Ω_{n} converges to Ω. We introduce u_{n} the solution of the Dirichlet problem

\{ \begin{matrix}−Δu_{n} = f in Ω_{n} \\ u_{n} = 0 on ∂Ω_{n} \end{matrix}

and we prove that, if Ω_{n} converges to Ω in the Hausdorff sense, the sequence u_{n} (extended by 0 outside Ω_{n}) converges strongly in H_{0}^{1} (D) to u solution of the Dirichlet problem

\{ \begin{matrix}−Δu = f in Ω \\ u = 0 on ∂Ω, \end{matrix}

(f ∈ H^{−1}(D)).

The proof relies on the notion of stability introduced by Keldyš and the convergence in the compact sense of Ω_{n} to Ω. It requires a particular study of the cusp points of ∂Ω, and a precise computation of the capacity of the exterior of Ω, in a neighbourhood of such a point in order to obtain the stability of the set Ω.

2. The geometric normal property

In this section, we introduce for an open subset Ω of ℝ^{N}, the geometrical property of the normal with respect to convex C (noted C-GNP), and we study its various properties. It will be shown first of all that the points of the boundary of Ω which are outside of C have a property of the cone. Then for the points of ∂Ω which are on C, we prove a geometrical result (see Proposition 2.2) expressing that the outside of Ω is sufficiently consistent in term of capacity, in the neighbourhood of those points. Finally, we give a characterization of C-GNP which does not utilize the normal.

Let D be an open ball of ℝ^{N} and C be a convex compact of D. All the open subsets with which we shall work will be included in D. For any point x of the boundary of an open subset Ω, we denote the inward normal vector to ∂Ω (if it exists) by ν(x) and set

N_{Ω} = {x ∈ ∂Ω : ν(x) exists}.

Finally, we shall denote by D(x, ν(x)), the half-line with origin x and director vector ν(x).

Definition 2.1 ([3]). Let C be a convex set in ℝ^{N} and c a point of its boundary. By the normal cone to C at c we mean the set:

CN_{c} = {y ∈ ℝ^{N} : (y − c) · (z − c) ≤ 0 ∀z ∈ C}.

CN_{c} can also be seen as the set of the points of ℝ^{N}, for which c is the projection on C.

• We call a half-normal to C at c, any half-line with origin c and contained in the normal cone CN_{c}.

• We call a normal to C at c, a line containing one half-normal.

Remark 2.1. If c is a regular point of ∂C, then CN_{c} is exactly the usual normal to C at c.

Rule. Throughout this paper it is supposed that for any point c of ∂C, a particular normal Δ_{c} was fixed. One calls it the selected normal to C at c.

Definition 2.2. We say that an open subset Ω has a geometric normal property with respect to C (or more simply Ω satisfies the C-GNP) if:

(P1) Ω contains the interior of C. (P2) ∂Ω is Lipschitz outside of C. (P3) ∀ x ∈ N_{Ω}⧹C, D(x, ν(x)) ∩ C ≠ ∅. (P4) For all selected normals, Δ, to C, Δ ∩ Ω is connected.

C is the class of all domains which satisfy C-GNP.

We begin by showing the following proposition which will be useful thereafter.

Proposition 2.1. Let Ω be an open subset which has C-GNP. Let c be some point of the boundary of C and Δ_{c} be a half-normal to C at c. Then:

1. Δ_{c} ∩ Ω cannot have a connected component ω such that, c ∉ \overline{ω}.

2. Δ_{c} ∩ \overline{Ω}^{c} cannot have a bounded connected component.

Proof. We show the first point. The proof of the second is the same while working with the extremity of the connected component of Δ_{c} ∩ \overline{Ω}^{c} further from C.

Suppose by contradiction that Δ_{c} ∩ Ω has a connected component ]x_{0}, y_{0}[ with c, x_{0} and y_{0} arranged in this order on Δ_{c} and x_{0} ≠ c. Let e be the unit vector of the half-line Δ_{c} and put the origin O in x_{0}. By hypothesis, O is not on C. Thus:

• On one hand, one can separate C and {O} by an hyperplane H orthogonal to Δ_{c}.

• On the other hand, as O is on ∂Ω, there exists a neighbourhood V = V′×] − α, α[ of O, an orthonormal cartesian coordinate system R := (O, e_{1}, …, e_{N}) and a Lipschitz function ϕ in V′ such that:

Ω ∩ V = {(y′, y_{N}) ∈ V | y_{N} < ϕ(y′)},
(\overline{Ω})^{c} ∩ V = \{(y′, y_{N}) ∈ V | y_{N} ≥ ϕ(y′)\}.

In the cartesian coordinate system R, we have

H = {x ∈ ℝ^{N} | x · e = δ_{0}}

with δ_{0} < 0, (x · e being the scalar product of x and e, in ℝ^{N}). According to what precedes,

C ⊂ {x ∈ ℝ^{N} | x · e < δ_{0}}.

Now, to continue the proof, we shall need the

Lemma 2.1. Let ψ be a Lipschitz function in a neighbourhood V′ of 0 (in ℝ^{N−1}). If ψ (t, 0, .., 0) > 0 for t > 0 and ψ (0, .., 0) = 0, then, there exists x′ ∈ V′ such that ∇ψ(x′) exists and \frac{∂ψ}{∂x_{1}} (x′) ≥ 0.

Proof. Since the 1-dimensional function f : s ↦ ψ(s, 0, …, 0) is (as ψ) Lipschitz in a neighbourhood of 0, it is differentiable for almost all s and

f(t) = f(0) + ∫_{0}^{t} f′(s) ds.

Using the function ψ, we can write

ψ(t, 0, …, 0) = ∫_{0}^{t} \frac{∂ψ}{∂x_{1}} (s, 0, …, 0) ds.

So, if ψ(t, 0, …, 0) > 0 then there exists at least one s ∈ ]0, t[ such that

\frac{∂ψ}{∂x_{1}} (s, 0, …, 0) > 0.

Now, if ∇ψ(s, 0) exists, the demonstration is achieved. If not, set

G = {x ∈ V′ : ∇ψ (x) exists} .

(As ψ is Lipschitz, G is not empty and not negligible). We show that there exists v ∈ G such that \frac{∂ψ}{∂x_{1}} (v) ≥ 0.

Suppose by contradiction that no such point exists, i.e., ∀ x ∈ G, \frac{∂ψ}{∂x_{1}} (x) < 0. This implies that, for h > 0 small enough and y in a neighbourhood of 0 (in ℝ^{N−2}),

ψ (s + h, y) − ψ(s, y) = ∫_{s}^{s+h} \frac{∂ψ}{∂x_{1}} (t, y) dt < 0.

Tending y to 0, ψ (s + h, 0) − ψ(s, 0) ≤ 0. Then, dividing by h tending h to 0, we obtain \frac{∂ψ}{∂x_{1}} (s, 0) ≤ 0, which is absurd. □

End of the proof of Proposition 2.1.

Case 1: e = −e_{N} (by construction, e = e_{N} is not possible). In this case, the convex C is included in the half-space {x_{N} > δ_{0}}. As the inward normal to ∂Ω at a point x, with director vector (\frac{∂ϕ}{∂x_{1}} (x′), …, \frac{∂ϕ}{∂x_{N−1}} (x′), −1) is in the opposite direction to C, it cannot meet C and one has the desired contradiction.

Case 2: The vector e is not parallel to e_{N}. Note e′ the orthogonal projection of e on the hyperplane orthogonal to e_{N}. One can without restriction of generality, choose a cartesian coordinate system in which e_{1} = \frac{1}{α_{1}} e′, α_{1} > 0. Set e = e′ + α_{N}e_{N} (e = α_{1}e_{1} + α_{N}e_{N}). The hypothesis ]O, y_{0}[⊂ Ω, becomes

tα_{N} < ϕ(tα_{1}, 0, …, 0) for t > 0, t small.

Applying Lemma 2.1 to the function

ψ(x_{1}, …, x_{N−1}) = ϕ(α_{1}x_{1}, x_{2}, …, x_{N−1}) − α_{N}x_{1},

there exists at least one point x = (x′, x_{N}) (x′ ∈ B(O, −δ_{0}/2)) where the inward normal directed by the vector (\frac{∂ϕ}{∂x_{1}} (x′), …, \frac{∂ϕ}{∂x_{N−1}} (x′), −1) exists and \frac{∂ψ}{∂x_{1}} (x′) ≥ 0. Hence

α_{1} \frac{∂ϕ}{∂x_{1}} (x′) − α_{N} ≥ 0. (1)

To conclude, show that the inward normal to ∂Ω at x cannot intersect C. Let y = x + tν(x) (t > 0) be a point of this inward normal. Since |x · e| ≤ ||x|| < −δ_{0}/2 and ν(x)·e has the sign of α_{1} \frac{∂ϕ}{∂x_{1}} (x′)−α_{N} (which is positive according to (1)), then y · e > δ_{0}/2; which proves the result since the convex C is included in the half-space {x · e < δ_{0}}. □

As a corollary of Proposition 2.1, we have:

Corollary 2.1. Let Ω be an open subset which has C-GNP. Let c be a point of ∂C and Δ_{c} be a half-normal to C at c, with director vector e. Let x be some point of Δ_{c} ∩ Ω. Then:

• The interval ]c, x[ does not meet the exterior of Ω.

