BEHAVIOR OF THE BEST SOBOLEV TRACE CONSTANT AND EXTREMALS IN DOMAINS WITH HOLES

JULIÁN FERNÁNDEZ BONDER, JULIO D. ROSSI AND NOEMI WOLANSKI

Abstract. We study the dependence of the Sobolev trace constant for functions defined in a bounded domain that vanish in a subset. First we find that there exists an optimal subset that makes the trace constant smaller among all the subsets with prescribed and positive Lebesgue measure. Second, we study the behavior of the best constant when the hole is allowed to have zero Lebesgue measure. We show that this constant depends continuously on the subset and we discuss when it is equal to the Sobolev trace constant without the vanishing restriction.

1. Introduction.

Given Ω ⊂ ℝ^{N} a smooth bounded domain, we study the best Sobolev trace constant corresponding to the embedding of W^{1,p}(Ω) into L^{q}(∂Ω) for functions that vanish on a fixed subset of Ω, that we will call A. We consider subcritical exponents 1 ≤ q < p_{∗} = p(N −1)/(N −p) so that the immersion W^{1,p}(Ω) ↪ L^{q}(∂Ω) is compact.

To begin with, we consider a subset A ⊂ Ω with positive measure and define the best Sobolev trace constant associated with this set, that is (1.1)

S_{A} = inf \{\frac{∫_{Ω}(|∇u|^{p} + |u|^{p}) dx}{( ∫_{∂Ω} |u|^{q} dS)^{p/q}} , u ∈ W^{1,p}(Ω) s.t. u ≢ 0 on ∂Ω and u = 0 a.e. A\}.

As a consequence of the compact immersion, there exist extremals for S_{A}. An extremal for S_{A} is a weak solution to

(1.2)  \begin{cases}−Δ_{p}u + |u|^{p−2}u = 0 & in Ω \ A, \\ |∇u|^{p−2} \frac{∂u}{∂ν} = λ|u|^{q−2}u & on ∂Ω, \\ u = 0 & in A, \end{cases}

where Δ_{p}u = div(|∇u|^{p−2}∇u) is the p−laplacian, \frac{∂}{∂ν} is the outer unit normal derivative and λ depends on the normalization of u. For instance, if ∥u∥_{L^{q}(∂Ω)} = 1, then λ = S_{A}. The existence of weak solutions to (1.2) when A = ∅ has been studied in [7]. Of special importance is the case q = p in which the equation and boundary condition in (1.2) have the same scaling. In this case, (1.2) can be considered as a nonlinear eigenvalue problem. This nonlinear eigenvalue problem, in the case A = ∅ has been studied in [7, 12]. The linear case p = 2 has been widely studied and is known as the Steklov eigenvalue problem in the literature. In this case, S_{A} is the first eigenvalue, see [14].

Key words and phrases. Sobolev trace constant, p−Laplacian, eigenvalue optimization problems, p−capacity.

2000 Mathematics Subject Classification. 35J20, 35P30, 49K20.

Supported by ANPCyT PICT No. 03-05009, CONICET PIP0660/98, UBA X431 and Fundación Antorchas. J.D. Rossi and N. Wolanski are members of CONICET.

So we are studying a nonlinear generalization of a very classical eigenvalue problem by adding the restriction that the functions involved vanish on a subset A. Up to our knowledge, this problem was not studied before. Our results are new even in the linear case.

Optimal design problems are usually formulated as problems of minimization of the energy, stored in the design under a prescribed loading. Solutions of these problems are unstable to perturbations of the loading. The stable optimal design problem is formulated as minimization of the stored energy of the project under the most unfavorable loading. This most dangerous loading is one that maximizes the stored energy over the class of admissible functions. The problem is reduced to minimization of Steklov eigenvalues. See [2].

In view of the above discussion, our first concern is to consider the following optimization problem: For a fixed 0 < α < |Ω|, find a set A_{0} of measure α that minimizes S_{A} among all subsets A ⊂ Ω of measure α. We prove that such a set exists and, in the case that Ω is a ball, we prove that there exists an optimal set that is spherically symmetric in the sense of [13] (see Section 2.3 for the definition). Moreover, in the case p = 2 we prove that every optimal set is spherically symmetric. On the other hand, we also prove that there does not exist a set A that maximizes S_{A}. Moreover, we prove that sup S_{A} = +∞ where the supremum is taken over all sets A of given measure α.

We also give a result about the continuity of S_{A} with respect to the “hole” A. Optimization problems for eigenvalues of elliptic operators has been widely stud-

ied in the past, and is still an area of intensive research. For a comprehensive description of the current developments in the field and very interesting open problems, we refer to [9]. In [3] the author studies an optimization problem for the second Neumann eigenvalue of the laplacian with A ⊂ ∂Ω. He proves the existence of an optimal window A_{0} and shows that – when Ω is a ball – A_{0} is spherically symmetric in the sense of [13]. Our approach to the optimization problem follows closely the one in [3]. Optimal design problems have been widely studied not only for eigenvalue problems. See for instance [1, 10, 11].

