The best constant for the Sobolev trace embedding from W^{1,1}(Ω) into L^{1}(∂Ω).

F. Andreu^{a} J.M. Mazón^{a} J.D. Rossi^{b}

aDepartamento de Análisis Matemático, Universitat de Valencia, andreu@uv.es, mazon@uv.es

bDept. de Matemáticas, Universidad Católica de Chile, jrossi@riemann.mat.puc.cl

Abstract

In this paper we study the best constant, λ_{1}(Ω) for the trace map from W^{1,1}(Ω) into L^{1}(∂Ω). We show that this constant is attained in BV (Ω) when λ_{1}(Ω) < 1. Moreover, we prove that this constant can be obtained as limit when p ↘ 1 of the best constant of W^{1,p}(Ω) ↪ L^{p}(∂Ω). To perform the proofs we will look at Neumann problems involving the 1−Laplacian, Δ_{1}(u) = div(Du/|Du|).

Key words: Sobolev trace embedding, functions of bounded variation AMS (MOS) subject classification: 35J65, 35B33.

1 Introduction

Let Ω be a bounded set in IR^{N} with Lipschitz continuous boundary ∂Ω. Of importance in the study of boundary value problems for differential operators in Ω are the Sobolev trace inequalities. In particular, W^{1,1}(Ω) ↪ L^{1}(∂Ω) and hence the following inequality holds:

λ∥u∥_{L^{1}(∂Ω)} ≤ ∥u∥_{1,1},

for all u ∈ W^{1,1}(Ω). The best constant for this embedding is the largest λ such that the above inequality holds, that is,

λ_{1}(Ω) = inf {∫_{Ω} |u| + ∫_{Ω} |∇u| : u ∈ W^{1,1}(Ω), ∫_{∂Ω} |u| = 1} . (1)

Our main interest in this paper is to study the dependence of the best constant λ_{1}(Ω) and extremals (functions where the constant is attained) on the

Preprint submitted to Elsevier Preprint 22 May 2004

domain. A related problem was studied by F. Demengel in [7] (see Remark 2). We remark that the existence of extremals is not trivial, due to the lack of compactness of the embedding.

For 1 < p ≤ N, let us consider the variational problem

inf {∫_{Ω} |u|^{p} + ∫_{Ω} |∇u|^{p} : u ∈ W^{1,p}(Ω), ∫_{∂Ω} |u|^{p} = 1} . (2)

If we denote by λ_{p}(Ω) the above infimum, we have that

λ_{p}(Ω) = inf \{\frac{∫_{Ω} |u|^{p} + ∫_{Ω} |∇u|^{p}}{∫_{∂Ω} |u|^{p}} : u ≢ 0 on ∂Ω, u ∈ W^{1,p}(Ω)\} (3)

is the best constant for the trace map from W^{1,p}(Ω) into L^{p}(∂Ω). Due to the compactness of the embedding W^{1,p}(Ω) ↪ L^{p}(∂Ω), it is well known (see for instance [10]) that problem (3) has a minimizer in W^{1,p}(Ω). These extremals are weak solutions of the following problem

\{ \begin{matrix}Δ_{p}u = |u|^{p−2}u & in Ω, \\ |∇u|^{p−2} \frac{∂u}{∂ν} = λ|u|^{p−2}u & on ∂Ω, \end{matrix} (4)

where Δ_{p}u = div(|∇u|^{p−2}∇u) is the p-Laplacian, \frac{∂}{∂ν} is the outer unit normal derivative and if we use the normalization ∥u∥_{L^{p}(∂Ω)} = 1, one can check that λ = λ_{p}(Ω), see [11].

Our first result says that λ_{1}(Ω) is the limit as p ↘ 1 of λ_{p}(Ω) and provides a bound for λ_{1}(Ω).

Theorem 1 We have that

lim_{p↘1} λ_{p}(Ω) = λ_{1}(Ω)

and

λ_{1}(Ω) ≤ min \{\frac{|Ω|}{P (Ω)}, 1\} ,

where P (Ω) stands for the perimeter of Ω.

Therefore, it seems natural to search for an extremal for λ_{1}(Ω) as the limit of extremals for λ_{p}(Ω) when p ↘ 1. Formally, if we take limit as p ↘ 1 in equations (4), we get

\{ \begin{matrix}Δ_{1}u := div (\frac{Du}{|Du|}) = \frac{u}{|u|} in Ω \\ \frac{Du}{|Du|} · ν = λ_{1}(Ω)\frac{u}{|u|} on ∂Ω. \end{matrix} (5)

Hence we will look at Neumann problems involving the 1−Laplatian, Δ_{1}(u) = div(Du/|Du|) in the context of bounded variation functions (the natural context for this type of problems). To our knowledge the results obtained here have independent interest.

We shall say that Ω has the trace-property if there exists a vector field z_{Ω} ∈ L^{∞}(Ω, IR^{N}), with ∥z_{Ω}∥_{∞} ≤ 1 such that div(z_{Ω}) ∈ L^{∞}(Ω) and

[z_{Ω}, ν] = λ_{1}(Ω) H^{N−1} − a.e. on ∂Ω.

Our main result states that for any domain having the trace property, the best Sobolev trace constant, λ_{1}(Ω), is attained by a function in L^{1}(Ω) whose derivatives in the sense of distributions are bounded measures on Ω, that is a function with bounded variation.

Theorem 2 Let Ω be a bounded open set in IR^{N} with the trace-property. Then, there exists a nonnegative function of bounded variation which is a minimizer of the variational problem (1) and a solution of problem (5).

We will see that every bounded domain Ω with λ_{1}(Ω) < 1, has the traceproperty. Hence we have proved that, if λ_{1}(Ω) < 1 then there exists an extremal. Moreover, using results from [12], we can find examples of domains (a ball or an annulus) such that λ_{1}(Ω) = 1 and verify the trace property (and therefore they have extremals). We also prove that every planar domain Ω with a point of curvature greater than 2 verifies λ_{1}(Ω) < 1.

Organization of the paper. In Section 2 we collect some preliminary results and prove Theorem 1. In Section 3 we deal with the Neumann problem for the equation div(z) = 1. Finally, in Section 4 we use these results to prove the main theorem, Theorem 2. Throughout this paper C and c denote constants that may change from one line to another.