• The half-line with origin x directed by e does not meet Ω.

Lemma 2.2. Let Ω be an open subset which has C-GNP and S be a similarity transformation (of ratio k > 0) then S(Ω) has the S(C)-GNP.

The proof of this lemma is trivial and therefore is omitted.

As we have said in the begining of this paper, if Ω satisfies C-GNP then ∂Ω ∩ ∂C can have cusps. We shall describe the behaviour of such points, this with the intention to prove that the eventual cusps of ∂Ω are regular in the sense of Wiener [17] (see Section 4).

Proposition 2.2. Let C be a convex set with a nonempty interior (int(C) ≠ ∅) and x_{0} ∈ ∂Ω ∩ ∂C. Let CN_{0} be the normal cone to C at x_{0}.

1. If Ω satisfies C-GNP, then CN_{0} ∩ Ω = ∅.

2. If CN_{0} is reduced to the half-line Δ_{0}, let H be the hyperplane orthogonal to Δ_{0} in x_{0} and H^{+} the open half-space limited by H and not containing C. Let R be a real number which is strictly superior to the diameter of C and ε_{0} be a small strictly positive number. Put B_{x_{0}} = B′(x_{0}, ε_{0}) × ℝ (B′(x_{0}, ε_{0}) being the (N − 1)-dimensional ball with center x_{0} and radius ε_{0}). Then

Ω ∩ B_{x_{0}} ∩ H^{+} ⊂ ⋃_{z∈H, |z−x_{0}|=R}B(z, R) = B_{x_{0}}.

Remark 2.2. If int(C) = ∅, the previous result remains true if we replace CN_{0} by CN_{0} ∩ E, where E is one of the half-spaces limited by an hyperplane containing C.

Remark 2.3. The interest of Proposition 2.2 is to describe the boundary ∂Ω of Ω, in the neighbourhood of x_{0}: the union of spheres centered in z and of radius R, form in x_{0} a (hyper)surface of revolution with a perfectly characterized cusp point. This proposition says that the eventual singularity of ∂Ω in the point x_{0} can be a cusp point which is included in the one of the revolution surface, described above. This geometric characterization will allow us to estimate the capacity of the exterior of Ω in the neighbourhood of the point x_{0} (see Section 4).

Proof of Proposition 2.2. We first prove that the normal cone CN_{0} does not intersect Ω. If CN_{0} is reduced to a half-line, this is exactly the selected normal to C and the result is then an immediate consequence of the condition (P4). Now, let x_{0} be a vertex of the convex C such that x_{0} ∈ ∂Ω and let Δ_{0} be the selected normal to C at x_{0}. Suppose that the open subset Ω meets the normal cone CN_{0} and let ω be a connected component of Ω∩CN_{0}. By hypothesis, ω does not meet Δ_{0}. Now among all the planes containing Δ_{0} which meet ω, one can find at least a plane P such that the 1-dimensional Lebesgue measure of the complement of N_{Ω} ∩P is null, i.e., almost all the points of ∂Ω∩P have a normal. For x ∈ ∂(ω∩P), ν_{x} (resp. n_{x}) is the inward normal vector to ∂ω (resp. to ∂(ω ∩ P)) at x. In the plane P, we have the following situation: ω ∩ P is a relative open subset such that ∂ (ω ∩ P) = γ_{1} ∪ γ_{2}, where γ_{1} ⊂ ∂Ω and γ_{2} is included in some half-normal Δ_{1}which limits CN_{0} ∩ P.

Let H be the supporting hyperplane orthogonal to Δ_{1} and passing by x_{0}. Let Δ be the line H ∩ P. Let R (x_{0}, e_{1}, …, e_{N}) be the cartesian coordinate system of ℝ^{N}, where x_{0} is the origin, e_{1} is parallel to Δ and e_{N} is parallel to Δ_{1}. By hypothesis, the convex C is in the half-space {x_{N} ≤ 0}. Since

0 = ∫_{∂} n · e_{N} = ∫_{γ_{1}} n · e_{N} + ∫_{γ_{2}} n · e_{N} = ∫_{γ_{1}} n · e_{N} (2)

(n is the inward normal vector to ∂(ω ∩ P)), then there exists a subset (of strictly positive measure) of γ_{1} on which n_{x} · e_{N} ≥ 0. Then, it is easy to see that using, for ∂Ω, the cartesian representation ψ(x_{1}, x_{2}, …, x_{N}) = 0, one can have ν_{x}·e_{N} ≥ 0, i.e., the inward normal vector at such point x, cannot intersect C (which is “below” the hyperplane H), contradicting thus, the property (P3).

Suppose now, that the cone CN_{0} is reduced to a half-line Δ_{0}. To simplify, we can put the origin at x_{0}. Let B_{0} = B′(O, ε_{0})×ℝ and suppose that (Ω∩B_{0} ∩H^{+})\B_{x_{0}} is not empty. Let ω be a connected component of this set. By hypothesis, ω does not intersect Δ_{0}. Let P be a plane containing the normal Δ_{0} such that the complement of N_{Ω}∩P has a null measure. Once again, for x ∈ ∂(ω∩P), let ν(x) (resp. n(x)) be the inward normal vector to ∂ω (resp. to ∂(ω ∩ P)) at x.

In the plane P, we have the following situation: ω ∩ P is a relative open subset such that ∂ (ω ∩ P) = γ_{1} ∪ γ_{2}, where γ_{1} ⊂ ∂Ω and γ_{2} ⊂ ∂B_{x_{0}}. Note Δ the line H ∩P. Introduce, as above, a cartesian coordinate system of ℝ^{N}, R (x_{0}, e_{1}, …, e_{N}), where x_{0} is the origin, e_{1}is parallel to Δ (and of same direction that ω ∩ P) and e_{N} is parallel to Δ_{0}.

By hypothesis, Ω satisfies the condition (P3) w.r.t. C. Thus there exists ε > 0 (small enough) such that Ω∩H^{+}∩B_{0} satisfies (P3) w.r.t. the (N − 1)-dimensional ball B(O, R − ε). This can be expressed by

∀X ∈ N_{Ω} ∩ H^{+} ∩ B_{0}, ν_{N}(X) < 0 and ∑_{i=1}^{N−1} (x_{i} − x_{N} \frac{ν_{i}(X)}{ν_{N}(X)})^{2} ≤ (R − ε)^{2}.

Therefore

(x_{1} − x_{N} \frac{ν_{1}(X)}{ν_{N}(X)})^{2} ≤ (R − ε)^{2}. (3)

Let n(X) = (n_{1}(X), n_{N}(X)), be the inward normal vector (if it exists) to ∂(ω ∩ P) at X ∈ γ_{1}. Using the cartesian representation ψ(x_{1}, …, x_{N}) = 0, one can have \frac{ν_{1}(X)}{ν_{N} (X)} = \frac{n_{1}(X)}{n_{N} (X)}. This together with the inequality (2) implies that

∀X = (x_{1}, x_{N}) ∈ γ_{1} ∩ N_{Ω}, n_{N}(X) < 0 and − R + ε ≤ x_{1} − x_{N} \frac{n_{1}(X)}{n_{N}(X)} ≤ R − ε, (4)

or again, if we introduce the tangential vector t(X) = (n_{N}(X), −n_{1}(X)) to ∂(ω∩P):

(R − ε)n_{N}(X) ≤ X · t(X) ≤ (−R + ε)n_{N}(X). (5)

Since n_{N} < 0, then X · t(X) > Rn_{N} and,

∫_{γ_{1}} X · t(X) ds > ∫_{γ_{1}} Rn_{N}(X) ds.

Now, as the points of γ_{2} belong to some circle of center z, we have

X · t(X) = Rn_{N}(X).

This implies that

0 = ∫_{∂ω} X · t(X) ds = ∫_{γ_{1}} X · t ds + ∫_{γ_{2}} X · t ds > ∫_{∂ω} Rn_{N}(X) ds = 0,

which is absurd. □

Remark 2.4. An immediate consequence of Proposition 2.2 above, is that an open subset which has the C-GNP is of Caratheodory type. Recall that an open subset Ω is of Caratheodory type if, for example, any point of its boundary is limit of a sequence of points of its exterior \overline{Ω}^{c}.

In fact as ∂Ω\C is Lipschitz then, all its points can be approached by a sequence of points of the exterior of Ω.

Now, if x_{0} ∈ ∂Ω ∩ C, according to Proposition 2.2, x_{0} is also in \overline{B_{x_{0}}} which is of Caratheodory type and its exterior is contained in the exterior of Ω. Consequently, there exists a sequence of points of the exterior of Ω which converges to x_{0}.

Remark 2.5. In two dimensions, an open subset which has C-GNP satisfies a property of the exterior segment.

Now, let us give a characterization of C-GNP where we don’t need the normal.

Definition 2.3. Let C be a convex set. We say that an open subset Ω has the C-SP, if the conditions (P1), (P2), (P4) of Definition 2.2 are satisfied and

(S) ∀ x ∈ ∂Ω⧹C K_{x} ∩ Ω = ∅,

where K_{x} is the closed cone defined by {y ∈ ℝ^{N} : (y − x).(z − x) ≤ 0, ∀ z ∈ C} .

Remark 2.6. K_{x} is the normal cone to the convex hull of C and {x}.

