REGULARITY OF THE FREE BOUNDARY IN AN OPTIMIZATION PROBLEM RELATED TO THE BEST SOBOLEV TRACE CONSTANT

JULIÁN FERNÁNDEZ BONDER, JULIO D. ROSSI AND NOEMI WOLANSKI

Abstract. In this paper we study the regularity properties of a free boundary problem arising in the optimization of the best Sobolev trace constant in the immersion H^{1}(Ω) ↪ L^{q}(∂Ω) for functions that vanish in a subset of Ω. This problem is also related to a minimization problem for Steklov eigenvalues.

1. Introduction.

The study of Sobolev inequalities and of optimal constants is a subject of interest in the analysis of PDE’s and related topics. It has been widely studied in the past by many authors and is still an area of intensive research. See for instance the book [1], and, for recent developments in this field, see the articles [6, 9, 10, 17] and the survey [7] among others.

The optimal Sobolev constant and its corresponding extremals (if they exist) are related to eigenvalue problems. In the case of the best Sobolev trace embedding H^{1}(Ω) → L^{q}(∂Ω) where Ω is a bounded smooth domain in ℝ^{N}, the best constant and the extremal (that exists for 1 ≤ q < 2_{∗} = 2(N −1)/(N −2) since the immersion is compact) give rise to the following elliptic problem with nonlinear boundary conditions

\{ \begin{matrix}−Δu + u = 0  in Ω, \\ \frac{∂u}{∂ν} = λu^{q−1}  on ∂Ω. \end{matrix}

The constant λ depends on the normalization of the extremal u. For instance if u is chosen so that ∥u∥_{L^{q}(∂Ω)} = 1, then λ = S the best Sobolev trace constant. In the linear case, q = 2, this problem becomes an eigenvalue problem that is known as the Steklov eigenvalue problem [19].

In this paper we are interested in the best Sobolev trace constant among functions that vanish in a subset of Ω. We try to optimize this best constant when varying the subset in the class of measurable sets with prescribed positive measure α. In a previous article [11], we proved that there exists an optimal set. In this paper we focus our attention on regularity properties of these optimal sets.

Key words and phrases. Sobolev trace constant, free boundaries, eigenvalue optimization problems.

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

Supported by ANPCyT PICT No. 03-05009 and 03-10608, CONICET PIP0660/98 and PEI6388/04, UBA X052 and X066 and Fundación Antorchas 13900-5. J.D. Rossi and N. Wolanski are members of CONICET.

More precisely, in [11] we studied the following problem. Let

J (v) = ∫_{Ω} |∇v|^{2} + v^{2} dx,
A_{α} = {v ∈ H^{1}(Ω) / ∥v∥_{L^{q}(∂Ω)} = 1 and |{v > 0}| = α}.

Then the problem is:

(P_{α})  Find ϕ_{0} ∈ A_{α} such that S(α) := inf_{v∈A_{α}} J (v) = J (ϕ_{0}).

In [11] we proved that there exists a solution ϕ_{0} to (P_{α}) but the approach in [11] does not give any regularity properties of ϕ_{0} nor of the hole {ϕ_{0} = 0}.

In this paper we consider a different approach. Instead of minimizing J (v) over A_{α} we penalize the functional and minimize without the measure restriction. This approach has been used with great success by many authors starting with the work [2] (see also [3, 15, 16, 20], etc.). So, let

(1.1)  J_{ε}(v) = ∫_{Ω} |∇v|^{2} + v^{2} dx + F_{ε}(|{v > 0}|),

where

F_{ε}(s) = \begin{cases}\frac{1}{ε}(s − α) & if s ≥ α \\ ε(s − α) & if s < α. \end{cases}

The penalized problem is to minimize J_{ε} over the class

K_{1} = {v ∈ H^{1}(Ω) / ∥v∥_{L^{q}(∂Ω)} = 1}.

For technical reasons, it is better to minimize in the class

K = {v ∈ H^{1}(Ω) / ∥v∥_{L^{q}(Γ_{N})} = 1, v = φ_{0} on Γ_{D}},

where ∅ ≠ Γ_{N} ⊂ ∂Ω, Γ_{D} = ∂Ω \ Γ_{N} is the closure of a relatively open set of the boundary and φ_{0} ∈ H^{1}(Ω), φ_{0} ≥ c_{0} > 0 on Γ_{D}. We will only need to assume that Γ_{D} ≠ ∅ at the end of our arguments. See Section 4, Lemma 4.3.

So the penalized problem is:

(P_{ε})  Find u_{ε} ∈ K such that J_{ε}(u_{ε}) = inf_{v∈K} J_{ε}(v).

Observe that minimizing J_{ε} over K gives a problem with mixed boundary conditions. We believe that this problem has independent interest.

The main idea is to prove that for ε small any minimizer u_{ε} of J_{ε} in K satisfies |{u_{ε} > 0}| = α, therefore the penalization term F_{ε} vanishes and hence we have a minimizer of our original problem. This allows us to avoid the passage to the limit (as ε → 0) where uniform bounds are needed. To prove regularity of the minimizers of J_{ε} and their free boundaries, ∂{u_{ε} > 0}, is easier than the original problem, thanks to the results of [4].

The main theorem in this article is:

Theorem 1.1. For every ε > 0 there exists a solution u_{ε} ∈ K to (P_{ε}). Moreover, any such solution is a locally Lipschitz continuous function and the free boundary ∂{u_{ε} > 0} is locally a C^{1,β} surface up to a set of H^{N−1}−measure zero. In the case N = 2 the free boundary is locally a C^{1,β} surface. Moreover, if Γ_{D} ≠ ∅, for ε small we have |{u_{ε} > 0}| = α.

Outline of the paper. In Section 2 we begin our analysis of problem (P_{ε}) for fixed ε. First we prove the existence of a minimizer, local Lipschitz regularity and nondegeneracy near the free boundary (Theorem 2.1). Then we prove that a minimizer u_{ε} of (P_{ε}) is a weak solution to the following free boundary problem

\{ \begin{matrix}−Δu + u = 0  in \{u > 0\} ∩ Ω, \\ \frac{∂u}{∂ν} = λ_{ε}  on ∂\{u > 0\} ∩ Ω, \end{matrix}

where λ_{ε} is a positive constant (Theorem 2.6).

In Section 3, again for fixed ε, we analyze the regularity of the free boundary and show that, up to a set of H^{N−1}−measure zero, ∂{u_{ε} > 0} is locally a C^{1,β} surface and, in the case N = 2, the free boundary has no exceptional points (Theorem 3.1). The proof of this result follows almost exactly the lines in [4], so we only remark the significant differences and refer to [4] for further details.

In Section 4 we analyze the behavior of the solutions to (P_{ε}) for small ε. We prove that, if Γ_{D} ≠ ∅, the positivity set of the minimizer u_{ε} has measure α (Theorem 4.1).

Finally, in Section 5, we go back to our original problem and show, under some mild assumptions on the solutions ϕ_{0} to (P_{α}), that they are also solutions to (P_{ε}) for small ε, so they inherit the properties of the solutions to (P_{ε}) (Theorem 5.1). These extra assumptions are satisfied, for instance, if Ω is a ball (Corollary 5.1). In the general case, without the assumption that Γ_{D} ≠ ∅, we prove that the set of α’s for which there is a solution to (P_{α}) with smooth free boundary is dense in (0, |Ω|) (Theorem 5.2). Then, we show that the minimizers of (P_{ε}) converge (up to a subsequence) to a solution to (P_{α}) (Theorem 5.3). We believe that this last result might be of interest in numerical approximations.

2. The penalized problem

In this section, we consider the penalized problem (P_{ε}) stated in the introduction and prove the existence of a minimizer and some regularity properties.

Theorem 2.1. There exists a solution to the problem (P_{ε}). Moreover, any such solution u_{ε} has the following properties:

(1) u_{ε} is locally Lipschitz continuous in Ω.

(2) For every D ⊂⊂ Ω, there exist constants C, c > 0 such that for every x ∈ D ∩ {u_{ε} > 0},

c dist(x, ∂{u_{ε} > 0}) ≤ u_{ε}(x) ≤ C dist(x, ∂{u_{ε} > 0}).