When trying to give sense to a best Sobolev trace constant for functions that vanish in a set of zero Lebesgue measure a different approach has to be made. We consider the space

(1.3)  W_{A}^{1,p}(Ω) = \overline{C_{0}^{∞}(\overline{Ω} \ A)},

where the closure is taken in W^{1,p}−norm. That is, W_{A}^{1,p}(Ω) stands for the set of functions of the Sobolev space W^{1,p}(Ω) that can be approximated by smooth functions that vanish in a neighborhood of A.

In this context the best Sobolev trace constant is defined as

(1.4)  S_{A} = inf_{u∈W_{A}^{1,p}(Ω)\W_{0}^{1,p}(Ω)} \frac{∫_{Ω} |∇u|^{p} + |u|^{p} dx}{(∫_{∂Ω} |u|^{q} dS)^{p/q}}.

In this case we prove that this problem only makes sense if A is a set with positive p−capacity (see (1.6)). More precisely, we prove that S_{A} = S_{∅} if and only if the p−capacity of A is zero. Note that S_{∅} is the usual Sobolev trace constant from W^{1,p}(Ω) onto L^{q}(∂Ω).

Observe that the constants S_{A} and S_{A} need not be the same. For a discussion on this relation, see the end of Section 3.

Finally, we address the problem of the continuity of S_{A} with respect to A. Here the natural topology for the sets A is the Hausdorff distance and in fact, we prove that S_{A} is continuous in this topology. Also, we prove the continuity of the extremals with respect to the Hausdorff distance.

In order to finish this introduction, let us comment briefly on related work. First we comment on works related to the dependence of the Sobolev trace constant with respect to variations of the domain. In [5] the authors analyze the behavior of extremals and best Sobolev trace constants in expanding domains for p = 2 and q > 2. They prove that the extremals develop a peak near the point where the mean curvature of the boundary attains a maximum. Also in [8] the authors analyze the dependance of the best Sobolev trace constant and extremals in expanding and contracting domains for p > 1 and 1 ≤ q < p_{∗}. Finally, in [6] the behavior of the Sobolev trace constant and extremals in thin domains is analyzed.

1.1. Main results. Now we state the main results of the paper.

Our first result is the sequential lower semicontinuity of S_{A}.

Theorem 1.1. Let A_{n} ⊂ Ω be sets of positive measure such that

χ_{A_{n}} ⇀^{∗} χ_{A_{0}} in L^{∞}(Ω),

where χ_{A} is the characteristic function of the set A. Then

S_{A_{0}} ≤ lim inf_{n→∞} S_{A_{n}},

where S_{A} is given by (1.1).

We remark that the continuity is not true in general. See Remark 2.1.

This semicontinuity result suggest that a minimizer for S_{A} among sets A of fixed positive Lebesgue measure exists. However, there is a major difficulty here because of the fact that sets of prescribed positive Lebesgue measure are not compact with respect to the topology of Theorem 1.1. We overcome this difficulty in the next theorem.

Our result concerning the existence of an optimal design for the constant S_{A} is as follows.

Theorem 1.2. Given 0 < α < |Ω|, let us define

(1.5)  S(α) := inf_{A⊂Ω, |A|=α} S_{A}.

Then, there exists a set A_{0} ⊂ Ω such that |A_{0}| = α and S_{A_{0}} = S(α).

On the other hand, there is no upper bound for S(α).

Theorem 1.3. Let 0 < α < |Ω|. Then,

sup_{A⊂Ω, |A|=α} S_{A} = ∞.

Next we study symmetry properties of optimal sets A_{0} in the special case where Ω is a ball. To this end, we need the definition of spherical symmetrization (see [13]). Given a measurable set A ⊂ ℝ^{N}, the spherical symmetrization A^{∗} of A is defined as follows: for each r, take A ∩ ∂B(0, r) and replace it by the spherical cap of the same area and center re_{N}. The union of these caps is A^{∗}.

We have the following result,

Theorem 1.4. Let Ω = B(0, 1) and 0 < α < |B(0, 1)|. Then, there exists an optimal hole of measure α which is spherically symmetric, that is A^{∗} = A. Moreover, when p = 2, every optimal hole is spherically symmetric.

Now we state the results that allow us to consider the case |A| = 0. For simplicity we will consider closed sets A.

First, we study when S_{A} is equal to the usual Sobolev trace constant, that is, when S_{A} = S_{∅}. For this purpose we recall de definition of p−capacity (see [4]). For A ⊂ Ω, define

(1.6) Cap_{p}(A) = inf{∫_{ℝ^{n}} |∇ϕ|^{p} dx | ϕ ∈ W^{1,p}(ℝ^{N})∩C^{∞}(ℝ^{N}) and A ⊂ {ϕ ≥ 1}^{∘}}.