Proposition 2.3. An open subset Ω has C-GNP if and only if it satisfies the C-SP.

To prove this proposition, we need to show the following lemma:

Lemma 2.3. Let c be a point of ∂C and CN_{c} be the normal cone to C at c. Then for all y ∈ CN_{c} and y ≠ c, the half-line Δ(y, \overrightarrow{cy}), with origin y and director vector \overrightarrow{cy} is contained in K_{y} (the normal cone of the convex hull of \{y\} and C).

Proof. Let z ∈ Δ(y, \overrightarrow{cy}), then there exists λ ∈ ℝ^{+} such that z − y = λ(y − c). Show that z ∈ K_{y}, that is to say:

∀ ψ ∈ C, (z − y) · (ψ − y) ≤ 0,

or again,

∀ ψ ∈ C, (y − c) · (ψ − y) ≤ 0.

Since c is the projection of y on C, one has

(y − c) · (ψ − c) ≤ 0.

Thus,

(y − c) · (ψ − y) = (y − c) · (ψ − c) − ∥y − c∥^{2} ≤ 0.
□

Proof of Proposition 2.3. We first show that C-SP implies C-GNP: Let x ∈ N_{Ω} \ C. Denote by H_{x} the tangential hyperplane to ∂Ω at x and by C_{x} the orthogonal projection of C on H_{x}. The C-GNP will be proved if we show that x belongs to C_{x}. In fact, if not, there would exist an hyperplane of H_{x} (therefore an affine space in ℝ^{N−2}) separating strictly x of C_{x}, therefore, there would exist a tangential vector τ⃗ such that one has,

∀ y ∈ C, τ⃗ · (y − x) > 0.

Now by C-SP, one can have, for all z ∈ Ω and all y ∈ C, the inequality (z − x) · (y − x) > 0. Choosing a sequence of points z_{n} ∈ Ω such that \frac{1}{∥z_{n}−x∥}(z_{n} − x) converges to −τ⃗, one obtains −τ⃗ · (y − x) ≥ 0, and thus the contradiction.

We now show that C-GNP implies C-SP: If x ∈ ∂Ω \ C, let K_{x} be the cone defined above. Let c be the projection of x on C. By Lemma 2.3, the halfline Δ(x, \overrightarrow{cx}) is in K_{x}. Therefore, according to the Corollary 2.1, Δ(x, \overrightarrow{cx}) does not intersect Ω. Then, we find ouerselves in the same conditions as in the demonstration of Proposition 2.2 by replacing CN_{0} by K_{x} and Δ_{0} by Δ(x, \overrightarrow{cx}). One can conclude that Ω ∩ K_{x} = ∅ (that returns to apply Proposition 2.2 to Ω and the convex hull of C and {x}). □

Remark 2.7. The definition of K_{x} implies that: if z ∈ int(K_{x}) and if H denotes the hyperplane passing by x and orthogonal to \overrightarrow{xz}, then the convex C is included in the open half-space limited by H and not containing z. Now, if z is a point (different to x) such that if H is the hyperplane defined above and C is included in the open half-space limited by H and not containing z, then z ∈ int(K_{x}).

This last remark allows us to state the following lemma which will be useful for later.

Lemma 2.4. Let x be a point which does not belong to C. Let H be an hyperplane which separate strictly {x} and C. If D_{x} is the half-line with origin in x, orthogonal to H and not intersecting this one, then D_{x} is in the interior of K_{x}.

Proof. In fact, if y is a point of the half-line D_{x}, applying Remark 2.7 to the hyperplane H_{x} passing by x and parallel to H, one deduces that y belongs to int(K_{x}). □

Lemma 2.4 allows us to show the following result which precises Proposition 2.1.

Corollary 2.2. Let Ω be an open subset which satisfies the condition (S). Let c be a point of ∂C and CN_{c} the normal cone to C at c. Then any half-line with origin in c which is contained in CN_{c}, intersects the boundary ∂Ω at most in one point.

Proof. Let Δ be a half-line with origin in c included in CN_{c}. Suppose there exists two points x and y in Δ ∩ ∂Ω (for example y ∈ [c, x]). According to Lemma 2.4, one can deduce that x ∈ int(K_{y}) and therefore K_{y} ∩ Ω is not empty, contradicting the property (S). □

Corollary 2.3. Let Ω be an open subset which contains the convex C. If Ω satisfies (P2) and (S), then it satisfies (P4).

Proof. Let Δ be a normal to C at some point c. Suppose that Δ ∩ Ω be not connected. As this one is a relative open subset of Δ, it has at least two connected components ]a_{1}, a_{2}[ and ]b_{1}, b_{2}[. Now, the points a_{1}, a_{2}, b_{1} and b_{2} are necessarily on ∂Ω. By hypothesis, at least one of the two intervals does not meet C (for example ]a_{1}, a_{2}[). Suppose that a_{1} be nearer to C than a_{2}, then the Corollary 2.2 applied to c and Δ gives the contradiction. □

Now, we shall show the existence of an intrinsic cartesian coordinate system such that the boundary of any open subset which satisfies C-GNP can be locally represented by some Lipschitz function.

Proposition 2.4. Let Ω be an open subset which satisfies C-GNP. Let x ∈ ∂Ω⧹C and x_{C} its projection on C. Then, ∂Ω admits in a neighbourhood of x, a Lipschitz representation in an orthonormal cartesian coordinate system.

Proof. Put R_{x} = (x, e′, e_{N}) with e_{N} = \frac{\overrightarrow{x_{C}x}}{∥\overrightarrow{x_{C}x}∥}. We shall show that, in a neighbourhood of x:

1) ∂Ω is a graph in R_{x}.

2) ∂Ω \ C is Lipschitz in R_{x}.

Let ε > 0 be sufficiently small and y ∈ ∂Ω ∩ B(x, ε). Put y = (y′, y_{N}) and c = (y′, 0).

We first show 1). Note D_{c} the half-line with origin in c and director vector e_{N}. One can show that

D_{c} ∩ ∂Ω = {y}.

By construction of K_{x}, the open half-line D(x, e_{N}) with origin in x and director vector e_{N} is contained in the interior of K_{x}. As K_{y} continuously varies with y, the half-line D(y, e_{N}) with origin in y and director vector e_{N} is in the interior of K_{y}, for y sufficiently close to x. Now, since Ω has C-GNP, Proposition 2.3 allows us to derive

Ω ∩ K_{y} = ∅. (6)

Therefore, if D_{c} = [c, y] ∪ D(y, e_{N}) contains some point η of ∂Ω, η ≠ y, then according to (6), η would be on the interval [c, y[. Applying the C-SP to η, one would have

Ω ∩ K_{η} = ∅,

but, by Lemma 2.4 y ∈ K_{η}, which gives the contradiction.

Now, we show 2). As Ω has C-GNP, Proposition 2.3 implies that Ω ∩ K_{y} = ∅ or again that K_{y} ⊂ Ω^{c}. So the graph of ∂Ω does not meet the interior of the cone K_{y} which contains the open half-line D(y, e_{N}) (see above). To conclude, it remains to find an analogous cone situated under the graph of ∂Ω. Put E_{y} = {η : y ∈ K_{η}}. Note that, according to Proposition 2.3, E_{y} does not contain any point of ∂Ω.

Introduce H_{N}, the hyperplane \{z_{N} = \frac{y_{N}}{2} \} and H_{N}^{+}, the closed half-space \{z_{N} ≥ \frac{y_{N}}{2} \}. As the closed intersection H_{N}^{+} ∩ C is empty, then there exists α > 0 such that for all cartesian coordinate system R(y, e′_{1}, …, e′_{N}) such that the angle { \atop (e′_{N}, e_{N})} < α, the convex C is in the half-space \{y′_{N} < \frac{y_{N}}{2} \}. Let η ∈ B(y, \frac{y_{N}}{2} ) ∩ C(y, α), by construction, C is in the open half-space limited by the hyperplane which is orthogonal to \overrightarrow{ηy} passing by η and not containing y. Then Remark 2.7 gives y ∈ K_{η}, and

B (y, \frac{y_{N}}{2} ) ∩ C(y, α) ⊂ E_{y}.

This implies the desired property. □

An immediate consequence to this proposition is the following:

Corollary 2.4. The intersection of two open subsets which have C-GNP is an open subset which satisfies C-GNP.