2 Preliminary results. Proof of Theorem 1.

Let us begin with some notation and definitions. Recall that a function u ∈ L^{1}(Ω) whose partial derivatives in the sense of distributions are measures with finite total variation in Ω is called a function of bounded variation. The class of such functions will be denoted by BV (Ω). Thus u ∈ BV (Ω) if and only if there are Radon measures μ_{1}, …, μ_{N} defined in Ω with finite total mass in Ω and

∫_{Ω} uD_{i}φ = − ∫_{Ω} φdμ_{i}

for all φ ∈ C_{0}^{∞}(Ω), i = 1, …, N. Thus the gradient of u is a vector valued measure with finite total variation

|Du|(Ω) = sup{∫_{Ω} u div φ : φ ∈ C_{0}^{∞}(Ω, IR^{N}), |φ(x)| ≤ 1 for x ∈ Ω}.

The set of bounded variation functions, BV (Ω), is a Banach space endowed with the norm

∥u∥_{BV (Ω)} := ∥u∥_{1} + |Du|(Ω) = ∫_{Ω} |u| + ∫_{Ω} |Du|.

A measurable set E ⊂ IR^{N} is said to be of finite perimeter in Ω if χ_{E} ∈ BV (Ω), in this case the perimeter of E in Ω is defined as P (E, Ω) := |Dχ_{E}|. We shall use the notation P (E) := P (E, IR^{N}), and P_{f}(Ω) shall denote the set of all subset of Ω of finite perimeter. For a set of finite perimeter E one can defined the essential boundary ∂^{∗}E, which is countably (N −1) rectificable with finite H^{N−1} measure, and compute the unit normal ν^{E}(x) at H^{N−1} almost all points x of ∂^{∗}E, where H^{N−1} is the (N − 1) dimensional Hausdorff measure, and |Dχ_{E}| coincides with the restriction of H^{N−1} to ∂^{∗}E.

It is well known (see, [1], [9] or [13]) that for a given function u ∈ BV (Ω) there exists a sequence u_{n} ∈ W^{1,1}(Ω) such that u_{n} strict converges to u, i.e.,

u_{n} → u in L^{1}(Ω) and ∫_{Ω} |∇u_{n}| → ∫_{Ω} |Du|.

Moreover, there is a trace operator τ : BV (Ω) → L^{1}(∂Ω) such that

∥τ(u)∥_{L^{1}(∂Ω)} ≤ C∥u∥_{BV (Ω)} ∀ u ∈ BV (Ω) (6)

for some constant C depending only on Ω. The trace operator τ is continuous between BV (Ω), endowed with the topology induced by the strict convergence, and L^{1}(∂Ω). In the sequel we write τ(u) = u.

From the above results, we get

λ_{1}(Ω) = inf \{\frac{∫_{Ω} |u| + ∫_{Ω} |Du|}{∫_{∂Ω} |u|} : u ≢ 0 on ∂Ω, u ∈ BV (Ω)\} .

Let us recall a generalized Green’s formula given in [5] (see also [4]). For 1 ≤ p < ∞, let

X_{p}(Ω) = {z ∈ L^{∞}(Ω, IR^{N}) : div(z) ∈ L^{p}(Ω)}.

If z ∈ X_{p}(Ω) and w ∈ BV (Ω) ∩ L^{p′}(Ω) the functional (z, Dw) : C_{0}^{∞}(Ω) → IR is defined by the formula

⟨(z, Dw), φ⟩ = − ∫_{Ω} w φ div(z) − ∫_{Ω} w z · ∇φ.

Then (z, Dw) is a Radon measure in Ω,

∫_{Ω} (z, Dw) = ∫_{Ω} z · ∇w

for all w ∈ W^{1,1}(Ω) ∩ L^{∞}(Ω) and

| ∫_{B} (z, Dw)| ≤ ∫_{B} |(z, Dw)| ≤ ∥z∥_{∞} ∫_{B} |Dw| (7)

for any Borel set B ⊆ Ω.

In [5], a weak trace on ∂Ω of the normal component of z ∈ X_{p}(Ω) is defined. Concretely, it is proved that there exists a linear operator γ : X_{p}(Ω) → L^{∞}(∂Ω) such that

∥γ(z)∥_{∞} ≤ ∥z∥_{∞}
γ(z)(x) = z(x) · ν(x) for all x ∈ ∂Ω if z ∈ C^{1}(\overline{Ω}, IR^{N}).

We shall denote γ(z)(x) by [z, ν](x). Moreover, the following Green’s formula, relating the function [z, ν] and the measure (z, Dw), for z ∈ X_{p}(Ω) and w ∈ BV (Ω) ∩ L^{p′}(Ω), is established:

∫_{Ω} w div(z) dx + ∫_{Ω} (z, Dw) = ∫_{∂Ω} [z, ν]w dH^{N−1}. (8)

Now we can prove Theorem 1.

Proof of Theorem 1. We have to prove that

lim_{p↘1} λ_{p}(Ω) = λ_{1}(Ω) (9)

and

λ_{1}(Ω) ≤ min \{\frac{|Ω|}{P (Ω)}, 1\} . (10)

Since

λ_{p}(Ω)∥u∥_{L^{p}(∂Ω)}^{p} ≤ ∥u∥_{1,p}^{p} ∀ u ∈ W^{1,p}(Ω),

if we set

λ^{∗} := lim sup_{p↘1} λ_{p}(Ω)

we have

λ^{∗}∥u∥_{L^{1}(∂Ω)} ≤ ∥u∥_{1,1} ∀ u ∈ W^{1,1}(Ω),

from where it follows that λ^{∗} ≤ λ_{1}(Ω), that is

lim sup_{p↘1} λ_{p}(Ω) ≤ λ_{1}(Ω). (11)

Let v_{p} be a minimizer of problem (3). Then, if u_{p} := a_{p}v_{p}, with a_{p} satisfying

a_{p} = (∫_{∂Ω} |v_{p}|)^{\frac{1}{p−1}} (∫_{∂Ω} |v_{p}|^{p})^{\frac{1}{1−p}} ,

we get

∫_{∂Ω} |u_{p}| = ∫_{∂Ω} |u_{p}|^{p},

and consequently

λ_{p}(Ω) = \frac{∫_{Ω} |u_{p}|^{p} + ∫_{Ω} |∇u_{p}|^{p}}{∫_{∂Ω} |u_{p}|} . (12)

Applying Hölder inequality and (12), we have

λ_{1}(Ω) ≤ \frac{∫_{Ω} |u_{p}| + ∫_{Ω} |∇u_{p}|}{∫_{∂Ω} |u_{p}|} ≤ |Ω|^{\frac{1}{p′}} \frac{(∫_{Ω} (|u_{p}| + |∇u_{p}|)^{p})^{\frac{1}{p}}}{∫_{∂Ω} |u_{p}|}

≤ |Ω|^{\frac{1}{p′}} 2^{\frac{p−1}{p}} \frac{(∫_{Ω} |u_{p}|^{p} + ∫_{Ω} |∇u_{p}|^{p})^{\frac{1}{p}}}{∫_{∂Ω} |u_{p}|} = |Ω|^{\frac{1}{p′}} 2^{\frac{p−1}{p}} \frac{λ_{p}(Ω)^{\frac{1}{p}}}{(∫_{∂Ω} |u_{p}|)^{\frac{1}{p′}}} .

