On Shape Optimization Theory with Fractional Laplacian

: The fractional Laplacian is a nonlocal operator that appears in biology, in physic, in ﬂuids dynamic, in ﬁnancial mathematics and probability. This paper deals with shape optimization problem associated to the fractional laplacian ∆ s , 0 < s < 1 . We focus on functional of the form J (Ω) = j (Ω , u Ω ) where u Ω is solution to the fractional laplacian. A brief review of results related to fractional laplacian and fractional Sobolev spaces are ﬁrst given. By a variational approach, we show the existence of a weak solution u Ω belonging to the fractional Sobolev spaces D s, 2 (Ω) of the boundary value problem considered. Then


Introduction
In this paper, we are interested for shape optimization problems using fractional laplacian problems. In other words, we look for a domain Ω ⊂ R N , N ≥ 2 and a function u Ω solutions to the problem inf Ω⊂R N , vol(Ω)=c, ∂Ω∈C 2 where and u Ω is solution to where 0 < s < 1, Ω is an open bounded set of R N , N ≥ 2. Shape optimization problems have always interested the research community.
A lot of work related to shape optimization is topical today [14], [2], [4], [5], [7], [6]. Allaire and Henrot [2] give a review on recent development in shape optimization. In general, the functional J depends on Ω and u Ω solution to a partial differential equation. In most of his papers, the authors consider a domain-dependent functionals with constraint a partial differential equation posed in Ω. In general, the solution u Ω of this PDE belongs to a Sobolev space. In this paper, we consider a functional J(Ω) depending on Ω and u Ω solution to the fractional Laplacian. Dalibard and Gerad-Varet in [12], showed that it is possible to calculate the shape derivative of the functional considered in the case s = 1 2 . In this work, we try to generalize the results for all 0 < s < 1.
We have the following the main result.
Theorem 1.1. Let J(Ω) be a functional given by (2) where u Ω is solution to (3). Let's consider, a small perturbation of the domain Ω in the form Ω t = φ t (Ω) where φ t is a C 1 diffeomorphism such that φ 0 = Id and ∂φt ∂t = V, where V ∈ W 1,∞ (R N , R N ). The shape derivative of the function (2) is given by the following result.
Theorem 1.2. Let Ω ⊂ R 2 be an open set of class C 2 , and Ω t = φ t (Ω) as below. Then the function J defined in Ω t by (2) is differentiable and we have where u 0 is solution to (26) andu 0 , the shape derivative of u 0 is solution to The third result is given by Let V ∈ C ∞ 0 (R 2 ), and (φ t ) be the flow associated with V , namelyφ Let Ω be an open set with C ∞ boundary, and let u Ω,f be the unique minimizer of J f (Ω), namely.
Then ∂ s n u Ω,f exists, and there exists an explicit constant k such that More over, the optimal condition is given the following result: Theorem 1.4. Let Ω be the solution of the shape optimization problem min{J(Ω, ω ∈ O} under the constraint u ω solution to (22).
Then, there exists a Lagrange multiplier λ = λ(Ω) such that where k is a constant. The paper is organized as follows. In section 2, some preliminaries results concerning the fractional Laplacian problem and fractional Sobolev spaces are given. In section 3 , we give the main results of this paper and its proofs: existence result for the shape optimization problem, shape derivative of the functional and optimality condition. In section 4, we give some concluding remarks and possible extension.

Preliminaries
In this section, we recall some results that will be useful to in the following of the work.

On the Fractional Problem
where the term is the so-called Gagliardo (semi) norm of f .