(3) For every D ⊂⊂ Ω, there exists a constant c > 0 such that for x ∈ ∂{u > 0} and B_{r}(x) ⊂ D,

c ≤ \frac{|B_{r}(x) ∩ \{u_{ε} > 0\}|}{|B_{r}(x)|} ≤ 1 − c.

The constants may depend on ε.

The proof will be divided into a series of steps for the reader’s convenience.

Proof of existence. Let (u_{n}) ⊂ K be a minimizing sequence for J_{ε}. Then J_{ε}(u_{n}) is bounded and so ∥u_{n}∥_{H^{1}(Ω)} ≤ C. Therefore there exists a subsequence (that we still call u_{n}) and a function u_{ε} ∈ H^{1}(Ω) such that

u_{n} ⇀ u_{ε} weakly in H^{1}(Ω),
u_{n} → u_{ε} strongly in L^{q}(∂Ω),
u_{n} → u_{ε} a.e. Ω.

Thus,

∥u_{ε}∥_{L^{q}(Γ_{N})} = 1, u_{ε} = φ_{0} on Γ_{D},

|{u_{ε} > 0}| ≤ lim inf_{n→∞} |{u_{n} > 0}| and

∥u_{ε}∥_{H^{1}(Ω)} ≤ lim inf_{n→∞} ∥u_{n}∥_{H^{1}(Ω)}.

Hence u_{ε} ∈ K and

J_{ε}(u_{ε}) ≤ lim inf_{n→∞} J_{ε}(u_{n}) = inf_{v∈K} J_{ε}(v),

therefore u_{ε} is a minimizer of J_{ε} in K. □

In the remaining of the section we will remove the subscript ε from the solution of (P_{ε}).

For the proof of properties (1)–(3), we apply the ideas developed in [4]. To this end, we need a series of lemmas.

Lemma 2.1. Let u ∈ K be a solution to (P_{ε}). There exists a constant C = C(N, Ω, ε) such that for every ball B_{r} ⊂⊂ Ω

\frac{1}{r} – ∫–_{∂B_{r}} u ≥ C implies u > 0 in B_{r}.

Proof. The idea is similar to that of Lemma 3.2 in [4]. Let v be the solution to

(2.1)  \begin{cases}v = u & in \overline{Ω \ B_{r}}, \\ Δv = v & in B_{r}. \end{cases}

Then v ∈ K, v > 0 in B_{r}. We claim that

(2.2)  ∥u − v∥_{H^{1}(Ω)}^{2} = ∥u∥_{H^{1}(Ω)}^{2} − ∥v∥_{H^{1}(Ω)}^{2}.

In fact,

∫_{B_{r}} ∇v∇(v − u) + v(v − u) dx = 0

since v − u ∈ H_{0}^{1}(B_{r}). This implies

(2.3)  ∫_{B_{r}} ∇u∇v + uv dx = ∫_{B_{r}} |∇v|^{2} + v^{2} dx.

This equality implies the claim since u = v in Ω \ B_{r}.

REGULARITY IN AN OPTIMIZATION PROBLEM 5

By (2.2), since u is a minimizer and u ≤ v in B_{r}, u = v in Ω \ B_{r}, we have

(2.4)  \begin{aligned}∫_{Ω} |∇(u − v)|^{2} + (u − v)^{2} dx & ≤ −F_{ε}(|\{u > 0\}|) + F_{ε}(|\{v > 0\}|) \\ & ≤ C_{ε} |\{u = 0\} ∩ B_{r}|. \end{aligned}

Now, as in [4], the idea is to control |{u = 0}∩B_{r}| from above by the left hand side of (2.4). We can assume that B_{r} = B_{1}(0). For |z| ≤ \frac{1}{2} we consider the change of variables from B_{1} into itself such that z becomes the new origin. We call u_{z}(x) = u((1 − |x|)z + x), v_{z}(x) = v((1 − |x|)z + x) and define

r_{ξ} = inf \{r / \frac{1}{8} ≤ r ≤ 1 and u_{z}(rξ) = 0\},

if this set is nonempty. Observe that this change of variables leaves the boundary fixed. Now, for almost every ξ ∈ ∂B_{1} we have

(2.5)  v_{z}(r_{ξ}ξ) = ∫_{r_{ξ}}^{1} \frac{d}{dr} (u_{z} − v_{z})(rξ) dr ≤ \sqrt{1 − r_{ξ}} (∫_{r_{ξ}}^{1} |∇(u_{z} − v_{z})(rξ)|^{2} dr)^{1/2} .

Let us see that

(2.6)  v_{z}(r_{ξ}ξ) ≥ C(N, Ω)(1 − r_{ξ}) – ∫–_{∂B_{1}} u.

In fact v_{z}(r_{ξ}ξ) = v((1 − r_{ξ})z + r_{ξ}ξ) and if |(1 − r_{ξ})z + r_{ξ}ξ| ≤ \frac{3}{4}, by Harnack inequality,

v_{z}(r_{ξ}ξ) ≥ C_{N}v(0).

Clearly (2.6) follows from

(2.7)  v(0) ≥ α(N, Ω) – ∫–_{∂B_{1}} u.

Observe that by (2.3), ∥v∥_{H^{1}(B_{1})} ≤ C∥u∥_{H^{1}(B_{1})} ≤ C = C(Ω).

In order to prove (2.7), we see that we can assume that – ∫–_{∂B_{1}} u = – ∫–_{∂B_{1}} v = 1. So that

\{ ; \begin{matrix}Δv = v & in B_{1}, \\ v > 0 & in B_{1}, \\ – ∫–_{∂B_{1}} v = 1. \end{matrix}

Assume that (2.7) does not hold, then there exists a sequence v_{n} ∈ H^{1}(B_{1}) with ∥v_{n}∥_{H^{1}(B_{1})} ≤ C,

– ∫–_{∂B_{1}} v_{n} = 1, Δv_{n} = v_{n}, v_{n} ≥ 0 in B_{1} and v_{n}(0) ≤ \frac{1}{n}. Then, for a subsequence, v_{n} → v weakly in H^{1}(B_{1}), strongly in L^{1}(∂B_{1}) and uniformly on compact sets of B_{1} with v(0) = 0. As Δv = v in B_{1} and v ≥ 0 there holds that v ≡ 0 and this contradicts the fact that – ∫–_{∂B_{1}} v = 1.

If |(1 − r_{ξ})z + r_{ξ}ξ| ≥ \frac{3}{4} we prove by a comparison argument that inequality (2.6) also holds. In fact, first observe that we can assume that – ∫–_{∂B_{1}} v = – ∫–_{∂B_{1}} u = 1. So that, by (2.7), v ≥ C_{N}α in B_{3/4}. Let w(x) = e^{−λ|x|^{2}} − e^{−λ}. There exists λ = λ(N, α) such that

\begin{cases}Δw ≥ w & in B_{1} \ B_{3/4}, \\ w ≤ C_{N}α & in ∂B_{3/4}, \\ w = 0 & in ∂B_{1}, \end{cases}

so v ≥ w ≥ C(1 − |x|) in B_{1} \ B_{3/4}. Therefore

v_{z}(r_{ξ}ξ) ≥ C(1 − |(1 − r_{ξ})z + r_{ξ}ξ|) – ∫–_{∂B_{1}} u ≥ C(1 − r_{ξ}) – ∫–_{∂B_{1}} u

since |z| ≤ \frac{1}{2}. So that (2.6) holds for every r_{ξ} ≥ \frac{1}{8}.

By (2.5) and (2.6) we have

c\sqrt{1 − r_{ξ}} – ∫–_{∂B_{1}} u ≤ (∫_{r_{ξ}}^{1} |∇(u_{z} − v_{z})|^{2}(rξ) dr)^{1/2} .

Hence

c^{2} ∫_{∂B_{1}}(1 − r_{ξ}) dS_{ξ} (– ∫–_{∂B_{1}} u)^{2} ≤ ∫_{∂B_{1}} ∫_{r_{ξ}}^{1} |∇(u_{z} − v_{z})|^{2}(rξ) dr dS_{ξ}
≤ C ∫_{B_{1}} |∇(u_{z} − v_{z})|^{2} dx.

