Necessary and Sufficient Condition of Existence for the Quadrature Surfaces Free Boundary Problem

Performing the shape derivative (Sokolowski and Zolesio, 1992) and using the maximum principle, we show that the so-called Quadrature Surfaces free boundary problem QS ( f , k) ⎪⎪⎪⎨⎪⎪⎪⎩ −ΔuΩ = f in Ω uΩ = 0 on ∂Ω |∇uΩ| = k (constant) on ∂Ω. has a solution which contains strictly the support of f if and only if ∫


Introduction
Assuming throughout that: D ⊂ R N (N ≥ 2) is a bounded ball which contains all the domains we use. If ω is an open subset of D, let ν be the outward normal to ∂ω and let |∂ω| (respectively |ω|) be the perimeter (respectively the volume) of ω.
Let k > 0 and let f be a positive function belonging to L 2 (R N ) and having a compact support K with nonempty interior. Denote by C the convex hull of K and consider the following free boundary problem. Find an open set Ω ⊂ D which contains strictly C and such that the following problem has a solution: Notice that since u Ω = 0 on ∂Ω then |∇u Ω | = − ∂u Ω ∂ν , where ν is the outward normal vector to ∂Ω.
Imposing boundary conditions for both u Ω and |∇u Ω | on ∂Ω makes problem Q S ( f, k) overdetermined, so that in general without any assumptions on data this problem has no solution.
The problem Q S ( f, k) is called the quadrature surfaces free boundary problem and arises in many areas of physics (free streamlines, jets, Hele-show flows, electromagnetic shaping, gravitational problems etc.) It has been intensively studied from different points of view, by several authors. For more details about the methods used for solving this problem see the (Gustafsson and Shahgholian, Introduction, 1996). Using the maximum principle together with the compatibility condition of the Neumann problem, the authors gave sufficient condition of existence for problem Q S ( f, k) (Barkatou and al., 2005).
Thegoal of this paper is to prove the following To see that (NS ) is a necessary condition of existence, let us suppose that Ω is a smooth solution of Q S ( f, k). Integrating the equation (1) over Ω and using the fact that u Ω vanishes on ∂Ω, the Green formula gives: Now, since the convex C is strictly contained in Ω, we obtain (NS ).
To prove that (NS ) is sufficient to get a solution of Q S ( f, k), we proceed as follows.
By using the shape derivative (Sokolowski and Zolesio, 1992), the problem Q S ( f, k) seems to be the Euler equation of the following optimization problem. Put where O ε is the class of the domains which satisfy the ε-cone property (Chenais, 1975).
u ω is the solution of the following Dirichlet problem.
We begin by proving the following propositions.
2. If Ω is of class C 2 , then

Now, put
u ω being the solution of the problem P( f, ω). We prove Next, we prove by contradiction that (NS ) is sufficient to solve Q S ( f, k). The contradiction is obtained according to (I), (II) and by applying the maximum principle to Ω and Ω * . The paper is ended by two sections. Section 5 is concerned by the special case of the uniform density supported by a segment for which we obtain a necessary and sufficient condition of existence. Section 6 contains some remarks.

Definitions and lemmas
Definition 2.1 Let K 1 and K 2 be two compact subsets of D. We call a Hausdorff distance of K 1 and K 2 (or briefly d H (K 1 , K 2 )), the following positive number:

Definition 2.2
Let ω n be a sequence of open subsets of D and let ω be an open subset of D. Let K n and K be their complements inD.
We say that the sequence ω n converges in the Hausdorff sense, to ω (or briefly ω n

Definition 2.3
Let {ω n , ω} be a sequence of open subsets of D. We say that the sequence ω n converges in the compact sense, to ω (or • every compact subset of ω is included in ω n , for n large enough, and • every compact subset ofω c is included in ω c n , for n large enough.

Definition 2.4
Let {ω n , ω} be a sequence of open subsets of D. We say that the sequence ω n converges in the sense of characteristic (Chenais, 1975) We say that a domain ω satisfies the ε-cone property if for all x ∈ ∂ω there exist a direction vector ξ ∈ R N such that the cone C(y, ξ, ε) ⊂ ω for all y ∈ B(x, ε) ∩ ω . ε denotes both the angle and the hight of the cone.
Denoting by O ε the class of domains which have the ε-cone property, we can have this lemma.
Lemma 2.6 (Chenais, 1975) If ω n ∈ O ε , then there exist an open subset ω ⊂ D and a subsequence (again denoted by ω n ) such that (i) ω n Lemma 2.7 (Pironneau, 1984) Let ω n be a sequence of open and bounded subsets of D. There exist a subsequence (again denoted by ω n ) and some open subset ω of D such that 1. ω n converges to ω in the Hausdorff sense, and 2. |∂ω| ≤ lim inf n→+∞ |∂ω n |.

Shape derivative
In this subsection, we use the standard tool of the domain derivative to write down the optimality conditions. Before doing this, recall the definition of the domain derivative (Sokolowski and Zolesio, 1992). Suppose that ω is of class C 2 . Consider a deformation field V ∈ C 2 R N ; R N and set ω t = {x + tV(x) : x ∈ ω}, t > 0. The application Id + tV (a perturbation of the identity) is a Lipschitz diffeomorphism for t small enough and by definition, the derivative of J at ω in the direction V is As the functional J depends on the domain ω through the solution of some Dirichlet problem, we need to define also the domain derivative u ω of u ω : Furthermore, u ω is the solution of the following problem (Sokolowski and Zolesio, 1992): Now to compute the derivative of the functionals we consider in the sequel, recall the following (Sokolowski and Zolesio, 1992).
1. The shape derivative of the volume is 2. If G(ω) = ω |∇u ω | 2 dx, by the Hadamard formula we get Since the domain ω satisfies ε-cone property, the deformation domain ω t satisfies the same property (for t sufficiently small).