Sobolev Inequalities
We need the following results whose proof can be found in [15] and [23].
Lemma 2.1. Let s ∈]0, 1[ and p ∈ [1, +∞[ such that sp < N. Fixe T > 1, let N ∈ Z and (a k ) k a bounded non-negative sequence with a k = 0 for any k ≥ N . Then : Then: Theorem 2.1. : Let s ∈]0, 1[ and p ∈ [1, +∞[ such that sp < N. Then there exists a positive constant C = C(N, s, p) such that, for any mesurable and compactly supported function Where p * = p * (N, s) is the so-called fractional critical exponent and it is equal to N p N −sp . Consequently, the space then nothing to show. We then assume that we have two cases: * If f ∈ L ∞ (R N ) so we set A k = {|f | > 2 k } and a k = |A k |, we have: then we have We set T = 2 p and we apply the lemma 2.1, to obtain: with C = C(N, s, p). Finally according to the lemma 2.2 we have: we set : f n := max(min(f (x), n) − n) ∀x ∈ R N . So, the sequence (f n ) n is bounded, and moreover: So according to the first case we have: Moreover: and according to the dominated convergence theorem we have: which implies: The following theorems are useful for the proof of the results in the next section. Their proofs are given in [15] and [23].
Theorem 2.2. Let s ∈]0, 1[ and p ∈ [1, +∞[ such that sp < N. Let Ω ⊆ R N be a domain for W s,p . Then there exists a positive constant C = C(N, s, p, Ω) such that for any function f ∈ W s,p (Ω) we have for any q ∈ [p, p * ]; i.e., W s,p (Ω) is a continuous injection for L q (Ω) for any q ∈ [p, p * ].
Then T is pre-compact in L q .
Lemma 2.4. Let 0 < s < 1 and let (−∆) s be the fractional Laplacian operator defined by (2.2) . Then for any f ∈ S(R N ), We define Saying that (−∆) s u = f in D (Ω), is equivalent to the very weak formulation . H s , as the completion of C ∞ c (Ω), which is an Hilbert espace with respect to the norm: If u ∈ D s,2 (Ω) ⊂ L 1 s satisfies: (−∆) s u = f in D (Ω), we have the weak formulation: where Let Ω ⊂ R N a bounded open set with Lipschitz boundary, and 0 < s < 1. Note here for the space of smooth functions with compact support, we take the notation C ∞ c instead of C ∞ 0 . Consider the bilinear form: which is a scaler product on C ∞ c (Ω). We recall that the Hilbert space D s,2 (Ω) the completion of C ∞ c (Ω).
We denote by θ the following set: (13) In what the follows, we denote O the set of all open bounded sets Ω satisfying the − cône property. We have also the following compactness result.

Shape Derivative of the Functional
The objective of this section is to prove that the shape optimization problem (1)-(2) admits a solution Ω, when u Ω is solution to We get also optimal condition. Before going further, we first prove existence of uniqueness of the solution u Ω to (3).

Existence of Solution to (3)
These types of problems were first studied by Caffarelli and Sylvestre [9] and references there in, in the case s = 1 2 . The regularity of the solution to this problem is also studied by many authors. Cafarelli et al. [11] prove interior and boundary Schauder regularity estimates depending on the smoothness of the coefficients. Niang [16], proved in his thesis , by using a blow up and compactness analysis, a boundary regularity for the solution to the mixed boundary degenerate elliptic equation. Silvestre in [21] proved also some regularities results of the obstacle problems for a fractionnal power. In this work, we propose to generalize these problems by using a variational approach in the case where 0 < s < 1.
We begin this section by proving that it exists a solution to (3). We first use the Euler Lagrange equation of (3) in order to transform it into a functional J(u). We have the following theorem be an open set of class C 2 , and s ∈]0, 1[. Then there exist a unique weak solution u ∈ D s,2 (Ω) of (3).
In a addition, this solution satisfies the following problem with Before giving the proof of this theorem, we give the following lemma, which is usefull for the proof.
Lemma 3.1. Let (u k ) k≥1 ⊂ D s,2 (Ω) be a minimizing sequence of (15), i.e. (16). Then, there exists n 0 ∈ N * such that for k ≥ n 0 , From (15) we have by using schwartz inequality's From this last inequality, we get Taking into account the fact that f ∈ L 2 (Ω), the sequence u k ∈ L 2 (Ω) and the functional J(u k , u k ) ≤ m + 1 k , then for k large enough, we show that the quantity on the right hand side of (17) is bounded. Thus, we show that the term in the left hand side of (17) is the norm ||u|| D s,k (Ω is bounded by a constant which depends only on f and m. Proof. of Theorem 3.1: Multiplying (3) by a test function v ∈ D s,2 (Ω) and integrating over Ω we get Let and It is therefore very difficult to find the existence of a solution u ∈ D s,2 (Ω) such that a(u, v) = l(v) for all v ∈ D s,2 (Ω), using the Lax Milgram theorem, because of the nonlinearity of the term (−∆) s . To overcome this difficulty, we consider the following functional.
The objective will be to show that the functional J defined on D s,2 (Ω) × D s,2 (Ω) by (21) is well defined and a solution u of the problem min{J(u Ω , u Ω ), u Ω ∈ D s,2 (Ω)} is a weak solution of (3). For that, let's start by showing the functional J is reduced and does not reach −∞. We have: We recall that, there exists a constant C(N, s, Ω) such that u L 2 (Ω) ≤ C(N, s, Ω) u D s,2 (Ω) Then we have If u Ω k = u k is a minimizing sequence, then by lemma 3.1, u k is bounded in D s,2 , there exists a sub-sequence (u k l ) l≥1 of (u k ) k≥1 such that u k l u ∈ D s,2 (Ω), u k l −→ u ∈ L 2 (Ω) and u k l u ∈ L 2 (Ω), when l −→ ∞.
In consequence: Passing to the limit in the second member of the above inequality, we get Therefore, passing to the milit, wheh k → +∞ we have From this inequality, we have Then, the functional J admits a minimum u ∈ D s,2 (Ω). In the following, we calculate the Frechet derivative of the functional J(u, u). Let t ∈ R and u, v ∈ D s,2 (Ω) : Taking the limit as t → 0 we get: Moreover, J(u) = min (J(v)) then J (u) = 0, then