Since

∫_{∂B_{1}}(1 − r_{ξ}) dS_{ξ} ≥ ∫_{B_{1}\B_{1/4}(z)} χ_{{u=0}} dx,

we have

c^{2}|{x ∈ B_{1} \ B_{1/4}(z) / u(x) = 0}| (– ∫–_{∂B_{1}} u)^{2} ≤ C ∫_{B_{1}} |∇(u_{z} − v_{z})|^{2} dx
≤ K ∫_{B_{1}} |∇(u − v)|^{2} dx.

Finally, we integrate over z ∈ B_{1/2}(0) and use (2.4) to obtain

(2.8)  |B_{1} ∩ {u = 0}| (– ∫–_{∂B_{1}} u)^{2} ≤ K ∫_{B_{1}} |∇(u − v)|^{2} dx
≤ KC_{ε}|B_{1} ∩ {u = 0}|.

Therefore we either have u > 0 almost everywhere in B_{1} or else – ∫–_{∂B_{1}} u ≤ \sqrt{KC_{ε}}.

Hence we deduce that if

– ∫–_{∂B_{1}} u ≥ \sqrt{KC_{ε}} = C(N, Ω, ε)

then |B_{1} ∩ {u = 0}| = 0. So that by (2.4) u = v > 0 in B_{1}. □ Now we can prove the Lipschitz continuity of the minimizer u.

Proof of Theorem 2.1 (1). The proof follows as in [4] Lemma 3.3. In fact, let D ⊂⊂ D′ ⊂⊂ Ω and x ∈ D. Let r > 0 be the largest number such that B_{r}(x) ⊂ {u > 0} ∩ D′. As in [4] we prove by using Lemma 2.1 that {u > 0} is open and

\frac{1}{r} – ∫–_{∂B_{r}(x)} u ≤ C

with C independent of either u or x. Since u > 0 in B_{r}(x), it is a solution to

Δu = u in B_{r}(x).

Hence, there is a universal constant such that

|∇u(x)| ≤ C\{r∥u∥_{L^{∞}(B_{r}(x))} + \frac{1}{r} – ∫–_{∂B_{r}(x)} u\}.

Now, since u is subharmonic in Ω and D′ ⊂⊂ Ω, there holds that u is bounded in D′ by a constant that depends on the H^{1} norm of u in Ω which is bounded by a constant that depends only on Ω and ε. Therefore,

|∇u(x)| ≤ C

with C depending only on N, Ω, ε, D and D′. □

In order to prove the nondegeneracy of u we need the following Lemma (see [4], Lemma 3.4).

Lemma 2.2. Let u ∈ K be a solution to (P_{ε}). For 0 < κ < 1 there exists a constant c = c(κ, N, Ω, ε) such that for every ball B_{r}(x) ⊂⊂ Ω,

\frac{1}{r} – ∫–_{∂B_{r}} u ≤ c implies that u = 0 in B_{κr}.

Proof. As in [4] Lemma 3.4, we consider the function

(2.9)  ϕ_{r}^{N}(x) = \begin{cases}\frac{r}{N − 2} ((\frac{r}{|x|})^{N−2} − 1) & for N ≥ 3, \\ r log \frac{r}{|x|} & for N = 2, \\ r − |x| & for N = 1. \end{cases}

Without loss of generality we assume that r = 1 and x = 0. Then, we take

v(x) = \frac{γ\sqrt{κ}}{−ϕ_{κ}^{N}(\sqrt{κ})} max(−ϕ_{κ}^{N}(x), 0).

where

γ = \frac{1}{\sqrt{κ}} sup_{B\sqrt{κ}} u ≤ C(N, κ) – ∫–_{∂B_{1}} u,

since u is subharmonic. Hence, v ≥ u on ∂B\sqrt{κ}, and therefore if

w = \begin{cases}min(u, v) & in B\sqrt{κ}, \\ u & in Ω \ B\sqrt{κ}, \end{cases}

8 J. FERNANDEZ BONDER, J.D. ROSSI & N. WOLANSKI

there holds that

∫_{B_{κ}} |∇u|^{2} + u^{2} dx + |B_{κ} ∩ {u > 0}|

≤ J_{ε}(u) − ∫_{Ω\B_{κ}} |∇u|^{2} + u^{2} dx + |B_{κ} ∩ {u > 0}| − F_{ε}(|Ω ∩ {u > 0}|)
≤ J_{ε}(w) − ∫_{Ω\B_{κ}} |∇u|^{2} + u^{2} dx + |B_{κ} ∩ {u > 0}| − F_{ε}(|Ω ∩ {u > 0}|)
= ∫_{B\sqrt{κ}\B_{κ}} |∇w|^{2} + w^{2} dx − ∫_{B\sqrt{κ}\B_{κ}} |∇u|^{2} + u^{2} dx + |B_{κ} ∩ \{u > 0\}|
∫ + F_{ε}(|Ω ∩ {w > 0}|) − F_{ε}(|Ω ∩ {u > 0}|)
≤ ∫_{B\sqrt{κ}\B_{κ}} |∇w|^{2} + w^{2} dx − ∫_{B\sqrt{κ}\B_{κ}} |∇u|^{2} + u^{2} dx + (1 − ε) |B_{κ} ∩ \{u > 0\}|.

since w = 0 in B_{κ}, w = u in Ω \ B\sqrt{κ} so that, in particular, \{w > 0\} ⊂ \{u > 0\}. We have also used that F_{ε}(A) − F_{ε}(B) ≥ ε(A − B) if A ≥ B. Thus,

∫_{B_{κ}} |∇u|^{2} + u^{2} dx + ε|B_{κ} ∩ {u > 0}|

≤ ∫_{B\sqrt{κ}\B_{κ}} |∇w|^{2} + w^{2} dx − ∫_{B\sqrt{κ}\B_{κ}} |∇u|^{2} + u^{2}
= ∫_{B\sqrt{κ}\B_{κ}} |∇u − ∇(u − v)^{+}|^{2} − |∇u|^{2} dx + ∫_{B\sqrt{κ}\B_{κ}} (u − (u − v)^{+})^{2} − u^{2} dx
= − ∫_{B\sqrt{κ}\B_{κ}} ∇(u − v)^{+}∇(u + v) dx − ∫_{B\sqrt{κ}\B_{κ}}(u − v)^{+}(u + v) dx
= − ∫_{B\sqrt{κ}\B_{κ}} ∇(u − v)^{+}∇u dx − ∫_{B\sqrt{κ}\B_{κ}}(u − v)^{+}u dx
− ∫_{B\sqrt{κ}\B_{κ}} ∇(u − v)^{+}∇v dx − ∫_{B\sqrt{κ}\B_{κ}}(u − v)^{+}v dx
≤ −2 ∫_{B\sqrt{κ}\B_{κ}} ∇(u − v)^{+}∇v dx − 2 ∫_{B\sqrt{κ}\B_{κ}}(u − v)^{+}v dx
≤ 2 ∫_{∂B_{κ}} u∇v η dS ≤ C(N, κ) γ ∫_{∂B_{κ}} u.

Therefore,

(2.10)  ∫_{B_{κ}} |∇u|^{2} + u^{2} dx + ε|B_{κ} ∩ {u > 0}| ≤ C(N, κ) γ ∫_{∂B_{κ}} u.

Here we have used that min(u, v) = u − (u − v)^{+}, Δv = 0 in B\sqrt{κ} \ B_{κ}, v = 0 on ∂B_{κ}, and (u − v)^{+} = 0 on ∂B\sqrt{κ}.

Recall that γ is controlled by ∫_{∂B_{1}} u, so that γ will be small if ∫_{∂B_{1}} u is small.

By standard estimates,

\begin{matrix}∫_{∂B_{κ}} u ≤ C(N, κ) ∫_{B_{κ}} |∇u| + u dx \\ ≤ \frac{C}{2} ∫_{B_{κ}} |∇u|^{2} + u^{2} + \frac{C}{2} |B_{κ} ∩ \{u > 0\}|. \end{matrix}

So that, by (2.10), if γ is small enough we deduce that |B_{κ} ∩ {u > 0}| = 0. This is, u = 0 in B_{κ} and the lemma is proved. □

We can now prove the nondegeneracy of u.