Proof. Let Ω_{1} and Ω_{2} be two open subsets which contain the interior of C. If Δ is a selected normal to C. By hypothesis, Δ ∩ Ω_{1} and Δ ∩ Ω_{2} are connected (there are intervals of Δ) then their intersection is an interval of Δ and is, therefore, connected and Ω_{1} ∩ Ω_{2} satisfies (P4). It remains now to prove that if Ω_{1} and Ω_{2} satisfy (P2) and (P3), then Ω_{1} ∩ Ω_{2} also. Let Γ_{I} be the boundary of Ω_{1} ∩ Ω_{2}. If x ∈ Γ_{I} where the inward normal exists, this one is necessarily the inward normal at x to ∂Ω_{1} or to ∂Ω_{2}, so (P3) will be verified. To conclude, it suffices to show that if Ω_{1} and Ω_{2} have C-GNP, then Γ_{I} \ C is Lipschitz (in general, if Ω_{1} and Ω_{2} are two Lipschitz open subsets, their intersection is an open subset which is not necessarily Lipschitz). Let x ∈ Γ_{I}, then in the cartesian coordinate system R_{x} defined by Proposition 2.4, there exists a neighbourhood V_{x} of x and two Lipschitz functions ϕ_{1} and ϕ_{2} representing respectively ∂Ω_{1} ∩ V_{x} and ∂Ω_{2} ∩ V_{x} in R_{x}. Consequently if we put Φ_{I} = inf(ϕ_{1}, ϕ_{2}), then Φ_{I} is a Lipschitz representation of Γ_{I} ∩ V_{x} in R_{x}. □

Remark 2.8. In general, the union of two open subsets which satisfy the property (P4) does not satisfy it. One can consider in two dimensions the convex C = [−1, 1] × {0}. Let Ω be the union of the two open discs D_{−1} and D_{1} centred respectively in −1 and 1 and of radius 1/2. If the selected normal to C at the two extremities is the line y = 0 then D_{−1} and D_{1} satisfy (P4) while Ω does not.

Now we shall precise the result obtained above. We want to prove that for an open subset Ω which has C-GNP, ∂Ω⧹C satisfies the cone property. More precisely we shall show the following proposition.

Proposition 2.5. Let Ω be an open subset which has C-GNP and x_{0} a point of ∂Ω⧹C. Then, there exists an unitary vector η and a real number ε (strictly positive) which depend only on x_{0} and C and such that

∀ y ∈ B(x_{0}, ε) ∩ \overline{Ω} C(y, η, ε) ⊂ Ω,

where C(y, η, ε) is the cone with vertex y, of direction η and angle to the vertex and of height ε:

C(y, η, ε) = {x ∈ ℝ^{N} ; |x − y| ≤ ε and |(x − y) · η| ≥ |x − y| cos ε} .

Proof. Denote by δ, the distance of x_{0} to C. The C-GNP allows us to work in the cartesian coordinate system R_{0} with origin in O (the projection of x_{0} on C) and which has the last vector of coordinates, \overrightarrow{e_{N}} = \overrightarrow{Ox_{0}}/δ such that \{x_{N} = 0\} is the supporting hyperplane to C at O. One can complete the base by choosing an orthonormal base of {x_{N} = 0}.

Let ϕ be the Lipschitz representation of ∂Ω ∩ B(x_{0}, α) in R_{0}. One can suppose that α < δ/2. Then

∂Ω ∩ B(x_{0}, α) = {(x′, x_{N}) ∈ B(x_{0}, α) ; x_{N} = ϕ(x′)},

and

Ω ∩ B(x_{0}, α) = {(x′, x_{N}) ∈ B(x_{0}, α) ; x_{N} < ϕ(x′)}.

Let R > 0 be large enough in order that the intersection between the cone with vertex y supported by C and the hyperplane x_{N} = 0 will be contained in the (N −1)-dimensional ball B′(O, R), for all y ∈ B(x_{0}, δ/2). Start with characterizing analytically the geometric property of the normal. At every point ξ = (ξ′, ξ_{N}) ∈ ∂Ω ∩ B(x_{0}, α) where the normal ν_{ξ} exists, the half-line with origin ξ and directed vector ν_{ξ} intercects the hyperplane {x_{N} = 0} inside the ball B′(O, R):

∀ξ ∈ N_{Ω} ∩ B(x_{0}, α), |ξ′ + ϕ(ξ′)∇ϕ(ξ′)| ≤ R. (7)

This implies that the function

ξ ⟼ ϕ^{2}(ξ′) + |ξ′|^{2}

is 2R-Lipschitz in B′(O, α). Hence ϕ^{2} is (2R + 2α)-Lipschitz and if ϕ ≥ δ/2 we obtain that ϕ is (2R + 2α) /δ-Lipschitz and therefore satisfies an uniform cone property (see [7]). But as it is not absolutely evident that the geometric characteristics of the cone can be chosen independently of Ω, we continue the demonstration.

Let us fix ε > 0 sufficiently small that

2ε + (1 + tan^{2} ε)ε + 2 tan ε < δ,

(it is clear that ε depends only on x_{0} and C, by the intermediary of R and δ).

Choose, as direction of the cone, η = −e_{N}. Let y ∈ B(x_{0}, ε) ∩ Ω, y = (y′, δ + y_{N}) with

|y′|^{2} + y_{N}^{2} < ε^{2} and δ + y_{N} ≤ ϕ(y′). (8)

Now, let z ∈ C(y, η, ε):

(9) z = y + (h′, −h_{N}) with, |h′|^{2} + h_{N}^{2} < ε^{2}, |h′| ≤ (tan ε)h_{N} and h_{N} > 0.

We shall prove the uniform cone property, that is to say that, z ∈ Ω, or again that,

(δ + y_{N} − h_{N})^{2} < (ϕ(y′ + h′))^{2}.

Using (7), we can have

(ϕ(y′ + h′))^{2} ≥ (ϕ(y′))^{2} − 2R|h′| + |y′|^{2} − |y′ + h′|^{2}.

And from (8),

(ϕ(y′ + h′))^{2} ≥ (δ + y_{N})^{2} − 2R|h′| + |y′|^{2} − |y′ + h′|^{2}
≥ (δ + y_{N})^{2} − 2R|h′| − |h′|^{2} − 2ε|h′|.

Now, according to (9), one gets

(ϕ(y′ + h′))^{2} ≥ (δ + y_{N})^{2} − 2(R + ε) tan(ε)h_{N} − tan^{2}(ε)h_{N}^{2}.

Using the definition of ε, we obtain

(ϕ(y′ + h′))^{2} ≥ (δ + y_{N})^{2} + h_{N}^{2} − 2h_{N}(δ − ε) ≥ (δ + y_{N} − h_{N})^{2}.

which is the result. □

Let us now see if the class C is stable by Steiner symmetrization.

Definition 2.4. Let Ω be an open subset of ℝ^{2}. Assume that Ω is convex in the direction Oy. For α ∈ ℝ, let Ω_{α} be the segment{(α, y) ∈ ℝ^{2} ; (α, y) ∈ Ω}. The Steiner symmetrization of Ω is

Ω^{∗} = \{(α, t) ∈ ℝ^{2} ; α ∈ ℝ and |t| < \frac{|Ω_{α}|}{2}\} .

Lemma 2.5. Let Ω be an open subset of ℝ^{2} which contains the convex C = [−1, 1]× {0}. If Ω satisfies C-GNP, then Ω is convex in the direction Oy.

Proof. The proof is an immediate consequence of Definition 2.2. Let H^{+} and H^{−} be the half-planes separated by the axis Ox. Let x ∈ ∂Ω. As Ω satisfies C-GNP from Proposition 2.3, we deduce that the vertical segment over x is in the closed cone K_{x}. In the same way, if there exists z in the vertical segment under x we have x ∈ K_{z} which contradicts Lemma 2.4. Therefore Ω is convex in the direction Oy and ∂Ω ∩ H^{+} and ∂Ω ∩ H^{−} are two graphs. □

Proposition 2.6. There exists an open subset Ω ⊂ ℝ^{2} which satisfies C-GNP and such that its Steiner symmetrization doesn’t satisfy C-GNP.

Proof. Using the notations of the previous proof, let φ_{1} (resp. φ_{2}) be the representation of ∂Ω∩H^{+} (resp. of ∂Ω∩H^{−}). It is easy to see that C-GNP is equivalent to

−1 ≤ x + φ_{i}(x)φ′_{i}(x) ≤ 1, i = 1, 2,

for all x such that the derivative φ′_{i}(x) exists.

Let r ∈ ]0, 1[ and put

φ_{1}(x) = \sqrt{(2 + r)^{2} − (x − 1)^{2}}
φ_{2}(x) = \{ \begin{matrix}\sqrt{r^{2} − (x + 1)^{2}} if x ∈ [−1 − r, −1] \\ \sqrt{4 − (x − 1)^{2}} if x ∈ [−1, +∞[ . \end{matrix}

The symmetrized Ω^{∗} (of Ω) is limited by the graph of \frac{φ_{1}+φ_{2}}{2} in H^{+} and the graph of −\frac{φ_{1}+φ_{2}}{2} in H^{−}. Now, if x = −1 + r with r sufficiently small, a simple calculation shows that

x + (\frac{φ_{1} + φ_{2}}{2} ) (x) (\frac{φ_{1} + φ_{2}}{2} )′ (x) ≃ −1 + \frac{1}{2r} (r + \sqrt{4r + r^{2}}) ,

which tends to +∞ when r tends to 0. Therefore, Ω^{∗} doesn’t satisfy C-GNP. □

When we symmetrize an open set Ω (which satisfies C-GNP) by the Steiner continuous symmetrization, the same property is satisfied by its symmetrized Ω_{t}, for t small, at the points of the boundary ∂Ω whose normal meets the relative interior of C.