Hence

λ_{p}(Ω) ≥ λ_{1}(Ω)^{p} \frac{2^{1−p}}{|Ω|^{\frac{p}{p′}}} (∫_{∂Ω} |u_{p}|)^{\frac{p}{p′}} ,

from where it follows that lim inf_{p↘1} λ_{p}(Ω) ≥ λ_{1}(Ω). (13)

Now, (9) follows from (11) and (13). Taking u = χ_{Ω}, we obtain that

λ_{1}(Ω) ≤ \frac{∫_{Ω} |χ_{Ω}| + ∫_{Ω} |Dχ_{Ω}|}{∫_{∂Ω} |χ_{Ω}|} = \frac{|Ω|}{P (Ω)}.

On the other hand, if Ω_{ε} := {x ∈ Ω : d(x, ∂Ω) < ε}, we have

λ_{1}(Ω) ≤ \frac{∫_{Ω} |χ_{Ω_{ε}}| + ∫_{Ω} |Dχ_{Ω_{ε}}|}{∫_{∂Ω} |χ_{Ω_{ε}}|} = \frac{|Ω_{ε}| + P (Ω_{ε}, Ω)}{P (Ω)}.

Hence, taking ε → 0^{+}, it follows that λ_{1}(Ω) ≤ 1. Therefore, (10) holds. □ It is well known (see for instance [11]) that, for every p > 1, there exist an extremal for the embedding W^{1,p}(Ω) ↪ L^{p}(∂Ω) (this embedding is compact). This is a solution 0 ≤ u_{p} ∈ W^{1,p}(Ω) of the equation

λ_{p}(Ω) = ∫_{Ω} |u_{p}|^{p} + ∫_{Ω} |∇u_{p}|^{p}, ∫_{∂Ω} |u_{p}| = 1, (14)

such that u_{p} > 0 and satisfies in the weak sense

\{ \begin{matrix}Δ_{p}u_{p} := div (|∇u_{p}|^{p−2}∇u_{p}) = |u_{p}|^{p−2}u_{p} in Ω \\ |∇u_{p}|^{p−2}∇u_{p} · ν = λ_{p}(Ω)|u_{p}|^{p−2}u_{p} on ∂Ω. \end{matrix} (15)

Now, by Young’s inequality we have

λ_{1}(Ω) ≤ ∫_{Ω} |u_{p}| + ∫_{Ω} |∇u_{p}| ≤ \frac{1}{p} (∫_{Ω} |u_{p}|^{p} + ∫_{Ω} |∇u_{p}|^{p}) + \frac{2}{p′} |Ω|

= \frac{1}{p}λ_{p}(Ω) + \frac{2}{p′} |Ω|.

Then, by (9) it follows that

λ_{1}(Ω) = lim_{p↓1} ∫_{Ω} |u_{p}| + ∫_{Ω} |∇u_{p}|. (16)

Moreover, by the compact embedding of BV (Ω) into L^{1}(Ω), we can suppose that

u_{p} → u ∈ BV (Ω) in the L^{1} − norm and a.e. in Ω, (17)

and

∇u_{p} → Du weakly^{∗} as measures. (18)

Then, we have

∃ A := lim_{p↓1} ∫_{Ω} |∇u_{p}| = λ_{1}(Ω) − ∫_{Ω} |u|. (19)

On the other hand, by the lower-semi-continuity of the total variation respect the L^{1}-norm, we get

∫_{Ω} |Du| ≤ lim inf_{p↓1} ∫_{Ω} |∇u_{p}| = λ_{1}(Ω) − ∫_{Ω} |u|.

Hence,

∫_{Ω} |Du| + ∫_{Ω} |u| ≤ λ_{1}(Ω). (20)

We are interested in the problem: When is u a minimizer of the variational problem (1)? In these cases we would find an extremal for our minimization problem (1).

Formally, if we take limit as p ↘ 1 in equations (15), we get

\{ \begin{matrix}Δ_{1}u := div (\frac{Du}{|Du|}) = \frac{u}{|u|} in Ω \\ \frac{Du}{|Du|} · ν = λ_{1}(Ω)\frac{u}{|u|} on ∂Ω. \end{matrix} (21)

Following [2] and [6] (see also [4]), we give the following definition of solution of problem (21).

Definition 1 A function u ∈ BV (Ω) is said to be a solution of problem (21) if there exists z ∈ X_{1}(Ω) with ∥z∥_{∞} ≤ 1, τ ∈ L^{∞}(Ω) with ∥τ∥_{∞} ≤ 1 and θ ∈ L^{∞}(∂Ω) with ∥θ∥_{∞} ≤ 1 such that

div(z) = τ  in D′(Ω)  (22)

τu = |u| a.e. in Ω, and (z, Du) = |Du| as measures, (23)

[z, ν] = λ_{1}(Ω)θ and θu = |u| H^{N−1} − a.e. on ∂Ω. (24)

Another problem we are interested in is: In what cases is u a solution of problem (21)?

Note that if v is a solution of problem (21) and ∫_{∂Ω} |v| ≠ 0, then w := \frac{v}{∫_{∂Ω} |v|}

is a minimizer of the variational problem (1). Indeed, multiplying (22) by v and integrating by parts, we have

∫_{Ω} |v| = ∫_{Ω} τv = ∫_{Ω} div(z)v = − ∫_{Ω} (z, Dv) + ∫_{∂Ω} [z, ν]v = − ∫_{Ω} |Dv| + λ_{1}(Ω) ∫_{∂Ω} |v|.

From where it follows that

λ_{1}(Ω) = ∫_{Ω} |w| + ∫_{Ω} |Dw|.

Before solve the above problems, let us study first the equation div(z) = 1 with Neumann boundary conditions.