Proof of Theorem 2.1, (2). Let x ∈ {u > 0} and r = dist (x, {u = 0}). As we proved in (2.7), since Δu = u in B_{r}(x), there holds that

u(x) ≥ α(N, Ω) – ∫–_{∂B_{r}(x)} u.

Since u(x) > 0,

\frac{1}{r} – ∫–_{∂B_{r}(x)} u ≥ c

where c is the constant in Lemma 2.2 for κ = 1/2. Thus,

u(x) ≥ cα r.

The upper bound clearly follows from the Lipschitz continuity of u. Hence (2) is proved. □

Proof of Theorem 2.1, (3). In order to prove the uniform positive density of {u > 0} and {u = 0} at every free boundary point we proceed as in [4], Lemma 3.7. The only difference being that the function v that we have to take is the one in (2.1).

This ends the proof of Theorem 2.1. □

Corollary 2.1. Let u ∈ K be a solution to (P_{ε}). Let D ⊂⊂ Ω. There exist constants c, C > 0 depending only on N, Ω, D and ε such that for B_{r}(x) ⊂ D and x ∈ ∂{u > 0},

(2.11)  c ≤ \frac{1}{r} – ∫–_{∂B_{r}(x)} u ≤ C.

Proof. It follows easily from Lemmas 2.1 and 2.2. □

Lemma 2.3. Let u ∈ K be a solution to (P_{ε}). Then u satisfies for every φ ∈ C_{0}^{∞}(Ω) such that supp φ ⊂ {u > 0},

(2.12)  ∫_{Ω} ∇u∇φ + uφ dx = 0.

Moreover, the application

λ(φ) := − ∫_{Ω} ∇u∇φ + uφ dx

from C_{0}^{∞}(Ω) into ℝ defines a nonnegative Radon measure with support on Ω ∩ ∂{u > 0}.

Proof. The proof follows exactly as in [4], Lemma 4.2. □

Theorem 2.2. Let u ∈ K be a solution to (P_{ε}). Let D ⊂⊂ Ω. Then, there exist constants C, c > 0 such that for B_{r}(x) ⊂ D and x ∈ ∂{u > 0},

c r^{N−1} ≤ ∫_{B_{r}(x)} dλ ≤ C r^{N−1}.

Proof. For n large enough, let u_{n} = u ∗ ρ_{n} where ρ_{n} are the standard mollifiers. Then,

\begin{matrix}∫_{B_{r}(x)} λ ∗ ρ_{n} dx = ∫_{B_{r}(x)} Δu_{n} − u_{n} dx = ∫_{∂B_{r}(x)} ∇u_{n} · ν dS − ∫_{B_{r}(x)} u_{n} \\ ≤ ω_{N−1} sup_{∂B_{r}(x)} |∇u_{n}| r^{N−1} ≤ C r^{N−1} \end{matrix}

since |∇u_{n}| ≤ |∇u| ≤ C for a certain constant C depending on D. By taking limit for n → ∞ we get

∫_{B_{r}(x)} dλ ≤ C r^{N−1}.

The other inequality follows as in the proof of Theorem 4.3 in [4] by taking as G_{y}(z) the (positive) Green function of −Δ + Id with homogeneous Dirichlet boundary conditions in the ball B_{r}(x). Then, for 0 < κ < 1/2 and y ∈ B_{κr}(x) one uses the inequality

v(y) ≥ Cv(x) ≥ Cα – ∫–_{∂B_{r}(x)} u

for v the solution to Δv −v = 0 in B_{r}(x), v = u on ∂B_{r}(x), that follows from Harnack inequality and (2.7). □

Theorem 2.3 (Representation Theorem). Let u ∈ K be a solution to (P_{ε}). Then,

(1) H^{N−1}(D ∩ ∂{u > 0}) < ∞ for every D ⊂⊂ Ω.

(2) There exists a Borel function q_{u} such that

Δu − u = q_{u} H^{N−1}⌊∂{u > 0}.

(3) For D ⊂⊂ Ω there are constant 0 < c ≤ C < ∞ depending on N, Ω, D and the constants in (2.11) such that for B_{r}(x) ⊂ D and x ∈ ∂{u > 0},

c ≤ q_{u}(x) ≤ C, c r^{N−1} ≤ H^{N−1}(B_{r}(x) ∩ ∂{u > 0}) ≤ C r^{N−1}.

Proof. The proof follows exactly as that of Theorem 4.5 in [4]. □

Remark 2.1. Let u ∈ K be a solution to (P_{ε}) and D ⊂⊂ Ω. Then D ∩ ∂{u > 0} has finite perimeter. Thus, the reduce boundary ∂_{red}{u > 0} is defined as well as the measure theoretic normal ν(x) for x ∈ ∂_{red}{u > 0}. See [8].

If the free boundary ∂{u > 0} is a regular surface then q_{u} = −∂_{ν}u. In Theorem 2.4 it is shown that this is true for almost all points in the reduce boundary.

Proposition 2.1. Let u ∈ K be a solution to (P_{ε}) and let B_{ρ_{k}}(x_{k}) ⊂ Ω be a sequence of balls with ρ_{k} → 0, x_{k} → x_{0} ∈ Ω and u(x_{k}) = 0. Let

u_{k}(x) := \frac{1}{ρ_{k}} u(x_{k} + ρ_{k}x).

We call u_{k} a blow-up sequence with respect to B_{ρ_{k}}(x_{k}). Since u is locally Lipschitz continuous, there exists a blow-up limit u_{0} : ℝ^{N} → ℝ satisfying (2.11) with the same constants, when x_{k} ∈ ∂{u > 0}, and such that, for a subsequence,

u_{k} → u_{0} in C_{loc}^{α}(ℝ^{N}) for every 0 < α < 1,
∇u_{k} → ∇u_{0} weakly star in L_{loc}^{∞}(ℝ^{N}),
∂{u_{k} > 0} → ∂{u_{0} > 0} locally in Hausdorff distance,
χ_{{u_{k}>0}} → χ_{{u_{0}>0}} in L_{loc}^{1}(ℝ^{N}),
Δu_{0} = 0 in {u_{0} > 0}.

Moreover if x_{k} ∈ ∂{u > 0}, then 0 ∈ ∂{u_{0} > 0}

Proof. It follows as in [4], Section 4.7 observing that Δu_{k} − ρ_{k}^{2}u_{k} = 0 in {u_{k} > 0}. □

Theorem 2.4 (Identification of q_{u}). Let u ∈ K be a solution to (P_{ε}). Then, for almost every x_{0} ∈ ∂_{red}{u > 0},

u(x_{0} + x) = q_{u}(x_{0})⟨x, ν(x_{0})⟩^{−} + o(|x|) for x → 0

with ν(x_{0}) the outward unit normal de ∂{u > 0} in the measure theoretic sense.

Proof. It follows exactly as Theorem 4.8 and Remark 4.9 in [4]. □

Remark 2.2. Observe that by Theorem 2.1, (3)

H^{N−1}(∂{u > 0} \ ∂_{red}{u > 0}) = 0.

See [8].

Now we get a more precise identification of q_{u}.

Theorem 2.5. Let u ∈ K be a solution to (P_{ε}) and q_{u} the function in Theorem 2.4. Then there exists a constant λ_{u} such that

(2.13)  lim sup_{u(x)>0^{x→x_{0}}} |∇u(x)| = λ_{u},  for every x_{0} ∈ Ω ∩ ∂{u > 0}

q_{u}(x_{0}) = λ_{u}, H^{N−1} − a.e x_{0} ∈ Ω ∩ ∂{u > 0}. (2.14)

Moreover, if B is a ball contained in {u = 0} touching the boundary ∂{u > 0} at x_{0}. Then

(2.15)  lim sup_{u(x)>0^{x→x_{0}}} \frac{u(x)}{dist(x, B)} = λ_{u}.

Proof. We follow the ideas of [2] Theorem 3 and [16] Theorem 5.1 and Lemma 5.2.

Let x_{0}, x_{1} ∈ ∂{u > 0} and ρ_{k} → 0^{+}. For i = 0, 1 let x_{i,k} → x_{i} with u(x_{i,k}) = 0 such that B_{ρ_{k}}(x_{i,k}) ⊂ Ω and such that the blow-up sequence

u_{i,k}(x) = \frac{1}{ρ_{k}} u(x_{i,k} + ρ_{k}x)

has a limit u_{i}(x) = λ_{i}⟨x, ν_{i}⟩^{−}, with 0 < λ_{i} < ∞ and ν_{i} a unit vector. We will prove that λ_{0} = λ_{1}. From this, the Theorem will follow as in [16].