Definition 2.5. Let Ω be an open set in ℝ^{2} which is convex in the direction Oy. The Steiner continuous symmetrization consists in centering each segment [y_{1}, y_{2}] parallel with the axis Oy (y_{1} and y_{2} belong to ∂Ω) with a speed equal to the distance from the center of [y_{1}, y_{2}] to the axis {x = 0}, i.e., that if ∂Ω is given by two functions ϕ_{1} and ϕ_{2} then for all t (t ∈ [0, 1]) the boundary ∂Ω_{t} of its symmetrized Ω_{t} will be given by the functions ϕ_{1}^{t}and ϕ_{2}^{t} such that

\begin{cases}ϕ_{1}^{t} = ϕ_{1} − \frac{t}{2}(ϕ_{1}−ϕ_{2}) \\ ϕ_{2}^{t} = ϕ_{2} + \frac{t}{2}(ϕ_{1}−ϕ_{2}). \end{cases}

Definition 2.6. An arc γ centered in (−1, 0) or (1, 0) is said to be of Type I if it is not included in {(x, y) ∈ ℝ^{2} : x ≤ −1} ∪ {(x, y) ∈ ℝ^{2} : x ≥ 1} .

Proposition 2.7. Let Ω be an open set which strictly contains the segment C and satisfying C-GNP. If ∂Ω does not contain arcs of Type I, then for t sufficiently small, Ω_{t} satisfies also C-GNP.

Proof. Suppose that ∂Ω is given by two functions ϕ_{1} and ϕ_{2}, then, for t small enough, its symmetrized ∂Ω_{t} will be given by the functions ϕ_{1}^{t} and ϕ_{2}^{t} such that

\begin{cases}ϕ_{1}^{t} =ϕ_{1} − \frac{t}{2}(ϕ_{1}−ϕ_{2}) \\ ϕ_{2}^{t} =ϕ_{2} + \frac{t}{2}(ϕ_{1}−ϕ_{2}). \end{cases}

For x ∈ ∂Ω_{t}, the inward normal meets C if and only if

−1 ≤ x + ϕ_{i}^{t}(x)(ϕ_{i}^{t})′(x) ≤ 1 , i = 1, 2.

It is thus a question of checking if

\begin{matrix}−1 ≤ x + ϕ_{1}(x)ϕ′_{1}(x) − \frac{t}{2}[2ϕ_{1}(x)ϕ′_{1}(x) − (ϕ′_{1}(x)ϕ_{2}(x) + ϕ_{1}(x)ϕ′_{2}(x))] \\ +\frac{t^{2}}{4} (ϕ_{1}(x) − ϕ_{2}(x))(ϕ′_{1}(x) − ϕ′_{2}(x)) ≤ 1. \end{matrix}

Now we place ourselves in a point of the boundary ∂Ω, of x-coordinate x_{0} and such that the interior normal cuts the segment C, for example, at (−1, 0). One has

x_{0} + ϕ_{1}(x_{0})ϕ′_{1}(x_{0}) = −1,

and

ϕ′_{2}(x_{0}) ≥ −\frac{1 + x_{0}}{ϕ_{2}(x_{0})}.

One thus deduces that at the point of x-coordinate x_{0},

ϕ′_{1}(x_{0})ϕ_{2}(x_{0}) + ϕ_{1}(x_{0})ϕ′_{2}(x_{0}) ≥ −(1 + x_{0}) (\frac{ϕ_{2}(x_{0})}{ϕ_{1}(x_{0})} + \frac{ϕ_{1}(x_{0})}{ϕ_{2}(x_{0})}) .

But

(\frac{ϕ_{2}(x_{0})}{ϕ_{1}(x_{0})} + \frac{ϕ_{1}(x_{0})}{ϕ_{2}(x_{0})}) ≥ 2

(and even > 2 if ϕ_{1}(x_{0}) ≠ ϕ_{2}(x_{0}), the equality corresponds to the case where we don’t move). Therefore if −(1 + x_{0}) > 0 one has

ϕ′_{1}(x_{0})ϕ_{2}(x_{0}) + ϕ_{1}(x_{0})ϕ′_{2}(x_{0}) ≥ −2(1 + x_{0}) = 2ϕ_{1}(x_{0})ϕ′_{2}(x_{0})

and the term between brackets in the previous inequality is negative and consequently this one is checked at the point of x-coordinate x_{0}. In addition, since x_{0} + ϕ(x_{0})ϕ′_{1}(x_{0}) = −1 and t is rather small then

\begin{matrix}x_{0} + ϕ_{1}(x_{0})ϕ′_{1}(x_{0}) − \frac{t}{2}[2ϕ_{1}(x_{0})ϕ′_{1}(x_{0}) − (ϕ′_{1}(x_{0})ϕ_{2}(x_{0}) + ϕ_{1}(x_{0})ϕ′_{2}(x_{0}))] \\ +\frac{t^{2}}{4} (ϕ_{1}(x_{0}) − ϕ_{2}(x_{0}))(ϕ′_{1}(x_{0}) − ϕ′_{2}(x_{0})) ≤ 1. \end{matrix}

Which gives the result. □

3. The geometric normal property and the convergence of domains

In this section, we start by recalling three notions of convergence we can define on the open subsets of ℝ^{N}. Next, we prove that the class C of all open subsets satisfying C-GNP is compact for the Hausdorff convergence. We finish by showing that the three considered convergences are equivalent on C.

In all the following, we consider a fixed ball D centered to the origin and of sufficiently large radius to be able to contain the convex compact C, and all the open subsets we shall use.

Definition 3.1. Let K_{1} and K_{2} be two compact subsets of D. One calls a Hausdorff distance of K_{1} and K_{2} and we denote by it d_{H}(K_{1}, K_{2}), the following positive number:

d_{H}(K_{1}, K_{2}) = max [ρ(K_{1}, K_{2}), ρ(K_{2}, K_{1})] ,

where ρ(K_{i}, K_{j}) = max_{x∈K_{i}} d(x, K_{j}) i, j = 1, 2 and d(x, K_{j}) = min_{y∈K_{j}} |x − y| .

Definition 3.2. Let Ω_{n} be a sequence of open subsets of D and Ω be an open subset of D. Let K_{n} and K be their complements in D. One says that the sequence Ω_{n} converges in the Hausdorff sense to Ω and we denote by Ω_{n} ⟶^{H} Ω if

lim_{n→+∞}d_{H}(K_{n}, K) = 0.

The notion of convergence that follows is less classical than the Hausdorff convergence, but it is very useful for the stability of the solution of elliptic problems when the domain varies, e.g., [10] and [12].

Definition 3.3. Let Ω_{n} be a sequence of open subsets of D and Ω be an open subset of D. One says that the sequence Ω_{n} converges in the compact sense to Ω and we denote by Ω_{n} ⟶^{K} Ω if:

• Every compact subset of Ω is included in Ω_{n}, for n large enough.

• Every compact subset of \overline{Ω}^{c} is included in \overline{Ω}_{n}^{c}, for n large enough.

Definition 3.4. Let Ω_{n} be a sequence of open subsets of D and Ω be an open subset of D. One says that the sequence Ω_{n} converges in the sense of characteristic functions to Ω and we denote by Ω_{n} ⟶^{L} Ω if χ_{Ω_{n}} converges to χ_{Ω} in L_{loc}^{p}(ℝ^{N}), p ≠ ∞, (χ_{Ω} is the characteristic function of Ω).

Now, we recall some elementary results concerning the Hausdorff convergence. First of all, the three following propositions which are very classical, see for example [4] or [15].

Proposition 3.1. The set of open subsets confined in the ball D is relatively compact for the Hausdorff convergence.

Remark 3.1. The previous proposition is very important because it allows us to extract from each sequence of open subsets of D, a subsequence which converges in the Hausdorff sense.

Proposition 3.2. If Ω_{n} is a sequence of open subsets of D and Ω is an open subset of D such that Ω_{n} ⟶^{H} Ω, then:

(i) Every compact subset of Ω is included in Ω_{n} for n large enough. (ii) For all x in ∂Ω, lim_{n→+∞}d(x, ∂Ω_{n}) = 0.

Proposition 3.3. If Ω_{n} is a sequence of open subsets of D and Ω is an open subset of D such that Ω_{n} ⟶^{H} Ω, then: (iii) lim meas (Ω⧹Ω_{n}) = 0. (iv) χ_{Ω} ≤ lim inf_{n→+∞}χ_{Ω_{n}}.

Show now the following theorem.

Theorem 3.1. Let Ω_{n} be a sequence of open subsets of D satisfying the C-GNP. Then there exists an open subset Ω of D and a subsequence (still denoted by Ω_{n}) such that:

• Ω_{n} converges to Ω in the Hausdorff sense.

• Ω_{n} converges to Ω in the compact sense.

• Ω_{n} converges to Ω in the sense of characteristic functions.

• Ω satisfies C-GNP.

We start by showing that Hausdorff convergence implies compact convergence.

Proposition 3.4. Let Ω_{n} be a sequence of open subsets of D, which have C-GNP. If the sequence Ω_{n} converges in the Hausdorff sense to an open subset Ω, then it converges in the compact sense to Ω.

Proof. One knows by Proposition 3.2 that for all compact K ⊂ Ω, one has K ⊂ Ω_{n}, for n large enough. It remains to show an analogous property for the compacts situated in the exterior of Ω. Let L be a compact set of D (one can always suppose that the interior of L is not empty) L ⊂ \overline{Ω}^{c}. Let Ω_{n} be an eventual open subset of the sequence such that \overline{Ω}_{n} ∩L ≠ ∅. In the case where L ⊂ Ω_{n}, one has immediately d_{H}(Ω_{n}, Ω) ≥ α, where α is the radius of a ball included in L. As lim_{n→+∞}d_{H}(K_{n}, K) =

0, the set of n such that L ⊂ Ω_{n}, is finite.