We have

Theorem 1.5. Let A ⊂ Ω then W_{A}^{1,p}(Ω) = W^{1,p}(Ω) if and only if Cap_{p}(A) = 0.

As a corollary we obtain

Corollary 1.1. Cap_{p}(A) = 0 if and only if S_{A} = S_{∅}.

Next, we look for the dependence of S_{A} under perturbations of A. We find that S_{A} is continuous with respect to A in the topology given by the Haussdorff distance.

Theorem 1.6. Let A, A_{n} ⊂ Ω be closed sets such that d(A_{n}, A) → 0 as n → ∞ where d(A_{n}, A) is the Hausdorff distance between A_{n} and A. Then

|S_{A_{n}} − S_{A}| → 0 when n → ∞

and if we denote by u_{n} an extremal for S_{A_{n}} normalized such that ∥u_{n}∥_{L^{q}(∂Ω)} = 1, there exists a subsequence u_{n_{k}} such that

(1.7)  lim_{k→∞} u_{n_{k}} = u,  strongly in W^{1,p}(Ω),

and u is an extremal for S_{A}.

The rest of the paper is organized as follows, in Section 2 we prove the existence of an optimal set A and its symmetry properties when Ω is a ball. In Section 3 we prove our results involving the p−capacity of a subset and the continuous dependence of the Sobolev constant on the subset A.

Throughout the paper, by C we mean a constant that may vary from line to line but remains independent of the relevant quantities.

2. The optimization problem

In this section, we prove the sequential lower semicontinuity of the best Sobolev trace constant S_{A} with respect to A (Theorem 1.1). Then following ideas from [3], we prove that if we consider holes with positive and fixed measure a minimizing hole does exist (Theorem 1.2). Moreover, we prove that a maximizing hole does not exist (Theorem 1.3). Finally, we study the symmetry properties of minimizing holes when Ω is a ball (Theorem 1.4).

2.1. Semicontinuity result.

Proof of Theorem 1.1. Let A_{n}, A ⊂ Ω such that χ_{A_{n}} ⇀^{∗} χ_{A} in L^{∞}(Ω).

Let u_{n} ∈ W^{1,p}(Ω) be an extremal for S_{A_{n}} normalized such that ∥u_{n}∥_{L^{q}(∂Ω)} = 1 and u_{n} ≥ 0. Let α = lim inf S_{A_{n}}. Without loss of generality, we may assume that α < ∞ and that S_{A_{n}} → α.

Then, there exists a constant C such that

∥u_{n}∥_{W^{1,p}(Ω)} ≤ C.

Thus, there exists a function u ∈ W^{1,p}(Ω) such that, for a subsequence that we still call u_{n},

(2.1)

u_{n} ⇀ u weakly in W^{1,p}(Ω),
u_{n} → u strongly in L^{p}(Ω),
u_{n} → u strongly in L^{q}(∂Ω).

In particular, ∥u∥_{L^{q}(∂Ω)} = 1, u ≥ 0 and

∥u∥_{W^{1,p}(Ω)} ≤ lim inf ∥u_{n}∥_{W^{1,p}(Ω)}.

We claim that u is an admissible function in the characterization of S_{A}. To this end, observe that by (2.1) and by our hypotheses on A_{n} and A,

∫_{A} u dx = lim ∫_{A_{n}} u_{n} dx = 0.

As u ≥ 0, the claim follows.

Now,

S_{A} ≤ ∥u∥_{W^{1,p}(Ω)}^{p} ≤ lim inf ∥u_{n}∥_{W^{1,p}(Ω)}^{p} = lim inf S_{A_{n}}.

The proof is now complete. □

Remark 2.1. The continuity of S_{A} with respect to A in the sense of Theorem 1.1 does not hold in general.

For instance, take A_{n} = B_{r}(x^{0})∪B_{1/n}(x^{1}) and A = B_{r}(x^{0}). It is easy to see that χ_{A_{n}} ⇀^{∗} χ_{A}. If p > N, as W^{1,p}(Ω) ↪ C(\overline{Ω}) and an extremal u_{n} for S_{A_{n}} vanishes in x^{1}, then u_{n} → u uniformly and so u(x^{1}) = 0. On the other hand, if S_{A} = lim S_{A_{n}} then u would be an extremal for S_{A}. Thus u is a weak solution of (1.2) which implies, by the maximum principle (see [16]), that u > 0 in Ω \ A. A contradiction.

For general p, we can take A_{n} = B_{r}(x^{0}) ∪ (0, 1/n) × Σ where Σ is a portion of the hyperplane {x_{1} = 0} and A = B_{r}(x^{0}). Arguing in a similar fashion as before, using W^{1,p}(Ω) ↪ L^{p}(Σ), S_{A} < lim S_{A_{n}}.