Assume that λ_{1} < λ_{0} then we will perturb the minimizer u near x_{0} and x_{1} and get an admissible function with less energy. To this end, we take a nonnegative C_{0}^{∞} function Φ supported in the unit interval that we will choose later on. For t > 0 small, define

τ_{k}(x) = \begin{cases}x + tρ_{k}Φ(\frac{|x − x_{0,k}|}{ρ_{k}})ν_{0} & for x ∈ B_{ρ_{k}}(x_{0,k}), \\ x + tρ_{k}Φ(\frac{|x − x_{1,k}|}{ρ_{k}})ν_{1} & for x ∈ B_{ρ_{k}}(x_{1,k}), \\ x & elsewhere, \end{cases}

which is a diffeomorphism if t is small enough. Now let

v_{k}(x) = u(τ_{k}^{−1}(x)),

that are admissible functions. Let us also define

η_{i}(y) = (−1)^{i}Φ(|y|)ν_{i}.

We have

(2.16)  F_{ε}(|{v_{k} > 0}|) − F_{ε}(|{u > 0}|) = o(ρ_{k}^{N}),

(2.17)  \begin{matrix}∫_{Ω} |∇v_{k}|^{2} dx − ∫_{Ω} |∇u|^{2} dx = & ρ_{k}^{N}(t(λ_{1}^{2} − λ_{0}^{2}) ∫_{B_{1}(0)∩\{y_{1}=0\}} Φ(|y|) dH_{y}^{N−1} + o(t)) \\ + o(ρ_{k}^{N}). \end{matrix}

It remains to estimate the difference of the L^{2} norms.

ρ_{k}^{−N} ∫_{B_{ρ_{k}} (x_{i,k})} v_{k}^{2} − u^{2} = ∫_{B_{1}(0)∩{u_{i,k}>0}} (u_{i,k}^{2}(1 + t div η_{i}) − u_{i,k}^{2}) + o(t)
= t ∫_{B_{1}(0)∩{u_{i,k}>0}} u_{i,k}^{2} div η_{i} + o(t)
→ tλ_{i}^{2} ∫_{B_{1}(0)∩{y·ν_{i}<0}}(y · ν_{i})^{2} div η_{i} + o(t)
= −2t(−1)^{i}λ_{i}^{2} ∫_{B_{1}(0)∩{y·ν_{i}<0}} Φ(|y|)(y · ν_{i})
+ tλ_{i}^{2} ∫_{B_{1}(0)∩{x·ν_{i}=0}}(y · ν_{i})^{2}η_{i} · ν_{i} dH^{N−1} + o(t)
= −2t(−1)^{i}λ_{i}^{2} ∫_{B_{1}(0)∩{y_{1}<0}} Φ(|y|)y_{1} dy + o(t).

Hence

(2.18)  ∫_{Ω} v_{k}^{2} dx − ∫_{Ω} u^{2} dx = ρ_{k}^{N}(t(λ_{1}^{2} − λ_{0}^{2}) ∫_{B_{1}(0)∩{y_{1}<0}} 2y_{1}Φ(|y|) + o(t)) + o(ρ_{k}^{N}).

Now if we choose Φ so that

∫_{{y_{1}=0}} Φ(|y|) dH^{N−1} + ∫_{{y_{1}<0}} 2y_{1}Φ(|y|) dy > 0

we get from (2.16), (2.17) and (2.18), for k large enough and t small enough that

J_{ε}(v_{k}) < J_{ε}(u),

a contradiction. □

Summing up, we have the following theorem,

Theorem 2.6. Let u ∈ K be a solution to (P_{ε}). Then u is a weak solution to the following free boundary problem

− Δu + u = 0 in {u > 0} ∩ Ω,

\frac{∂u}{∂ν} = λ_{u} on ∂\{u > 0\} ∩ Ω,

where λ_{u} is the constant in Theorem 2.5. More precisely, H^{N−1}−a.e. point x_{0} ∈ ∂{u > 0} belongs to ∂_{red}{u > 0} and

u(x_{0} + x) = λ_{u}⟨x, ν(x_{0})⟩^{−} + o(|x|) for x → 0.

Finally, we get an estimate of the gradient of u that will be needed in order to get the regularity of the free boundary.

Theorem 2.7. Let u ∈ K be a solution to (P_{ε}). Given D ⊂⊂ Ω, there exist constants C = C(N, ε, D) r_{0} = r_{0}(N, D) > 0 and γ = γ(N, ε, D) > 0 such that, if x_{0} ∈ D ∩ ∂{u > 0} and r < r_{0}, then

sup_{B_{r}(x_{0})} |∇u| ≤ λ_{u}(1 + Cr^{γ}).

Proof. The proof follows the lines of the proof of Theorem 4.1 in [5].

Let U_{k} = (|∇u| − λ_{u} − \frac{1}{k})^{+} and U_{0} = (|∇u| − λ_{u})^{+}. By (2.13) we know that U_{k} vanishes in a neighborhood of the free boundary. Also, the support of U_{k} is contained in {u > 0}. Therefore U_{k} satisfies

ΔU_{k} ≥ U_{k} in Ω ∩ {u > 0}

and vanishes in a neighborhood of the free boundary. We extend U_{k} by zero into {u = 0} and set

h_{k}(r) = sup_{B_{r}(x_{0})} U_{k}, h_{0}(r) = sup_{B_{r}(x_{0})} U_{0},

for any r < r_{0} = dist (D, ∂Ω) and x_{0} ∈ D ∩ ∂{u > 0}.

Then, h_{k}(r) − U_{k} is a supersolution of Δv = v in the ball B_{r}(x_{0}) and

\begin{aligned}h_{k}(r) − U_{k} & ≥ 0 & in B_{r}(x_{0}) \\ & = h_{k}(r) & in B_{r}(x_{0}) ∩ \{u = 0\}. \end{aligned}

Applying the weak Harnack inequality (see [12] p. 246) with 1 ≤ p < N/(N − 2), we get

inf_{B_{r/2}(x_{0})} (h_{k}(r) − U_{k}) ≥ cr^{−N/p}∥h_{k}(r) − U_{k}∥_{L^{p}(B_{r}(x_{0}))} ≥ ch_{k}(r),

since, by Theorem 2.1 (3), |B_{r}(x_{0}) ∩ {u = 0}| ≥ cr^{N}. Taking now k → ∞ we obtain

inf_{B_{r/2}(x_{0})} (h_{0}(r) − U_{0}) ≥ ch_{0}(r),

for some 0 < c < 1, which is the same as

sup_{B_{r/2}(x_{0})} U_{0} ≤ (1 − c)h_{0}(r).

Therefore

h_{0}(\frac{r}{2}) ≤ (1 − c)h_{0}(r),

from which it follows that h_{0}(r) ≤ Cr^{γ} for some C > 0, 0 < γ < 1 and now the conclusion of the Theorem follows. □

3. Regularity of the free boundary.

At this point we have that our minimizer u_{ε} meets the conditions of the regularity theory developed in [4]. The only difference being the equation satisfied by u_{ε} in {u_{ε} > 0}.

We will recall some definitions and we will point out the only significant difference with [4]. The rest of the proof of the regularity then follows as sections 7 and 8 of [4] with only minor modifications.

Throughout this section we will remove the subscript ε.

Definition 3.1 (Flat free boundary points). Let 0 < σ_{+}, σ_{−} ≤ 1 and τ > 0. We say that u is of class

F (σ_{+}, σ_{−}; τ) in B_{ρ} = B_{ρ}(0)

if

(1) 0 ∈ ∂{u > 0} and

\begin{matrix}u = 0 & for x_{N} ≥ σ_{+}ρ, \\ u(x) ≥ −λ(x_{N} + σ_{−}ρ) & for x_{N} ≤ −σ_{−}ρ. \end{matrix}

(2) |∇u| ≤ λ(1 + τ) in B_{ρ}.