Existence of Optimal Shape
In this section we are interested in the existence of an optimal shape Ω, minimizing the functional J defined by (2) according to the set of domain O. Concerning the questions of existence of optimal shape, one can refer to the work of Allaire [1], A. Henrot and M. Pierre [14], Allaire and Henrot [2], Allaire et al. [3], D. Bucur et al. [7], Buttazo et al. [6] and O. Pironneau [17]. In these various works, the authors cited above use various shape functionals. These functionals generally depend on a function u Ω solution of a certain partial differential equation. In this present work, u Ω is the solution of a non-fractional partial differential equation. We try to do the same work but in the fractional case using a functional J(Ω) and u Ω is a solution of a fractional type equation. Thus we have the following result: Proof. The functional J(Ω) defined by (2) is positive because u Ω i solution of (22 )belong to D s,2 (Ω). It also does not expect the +∞ value.
Hence J is bounded. Let m = inf Ω∈O J(Ω), so there exists a minimizing sequence (Ω n ) n∈N ⊂ O such that Since Ω n ⊂ O, there exists a compact set K such thatΩ n ⊂ K. Then according to the compactness lemma 2.7, there is an open set Ω, with |Ω n | = c and an extracted sequence Ω n k such that Ω n k H − −−− → Ω and χ Ωn k p.p − → χ Ω It remains to show that: In Ω n k , u Ω k is solution to Multiplying (23) by a test function v = v Ω ∈ D s,2 (Ω) and integrating, we get And from Lemma 3.1, the sequence u Ωn k is bounded in D s,2 (Ω n k ). Since (u Ωn k ) is bounded in D s,2 (Ω n k ), there exists u * Ω ∈ D s,2 (Ω) and an extracted subsequence (u Ωn k ) k≥1 of (u Ωn k ) still denoted by(u Ωn k ) k≥1 such that: (u Ωn k ) k≥1 u * Ω ∈ L 2 (Ω), if k −→ ∞. Passing to the limit when k −→ ∞ and using weak convergence, we get the following formulation which is the weak formulation of the following problem Finally by taking ϕ = u Ωn k in (24), we have In the other hand, we have Then taking the limits in the right hand side after equality, as k −→ ∞ Then We can conclude that there is an open Ω * which minimizes J and Ω * ∈ O