In the other case one has ∂Ω_{n} ∩ L ≠ ∅. Let x be some point of this intersection. According to the Proposition 3.2, there exists ε > 0 and an unitary vector η (both are independent of n) such that the cone C(x, η, ε) is included in Ω_{n}. As one can always suppose ε small enough so that C(x, η, ε) ∩ Ω = ∅, one can derive that d_{H}(Ω_{n}, Ω) ≥ β (where β is the radius of some ball included in the cone C(x, η, ε)). As lim_{n→+∞}d_{H}(Ω_{n}, Ω) = 0, the set of n such that ∂Ω_{n} ∩ L ≠ ∅ is finite.

This achieves the demonstration. □

The following proposition will be useful later.

Proposition 3.5. Let Ω_{n} be a sequence of open subsets which converges, in the compact sense, to an open subset Ω. If ∂Ω has a null measure, then Ω_{n} converges to Ω in the sense of chracteristic functions.

Proof. If χ_{n} is the characteristic function of Ω_{n}, χ_{n} is bounded in L^{∞}(D), so it converges to a function l ∈ L^{∞}(D) for the w*-L^{∞}(D) convergence. Let K be a compact set in Ω. By the compact convergence, there exists N_{K} ∈ ℕ such that K ⊂ Ω_{n}, for all n ≥ N_{K}. Therefore, χ_{n} = 1 on K for all n ≥ N_{K} and thus, l = 1 on K. As this is valid for all compact K of Ω, one has l = 1 on Ω. In the same way considering some compact L ⊂ (\overline{Ω})^{c}, one has l = 0 on (\overline{Ω})^{c}. As D = Ω ∪ (\overline{Ω})^{c} ∪ ∂Ω and meas(∂Ω) = 0, then one has l = χ_{Ω} almost everywhere (χ_{Ω} being the characteristic of Ω). Therefore, χ_{n} ⇀^{∗} χ_{Ω} and thus, as all the open subsets are confined in D, χ_{n} converges strongly, to χ_{Ω} in L^{1}(D). □

According to Proposition 3.1, one knows that C is relatively compact for the Hausdorff convergence. It remains to show the closure of C, that is to say that if Ω_{n} is a sequence of open sets in C which converges in the Hausdorff sense to an open set Ω, this one belongs to C. For this, we need to prove the following:

1. Ω is Lipschitz outside of C.

2. At points of ∂Ω, where the inward normal exists, this one intersects C.

3. For all selected normals Δ to C, Δ ∩ Ω is connected.

It is clear that if Ω_{n} ⟶^{H} Ω and Δ∩Ω_{n} is connected, then Δ∩Ω is also connected (because the limit, in the Hausdoff sense, for a sequence of intervals is an interval).

We now show the first point.

Theorem 3.2. Let Ω_{n} be a sequence of open subsets of D which have C-GNP and Ω be an open subset of D. If Ω_{n} ⟶^{H} Ω, then ∂Ω is Lipschitz outside of the convex C.

Proof. Let x ∈ ∂Ω\C. According to (ii) of Proposition 3.2, there exists a sequence of points x_{n} ∈ ∂Ω_{n} \C which converges to x. Denote by τ_{n} the translation of vector \overrightarrow{x_{n}x}. It is clear that if Ω_{n} ⟶^{H} Ω, τ_{n}(Ω_{n}) ⟶^{H} Ω. So we can suppose that x ∈ ∂Ω_{n}, for all n.

For each n, ∂Ω_{n} can be represented, in a neighbourhood of x, by some Lipschitz representation ϕ_{n}, in the cartesian coordinate system (O, e_{1}, …, e_{N}) where O is the projection of x on C and e_{N}=\frac{\overrightarrow{Ox}}{|\overrightarrow{Ox}|}.

According to Proposition 2.5, all the Ω_{n} satisfy a uniform property of the ε-cone, in the neighbourhood of x, with ε independent of n (it depends only on x and on C). Using the result obtained by D. Chenais in [7], one can fix a (N −1)-dimensional closed ball B′(O, α) on which all the functions ϕ_{n} are defined.

As in the proof of Proposition 2.5, if ϕ_{n} ≥ δ/2 then ϕ_{n} is (2R + 2α) /δ-Lipschitz, proving that the functions ϕ_{n} are uniformly Lipschitz. One can use Ascoli’s theorem to extract from ϕ_{n}, a subsequence (still denoted ϕ_{n}) which converges uniformly to a Lipschitz function ϕ. It remains now, to show that the function ϕ is a representation of ∂Ω in a neighborhood of x, that is to say that in a neighborhood V of x, one has

∂Ω ∩ V = {(y′, y_{N}) ∈ B′(O, α) × ℝ : y_{N} = ϕ(y′)} := Gra(ϕ) (10)

Ω ∩ V = {(y′, y_{N}) ∈ B′(O, α) × ℝ : y_{N} < ϕ(y′)}. (11)

Let y = (y′, y_{N}) ∈ ∂Ω ∩ V. By Proposition 3.2, there exists a sequence of points y^{n} = ((y^{n})′, y_{N}^{n}) ∈ ∂Ω_{n} which converges to y. Since y_{N}^{n} = ϕ_{n}((y^{n})′), the uniform convegence of ϕ_{n} to ϕ implies that y_{N} = ϕ(y′), i.e., ∂Ω ∩ V ⊂ Gra(ϕ). For the inverse inclusion, one can remark that, for all fixed point y′ in B′(O, α), there exists only one point of Gra(ϕ) which is “above” y′. This one is necessarily the point (y′, y_{N}) which belongs to ∂Ω ∩ V.

Now, let y = (y′, y_{N}) ∈ Ω. According to Proposition 3.2, one knows that there exists n_{0} ∈ ℕ such that y ∈ Ω_{n}, ∀n ≥ n_{0}. Therfore, y_{N} < ϕ_{n}(y′) and, tending n to infinity, y_{N} ≤ ϕ(y′). As y ∉ ∂Ω, y_{N} < ϕ(y′).

At last, let y = (y′, y_{N}) with y_{N} < ϕ(y′). One can prove that y ∈ Ω. If this is not true, then either:

1. y ∈ ∂Ω which is impossible according to what precedes, or

2. y is in the exterior of Ω. This is also impossible because Ω_{n} ⟶^{K} Ω which implies that y ∈ \overline{Ω}_{n}^{c} for n large enough, and so y_{N} > ϕ_{n}(y′) and, at the limit y_{N} ≥ ϕ(y′).

□

Theorem 3.3. Let Ω_{n} be a sequence of open subsets of D which have C-GNP and Ω be an open subset of D. If Ω_{n} ⟶^{H} Ω, then Ω satisfies C-GNP.

Proof. To prove that Ω satisfies C-GNP is equivalent to show that Ω satisfies the C-SP.

Let x ∈ ∂Ω⧹C. As Ω_{n} ⟶^{H} Ω, by Proposition 3.2 there exists a sequence x_{n} ∈ ∂Ω_{n}⧹C which converges to x. This implies in particular that the sequence of the closed cones K_{x_{n}}(defined in Definition 2.3) converges in the Hausdorff sense to the closed cone K_{x}. As the intersection is continuous for the Hausdorff convergence, Ω_{n}∩K_{x_{n}} ⟶^{H} Ω∩K_{x}. But Ω_{n}∩K_{x_{n}} = ∅ so Ω∩K_{x} = ∅ (according to Proposition 3.2) and Ω satisfies the C-SP. □

As a consequence of what precedes, we have the following propositions which prove that the three convergences are equivalent on C.

Proposition 3.6. Let Ω_{n} be a sequence of open subsets of D which have C-GNP and Ω be an open subset of D. If Ω_{n} ⟶^{K} Ω, then Ω_{n} ⟶^{H} Ω.

Proof. Since the Ω_{n} are confined in D, then Proposition 3.1, gives us the existence of an open subset Ω̃ and a subsequence (still denoted Ω_{n}) such that Ω_{n} converges in the Hausdorff sense to Ω̃. We now show that Ω̃ = Ω. Since Ω has C-GNP, then Ω is of Caratheodory type (int(\overline{Ω}) = Ω). To conclude, it suffices to prove that Ω ⊂ Ω̃ ⊂ \overline{Ω}. We start by showing the first inclusion. Suppose there exists x ∈ Ω̃ ∩ (\overline{Ω})^{c}, then:

1. On one hand, the compact convergence of Ω_{n} to Ω implies the existence of an integer N_{x} ∈ ℕ such that

∀ n ≥ N_{x}, x ∈ (\overline{Ω}_{n})^{c}.

2. On the other hand, as Ω_{n} ⟶^{H} Ω̃, then there exists N′_{x} ∈ ℕ s.t. ∀ n ≥ N′_{x}, x ∈ Ω_{n}. Consequently,

∀ n ≥ max(N_{x}, N′_{x}) , x ∈ Ω_{n} ∩ (\overline{Ω}_{n})^{c},

which is absurd.