If the origin is replaced by x_{0} and the direction e_{N} by the unit vector ν we say that u is of class F (σ_{+}, σ_{−}; τ) in B_{ρ}(x_{0}) in direction ν.

Observe that the results in Section 2 imply that the minimizer u of J_{ε} is in the class F (σ, 1; σ) in B_{ρ}(x_{0}) in direction ν_{u}(x_{0}) for every x_{0} ∈ ∂_{red}{u > 0} with σ = σ(ρ) → 0 as ρ → 0.

The following lemma (Lemma 7.2 in [4]) is the only one that requires a non obvious modification.

Lemma 3.1. There is a constant C = C(N) such that u ∈ F (σ, 1; σ) in B_{ρ}(x_{0}) in direction ν implies u ∈ F (2σ, Cσ; σ) in B_{ρ/2}(x_{0}) in direction ν.

Proof. Clearly, by a change of variables, we may assume that x_{0} = 0 and ν = e_{N}. Let ū(x) = u(ρx)/λρ, then |∇ū| ≤ 1 + σ and ū ∈ F (σ, 1; σ) in B_{1}. That is ū = 0 if x_{N} > σ. Define

η(x′) = \begin{cases}exp (−\frac{9|x′|^{2}}{1 − 9|x′|^{2}} ) & for |x′| < \frac{1}{3} \\ 0 & otherwise \end{cases}

and choose s ≥ 0 maximal with the property that ū = 0 in x_{N} > σ − sη(x′).

Now, the proof follows as in Lemma 7.2 of [4] with the only difference that the comparison function v must be the solution to Δv = ρ^{2}ū in D = B_{1} ∩ {x_{N} < σ − sη(x′)} instead of a harmonic function. The estimate

∂_{−ν}v ≤ 1 + Cσ

follows from

|∇(v + x_{N})| ≤ C[ sup_{D}(v + x_{N}) + ρ^{2}] ≤ Cσ

in D ∩ B_{1/2} if ρ^{2} ≤ Cσ, since

Δ(v + x_{N}) = ρ^{2}ū in D,

v + x_{N} ≤ Cσ in D and |ū(x)| ≤ 2. □

Once this lemma is established the following regularity result follows.

Theorem 3.1. Let u ∈ K be a solution to (P_{ε}). Then ∂_{red}{u > 0} is a C^{1,β} surface locally in Ω and the remainder of the free boundary has H^{N−1}–measure zero. Moreover, if N = 2 then the whole free boundary is a C^{1,β} surface.

4. Behavior of the minimizer for small ε.

To complete the analysis of the problem, we will now show that if ε is small enough, then

|{u_{ε} > 0}| = α.

To this end, we need to prove that the constant λ_{ε} := λ_{u_{ε}} is bounded from above and below by positive constants independent of ε. We perform this task in a series of lemmas.

Lemma 4.1. Let u_{ε} ∈ K be a solution to (P_{ε}). Then, there exist constants C, c > 0 independent of ε such that

(4.1)  c ≤ |{u_{ε} > 0}| ≤ α + Cε.

Proof. As J_{ε}(u_{ε}) is bounded from above uniformly in ε we obtain

F_{ε}(|{u_{ε} > 0}|) ≤ C.

Hence

|{u_{ε} > 0}| ≤ α + Cε.

For the lower bound, we proceed as follows; by the Sobolev trace embedding, for some 1 < p < 2, such that p(N − 1)/(N − p) > q,

1 ≤ ∥u_{ε}∥_{L^{q}(∂Ω)} ≤ C∥u_{ε}∥_{W^{1,p}(Ω)} ≤ C∥u_{ε}∥_{H^{1}(Ω)}|{u_{ε} > 0}|^{θ},

for some exponent θ that depends only on p. Since ∥u_{ε}∥_{H^{1}(Ω)} is uniformly bounded, the lower bound follows. □

Lemma 4.2. Let u_{ε} ∈ K be a solution to (P_{ε}). Then, there exists a constant C > 0 independent of ε such that

λ_{ε} := λ_{u_{ε}} ≤ C.

Proof. For D ⊂⊂ Ω, we take φ ∈ C_{0}^{∞}(Ω) as a test function in Lemma 2.3 such that 0 ≤ φ ≤ 1, φ ≡ 1 in D and ∥∇φ∥_{∞} ≤ C = C(dist(D, ∂Ω)) to get, since ∥u_{ε}∥_{H^{1}(Ω)} is bounded independently of ε,

C ≥ ∫_{Ω} ∇u^{ε}∇φ dx + ∫_{Ω} u_{ε}φ dx = λ_{ε}(φ) ≥ λ_{ε}H^{N−1}(D ∩ ∂_{red}{u_{ε} > 0}).

Now, by the isoperimetric inequality and by Lemma 4.1 we have

H^{N−1}(D ∩ ∂_{red}{u_{ε} > 0}) ≥ c(|D ∩ {u^{ε} > 0}|)^{(N−1)/N} ≥ c > 0.

This completes the proof of the lemma. □

The uniform lower bound follows similarly to Lemma 6 in [2]. We only make a sketch of the proof for the reader’s convenience. It is at this point where we need the hypothesis that Γ_{D} ≠ ∅.

Lemma 4.3. Let Γ_{D} ≠ ∅ be the closure of a relatively open subset of ∂Ω. Let φ_{0} ∈ H^{1}(Ω) with φ_{0} ≥ c_{0} > 0 in Γ_{D}. Let u_{ε} ∈ K be a solution to (P_{ε}). Then

(1) u_{ε} is positive in a neighborhood of Γ_{D} (depending on ε).

(2) There exists a constant c > 0 independent of ε such that

c < λ_{ε} := λ_{u_{ε}}.

Proof. Let us first prove (1). In fact, arguing as in (2.8), given y_{0} ∈ Γ_{D} there exists a constant K > 0 independent of ε such that

|Ω_{r} ∩ \{u = 0\}| (\frac{1}{r} – ∫–_{∂Ω_{r}} u)^{2} ≤ K ∫_{Ω_{r}} |∇(u − v)|^{2} dx,

where Ω_{r} = Ω ∩ B_{r}(y_{0}) and v is the solution of

\begin{cases}Δv = v & in Ω_{r} \\ v = u & on ∂Ω_{r}. \end{cases}

Therefore,

(\frac{c_{0}}{r} )^{2} |Ω_{r} ∩ \{u = 0\}| ≤ K(∥u∥_{H^{1}(Ω_{r})}^{2} − ∥v∥_{H^{1}(Ω_{r})}^{2}) ≤ \frac{C}{ε} |Ω_{r} ∩ \{u = 0\}|.

So that, u > 0 in Ω_{r} for small r depending on ε.

In order to see (2) we proceed as in [2] Lemma 6. Let y_{0} ∈ Γ_{D} and let D_{t} with 0 ≤ t ≤ 1 be a family of open sets with smooth boundary and uniformly (in ε and t) bounded curvatures such that D_{0} is an exterior tangent ball at y_{0}, D_{1} contains a free boundary point, D_{t} ∩ ∂Ω ⊂ Γ_{D} and D_{0} ⊂⊂ D_{t} for t > 0.

Let t ∈ (0, 1) be the first time such that D_{t} touches the free boundary and let x_{0} ∈ ∂D_{t}∩∂{u_{ε} > 0\} ∩ Ω. Now, take w the harmonic function in D_{t} \ \overline{D}_{0} with w = c_{0} on ∂D_{0} and w = 0 on ∂D_{t}. Thus w ≤ u_{ε} in D_{t} ∩ Ω and ∂_{−ν}w(x_{0}) ≥ c c_{0} with c independent of ε, therefore, for r small enough

\frac{1}{r} – ∫–_{∂B_{r}(x_{0})} u_{ε} ≥ \frac{1}{r} – ∫–_{∂B_{r}(x_{0})} w ≥ c̄c_{0}

with c̄ is independent of ε.