2.2. Existence of an optimal hole. In order to prove Theorem 1.2, we define the functional

J(u) = ∫_{Ω} |∇u|^{p} + |u|^{p} dx

and try to minimize it on the set

D(Ω, α) ={(u, χ) ∈ W^{1,p}(Ω) × L^{∞}(Ω) ;
∫_{∂Ω} |u|^{q} = 1, χ^{2} = χ, ∫_{Ω} χ = α, u · χ = 0 a.e. Ω}.

It will be useful, to consider the larger set

E(Ω, α) ={(u, ϕ) ∈ W^{1,p}(Ω) × L^{∞}(Ω) ;
∫_{∂Ω} |u|^{q} = 1, 0 ≤ ϕ ≤ 1, ∫_{Ω} ϕ ≥ α, u · ϕ = 0 a.e. Ω},

to minimize J there.

Theorem 1.2 follows from the following theorem.

Theorem 2.1. There exists (u, χ) ∈ D(Ω, α) that minimizes J(u) on D(Ω, α). Moreover, every minimizer of J(u) on E(Ω, α) actually belongs to D(Ω, α).

Proof. Let (u_{n}, ϕ_{n}) be a minimizing sequence of J(u) on E(Ω, α). Observe that by taking (|u_{n}|, χ_{supp(ϕ_{n})}) we may assume that u_{n} ≥ 0 and ϕ_{n} is a characteristic function. Moreover, as (u_{n}, ϕ_{n}) is a minimizing sequence, we get

∥u_{n}∥_{W^{1,p}(Ω)} ≤ C.

Hence, taking a subsequence if necessary, we may assume that there exists u ∈ W^{1,p}(Ω) such that

(2.2)

u_{n} ⇀ u, weakly in W^{1,p}(Ω),
u_{n} → u, strongly in L^{q}(∂Ω),
u_{n} → u, strongly in L^{p}(Ω).

For the functions ϕ_{n}, we have that ∥ϕ_{n}∥_{L^{p′} (Ω)} ≤ C, and so

(2.3)  ϕ_{n} ⇀ ϕ, weakly in L^{p′}(Ω),

for some function ϕ ∈ L^{p′}(Ω), 0 ≤ ϕ ≤ 1, ∫_{Ω} ϕ ≥ α.

Observe that, by (2.2) and (2.3), u ≥ 0, ∫_{∂Ω} u^{q} = 1 and ∫_{Ω} uϕ = 0. So uϕ = 0 a.e. Ω. Therefore, the pair (u, ϕ) is a minimizer for J on E(Ω, α).

Now, we want to construct a minimizer on D(Ω, α). To this end let us replace ϕ by χ = χ_{supp(ϕ)}.

In order to see that (u, χ) ∈ D(Ω, α) we only need to check that ∫_{Ω} χ = α. This will follow from the fact that S(α) is a strictly decreasing function of α. Recall that S(α) is defined as

S(α) := inf_{A⊂Ω, |A|=α} S_{A}.

In fact, if ∫_{Ω} χ = α_{1} > α, we have that (u, χ) ∈ D(Ω, α_{1}) and so S(α_{1}) ≤ J(u) = S(α). This contradicts the strict monotonicity.

It remains to show the strict monotonicity property. The fact that S(α) is nondecreasing is a direct consequence of the inclusions

{u ; there exists χ such that χ^{2} = χ, ∫_{Ω} χ dx = α, uχ = 0}
⊂ {u ; there exists χ such that χ^{2} = χ, ∫_{Ω} χ dx = α_{1}, uχ = 0},

when α > α_{1}.

In order to show that the monotonicity is strict let us consider α_{0} < α_{2}. From our previous calculations it follows that there exists (u, χ) ∈ E(Ω, α_{2}) that minimizes J in E(Ω, α_{2}) with χ^{2} = χ. By replacing χ by the characteristic function of a smaller set of measure α_{2} we may assume that (u, χ) ∈ D(Ω, α_{2}) and

S(α_{2}) = J(u).

Let D = supp χ and let δ < α_{2} − α_{0}. We may replace D by a closed subset of measure α_{2} − δ > α_{0}, that we will still call D. Hence, we will assume that D is closed.

Let Ω̃ be the connected component of Ω \ D that contains the boundary of Ω in its boundary. Let us cover ∂Ω̃ \ ∂Ω with balls B_{ε}(x_{j}) of radius ε (ε small, such that |B_{2ε}| < α_{2} − α_{0} − δ). If the measure of B_{ε}(x_{j}) ∩ D is zero for every j we may replace D by a smaller closed set of the same measure by removing B_{ε}(x_{j}) ∩ D for every j. Therefore we may assume that, for some j, B_{ε}(x_{j}) ∩ D has positive measure (otherwise repeating this argument a finite number of times we will obtain that D has measure zero, a contradiction). So, let x_{0} ∈ ∂Ω̃ \ ∂Ω be such that |B_{ε}(x_{0}) ∩ D| > 0 and let D_{1} = D \ B_{2ε}(x_{0}). Then α_{0} < |D_{1}| < α_{2} − δ.