We now show that Ω ⊂ Ω̃. Let x ∈ Ω, there exists a closed ball B(x) centered in x s.t. B(x) ⊂ Ω_{n}. But Ω_{n} ⟶^{H} Ω̃, so B(x) ⊂ Ω̃ and thus, x ∈ Ω̃. □

Proposition 3.7. Let Ω_{n} be a sequence of open subsets of D which have C-GNP and Ω be an open subset of D. If Ω_{n} ⟶^{H} Ω, then Ω_{n} ⟶^{L} Ω.

Proof. As Ω_{n} ⟶^{H} Ω, then according to Proposition 3.4, Ω_{n} ⟶^{K} Ω. Since ∂Ω ⊂ (∂Ω \ C) ∪ ∂C and ∂C and ∂Ω \ C have null measures (since they are Lipschitz), then meas(∂Ω) = 0. Then, Proposition 3.5 gives the result. □

Proposition 3.8. Let Ω_{n} be a sequence of open subsets of D which have C-GNP and Ω be an open subset of D. If Ω_{n} ⟶^{L} Ω, then Ω_{n} ⟶^{H} Ω and Ω has C-GNP.

Proof. Let Ω_{n} be a sequence of open subsets of D converging in the sense of characteristic functions to Ω. The Ω_{n} are confined in D, so, even if to take a subsequence of Ω_{n}, Ω_{n} ⟶^{H} Ω̃ and Ω̃ has C-GNP. Now, as the Ω_{n} have C-GNP, Ω_{n} ⟶^{K} Ω̃ (see Proposition 3.4). Because Ω and Ω̃ are of Caratheodory type, Proposition 3.5 implies that Ω̃ = Ω, and thus Ω satisfies C-GNP. □

4. Stability of the solution of Dirichlet problem

By Remark 2.5 in two dimensions, an open subset which has C-GNP satisfies the exterior segment property. Therefore, it is stable in the sense of Keldyš (see Theorem 4.1). In all the following we suppose that N ≥ 3.

Generally, Hausdorff convergence is not sufficient to get the stability of the solution of a boundary value problem like the Dirichlet one, when the domain varies. In our case, nevertheless, we were able to prove (Proposition 3.4) that in the class C, Hausdorff convergence implies compact convergence which is the main tool for proving the stability of the solution of Dirichlet problem. However, this is not sufficient to assure a priori the desired result of stability. In fact, it is necessary that the limit domain has a minimum of regularity in order to get this stability result. Now, if an open subset Ω satisfies C-GNP, it is Lipschitz outside of C that is more than the wished regularity. Unfortunatly, we have seen, in Proposition 2.1, that the boundary of Ω could have cusps which are on ∂C. In general, at cusp point, an open subset can be regular or irregular in the sense of Wiener (Definition 4.2). This depends on the geometric shape of cusps and, in particular, it depends on the “quantity of material” (in terms of capacity) contained in the exterior of Ω, in a neighbourhood of such points. We will make a precise computation of the capacity of the exterior of Ω in a neighbourhood of cusps in order to be sure that C-GNP implies their regularity. Before that, let us recall some definitions and results which are very useful for proving the stability of the solution of the Dirichlet problem, when the domain varies.

Definition 4.1. The Sobolev capacity is defined as follows:

• For a compact subset K of ℝ^{N}

Cap(K) = inf {∫_{ℝ^{N}} |∇φ|^{2} ; φ ∈ C_{0}^{∞}(ℝ^{N}), φ ≥ 1 on K} .

• For an open subset G of ℝ^{N}

Cap(G) = sup {Cap(K) : K ⊆ G, K is compact} .

• For any set E of ℝ^{N}

Cap(E) = inf {Cap(G) : G ⊇ E, G is open} .

One says that a property takes place quasi everywhere (or more simply q.e.) if it takes place in the complement of a set of null capacity.

We now give a regularity criterion called the Wiener criterion [17].

Definition 4.2 (Wiener criterion). Let Ω be an open subset of ℝ^{N}, N ≥ 3, and x_{0} a point of ∂Ω. For all q, q ∈ ]0, 1[, let

G_{n} = \overline{Ω}^{c} ∩ \{x : q^{n+1} ≤ |x − x_{0}| ≤ q^{n}\} .

Then, x_{0} is regular if and only if the series of general term \frac{Cap(G_{n})}{q^{n(N−2)}} is divergent.

Now we return to our question concerning the stability of the solution of the Dirichlet problem when the domain varies. This stability is related to the stability (in the sense of Keldyš) of the domain Ω intervening in such a problem. The fundamental question now, is to know which condition the domain Ω must satisfy in order that a function which vanishes q.e. on the exterior of Ω, vanishes in the complement of Ω (that is to say, that such a function belongs to H_{0}^{1}(Ω)). This is exactly what the Wiener criterion says. To be more precise let us give the following results, e.g., [10], [12] and [14].

Definition 4.3 (Keldyš). An open set Ω is said to be stable if and only if all points of its boundary (except a set of null capacity) are regular in the Wiener sense.

Theorem 4.1. Let Ω_{n} be a sequence of open subsets of D and let u_{n} be the solution of the Dirichlet problem

(P_{n}) \{ \begin{matrix}−Δu_{n} = f in Ω_{n} \\ u_{n} = 0 on ∂Ω_{n}. \end{matrix}

If Ω_{n} ⟶^{K} Ω and Ω is stable in the sense of Keldyš, then u_{n} converges strongly in H_{0}^{1}(D) to a function u, the solution of the Dirichlet problem on Ω :

(P) \{ \begin{matrix}−Δu = f in Ω \\ u = 0 on ∂Ω \end{matrix}

(the functions u_{n} and u are extended by zero in D, f ∈ H^{−1}(D)).

Before applying this result to our problem, we shall give a necessary and sufficient condition so that a cusp of surface of revolution will be regular in Wiener sense. Without loss of generality, one can suppose this point in the origin. Such point is given, in spherical coordinates, by

(P R) \{ \begin{matrix}θ = f(ρ), ρ ≤ 1, f(0) = 0 and f(1) < π, \\ where f(ρ) is an increasing function. \end{matrix}

Theorem 4.2. Let Ω be an open subset of ℝ^{N} and O ∈ ∂Ω be given by (P R).

• If N = 3, then O is regular if and only if the series of general term \frac{1}{ln f(q^{n})} is divergent.

• If N ≥ 4, then O is regular if and only if the series of general term [f(q^{n})]^{N−3} is divergent.

Proof. Suppose first, that N = 3. Let q be a real number such that 0 < q < 1, and let θ_{n} = f(q^{n}). Then the set

G_{n} = \overline{Ω}^{c} ∩ \{x : q^{n+1} ≤ |x| ≤ q^{n}\}

is on one hand included in the right circular cylinder with height q^{n} and of basis radius q^{n} sin θ_{n}, or again in the ellipsoid E_{n} of half-axes 2q^{n} sin θ_{n}, 2q^{n} sin θ_{n}, 2q^{n}. On the other hand, it contains the right circular cylinder with height q^{n} − q^{n+1} and of basis radius q^{n+1} sin θ_{n+1}, or again, it contains the ellipsoid F_{n} of half-axes q^{n+1} sin θ_{n+1}, q^{n+1} sin θ_{n+1}, q^{n}(1 − q)/2.

Now, compute the capacity of these two ellipsoids. For this, let us consider a general ellipsoid E

E := \{X = (x, y, z) ∈ ℝ^{3} | \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} + \frac{z^{2}}{c^{2}} < 1\} . (12)

Introduce the capacitary potential

U(x, y, z) = p ∫_{λ}^{∞} \frac{dt}{[(a^{2} + t)(b^{2} + t)(c^{2} + t)]^{\frac{1}{2}}} , (13)

where λ is the biggest root of the equation

\frac{x^{2}}{a^{2} + λ} + \frac{y^{2}}{b^{2} + λ} + \frac{z^{2}}{c^{2} + λ} = 1, (14)

and p is such that U = 1 on ∂E, that is to say

\frac{1}{p} = ∫_{0}^{∞} \frac{dt}{[(a^{2} + t)(b^{2} + t)(c^{2} + t)]^{\frac{1}{2}}} . (15)

We recall that U is the solution of

\{ \begin{matrix}ΔU = 0 & in & ℝ^{3} \ E \\ U = 1 on ∂E \\ U = O(\frac{1}{r}) & if & r → ∞. \end{matrix}

Note that the capacity of the set E is given by

Cap(E) = 8π (∫_{0}^{∞} \frac{dt}{[(a^{2} + t)(b^{2} + t)(c^{2} + t)]^{\frac{1}{2}}} )^{−1} .

In fact,

Cap(E) = ∫_{∂E} |∇U|.

By (14), one gets

(\frac{∂λ}{∂x})_{|λ=0} = \frac{2x}{a^{2}(x^{2}/a^{4} + y^{2}/b^{4} + z^{2}/c^{4})}.

(one can do the same for y and z).

Therefore on ∂E, one has

\frac{∂u}{∂x} = −p\frac{\frac{∂λ}{∂x}}{abc} = −\frac{p}{abc} \frac{2x}{a^{2}(x^{2}/a^{4} + y^{2}/b^{4} + z^{2}/c^{4})}

(one can do the same for y and z).

Now, if X = (x, y, z) is on ∂E and n⃗(X) is the normal vector to ∂E at X, then

n⃗(X) = \frac{1}{\sqrt{x^{2}/a^{4} + y^{2}/b^{4} + z^{2}/c^{4}}} (x/a^{2}, y/b^{2}, z/c^{2}) .