If v_{0} is the solution to

\begin{cases}Δv = v & in B_{r}(x_{0}) \\ v = u & on ∂B_{r}(x_{0}), \end{cases}

then, by (2.8), we have

c|B_{r}(x_{0}) ∩ \{u_{ε} = 0\}| ≤ |B_{r}(x_{0}) ∩ \{u_{ε} = 0\}| (\frac{1}{r} – ∫–_{∂B_{r}(x_{0})} u_{ε})^{2}

≤ K ∫_{B_{r}(x_{0})} |∇(u_{ε} − v_{0})|^{2} dx ≤ K(∥u_{ε}∥_{H^{1}(Ω_{r})}^{2} − ∥v_{0}∥_{H^{1}(Ω_{r})}^{2}).

Let now δ_{r} = |B_{r}(x_{0}) ∩ {u_{ε} = 0}| and let x_{1} ∈ ∂{u_{ε} > 0} be such that the free boundary is smooth in a neighborhood of x_{1}. We perturb {u_{ε} > 0} in a neighborhood of x_{1} so that the measure of the perturbed set is increased an amount δ_{r} (cf. with Theorem 2.5).

Let Φ be a smooth nonnegative function supported in B_{κ}(x_{1}) with κ > 0 small. For x ∈ B_{κ}(x_{1}) we write x = σ + sν(σ) with σ ∈ ∂{u_{ε} > 0} and s ∈ ℝ where ν(σ) is the outer unit normal to the free boundary at σ. We define the change of variables y = x − Φ(σ)τν(σ) with τ > 0 small and the deformed set D_{δ_{r}} such that D_{δ_{r}} ∩ B_{κ}(x_{1}) = {y / x ∈ {u_{ε} > 0} ∩ B_{κ}(x_{1})}. Observe that if r is small we can perform this perturbation in such a way that it decreases the measure of {u_{ε} > 0} in exactly δ_{r}. Also, observe that δ_{r} → 0 as r → 0.

Now let v_{r} be the solution of

(4.2)  \begin{cases}Δv = v & in D_{δ_{r}}, \\ v = 0 & on ∂D_{δ_{r}} ∩ B_{κ}(x_{1}), \\ v = u_{ε} & on ∂B_{κ}(x_{1}) ∩ \overline{D}_{r}, \end{cases}

then v_{r} verifies

\frac{∂v_{r}}{∂ν} = −λ_{ε} + o(δ_{r}).

On the other hand,

u_{ε} = λ_{ε}δ_{r} + o_{ε}(δ_{r}), on ∂{v_{r} > 0} ∩ B_{κ}(x_{1}).

Thus

∫_{B_{κ}(x_{1})} |∇v_{r}|^{2} + v_{r}^{2} dx − ∫_{B_{κ}(x_{1})} |∇u_{ε}|^{2} + (u_{ε})^{2} dx = ∫_{B_{κ}(x_{1})} |∇(u_{ε} − v_{r})|^{2} + (u_{ε} − v_{r})^{2} dx

= − ∫_{∂\{v_{r}>0\}∩B_{κ}(x_{1})} \frac{∂v_{r}}{∂ν} u_{ε} dS

= λ_{ε}^{2}δ_{r} + o_{ε}(δ_{r}).

Now we extend v_{r} by zero to B_{κ}(x_{1}) \ D_{δ_{r}} and define

w_{r} = \begin{cases}v_{r} & in B_{κ}(x_{1}), \\ v_{0} & in B_{r}(x_{0}), \\ u & elsewhere. \end{cases}

Then |{w_{r} > 0}| = |{u_{ε} > 0}| and w_{r} = u_{ε} on ∂Ω, thus

0 ≤ J_{ε}(w_{r}) − J_{ε}(u_{ε}) = ∫_{Ω} |∇w_{r}|^{2} + w_{r}^{2} dx − ∫_{Ω} |∇u_{ε}|^{2} + (u_{ε})^{2} dx
= ∫_{B_{r}(x_{0})} |∇v_{0}|^{2} + v_{0}^{2} dx − ∫_{B_{κ}(x_{1})} |∇u_{ε}|^{2} + (u_{ε})^{2} dx
+ ∫_{B_{κ}(x_{1})} |∇v_{r}|^{2} + v_{r}^{2} dx − ∫_{B_{κ}(x_{1})} |∇u_{ε}|^{2} + (u_{ε})^{2} dx
≤ −cδ_{r} + λ_{ε}^{2}δ_{r} + o_{ε}(δ_{r}),

for every r > 0 small. Therefore, λ_{ε}^{2} ≥ c/2. □

Now we are in a position to prove the main result of this section, mainly that for ε small the measure of the positivity set is exactly α.

Theorem 4.1. Let Γ_{D} ≠ ∅ be the closure of a relatively open subset of ∂Ω. Let φ_{0} ∈ H^{1}(Ω) with φ_{0} ≥ c_{0} > 0 in Γ_{D}. Let u_{ε} ∈ K be a solution to (P_{ε}). Then, for ε small

(4.3)  |{u_{ε} > 0}| = α.

Proof. Arguing by contradiction, assume first that |{u_{ε} > 0}| > α. Let x_{1} ∈ ∂{u_{ε} > 0} ∩ Ω be a regular point. We will proceed as in the proof of the previous lemma. Given δ > 0, we perturb the domain {u_{ε} > 0} in a neighborhood of x_{1}, B_{κ}(x_{1}), decreasing its measure by δ. We choose δ small so that the measure of the perturbed set is still larger than α. Then we let v be the solution to (4.2) extended by zero to the rest of B_{κ}(x_{1}) and equal to u in the rest of Ω. We have

0 ≤ J_{ε}(v) − J_{ε}(u_{ε}) = ∫_{Ω} |∇v|^{2} + v^{2} − ∫_{Ω} |∇u_{ε}|^{2} + (u_{ε})^{2} + F_{ε}(|{v > 0}|) − F_{ε}(|{u_{ε} > 0}|)

≤ λ_{ε}^{2}δ + o_{ε}(δ) − \frac{1}{ε}δ ≤ (C^{2} − \frac{1}{ε})δ + o_{ε}(δ) < 0,

if ε < ε_{0} and then δ < δ_{0}(ε). A contradiction.

Now assume that |{u_{ε} > 0}| < α. We proceed as in the previous case but this time we perturb in a neighborhood of x_{1} the set {u_{ε} > 0} increasing the measure by δ. Then we construct the function v as before, and if δ is small enough |{v > 0}| < α. Then

0 ≤ J_{ε}(v) − J_{ε}(u_{ε}) = ∫_{Ω} |∇v|^{2} + v^{2} − ∫_{Ω} |∇u_{ε}|^{2} + (u_{ε})^{2} + F_{ε}(|{v > 0}|) − F_{ε}(|{u_{ε} > 0}|)
≤ −λ_{ε}^{2}δ + o_{ε}(δ) + εδ ≤ (−c^{2} + ε)δ + o_{ε}(δ) < 0,

if ε < ε_{1} and then δ < δ_{0}(ε). Again a contradiction that ends the proof. □

As a consequence of the previous theorem, we get

Corollary 4.1. Let Γ_{D} ≠ ∅ be the closure of a relatively open subset of ∂Ω. Let φ_{0} ∈ H^{1}(Ω) with φ_{0} ≥ c_{0} > 0 in Γ_{D}. Then, there exists a minimizer u of J (v) in the set

K_{α} = {v ∈ H^{1}(Ω) / ∥v∥_{L^{q}(Γ_{N})} = 1, v = φ_{0} on Γ_{D}, |{v > 0}| = α}.

This minimizer can be chosen in such a way that it is locally Lipschitz continuous in Ω and the free boundary ∂{u > 0} ∩ Ω is locally a C^{1,β} surface up to a set of H^{N−1} measure zero. In the case N = 2 the free boundary is locally a C^{1,β} surface.

Proof. From our previous results we have (4.3) for every ε small enough. Therefore we can take u = u_{ε} and the desired regularity of u and its free boundary follows from the results of Sections 2 and 3. □

5. Main results.

In this last section we go back to our original minimization problem related to the best Sobolev trace constant. Here we prove that any extremal is a locally Lispchitz continuous function and the boundary of the hole ∂{u > 0} ∩ Ω is locally C^{1,β} up to a set of H^{N−1} measure zero.

We begin with the following.