Let u_{1} be such that J(u_{1}) = S_{D_{1}}. As u_{1} is a nonnegative solution of Δ_{p}u_{1} = u_{1}^{p−1} in Ω \ D_{1}, we obtain that u_{1} > 0 in B_{ε}(x_{0}). If not, using Harnack inequality we find that u_{1} = 0 in Ω̃ a contradiction with the fact that ∫_{∂Ω} |u_{1}|^{q} = 1.

On the other hand, J(u) ≥ J(u_{1}) and if J(u) = J(u_{1}) = S_{D_{1}} we would have that u is a solution of Δ_{p}u = u^{p−1} in Ω \ D_{1}. As before we deduce that u is positive in B_{ε}(x_{0}), which contradicts the fact that u = 0 in D. Hence J(u) > J(u_{1}).

Therefore

S(α_{0}) ≤ S(α_{1}) ≤ S_{D_{1}} = J(u_{1}) < J(u) = S(α_{2}),

and the strict monotonicity is proved.

This ends the proof of the Theorem. □

Now we prove that a maximal hole does not exist.

Proof of Theorem 1.3. Let ε > 0 and let δ = δ(ε) be such that

A_{ε} = {x ∈ Ω / ε < dist (x, ∂Ω) < δ}

has measure α. We will see that S_{A_{ε}} → ∞ as ε → 0. In fact, let u_{ε} ∈ W_{A_{ε}}^{1,p}(Ω) be an extremal for S_{A_{ε}} normalized so that ∥u_{ε}∥_{L^{q}(∂Ω)} = 1.

Let

Ω_{ε} = {x ∈ Ω / dist (x, ∂Ω) > ε}.

Observe that u_{ε} is a solution to

\begin{cases}Δ_{p}v = |v|^{p−2}v & in Ω_{ε}, \\ v = 0 & on ∂Ω_{ε}. \end{cases}

Thus, u_{ε} = 0 in Ω_{ε} and so u_{ε} → 0 a.e. Ω.

If S_{A_{ε}} were bounded then, up to a subsequence, there exists a function u ∈ W^{1,p}(Ω) such that

u_{ε} ⇀ u weakly in W^{1,p}(Ω),
u_{ε} → u strongly in L^{p}(Ω) and a.e. in Ω,
u_{ε} → u strongly in L^{q}(∂Ω).

Since u_{ε} → 0 a.e. Ω and ∥u_{ε}∥_{L^{q}(∂Ω)} = 1 we arrive at a contradiction. □

2.3. Symmetry properties of optimal holes. In this subsection, we consider the case Ω = B(0, 1) and investigate if the optimal hole constructed in the previous section inherits some symmetry from the domain.

To this end, we recall the definition of spherical symmetrization that we have given in the introduction. Given a measurable set A ⊂ ℝ^{N}, the spherical symmetrization A^{∗} of A is constructed as follows: for each r, take A ∩ ∂B(0, r) and replace it by the spherical cap of the same area and center re_{N}. This can be done for almost every r. The union of these caps is A^{∗}. Now, the spherical symmetrization u^{∗} of a measurable function u ≥ 0 is constructed by symmetrizing the super-level sets so that, for every t, {u^{∗} ≥ t} = {u ≥ t}^{∗}. See [13] for more details.

The following theorem is proved in [13].

Theorem 2.2. Let u ∈ W^{1,p}(B(0, 1)) and let u^{∗} be its spherical symmetrization. Then u^{∗} ∈ W^{1,p}(B(0, 1)) and

∫_{B(0,1)} |∇u^{∗}|^{p} dx ≤ ∫_{B(0,1)} |∇u|^{p} dx, (2.4)

∫_{B(0,1)} |u^{∗}|^{p} dx = ∫_{B(0,1)} |u|^{p} dx, (2.5)

∫_{∂B(0,1)} |u^{∗}|^{q} dS = ∫_{∂B(0,1)} |u|^{q} dS. (2.6)

With these preliminaries, we are ready to prove Theorem 1.4.

Proof of Theorem 1.4. Let (u, A) be an extremal for J and u^{∗} the spherical symmetrization of u. Now, by Theorem 2.2, (u^{∗}, A^{∗}) is an admissible pair for the minimization problem and J(u^{∗}) ≤ J(u). So the first part of the theorem follows.

Assume now that p = 2. In this case, it is proved in [3] that if equality holds in (2.4) then for each 0 < r ≤ 1 there exists a rotation R_{r} such that

(2.7)  u |_{∂B(0,r)}= (u^{∗} ∘ R_{r}) |_{∂B(0,r)} .

We can assume that the axis of symmetry e_{N} was taken so that R_{1} = Id.