3 The equation div(z) = 1 with Neumann boundary conditions

Throughout this section we shall denote by Ω a bounded connected open set in IR^{N}, N ≥ 2, with Lipschitz continuous boundary ∂Ω. Given g ∈ L^{∞}(∂Ω) with ∥g∥_{∞} < 1, consider the functional E_{g} : L^{2}(Ω) →] − ∞, +∞] defined by

E_{g}(u) := \{ \begin{matrix}∫_{Ω} |Du| − ∫_{∂Ω} gu if u ∈ BV (Ω) ∩ L^{2}(Ω) \\ +∞ if u ∉ BV (Ω). \end{matrix}

In [6] it is proved that

∂E_{g} = A_{g}, (25)

where A_{g} is the operator in L^{2}(Ω) defined by (u, v) ∈ A_{g} ⟺ u ∈ BV (Ω)∩L^{2}(Ω), v ∈ L^{2}(Ω) and ∃ z ∈ X_{2}(Ω), ∥z∥_{∞} ≤ 1 such that

−div(z) = v in D′(Ω)
(z, Du) = |Du| as measures

and

[z, ν] = g H^{N−1} − a.e. on ∂Ω.

Let ψ : L^{2}(Ω) → IR the operator defined by

ψ(u) := \frac{1}{2} ∫_{Ω} (u(x) + 1)^{2} dx.

We have

0 = argmin(E_{g} + ψ) ⟺ 0 ∈ ∂(E_{g} + ψ)(0)
⟺ −1 ∈ ∂E_{g}(0) ⟺ (0, −1) ∈ A_{g}.

Then, by (25), it follows that

0 = argmin(E_{g} + ψ) ⟺ ∃ z ∈ X_{2}(Ω), ∥z∥_{∞} ≤ 1, such that

div(z) = 1 in D′(Ω)
[z, ν] = g H^{N−1} − a.e. on ∂Ω.

On the other hand, 0 = argmin(E_{g} + ψ) if and only if

∫_{Ω} |Du| − ∫_{∂Ω} gu + \frac{1}{2} ∫_{Ω} (u + 1)^{2} ≥ \frac{1}{2}|Ω| ∀ u ∈ BV (Ω) ∩ L^{2}(Ω). (26)

Replacing u by εu (where ε > 0), expanding the L^{2} norm, dividing by ε, and letting ε → 0^{+} we get

∫_{∂Ω} gu ≤ ∫_{Ω} |Du| + ∫_{Ω} u ∀ u ∈ BV (Ω) ∩ L^{2}(Ω). (27)

Consequently we have obtained the following result Lemma 1 Let g ∈ L^{∞}(∂Ω) with ∥g∥_{∞} < 1. Then the following are equivalent:

(i) there exists z ∈ X_{2}(Ω), ∥z∥_{∞} ≤ 1, such that

div(z) = 1 in D′(Ω)

[z, ν] = g H^{N−1} − a.e. on ∂Ω.

(ii) (27) holds.

Working as above but using the functional

ϕ(u) := \frac{1}{2} ∫_{Ω} (|u(x)| + 1)^{2} dx

instead of ψ, we obtain the following result. Lemma 2 Let g ∈ L^{∞}(∂Ω) with ∥g∥_{∞} < 1. Then the following are equivalent:

(i) there exist z ∈ X_{2}(Ω), ∥z∥_{∞} ≤ 1, and τ ∈ L^{∞}(Ω), ∥τ∥_{∞} ≤ 1 such that

div(z) = τ in D′(Ω)

[z, ν] = g H^{N−1} − a.e. on ∂Ω.

(ii) the following inequality holds

∫_{∂Ω} gu ≤ ∫_{Ω} |Du| + ∫_{Ω} |u|  ∀ u ∈ BV (Ω) ∩ L^{2}(Ω).  (28)

We state now the main result of this section. Theorem 3 Let Ω be a bounded connected open set in IR^{N}, N ≥ 2, with Lipschitz continuous boundary ∂Ω. Assume that \frac{|Ω|}{P (Ω)} ≤ λ < 1. Then the following are equivalent:

(i) there exist z ∈ X_{2}(Ω), ∥z∥_{∞} ≤ 1, such that

div(z) = 1 in D′(Ω)

[z, ν] = λ H^{N−1} − a.e. on ∂Ω.
(ii)  λ ∫_{∂Ω} u ≤ ∫_{Ω} |Du| + ∫_{Ω} u  ∀ u ∈ BV (Ω) ∩ L^{2}(Ω).  (29)
(iii)  λ ∫_{∂Ω} u ≤ ∫_{Ω} |Du| + ∫_{Ω} |u|  ∀ u ∈ BV (Ω) ∩ L^{2}(Ω).  (30)
(iv)  λ ≤ λ_{1}(Ω).  (31)
(v)  ||E| − λH^{N−1}(∂^{∗}E ∩ ∂Ω)| ≤ P (E, Ω) ∀ E ∈ P_{f}(Ω).  (32)

Proof. By Lemma 1 with g = λ, (i) and (ii) are equivalent. Obviously, (iii) and (iv) are equivalent, and (ii) implies (iii). Let us see that (iii) implies (v). Taking u = χ_{E} in (30), with E ⊂ Ω a set of finite perimeter, it follows that

−[|E| − λH^{N−1}(∂^{∗}E ∩ ∂Ω)] ≤ P (E, Ω),

and taking u = χ_{Ω} \ χ_{E}, we get

λH^{N−1}(∂^{∗}(Ω \ E) ∩ ∂Ω) ≤ P (Ω \ E, Ω) + |Ω \ E|.

Now, since P (Ω \ E, Ω) = P (E, Ω) and H^{N−1}(∂^{∗}(Ω \ E) ∩ ∂Ω) = P (Ω) − H^{N−1}(∂^{∗}E ∩ ∂Ω), we have

λ[P (Ω) − H^{N−1}(∂^{∗}E ∩ ∂Ω)] ≤ P (E, Ω) + |Ω| − |E|.