One obtains therefore,

|∇u| = \frac{2p}{abc}X · n⃗(X).

Hence, (see also [17], [12], [11]),

Cap(E) = ∫_{∂E} |∇U| = \frac{2p}{abc} ∫_{∂E} X · n = \frac{2p}{abc}3Vol(E) = 8πp.

Where Vol(E), denotes the volume of E.

Now, as we work with ellipsoids which have two equal axes, we are in the case where a = c ≤ b. We have to compute an integral of the form

I_{3} := ∫_{0}^{∞} \frac{dt}{(a^{2} + t)(b^{2} + t)^{\frac{1}{2}}} .

Using the variable change s^{2} = b^{2} + t, we obtain

I_{3} = ∫_{b}^{∞} \frac{2ds}{s^{2} − (b^{2} − a^{2})}.

If we put, w^{2} = b^{2} − a^{2}, we get

I_{3} = \frac{1}{w} ln \frac{b − w}{b + w} .

If E = E_{n}, a = 2q^{n} sin θ_{n}, b = 2q^{n} and w = 2q^{n} cos θ_{n}, I_{3} becomes

I_{3} = \frac{1}{2q^{n} cos θ_{n}} ln \frac{1 − cos θ_{n}}{1 + cos θ_{n}} ,

and thus,

Cap(E_{n}) = c_{3} \frac{2q^{n} cos θ_{n}}{ln cot \frac{θ_{n}}{2}} .

In the same way, the capacity of F_{n} is given by

Cap(F_{n}) = c_{3} \frac{2q^{n+1} cos θ_{n+1}}{ln cot \frac{θ_{n+1}}{2}} ,

One can, therefore, deduces

c_{3} \frac{2q cos θ_{n+1}}{ln cot \frac{θ_{n+1}}{2}} < \frac{Cap(G_{n})}{q^{n}} < c_{3} \frac{2 cos θ_{n}}{ln cot \frac{θ_{n}}{2}} .

This shows that the point O is regular if and only if the series of general term _{\frac{2 cos θ_{n}}{ln cot \frac{θn}{2}}} diverges. As θ_{n} must to tend to zero, the divergence of this series is equivalent to the divergence of the series of general term \frac{1}{ln θ_{n}} = \frac{1}{ln f(q^{n})}.

Remark 4.1. According to what precedes, if f(ρ) = Aρ^{m}, the cusp point is regular and if f(ρ) = A exp(\frac{−m}{ρ}), it is irregular.

Now, suppose that N ≥ 4 and consider the ellipsoid

E_{N} := \{X = (x_{1}, …, x_{N}) ∈ ℝ^{N} | ∑_{i=1}^{i=N} \frac{x_{i}^{2}}{a_{i}^{2}} < 1\} ,

and the capacitary potential

U(X) = p ∫_{λ}^{∞} \frac{dt}{[(a_{1}^{2} + t)(a_{2}^{2} + t) … (a_{N}^{2} + t)]^{\frac{1}{2}}} ,

where λ is the biggest root of the equation

∑_{i=1}^{i=N} \frac{x_{i}^{2}}{a_{i}^{2} + λ} = 1,

and

\frac{1}{p} = ∫_{0}^{∞} \frac{dt}{[(a_{1}^{2} + t)(a_{2}^{2} + t) … (a_{N}^{2} + t)]^{\frac{1}{2}}} .

As seen above, the set G_{n} is contained in the ellipsoid E_{n} of (N − 1) equal halfaxes 2q^{n} sin θ_{n}, …, 2q^{n} sin θ_{n} and 2q^{n}. It contains the ellipsoid F_{n} of (N − 1) equal half-axes 2q^{n+1} sin θ_{n+1}, … 2q^{n+1} sin θ_{n+1} and 2q^{n+1}. To get the capacity of the ellipsoid

E_{N} := \{X = (x_{1}, …, x_{N}) ∈ ℝ^{N} | ∑_{i=1}^{i=N−1} \frac{x_{i}^{2}}{a^{2}} + \frac{x_{N}^{2}}{b^{2}} < 1\} ,

we need to compute the following intgral

I_{N} = ∫_{0}^{∞} \frac{dt}{(a^{2} + t)^{\frac{N−1}{2}} (b^{2} + t)^{\frac{1}{2}}} .

Using the variable change s^{2} = b^{2} − t and putting w^{2} = b^{2} − a^{2}, I_{N} becomes

I_{N} = 2 ∫_{b}^{∞} \frac{ds}{(s^{2} − w^{2})^{\frac{N−1}{2}}} .

Putting s = wx, x = cosh θ and t = tanh \frac{θ}{2}, we get

I_{N} = \frac{2^{2}}{(2w)^{N−2}} ∫_{α}^{1} \frac{(1 − t^{2})^{N−3}}{t^{N−2}} ,

with

α = tanh \frac{argch\frac{b}{w}}{2} = \sqrt{\frac{b − w}{b + w}} .

Thus,

I_{N} = \frac{2^{2}}{(2w)^{N−2}} ∑_{k=0}^{k=N−3} C_{N−3}^{k} \frac{(−1)^{k}}{2k − N + 3} [1 − (\frac{b − w}{b + w} )^{k− \frac{N−3}{2}} ]

Now, for the ellipsoid E_{n}, one has a = 2q^{n} sin θ_{n}, b = 2q^{n} and w = 2q^{n} cos θ_{n}, therefore

I_{N} = \frac{1}{2^{2N−6}q^{n(N−2)}} ∑_{k=0}^{k=N−3} C_{N−3}^{k} \frac{(−1)^{k}}{2k − N + 3} [1 − (\frac{1 − cos θ_{n}}{1 + cos θ_{n}} )^{k− \frac{N−3}{2}} ] ,

and,

\frac{Cap(E_{n})}{q^{n(N−2)}} = \frac{C_{N}(cos θ_{n})^{N−2}}{∑_{k=0}^{k=N−3} C_{N−3}^{k} \frac{(−1)^{k}}{2k−N+3} [1 − (\frac{1−cos θ_{n}}{1+cos θ_{n}} )^{k− \frac{N−3}{2}} ]}.

In the same way,

\frac{Cap(F_{n})}{q^{n(N−2)}} = \frac{C_{N}q(cos θ_{n+1})^{N−2}}{∑_{k=0}^{k=N−3} C_{N−3}^{k} \frac{(−1)^{k}}{2k−N+3} [1 − (\frac{1−cos θ_{n+1}}{1+cos θ_{n+1}} )^{k− \frac{N−3}{2}} ]}.

When θ_{n} tends to zero, \frac{Cap(E_{n})}{q^{n(N−2)}} and \frac{Cap(F_{n})}{q^{n(N−2)}} are equivalent to θ_{n}^{N−3}. As

\frac{Cap(F_{n})}{q^{n(N−2)}} < \frac{Cap(G_{n})}{q^{n(N−2}} < \frac{Cap(E_{n})}{q^{n(N−2)}},

we get

\frac{Cap(G_{n})}{q^{n(N−2)}} ∼ θ_{n}^{N−3},

or again,

\frac{Cap(G_{n})}{q^{n(N−2)}} ∼ f(q^{n})^{N−3}.

□

Remark 4.2. If f(ρ) = Aρ^{m} or f(ρ) = A exp(\frac{−m}{ρ}), then the considered cusp point

is irregular. On the other hand, if f(ρ) = [−\frac{1}{ln(ρ)}]^{\frac{1}{N−3}} , (ρ ≤ ρ_{0} < 1), it is regular.

Now, consider a point x_{0} ∈ ∂Ω ∩ ∂C. By Proposition 2.2, this eventual cusp point is contained in some cusp point of surface of revolution, which is given in spherical coordinates by,

θ = arcsin \frac{ρ}{2R}.

In particular, when ρ tends to zero, one has θ ∼ ρ/2R. By Theorem 4.2, this cusp point is regular in the sense of Wiener. One deduces therefore, that x_{0} is also regular since, in its neighbourhood, the capacity of the exterior of Ω is greater than the capacity of the set (B_{x_{0}})^{c} (of Proposition 2.2).

Now, Definition 4.3 and Theorems 3.1, and 4.1 allow us to state:

Theorem 4.3. Let Ω_{n} be a sequence of open subsets of ℝ^{N} (N = 2, 3), included in D and having C-GNP. If u_{n} is the solution of the Dirichlet problem

(P_{n}) \{ \begin{matrix}−Δu_{n} = f in Ω_{n} \\ u_{n} = 0 on ∂Ω_{n}, \end{matrix}

then there exists a subsequence of Ω_{n} (still denoted Ω_{n}) and an open subset Ω of D which is stable in the sense of Keldyš, such that:

• Ω_{n} ⟶^{H} Ω.

• Ω_{n} ⟶^{K} Ω.

• u_{n} converges strongly in H_{0}^{1}(D) to a function u, solution of the Dirichlet problem on Ω:

(P) \{ \begin{matrix}−Δu = f in Ω \\ u = 0 on ∂Ω \end{matrix}

(the functions u_{n} and u are extended by zero in D, f ∈ H^{−1}(D)).

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9-3405, Chemin des Quatre-Bourgeois, Sainte-Foy (Québec) G1W 2L1 Canada mohammed.barkatou@ramq.gouv.qc.ca

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