Let w = u − u^{∗}. Then w satisfies

\begin{cases}Δw = w & in B(0, 1) \ (A ∪ A^{∗}), \\ w = 0 & on ∂B(0, 1), \\ \frac{∂w}{∂ν} = 0 & on ∂B(0, 1). \end{cases}

Since u^{∗} = u ≢ 0 on ∂B(0, 1) there exists an open set D in B(0, 1)\(A∪A^{∗}) with ∂D ∩∂B(0, 1) ≠ ∅. By Holmgren’s uniqueness theorem there holds that w = 0 in D and by analytic continuation, w = 0 in the connected component of B(0, 1)\(A∪A^{∗}) that contains the set D.

Thus, w = 0 in every connected component D_{i} such that ∂D_{i} ∩ ∂B(0, 1) ≠ ∅.

Let K = ∂(B(0, 1) \ (⋃ D_{i})) ⊂ (A ∪ A^{∗}). Observe that K = K_{1} ∪ K_{2} where K_{1} ⊂ ∂B(0, 1) ∩ (A ∪ A^{∗}) and K_{2} ⊂ ∂(⋃ D_{i}) ∩ (A ∪ A^{∗}).

If x ∈ K_{1} then either u(x) = 0 or u^{∗}(x) = 0, but u = u^{∗} since K_{1} ⊂ ∂B(0, 1). Thus u = u^{∗} = 0 in K_{1}.

If x ∈ K_{2} then w(x) = 0 since K_{2} ⊂ ∪D_{i}. As either u(x) = 0 or u^{∗}(x) = 0 we have again that u = u^{∗} = 0 in K_{2}.

Finally, since both u and u^{∗} are extremals for J, if follows that u = u^{∗} = 0 in B(0, 1) \ (⋃ D_{i}). So we have proved that w = u − u^{∗} = 0 in B(0, 1) \ (A ∪ A^{∗}).

It remains to show that A = A^{∗} and that w = 0 in B(0, 1). To this end, assume that B = A^{∗} \ A ≠ ∅. Then ∂B = (∂A^{∗} \ A) ∪ (∂A ∩ A^{∗}), so w = 0 on ∂B. In the interior, w = u and so it verifies Δw = w. Therefore, w = 0 in B. Arguing in a similar way we conclude that w = 0 in A ∪ A^{∗}, as we wanted to show. □

3. Capacitary setting

In this section we consider general sets that may have zero Lebesgue measure. First, we analyze when the Sobolev trace constant sees the set A. This is the content of Theorem 1.5. Then, we prove the continuity of S_{A} with respect to A in Hausdorff distance. We end this section with a discussion about the relationship between S_{A} and S_{A}.

Proof of Theorem 1.5. Let Ω ⊂ ℝ^{N} be a bounded smooth domain and take A a subset of Ω such that Cap_{p}(A) = 0.

From (1.6), it follows that Cap_{p}(A) = 0 if and only if, for every ε > 0 there exists ϕ_{ε} ∈ W^{1,p}(ℝ^{N}) ∩ C^{∞}(ℝ^{N}) such that ∥∇ϕ_{ε}∥_{L^{p}(ℝ^{N})}^{p} < ε and ϕ_{ε} ≡ 1 in a neighborhood of A.

Take now u ∈ C^{∞}(\overline{Ω}) and define u_{ε} = (1 − ϕ_{ε})u ∈ W_{A}^{1,p}(Ω). Then we have

∥u − u_{ε}∥_{W^{1,p}(Ω)} = ∥ϕ_{ε}u∥_{W^{1,p}(Ω)}.

We will show that ∥ϕ_{ε}u∥_{W^{1,p}(Ω)} → 0 as ε → 0. From this fact, the result will follow since C^{∞}(\overline{Ω}) is dense in W^{1,p}(Ω).

Now,

\begin{aligned}∥ϕ_{ε}u∥_{W^{1,p}(Ω)}^{p} & ≤ C (∫_{Ω} |∇ϕ_{ε}|^{p}|u|^{p} dx + ∫_{Ω} |ϕ_{ε}|^{p}|∇u|^{p} dx + ∫_{Ω} |ϕ_{ε}|^{p}|u|^{p} dx) \\ & = C(I + II + III). \end{aligned}

We will bound each term separately.

(3.1)  I ≤ ∥u∥_{∞} ∫_{Ω} |∇ϕ_{ε}|^{p} dx < ∥u∥_{∞}ε.

The terms II and III are bounded in the same way. We perform the computations for II. By Hölder’s inequality we have

II ≤ (∫_{Ω} |ϕ_{ε}|^{pα} dx)^{1/α} (∫_{Ω} |∇u|^{pα′} dx)^{1/α′} .

So, if we take α = p^{∗}/p where p^{∗} = Np/(N − p) is the critical Sobolev exponent, it follows that

II ≤ (∫_{ℝ^{N}} |ϕ_{ε}|^{p^{∗}} dx)^{p/p^{∗}} (∫_{Ω} |∇u|^{N} dx)^{p/N} ≤ ∥∇u∥_{∞}^{p}|Ω|^{p/N} (∫_{ℝ^{N}} |ϕ_{ε}|^{p^{∗}} dx)^{p/p^{∗}} .