Proofs of Propositions 1.2 and 1.3
Let u D and u ω respectively denote the solution of P( f, D) and P( f, ω). The maximum principle implies: By using the variational formulation of P( f, ω), Then, Therefore, Hence, inf J and inf F exist.

Proposition 1.2
The first item is obtained by using item 1 of Lemma 2.7, and (iv) and (v) of Lemma 2.6. The continuity w.r.t. the domains for the Dirichlet problem P( f, ω) is obtained by (vi) of Lemma 2.6. For the item 2, using the same notations as in the subsection 2.2, to get int(C) in (Ω) t (for t small enough) the admissible directions V must satisfy Notice that for ∂Ω \ ∂C, each V is admissible. Now since u Ω vanishes on ∂Ω, (2) and (3) imply And since dJ(Ω, V) ≥ 0 for each admissible direction V, according to what precedes we obtain (I).

Proposition 1.3
By replacing k by M C , item 1 is obtained in the same way as in the previous proof. For the second item, on ∂Ω * \(∂Ω∪∂C), any direction V is admissible whereas V must satisfy V · ν ≥ 0 on ∂Ω * ∩ ∂C, and V · ν ≤ 0 on ∂Ω ∩ ∂Ω * .

Proof of Theorem 1.1
We would like to say that the minimum obtained in Proposition 1.2 is solution to the problem Q S ( f, k). It is not so simple. In general, without any assumptions on f and k, Q S ( f, k) does not have a solution : as we saw in the introduction, if Ω is a smooth solution of Q S ( f, k) then which shows that if f has a too small total mass or if k is too large, the perimeter of Ω will not be large enough so that Ω contains C. In such a case, the minimum that we found comes to intersect the convex C, i.e. ∂Ω and ∂C have a common part and so |∇u Ω | ≤ k on ∂Ω ∩ ∂C.

∂Ω ∂C and ∂Ω ∂Ω *
The aim of the sequel is to prove that each of the five cases above contradicts the condition (NS ).

The uniform density supported by a segment
Let a > 0 and put C = [−1, 1] × {0} ⊂ R 2 . Consider the following free boundary problem. Find an open and bounded set Ω ⊂ R 2 which contains strictly C and such that the following free boundary problem has a solution: Physically, this means that if the line segment in the complex plane is provided with a uniform density above a certain level, then there will exist a domain containing compactly the line segment such that the given measure on the line is equigravitational to the arc-length measure of the domain.

Theorem 5.1
The problem Q S (a, k) has a solution if and only if a > 2k.
We prove the theorem above in two propositions.

Proposition 5.2
Let (Ω a , u Ω a ) be the solution of the problem Q S (a, k). If Ω a is lipschitz and u Ω a ∈ H 2 (Ω a \ C) then a > 2k.

Proof
Writing u Ω a = h − a 2 |y| (where h is a harmonic function on Ω a ) and tending ε to 0, we obtain: C is strictly contained in Ω a thus |∂Ω a | > 4 and consequently a > 2k.

Proposition 5.3
If a > 2k, then the problem Q S (a, k) has a solution.

Proof
Let B be the unit ball of R 2 . Put u ω is the solution of the Dirichlet problem P(aδ C , ω). Arguing as in Section 3, we prove the existence of Ω a ∈ O ε,B which minimizes the functional J a on O ε,B . Then, supposing Ω a of class C 2 , we obtain the following optimality conditions: ) 2 )dx.
As above, there exists Ω * a ∈ O Ω a which minimizes the functional F a on O Ω a . If Ω * a is of class C 2 , then: Suppose by contradiction that ∂Ω a ∩ ∂B ∅. Replacing respectively in the proof of Theorem 1.1, C, Ω, Ω * , and M C by B, Ω a , Ω * a and a 2 , we obtain the desired contradiction. Remark 5.4 Barkatou and Khatmi (2008) proved that if a > 3.92k then Q S (a, k) has a solution while Shahgholian and Gustafsson (1996) showed that if a ≥ 24πk, then the problem Q S (a, k) admits a solution.
6. Final remarks Remark 6.1 For the p-Laplacian, the continuity with respect to the domain is a consequence of the γ p -convergence (Bucur and Trebeschi, 1998). So using Hopf's comparison principle and considering, for p > 1, and p 2, the functional J p (ω) = ω (|∇u ω (x)| p − p f (x)u ω (x) + k p )dx, one obtains the same necessary and sufficient condition of existence as in Theorem 1.1.
Remark 6.1 Gustafsson and Shahgholian (1996) showed the existence of a minimizer u to the functional over all 0 ≤ v ∈ H 1 (R N ). They prove that (Ω u , u) (Ω u = {u > 0}) is a solution to Q S ( f, k) but the overdetermined condition is given in a weak sense: lim ε 0 (|∇u| 2 − k 2 )η · vdH N−1 = 0, for every η ∈ C ∞ 0 (R N ; R N ). Then, they relate their minimization problem to quadrature domain Q D ( f, k) and show that They conclude their paper by giving (Gustafsson and Shahgholian, 1996, Theorem 4.7) the following sufficient condition: If Supp f ⊂ B R and if B R f (x)dx > ( 6 N N 3R |B R |)k, then Ω u ∈ Q D ( f, k) with B 3R ⊂ Ω u (B R being some ball of radius R).