Theorem 5.1. Let ϕ_{0} be a minimizer for (P_{α}). Assume that there exists a positive constant c such that ϕ_{0} > c in a ball B′_{0} ⊂ Ω (resp. on B′_{0} ∩ ∂Ω where B′_{0} is a ball centered at ∂Ω). Then ϕ_{0} is a minimizer of J_{ε} in

K_{2} = {v ∈ H^{1}(Ω) / ∥v∥_{L^{q}(∂Ω)} = 1, v = ϕ_{0} in B_{0}}

(resp. ϕ_{0}/k is a minimizer of J_{ε} in K = {v ∈ H^{1}(Ω) / ∥v∥_{L^{q}(Γ_{N})} = 1, v = ϕ_{0}/k on Γ_{D}} with Γ_{D} = ∂Ω∩B_{0}, Γ_{N} = ∂Ω\Γ_{D}). Here, B_{0} is a ball compactly contained in B′_{0} and k = ∥ϕ_{0}∥_{L^{q}(Γ_{N})}.

In particular, ϕ_{0} is locally Lipschitz continuous in Ω and the free boundary ∂{ϕ_{0} > 0} ∩ Ω is locally a C^{1,β} surface up to a set of zero H^{N−1} measure. In the case N = 2 the free boundary is locally a C^{1,β} surface.

Proof. We will make the proof for the first case, the second one follows in the same way.

Let ε be small enough so that any minimizer u_{ε} of J_{ε} in K_{2} verifies that |{u_{ε} > 0}| = α. Then, it follows that ϕ_{0} is one of such minimizers and so the conclusions of the theorem follow. In fact, as ϕ_{0} minimizes (P_{α}) we have

(5.1)  J_{ε}(ϕ_{0}) = ∫_{Ω} |∇ϕ_{0}|^{2} + |ϕ_{0}|^{2} dx ≤ ∫_{Ω} |∇v|^{2} + |v|^{2} dx

for any v ∈ H^{1}(Ω) such that ∥v∥_{L^{q}(∂Ω)} = 1 and |{v > 0}| = α. In particular (5.1) holds for v = u_{ε}. Thus

J_{ε}(ϕ_{0}) ≤ J_{ε}(u_{ε}) = inf_{v∈K_{2}} J_{ε}(v).

This ends the proof. □

In particular, by the symmetry results for minimizers of (P_{α}) in balls of [11] we have the following corollary.

Corollary 5.1. Let Ω = B(x_{0}, r) be a ball and let ϕ_{0} be a minimizer of (P_{α}). Then ϕ_{0} is locally Lipschitz continuous in B(x_{0}, r) and the free boundary ∂{ϕ_{0} > 0} ∩ B(x_{0}, r) is locally a C^{1,β} surface up to a set of zero H^{N−1} measure. In the case N = 2 the free boundary is locally a C^{1,β} surface.

Proof. The symmetry result proved in [11] states that any minimizer of (P_{α}) is spherically symmetric (see [14] or [18] for the definition and properties of this symmetrization). In particular, the positivity set of ϕ_{0} on ∂B(x_{0}, r) (which is nonempty because of the condition ∥ϕ_{0}∥_{L^{q}(∂Ω)} = 1) is a spherical cap. Hence the conditions of Theorem 5.1 are satisfied. □

In the general case, for the problem (P_{α}) we can prove that the set of α’s for which there exists minimizers with smooth free boundary is dense in (0, |Ω|). More precisely,

Theorem 5.2. For any 0 < α < |Ω| there exists α_{ε} → α as ε → 0 such that there exists a solution ϕ_{ε} of (P_{α_{ε}}) which is locally Lipschitz continuous in Ω and has a locally C^{1,β} free boundary up to a set of zero H^{N−1}−measure. In the case N = 2 the free boundary is a locally C^{1,β} surface.

Proof. Let u_{ε} be a minimizer of J_{ε}. We already know that α_{ε} := |{u_{ε} > 0}| ≤ α + Cε (see (4.1)). Let us see that α_{ε} → α as ε → 0. If not, there exists a sequence ε_{j} → 0 such that |{u_{ε_{j}} > 0}| ≤ θ < α. Let ϕ_{0} be a minimizer of (P_{α}). By the strict monotonicity of S(α) (see [11] Theorem 1.2) we have

J (ϕ_{0}) = S(α) < S(θ) ≤ J (u_{ε_{j}}) = J_{ε}(u_{ε_{j}}) − F_{ε}(α_{ε_{j}})
≤ J_{ε}(ϕ_{0}) − F_{ε}(α_{ε_{j}}) = J (ϕ_{0}) − F_{ε}(α_{ε_{j}}) ≤ J (ϕ_{0}) + Cε

a contradiction.

Now, taking ϕ_{ε} = u_{ε} we see that ϕ_{ε} is a minimizer of (P_{α_{ε}}). In fact, let v be an admissible function for (P_{α_{ε}}) then

J (v) + F_{ε}(α_{ε}) = J_{ε}(v) ≥ J_{ε}(ϕ_{ε}) = J (ϕ_{ε}) + F_{ε}(α_{ε})

therefore

J (v) ≥ J (ϕ_{ε}).

The theorem is proved. □

Finally, we have the following result.

Theorem 5.3. Let u_{ε} be a minimizer of J_{ε} in K_{1}. Then there exists ϕ_{0} ∈ H^{1}(Ω) a solution to (P_{α}) such that, up to a subsequence, u_{ε} → ϕ_{0} in H^{1}(Ω).

Proof. In the proof of Theorem 5.2 we showed that |{u_{ε} > 0}| → α as ε → 0.

It is easy to see that J_{ε}(u_{ε}) is bounded uniformly in ε and so u_{ε} is uniformly bounded in H^{1}(Ω). Therefore, passing to a subsequence if necessary, there exists u_{0} ∈ H^{1}(Ω) such that

u_{ε} ⇀ u_{0} weakly in H^{1}(Ω),
u_{ε} → u_{0} strongly in L^{q}(∂Ω),
u_{ε} → u_{0} a.e. Ω.

Thus,

∥u_{0}∥_{L^{q}(∂Ω)} = 1,

|{u_{0} > 0}| ≤ α = lim_{ε→0} |{u_{ε} > 0}| and

∥u_{0}∥_{H^{1}(Ω)} ≤ lim inf_{ε→0} ∥u_{ε}∥_{H^{1}(Ω)}.

Let us call ϕ_{0} = u_{0} and let us see that ϕ_{0} is a solution to (P_{α}). In fact, let v ∈ H^{1}(Ω) be such that |{v > 0}| = α and ∥v∥_{L^{q}(∂Ω)} = 1. Then

J (v) = J_{ε}(v) ≥ J_{ε}(u_{ε}).

Now, since lim inf_{ε→0} F_{ε}(|{u_{ε} > 0}|) ≥ 0 there holds that

(5.2)  J (v) ≥ lim inf_{ε→0} J_{ε}(u_{ε}) ≥ lim inf_{ε→0} J (u_{ε}) ≥ J (ϕ_{0}).

It remains to see that |{ϕ_{0} > 0}| = α. Assume not, then α_{1} := |{ϕ_{0} > 0}| < α. So, by the strict monotonicity of S(·) there holds that S(α) < S(α_{1}) but

S(α) = inf_{v} J (v) ≥ J (ϕ_{0}) ≥ S(α_{1}),

a contradiction.

Now, taking v = ϕ_{0} in (5.2),

J (ϕ_{0}) ≤ lim inf_{ε→0} J (u_{ε}) ≤ lim inf_{ε→0} J_{ε}(u_{ε}) ≤ J (ϕ_{0}).

Hence, ∥ϕ_{0}∥_{H^{1}(Ω)} = lim inf_{ε→0} ∥u_{ε}∥_{H^{1}(Ω)} and so, by taking a further subsequence if necessary, the convergence is actually strong. □

Remark 5.1. We believe that, as in the previous cases, the minimizers u_{ε} of J_{ε} in K_{1} will already be solutions to (P_{α}) for ε small. Nevertheless, despite the fact that the result of Theorem 5.3 does not give regularity of the minimizer ϕ_{0} we believe that it could be of interest in numerical approximations of the solution to (P_{α}).

Acknowledgements. We wish to thank Prof. Luis Caffarelli for suggesting this approach to our optimization problem and for providing us with useful references.

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Departamento de Matemática, FCEyN

UBA (1428) Buenos Aires, Argentina.

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