Then, by the Gagliardo–Nirenberg–Sobolev inequality,

(3.2)  II ≤ C∥∇u∥_{∞}^{p}|Ω|^{p/N} ∫_{ℝ^{N}} |∇ϕ_{ε}|^{p} dx < C∥∇u∥_{∞}^{p}|Ω|^{p/N}ε.

Analogously, for III we have

(3.3)  III ≤ C∥u∥_{∞}^{p}|Ω|^{p/N}ε.

Combining (3.1), (3.2) and (3.3) we have

(3.4)  ∥u − u_{ε}∥_{W^{1,p}(Ω)}^{p} = ∥ϕ_{ε}u∥_{W^{1,p}(Ω)}^{p} ≤ Cε.

Assume now that C_{0}^{∞}(\overline{Ω} \ A) is dense in W^{1,p}(Ω). We will show that Cap_{p}(A) = 0.

By hypotheses, for every ε > 0, there exists u_{ε} ∈ C^{∞}(\overline{Ω}) such that

A ∩ supp(u_{ε}) = ∅ and ∥1 − u_{ε}∥_{W^{1,p}(Ω)} < ε.

Take ϕ_{ε} = 1−u_{ε}, then ϕ_{ε} ∈ W^{1,p}(Ω). As Ω is smooth, it has the extension property. Then, there exists ϕ̄_{ε} ∈ W^{1,p}(ℝ^{N}) such that

∥ϕ̄_{ε}∥_{W^{1,p}(ℝ^{N})} ≤ E∥ϕ_{ε}∥_{W^{1,p}(Ω)}.

But

∫_{ℝ^{N}} |∇ϕ̄_{ε}|^{p} dx ≤ ∥ϕ̄_{ε}∥_{W^{1,p}(ℝ^{N})}^{p} ≤ E^{p}∥ϕ_{ε}∥_{W^{1,p}(Ω)}^{p} < E^{p}ε^{p},

as we wanted to show. □

With this theorem, we easily prove Corollary 1.1.

Proof of Corollary 1.1. We only need to see that if S_{A} = S_{∅} then Cap_{p}(A) = 0. Let u be an extremal for S_{A}. As S_{A} = S_{∅}, u is also an extremal for S_{∅}. Therefore, u is a weak solution to

\begin{cases}−Δ_{p}u + |u|^{p−2}u = 0 & in Ω, \\ |∇u|^{p−2} \frac{∂u}{∂ν} = λ|u|^{q−2}u & on ∂Ω. \end{cases}

By known regularity theory (see [15]) and the maximum principle (see [16]) it follows that u ∈ C^{1,α}(\overline{Ω}) and u > 0 in \overline{Ω}.

As u ∈ W_{A}^{1,p}(Ω), there exists a sequence u_{n} ∈ C_{0}^{∞}(\overline{Ω} \ A) such that u_{n} → u in W^{1,p}(Ω).

Let ϕ_{n} = 1 − \frac{u_{n}}{u}. Observe that ϕ_{n} ≡ 1 in a neighborhood of A. Moreover,

\begin{aligned}∥∇ϕ_{n}∥_{L^{p}(Ω)} & = ∥\frac{1}{u}∇(u_{n} − u) + \frac{1}{u^{2}} ∇u(u − u_{n})∥_{L^{p}(Ω)} \\ & ≤ C∥∇(u_{n} − u)∥_{L^{p}(Ω)} + C∥∇u∥_{L^{∞}(Ω)}∥u − u_{n}∥_{L^{p}(Ω)}, \end{aligned}

where we used the fact that u ≥ α > 0 in Ω. Then, for every ε > 0 there exists n_{0} such that

∥∇ϕ_{n}∥_{L^{p}(Ω)} < ε if n ≥ n_{0}.

Now the result follows by extending ϕ_{n} to ℝ^{N} and regularizing. □

Remark 3.1. If the set A is a regular surface of dimension k, then there exists a trace operator

T : W^{1,p}(Ω) → L^{p}(A)

if k > N − p. This relates to Theorem 1.5 by the fact that sets of Hausdorff dimension less than or equal to N − p have zero p−capacity (see [4]).

Now we prove the continuity of S_{A} with respect to A in Hausdorff distance. Recall that the Hausdorff distance is definded by,

d(A_{1}, A_{2}) = inf{r > 0 , A_{1} ⊂ B_{r}(A_{2}) and A_{2} ⊂ B_{r}(A_{1})}.

Here B_{r}(A) = ⋃_{x∈A} B_{r}(x) is the usual fattening of A.

Proof of Theorem 1.6. Let A_{ε} = B_{ε}(A). Assume that d(A_{n}, A) → 0 as n → ∞, then given ε > 0 there exists n_{0} such that A, A_{n} ⊂ A_{ε} if n ≥ n_{0} and it follows that

(3.5)  W_{A_{ε}}^{1,p}(Ω) ⊂ W_{A}^{1,p}(Ω) ∩ W_{A_{n}}^{1,p}(Ω).