Then, since λ ≥ \frac{|Ω|}{P (Ω)}, we obtain that

|E| − λH^{N−1}(∂^{∗}E ∩ ∂Ω) ≤ P (E, Ω),

and (v) holds. Finally, let us see that (v) implies (ii). Given u ∈ BV (Ω), since for all x ∈ Ω,

u(x) = ∫_{0}^{+∞}χ_{{u>t}}(x) dt − ∫_{−∞}^{0} χ_{{u≤t}}(x) dt,

using (32) and the coarea formula we get

∫_{Ω} u(x) dx = ∫_{0}^{+∞}∫_{Ω} χ_{{u>t}}(x) dx dt − ∫_{−∞}^{0} ∫_{Ω} χ_{{u≤t}}(x) dx dt

= ∫_{0}^{+∞}|{u > t}| dt − ∫_{−∞}^{0} |{u ≤ t}| dt

≥ ∫_{0}^{+∞}(λH^{N−1}(∂^{∗}{u > t} ∩ ∂Ω) − P ({u > t}, Ω)) dt

− ∫_{−∞}^{0} (λH^{N−1}(∂^{∗}{u ≤ t} ∩ ∂Ω) + P ({u ≤ t}, Ω)) dt

= λ ∫_{∂Ω} u dH^{N−1} − ∫_{−∞}^{+∞}P ({u > t}, Ω) dt

= λ ∫_{∂Ω} u dH^{N−1} − ∫_{Ω} |Du|,

and (29) holds. □

Taking λ = \frac{|Ω|}{P (Ω)} in the above theorem we obtain the following result.

Corollary 1 Let Ω be a bounded connected open set in IR^{N}, N ≥ 2, with Lipschitz continuous boundary ∂Ω. If \frac{|Ω|}{P (Ω)} < 1, then the following are equivalent:

(i)  \begin{matrix}∃ z ∈ X_{2}(Ω), ∥z∥_{∞} ≤ 1, such that \\ div(z) = 1 in D′(Ω) \\ [z, ν] = \frac{|Ω|}{P (Ω)} H^{N−1} − a.e. on ∂Ω. \end{matrix} \}  (33)

(ii)

λ_{1}(Ω) = \frac{|Ω|}{P (Ω)},

(iii)

||E| − \frac{|Ω|}{P (Ω)}H^{N−1}(∂^{∗}E ∩ ∂Ω)| ≤ P (E, Ω) ∀ E ∈ P_{f}(Ω).

We do not know if it is necessary the assumption Ω connected in Corollary 1. So, a natural question is the following: Is there a non-connected open bounded set Ω such that \frac{|Ω|}{P (Ω)} < 1, verifying (33) and λ_{1}(Ω) < \frac{|Ω|}{P (Ω)}?

There are open sets Ω for which (33) holds and \frac{|Ω|}{P (Ω)} = 1, as the following examples show.

Example 1 Let Ω = B_{R}(0) ⊂ IR^{N} the ball in IR^{N} centered in 0 of radius R. Then, if z(x) := \frac{x}{N}, we have

div(z) = 1 in D′(Ω)

and

[z, ν] = \frac{R}{N} = \frac{|Ω|}{P (Ω)} H^{N−1} − a.e. on ∂Ω.

Moreover,

∥z∥_{∞} = \frac{R}{N} ≤ 1 ⟺ \frac{|Ω|}{P (Ω)} ≤ 1.

Example 2 Let Ω = B_{R}(0) \ \overline{B_{r}(0)} ⊂ IR^{N} the annulus in IR^{N} centered in 0 of radius R and r. Then, it is easy to see that if

z(x) := [(R^{N−1} + r^{N−1}) − (R + r)\frac{r^{N−1}R^{N−1}}{∥x∥^{N}} ] \frac{x}{N (R^{N−1} + r^{N−1})}

we have

div(z) = 1 in D′(Ω)

and

[z, ν] = \frac{|Ω|}{P (Ω)} H^{N−1} − a.e. on ∂Ω.

Moreover,

∥z∥_{∞} ≤ 1 ⟺ \frac{|Ω|}{P (Ω)} ≤ 1.

Remark 1 M. Motron in [12] proves that if Ω = B_{R}(0) is the ball in IR^{N} centered in 0 of radius R or Ω = B_{R}(0) \ \overline{B_{r}(0)} the annulus in IR^{N} centered in 0 of radius R and r, then

∫_{∂Ω} |u| ≤ \frac{P (Ω)}{|Ω|} ∫_{Ω} |u| + ∫_{Ω} |∇u| ∀ u ∈ W^{1,1}(Ω), (34)

and equality holds in (34) if and only if u is constant.

From (10) and (34), it follows that if Ω = B_{R}(0) ⊂ IR^{N} or Ω = B_{R}(0)\\overline{B_{r}(0)} ⊂ IR^{N}, then

λ_{1}(Ω) = \{ \begin{matrix}\frac{|Ω|}{P (Ω)} if \frac{|Ω|}{P (Ω)} ≤ 1 \\ 1 if \frac{|Ω|}{P (Ω)} ≥ 1. \end{matrix}

Moreover, if \frac{|Ω|}{P (Ω)} ≤ 1, then u = \frac{1}{P (Ω)}χ_{Ω} is a minimizer of the variational problem (1), being the only minimizer in the case \frac{|Ω|}{P (Ω)} = 1, and if \frac{|Ω|}{P (Ω)} > 1, the variational problem (1) does not have minimizer.

In the following example we show that there exists bounded connected open sets Ω, with \frac{|Ω|}{P (Ω)} < 1, for which λ_{1}(Ω) < \frac{|Ω|}{P (Ω)}.

Example 3 For δ > 0 and 0 < α ≤ \frac{π}{2}, let

Ω_{δ,α} := B_{1}(0) ∪ (B_{2+δ}(0) \ \overline{B_{2}(0)})∪

\{(x, y) ∈ IR^{+} × IR^{+} : x^{2} + y^{2} ≤ 2, arctg (\frac{y}{x}) < α\} .

We have,

\frac{|Ω_{δ,α}|}{P (Ω_{δ,α})} = \frac{π + π(δ^{2} + 4δ) + \frac{3}{2}α}{10π + 2πδ + 2 − 3α}.

Thus, for 0 < δ ≤ 1 and 0 < α ≤ \frac{π}{2}, we have

\frac{|Ω_{δ,α}|}{P (Ω_{δ,α})} < 1.

Fig. 1. Triangle

Now, if we take u := χ_{B_{2+δ}(0)\\overline{B_{2}(0)}},

λ_{1}(Ω_{δ,α}) ≤ \frac{∫_{Ω_{δ,α}} |Du| + ∫_{Ω_{δ,α}} |u|}{∫_{∂Ω_{δ,α}} |u|} = \frac{2α + π(δ^{2} + 4δ)}{8π + 2δπ − 2α}.

Then, it is easy to see that for δ and α small enough, we get

λ_{1}(Ω_{δ,α}) < \frac{|Ω_{δ,α}|}{P (Ω_{δ,α})} < 1.

In the next example we will see that even we can take Ω convex.