Shape Derivative
Let Ω be a bounded open set of class C 2 . For t ≥ 0, let Ω t = φ t (Ω), where for all t, φ t associated for V is a diffeomorphism of R N , N ≥ 2 and satisfies the following properties: For all V ∈ C 1 ∩ W 1,∞ (R N ). Let u Ωt be the solution to the following problem Consider also then function (2) defined in Ω t , by where u Ωt is solution to (26). In this part we want to calculate the shape derivative of the functional (27). The calculation of shape derivative of the functional requires the knowledge of the shape and the material derivative of the solution u Ωt . In what follows, we recall some definitions and properties usefull for the following Definition 3.1. We transport the situation on the domain fixed by the change of variables defined by the following transformation : Id + tV . We look at the differentiability of t −→ u t • (Id + tV ). If this function can be differentiated into t = 0, we will be able to define the derivative of u t whereu(Ω, V ) and this derivative is called material derivative.
Definition 3.2. Let ω ⊂ Ω an open fixed ( strictly included in Ω ). So by definition we have ω ⊂ Ω t , for any t small enough. Therefore the function u t is well defined on ω and it is convenient to look at the limit of the differential quotient: If this limit exists for all ω, we define a function in the whole domain Ω noted u where u (Ω, V ) and u is called the form derivative of u.
We have the following theorem. Theorem 3.3. Let Ω ⊂ R 2 be an open set of class C 2 , and Ω t = φ t (Ω) as below. Then the function J defined by (27) is differentiable and we have where u 0 is solution to (26) andu 0 , the shape derivative of u 0 is solution to Proof. From the Theorem 3.1, the unique minimizer of the functional satisfies the following variational formulas: Then, for v = u we have So to study the functional (27) is equivalent to study the functional J defined by Then, the functional becomes Ωt v t f oφ t (y)j(t, y)dy and the functional J(Ω t ) becomes J(Ω t ) = Ω v t f oφ t (y)(y)j(t, y)dy Since v t ∈ D s,2 is differentiable, then t → J(Ω t ) is differentiable for t in a neiborhood of zero. Using Hadamard formula, we get, For k ∈ N large enough, we define χ k ∈ C ∞ 0 (R 2 ) by χ k (x) = χ(kr), where χ ∈ C ∞ (R) and χ(ρ) = 0 for ρ ≤ 1, χ(ρ) = 1 for ρ ≥ 2. Since u 0 = v 0 , we have: For fixed k and for t in a neigbourhood of zero, there exists a compact set K k such that K k ⊂ Ω and suppχ k oφ −1 Sinceu t ∈ L ∞ t (L 1 x ), we can use the chain rule and write : we have : With the definition of χ k we can prove that there exists C > 0 thus that: Since (−∆) s u 0 = f on the support of χ k , we have : Notice also that χ k (−∆) s (u 0 ) = 0. Indeed,u t is smooth on K k for t small enough. The integral formula makes sense and we have using the laplacian formulas given in Lemma 2.4 : Then, we have we obtain: Then: It's follows from Lemma 2.4 that From this last formulas, (29) becomes The functional used in our work corresponds to the energy functional relative to the fractional laplacian. In the case of Dirichlet energy, the shape derivative is known, see A. Henrot and M. Pierre [14] and Dalibard and Gerad-Varet [12]. By following the proof of the previous theorem, the functional J can be written in the form Since v t ∈ D s,2 is differentiable, then t → J(Ω t ) is differentiable for t in a neiborhood of zero. Using Hadamard formula, we get easily, For s = 1 2 , as in the case where the operator is the Laplacian, the shape derivative is known, see [12,14]. Using the same approach as in [12], we obtain also an expression of dJ f (Ωt) dt in terms on ∂ s u0 ∂n , i.e.
giving the proof of Theorem 1.3. The idea is to use an approximation of u 0 andu 0 .

Optimal Conditions
In this section, we are interested to an optimal condition. In other words, we look for the relation associated with the optimal condition Ω and the Lagrange multiplier λ(Ω).
Theorem 3.4. Let Ω be the solution of the shape optimization problem min{J(Ω, ω ∈ O} under the constraint u ω solution to (22).
Proof. Suppose that Ω is a minimizer of J under the contraint |Ω| = c, the theorem of Lagrange multipliers then implies that there exists a constant λ such that for any group of diffeomorphisms (φ t ) t∈R , d dt (J f (φ t ) + λ|φ t |) = 0 for t = 0.
From (31), we obtain In an other hand, From Theorem 1.3, we get By making the two preceding expressions equal, we have the following optimality conditions giving for all V, k (∂ s n u Ω ) 2 + λ = 0.
Since V is arbitrary, we infer that ∂ s n u Ω is constant on Ω. Moreover, since u Ω ≥ 0 on Ω, by maximum principle, ∂ s n u Ω is positive.

Conclusion
In this work, we have presented a shape optimization problem using a functional J dependent on the domain Ω and u Ω solution of the Fractional Laplacian. We have used some usual techniques to show an existence result of optimal shape and we calculate the shape derivative of the problem considered. Finally we found an optimality condition. It would be interesting in future work to consider the same functional under constraint −∆u s + u q = 0 in in Ω, for 0 < s < 1, q > 1.