First, observe that by (1.4), we have

(3.6)  S = S_{∅} ≤ S_{A}, S_{A_{n}} ≤ S_{A_{ε}},

if n ≥ n_{0}.

Now, let u ∈ W_{A}^{1,p}(Ω) be an extremal for S_{A} normalized such that ∥u∥_{L^{q}(∂Ω)} = 1. As u ∈ W_{A}^{1,p}(Ω), given δ > 0 there exists u_{δ} ∈ C_{0}^{∞}(\overline{Ω} \ A) such that

∥u − u_{δ}∥_{W^{1,p}(Ω)} ≤ δ

and, moreover, we can assume that

(3.7)  supp(u_{δ}) ⊂ \overline{Ω} \ A_{ε}

if ε is small enough.

Now, by (3.5) and (3.7), u_{δ} ∈ W_{A_{n}}^{1,p}(Ω) for n ≥ n_{0}, so

S_{A_{n}} ≤ \frac{∥u_{δ}∥_{W^{1,p}(Ω)}^{p}}{∥u_{δ}∥_{L^{q}(∂Ω)}^{p}} ≤ \begin{matrix}(∥u∥_{W^{1,p}(Ω)} + ∥u_{δ} − u∥_{W^{1,p}(Ω)}) & p \\ (∥u∥_{L^{q}(∂Ω)} − ∥u_{δ} − u∥_{L^{q}(∂Ω)}) & p \end{matrix}

≤ \frac{(S_{A}^{1/p} + δ)^{p}}{(1 − S^{−1/p}δ)^{p}} ≤ S_{A} + C_{1}δ,

where C_{1} is a constant that depends on S and S_{A}.

By symmetry, we get

S_{A} ≤ S_{A_{n}} + C_{2}δ,

where C_{2} depends on S and S_{A_{n}}, but by (3.6) it can be taken depending on S and S_{A_{ε}} only.

So

|S_{A_{n}} − S_{A}| ≤ Cδ if n ≥ n_{0},

as we wanted to show.

It remains to see the convergence of the extremals. Let u_{n} be an extremal for S_{A_{n}} normalized such that ∥u_{n}∥_{L^{q}(∂Ω)} = 1. Then, as

∥u_{n}∥_{W^{1,p}(Ω)} = S_{A_{n}}^{1/p} → S_{A}^{1/p},

it follows that u_{n} is bounded in W^{1,p}(Ω). So there exists a sequence (that we still denote u_{n}) and a function u ∈ W^{1,p}(Ω) such that

u_{n} ⇀ u weakly in W^{1,p}(Ω),
u_{n} → u strongly in L^{q}(∂Ω).

(3.8)

By the definition of the spaces W_{A_{n}}^{1,p}(Ω), W_{A}^{1,p}(Ω) and by (3.5), it is straightforward to see that u ∈ W_{A}^{1,p}(Ω).

Now, by (3.8), ∥u∥_{L^{q}(∂Ω)} = 1 and, also by (3.8),

S_{A} ≤ ∥u∥_{W^{1,p}(Ω)}^{p} ≤ lim_{n→∞} ∥u_{n}∥_{W^{1,p}(Ω)}^{p} = lim_{n→∞} S_{A_{n}} = S_{A}.

Therefore, u is an extremal for S_{A} and u_{n} → u strongly in W^{1,p}(Ω). The proof is now complete. □

3.1. Relationship between S_{A} and S_{A}. It is easy to see that S_{A} ≤ S_{A}. The other inequality is not true in general. Moreover, the constant S_{A} is not modified if we change the set A by a set of zero Lebesgue measure while S_{A} will be modified unless the variation on A is of zero p−capacity.

For example, if A is a hypersurface contained in Ω, then S_{A} = S_{∅}. On the other hand S_{A} > S_{∅} = S_{∅} = S_{A} since an extremal for S_{A} has zero trace on A.

However, if A is the closure of an open set with regular boundary, both constants agree.

This leads to the question of the existence of a representative of the set A for both constants to agree. We believe that this is an interesting problem.

Acknowledgements. We want to thank R. Duran for valuable help.

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Julián Fernández Bonder and Noemi Wolanski Departamento de Matemática, FCEyN UBA (1428) Buenos Aires, Argentina.

E-mail address: jfbonder@dm.uba.ar, wolanski@dm.uba.ar
Web-page:  http://mate.dm.uba.ar/∼jfbonder, http://mate.dm.uba.ar/∼wolanski

Julio D. Rossi Departamento de Matemática, Universidad Católica de Chile, Casilla 306, Correo 22, Santiago, Chile on leave from Departamento de Matemática, FCEyN UBA (1428) Buenos Aires, Argentina.

E-mail address: jrossi@mat.puc.cl
Web-page:  http://mate.dm.uba.ar/∼jrossi