Example 4 Let Ω be the set in IR^{2} with boundary the isosceles triangle with height k, base of length 2a and the two equal sides of length l. Let t the angle between the height and one of the equal side (see figure 1). Then,

\frac{|Ω|}{P (Ω)} = \frac{ak}{2(a + l)} = \frac{ak}{a2 (a + \frac{a}{sin t})} = \frac{k sin t}{2(1 + sin t)}.

Let E ⊂ Ω the set with boundary the isosceles triangle with height k −r, base of length 2b and the two equal sides of length l̃. Then, if u := χ_{E}, we have

λ_{1}(Ω) ≤ \frac{∫_{Ω} |Du| + ∫_{Ω} |u|}{∫_{∂Ω} |u|} = \frac{2b + b(k − r)}{2l̃} = \frac{b(k + 2 − r)}{\frac{2b}{sin t}} = \frac{sin t}{2}(k + 2 − r).

Hence,

λ_{1}(Ω) < \frac{|Ω|}{P (Ω)} < 1

if

k < min \{(r − 2)\frac{1 + sin t}{sin t}, 2\frac{1 + sin t}{sin t}\} .

Now, obviously, we can find k, r, and t satisfying the above inequality, and consequently, we can obtain a convex, bounded open set Ω satisfying

λ_{1}(Ω) < \frac{|Ω|}{P (Ω)} < 1.

The next example show the necessity of the assumption Ω connected in Lemma 2

Example 5 For 0 < ρ < r and δ > 0, let

Ω_{ρ,r,δ} := B_{ρ}(0) ∪ (B_{r+δ}(0) \ B_{r}(0)) ⊂ IR^{2}.

We have

\frac{|Ω_{ρ,r,δ}|}{P (Ω_{ρ,r,δ})} = \frac{δ^{2} + ρ^{2} + 2rδ}{2(2r + δ + ρ)}.

If we take u := χ_{B_{ρ}(0)} and v := χ_{B_{r+δ}(0)\\overline{B_{r}(0)}}, then

Φ(u) := \frac{∫_{Ω_{ρ,r,δ}} |Du| + ∫_{Ω_{ρ,r,δ}} |u|}{∫_{∂Ω_{ρ,r,δ}} |u|} = \frac{ρ}{2}.

and

Φ(v) := \frac{∫_{Ω_{ρ,r,δ}} |Dv| + ∫_{Ω_{ρ,r,δ}} |v|}{∫_{∂Ω_{ρ,r,δ}} |v|} = \frac{δ}{2}.

Suppose that 0 < ρ < δ ≤ 2. If we consider the vector field z in Ω_{ρ,r,δ} defined by

z(x, y) := \{ \begin{matrix}\frac{δ(x,y)}{2ρ} if (x, y) ∈ B_{ρ}(0) \\ [δ − (δ + 2r)\frac{r(r+δ)}{∥(x,y)∥}] \frac{(x,y)}{2(2r+δ)} if (x, y) ∈ B_{r+δ}(0) \ B_{r}(0), \end{matrix}

we have ∥z∥_{∞} ≤ 1,

div(z) = τ in D′(Ω_{ρ,r,δ}),

with

τ = \frac{δ}{ρ}χ_{∂B_{ρ}(0)} + χ_{B_{r+ρ}(0)\B_{r}(0)}

and

[z, ν] = \frac{δ}{2} H^{1} − a.e. on ∂Ω_{ρ,r,ρ}.

Now,

λ_{1}(Ω) ≤ Φ(u) = \frac{ρ}{2}.

Hence,

λ_{1}(Ω) < \frac{δ}{2}.

Consequently, in general, Lemma 2 it is not true if Ω is not connected.

If δ = ρ, we have

λ_{1}(Ω_{ρ,r,ρ}) ≤ Φ(u) = Φ(v) = Φ(χ_{Ω_{ρ,r,ρ}}) = \frac{|Ω_{ρ,r,ρ}|}{P (Ω_{ρ,r,ρ})} = \frac{ρ}{2}.

Suppose that ρ ≤ 1. Then, if we consider the vector field z in Ω_{ρ,r,ρ} defined by

z(x, y) := \{ \begin{matrix}\frac{(x,y)}{2} if (x, y) ∈ B_{ρ}(0) \\ 0 if (x, y) ∈ B_{r+ρ}(0) \ B_{r}(0), \end{matrix}

we have

div(z) = χ_{B_{ρ}(0)} in D′(Ω_{ρ,r,ρ}),

and

[z, ν] = \frac{ρ}{2}χ_{∂B_{ρ}(0)} H^{1} − a.e. on ∂Ω_{ρ,r,ρ}.

Therefore, u is a solution of the problem

\{ \begin{matrix}Δ_{1}w := div (\frac{Dw}{|Du|}) = \frac{w}{|w|} in Ω_{ρ,r,ρ} \\ \frac{Dw}{|Dw|} · ν = \frac{ρ}{2} \frac{w}{|w|} on ∂Ω_{ρ,r,ρ}. \end{matrix} (35)

Now, if we consider the vector field z in Ω_{ρ,r,ρ} defined by

z(x, y) := \{ \begin{matrix}0 if (x, y) ∈ B_{ρ}(0) \\ [ρ − (ρ + 2r)\frac{r(r+ρ)}{∥(x,y)∥}] \frac{(x,y)}{2(2r+ρ)} if (x, y) ∈ B_{r+ρ}(0) \ B_{r}(0), \end{matrix}

we have

div(z) = χ_{B_{r+ρ}(0)\B_{r}(0)} in D′(Ω_{ρ,r,ρ}),

and

[z, ν] = \frac{ρ}{2}χ_{∂(B_{r+ρ}(0)\B_{r}(0))} H^{1} − a.e. on ∂Ω_{ρ,r,ρ}.

Therefore, v is also a solution of the problem (35). Moreover, in this case also χ_{Ω_{ρ,r,ρ}} is a solution of the problem (35).

Problem Is λ_{1}(Ω_{ρ,r,ρ}) = \frac{ρ}{2}?

The next example shows that there are bounded connected open sets, Ω for which λ_{1}(Ω) < 1 and \frac{|Ω|}{P (Ω)} > 1.

Example 6 Let Ω :=] − k, 0[×]0, k[∪{0}×]0, δ[∪]0, 1[×]0, δ[⊂ IR^{2} be. Then

\frac{|Ω|}{P (Ω)} = \frac{k^{2} + δ}{4k + 2} > 1 if k > 2 + \sqrt{6 − δ}.

