On shape optimization problems involving the fractional laplacian

Abstract

Our concern is the computation of optimal shapes in problems involving (−∆) 1/2 . We focus on the energy J(Ω) associated to the solution u Ω of the basic Dirichlet problem (−∆) 1/2 u Ω = 1 in Ω, u = 0 in Ω c . We show that regular minimizers Ω of this energy under a volume constraint are disks. Our proof goes through the explicit computation of the shape derivative (that seems to be completely new in the fractional context), and a refined adaptation of the moving plane method.

On fractional elliptic equations in Lipschitz sets and epigraphs: regularity, monotonicity and rigidity results

“…The boundary regularity theory for fractional Laplacian, developed in [31,32,37,38], ensures that, for a solution u to (1.1) with Ω of class C 2 , the quantity (∂ ν ) s u is well defined. Natural Hopf's Lemmas were then proved in [24,Proposition 3.3] and [30,Lemma 1.2], and constituted the base point in the study of overdetermined problems for the fractional Laplacian, see [17,24,30,34,44]. In this paper we consider overdetermined problems of the type…”

Optimal control of systems governed by fractional Laplacian in the minimax framework

“…Proof From (A6), (A7) and classical theorems on semicontinuity of integral functional, see [22,Theorem 1.1] or [20,Theorem 5], we deduce that J is lower semicontinuous with respect to the strong topology in the space H α/2 0 and either the weak topology of L p (Ω, R m ) for p ∈ (2n/ (n + α) , ∞) or the weak * topology of L ∞ (Ω, R m ), since convergence of any sequence {u k } in H α/2 0 (Ω) implies the strong convergence of {u k } in L s (Ω) with s ∈ (1, 2 * α ) and the strong convergence of (−∆) α/4 u k in L 2 (Ω) and moreover we have the same implications for convergence of any sequence {v k } in H α/2 0 (Ω) . Next, let {(u k , v k , w k )} ⊂ D be a minimizing sequence for (16)…”

“…Specifically, we focus our attention on the of the solutions on the functional parameters and then on the existence of the optimal solutions minimizing some cost functional. For related results concerning optimal solution we refer the interested readers, for example, to papers [6,10,16,29]. The framework requires the minimax geometry (cf.…”

A class of shape optimization problems for some nonlocal operators

“…For other recent shape optimization problems where the state equation is nonlocal, see [7,11,20,21,28], and references therein.…”