Now, if we take u := χ_{]0,1[×]0,δ[}, we have

λ_{1}(Ω) ≤ \frac{∫_{Ω} |Du| + ∫_{Ω} |u|}{∫_{∂Ω} |u|} = \frac{2δ}{2 + δ} < 1 ⟺ 0 < δ < 2.

Therefore, for instance, if δ = 1 and k = 5, we have \frac{|Ω|}{P (Ω)} > 1 and λ_{1}(Ω) < 1.

4 Proof of Theorem 2.

Let Ω be a bounded connected open set in IR^{N}, N ≥ 2, with Lipschitz continuous boundary ∂Ω. By Lemma 2, if we assume that λ_{1}(Ω) < 1, there exist \overline{z} ∈ X_{2}(Ω), ∥\overline{z}∥_{∞} ≤ 1, and ∥div(\overline{z})∥_{∞} ≤ 1 such that

[\overline{z}, ν] = λ_{1}(Ω) H^{N−1} − a.e. on ∂Ω.

We recall the following definition Definition 2 Let Ω be a bounded open set in IR^{N} with Lipschitz continuous boundary ∂Ω. We shall say that Ω has the trace-property if there exists a vector field z_{Ω} ∈ L^{∞}(Ω, IR^{N}), with ∥z_{Ω}∥_{∞} ≤ 1 such that div(z_{Ω}) ∈ L^{∞}(Ω) and

[z_{Ω}, ν] = λ_{1}(Ω) H^{N−1} − a.e. on ∂Ω.

By the above, we have that every bounded connected open set Ω in IR^{N}, N ≥ 2, with Lipschitz continuous boundary and λ_{1}(Ω) < 1, has the traceproperty. Also, as a consequence of Examples 1, 2, and Remark 1, we have that if Ω = B_{R}(0) ⊂ IR^{N} or Ω = B_{R}(0) \ \overline{B_{r}(0)} ⊂ IR^{N}, and \frac{|Ω|}{P (Ω)} ≤ 1, then Ω has the trace-property. Therefore there exists Ω with λ_{1}(Ω) = 1 satisfying the trace-property.

Let us present some examples of planar domains that verify λ_{1}(Ω) < 1.

Example 7 Assume that Ω ⊂ IR^{2} is a bounded open set such that there exists some point, x_{0} ∈ ∂Ω (we may assume x_{0} = 0), with curvature of the boundary at that point greater than 2, we will show that in this case λ_{1}(Ω) < 1. So, let us assume that locally near the origin Ω can be described as Ω ∩ B_{r}(0) = {(x, y) : y > ax^{2}}. As we are assuming that the curvature at the origin is greater than 2 we have a > 1. Let us consider the function u_{ε} = χ_{Ω∩{y<ε}} as a test function to estimate λ_{1}. We have

λ_{1}(Ω) ≤ \frac{∫_{Ω} |Du_{ε}| + ∫_{Ω} |u_{ε}|}{∫_{∂Ω} |u_{ε}|} = \frac{\sqrt{ε/a} + ∫_{0}^{\sqrt{ε/a}}(ε − as^{2}) ds}{∫_{0}^{\sqrt{ε/a}}\sqrt{1 + (2a)^{2}s^{2}} ds}

= \frac{\sqrt{ε/a} + \frac{2}{3}ε\sqrt{ε/a}}{∫_{0}^{\sqrt{ε/a}}(1 + 2a^{2}s^{2} + O(s^{3})) ds} < 1,

if ε is small enough.

Remark that if Ω = B_{R}(0) we have that the curvature is 1/R and, by Example 1, we have λ_{1}(B_{R}(0)) < 1 if and only if R < 2. Hence, some restriction on the curvature must be imposed.

Next we prove that on every domain that enjoys the trace-property the best Sobolev trace constant is attained, Theorem 2.

Proof of Theorem 2. First let us see that the function u obtained in (17) is a minimizer of the variational problem (1). Let z_{Ω} be the vector field given in Definition 2. Then, by (18), we have

∫_{Ω} (z_{Ω}, Du) = lim_{p↘1} ∫_{Ω} (z_{Ω}, ∇u_{p}) = lim_{p↘1} (− ∫_{Ω} div(z_{Ω})u_{p} + ∫_{∂Ω} [z_{Ω}, ν]u_{p})

= − ∫_{Ω} div(z_{Ω})u + λ_{1}(Ω) = ∫_{Ω} (z_{Ω}, Du) − λ_{1}(Ω) ∫_{∂Ω} u + λ_{1}(Ω),

from where it follows that

∫_{∂Ω} |u| = 1. (36)

Therefore, by (20) we get that u is a minimizer of (1). Moreover, by (19), it follows that

lim_{p↘1} ∫_{Ω} |∇u_{p}| = λ_{1}(Ω) − ∫_{Ω} |u| = ∫_{Ω} |Du|. (37)

Hence u_{p} → u respect to the strict convergence, and consequently

u_{p} → u in L^{1}(∂Ω) as p ↘ 1. (38)

Let us see now that the function u is a solution of problem (21). By Hölder inequality, we have

∫_{Ω} |u_{p}|^{p−1} ≤ |Ω|^{\frac{1}{p}} (∫_{Ω} |u_{p}|^{p})^{\frac{p−1}{p}} ≤ |Ω|^{\frac{1}{p}} λ_{p}(Ω)^{\frac{p−1}{p}} ≤ M_{1}. (39)

On the other hand, if E is a measurable subset of Ω with |E| < 1, we have

|∫_{E} |u_{p}|^{p−2}u_{p}| ≤ ∫_{E} |u_{p}|^{p−1} ≤ M_{2}|E|^{\frac{1}{p}} . (40)

By (39) and (40), it follows that {|u_{p}|^{p−2}u_{p} : 1 < p ≤ 2} is a weakly relatively compact subset of L^{1}(Ω). Hence, we can assume that there exists τ ∈ L^{1}(Ω) such that

|u_{p}|^{p−2}u_{p} → τ weakly in L^{1}(Ω) as p ↘ 1. (41)

In a similar way, it is easy to see that there exists z ∈ L^{1}(Ω, IR^{N}) such that

|∇u_{p}|^{p−2}∇u_{p} → z weakly in L^{1}(Ω, IR^{N}) as p ↘ 1. (42)

Now, given φ ∈ D(Ω), from (41) and (42), it follows that

⟨div(z), φ⟩ = − ∫_{Ω} z · ∇φ = − lim_{p↘1} ∫_{Ω} |∇u_{p}|^{p−2}∇u_{p} · ∇φ

= lim_{p↘1} ∫_{Ω} div (|∇u_{p}|^{p−2}∇u_{p}) φ = lim_{p↘1} ∫_{Ω} |u_{p}|^{p−2}u_{p}φ = ∫_{Ω} τφ.

Thus,

div(z) = τ  in D′(Ω)  (43)

We claim now that

∥z∥_{∞} ≤ 1. (44)

In fact: for any k > 0, let

B_{p,k} := {x ∈ Ω : |∇u_{p}(x)| > k}.

As above, there exists some g_{k} ∈ L^{1}(Ω, IR^{N}) such that

|∇u_{p}|^{p−2}∇u_{p} χ_{B_{p,k}} → g_{k} weakly in L^{1}(Ω, IR^{N}) as p ↘ 1. (45)

Now, since

|B_{p,k}| = ∫_{Ω} χ_{B_{p,k}}(x) dx ≤ ∫_{Ω} \frac{|∇u_{p}(x)|^{p}}{k^{p}} dx ≤ \frac{λ_{p}(Ω)}{k^{p}},

for any ϕ ∈ L^{∞}(Ω) with ∥ϕ∥_{∞} ≤ 1, we have

|∫_{Ω} |∇u_{p}|^{p−2}∇u_{p} · ϕ χ_{B_{p,k}} | ≤ (∫_{Ω} |∇u_{p}|^{p})^{\frac{p−1}{p}} |B_{p,k}|^{\frac{1}{p}}

≤ λ_{p}(Ω)^{\frac{p−1}{p}} (\frac{λ_{p}(Ω)}{k^{p}})^{\frac{1}{p}} = \frac{λ_{p}(Ω)}{k}.

Letting p ↘ 1, we get that

∫_{Ω} |g_{k}| ≤ \frac{λ_{1}(Ω)}{k} for every k > 0. (46)

On the other hand, since we have

||∇u_{p}|^{p−2}∇u_{p}χ_{Ω\B_{p,k}} | ≤ k^{p−1} for any p > 1,

letting p ↘ 1, we obtain that there exists some f_{k} ∈ L^{1}(Ω, IR^{N}) with ∥f_{k}∥_{∞} ≤ 1 such that

|∇u_{p}|^{p−2}∇u_{p} χ_{Ω\B_{p,k}} → f_{k} weakly in L^{1}(Ω, IR^{N}) as p ↘ 1. (47)

Hence, for any k > 0, we may write z = f_{k} + g_{k}, with ∥f_{k}∥_{∞} ≤ 1 and g_{k} satisfying (46). From where (44) follows.

Since u_{p} → u a.e. in Ω, by (41) it follows that

τu = |u| a.e. in Ω and ∥τ∥_{∞} ≤ 1.

On the other hand, given a measurable subset E ⊂ ∂Ω, by Hölder inequality we have

∫_{E} u_{p}^{p−1} dH^{N−1} ≤ (∫_{∂Ω} u_{p} dH^{N−1})^{p−1} H^{N−1}(E)^{2−p} ≤ H^{N−1}(E)^{2−p},

from where it follows that {u_{p}^{p−1} : 1 < p ≤ 2} is a weakly relatively compact subset of L^{1}(∂Ω). Hence, we can assume that there exists θ ∈ L^{1}(∂Ω), such that

u_{p}^{p−1} → θ weakly in L^{1}(∂Ω). (48)

Moreover, by (48), (14) and applying Fatou’s Lemma, it is easy to see that

∥θ∥_{∞} ≤ 1. (49)

On the other hand, by (38) and (48), we get

θu = |u| H^{N−1} − a.e. on ∂Ω (50)

Now, since u_{p} is a weak solution of (15), having in mind (41), (9) and (48), if w ∈ W^{1,1}(Ω) ∩ C(Ω) ∩ L^{∞}(Ω), we have

⟨z, w⟩_{∂Ω} := ∫_{Ω} div(z)w + ∫_{Ω} z · ∇w = ∫_{Ω} τw + ∫_{Ω} z · ∇w

= lim_{p↘1} ∫_{Ω} u_{p}^{p−1}w + ∫_{Ω} |∇u_{p}|^{p−2}∇u_{p} · ∇w

= lim_{p↘1} ∫_{Ω} u_{p}^{p−1}w − ∫_{Ω} div (|∇u_{p}|^{p−2}∇u_{p}) w + ∫_{∂Ω} |∇u_{p}|^{p−2}∇u_{p} · νw

= lim_{p↘1} λ_{p}(Ω) ∫_{∂Ω} u_{p}^{p−1}w = λ_{1}(Ω) ∫_{∂Ω} θw.

Thus, having in mind the definition of the weak trace on ∂Ω of the normal component of z given in [5], we get

[z, ν] = λ_{1}(Ω)θ. (51)

Finally, since

∫_{Ω} |Du| = λ_{1}(Ω) − ∫_{Ω} |u| = λ_{1}(Ω) − ∫_{Ω} τu

= λ_{1}(Ω) − ∫_{Ω} div(z)u = λ_{1}(Ω) + ∫_{Ω} (z, Du) − ∫_{∂Ω} [z, ν]u

= λ_{1}(Ω) + ∫_{Ω} (z, Du) − λ_{1}(Ω) ∫_{∂Ω} θu = ∫_{Ω} (z, Du),

we have (z, Du) = |Du| as measures. □

Remark 2 Let us remark that as a consequence of Theorem 1 in [7] it is obtained the above theorem in the particular case that Ω is a bounded open subset of IR^{N}, whose boundary ∂Ω is at least piecewise C^{2} and λ_{1}(Ω) < 1.

Remark 3 Note that in the above theorem we have proved that if u is the limit as p ↘ 1 of the minimizers u_{p} of the variational problem

inf {∫_{Ω} |u|^{p} + ∫_{Ω} |∇u|^{p} : u ∈ W^{1,p}(Ω), ∫_{∂Ω} |u| = 1, } . (52)

then, if ∫_{∂Ω} |u| = 1, we have that u is a minimizer of the variational problem (1) and a solution of problem (21).

Acknowledgements. The first and second authors have been partially supported EC through the RTN Programme Nonlinear Partial Differential Equations Describing Front propagation and other Singular Phenomena, HPRN-CT-2002-00274, and by PNPGC project, reference BFM2002-01145. The third author acknowledges partial support by ANPCyT PICT No. 03-05009, CON-ICET PIP0660/98, UBA X431 and Fundación Antorchas.

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