On Shape Optimization Theory with Fractional Laplacian

M L Fall ,Ibrahima Faye,Alassane Sy,Diaraf Seck

doi:10.11648/j.acm.20211003.12

M L Fall , Ibrahima Faye + Show 2 more

Open Access

https://doi.org/10.11648/j.acm.20211003.12

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Abstract

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,uΩ) where uΩ 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 uΩbelonging to the fractional Sobolev spaces Ds, 2(Ω) of the boundary value problem considered. Then, we study the existence of an optimal shape of the functional J(Ω) on the class of admissible sets <img width="10" height="10" src="http://article.sciencepublishinggroup.com/journal/147/1471488/image001.png" /> under constraints volume. Finally, shape derivative of the functional is established by using Hadamard formula’s and an optimality condition is also given.

Highlights

We are interested for shape optimization problems using fractional laplacian problems
We look for a domain Ω ⊂ RN, N ≥ 2 and a function uΩ solutions to the problem inf
The functional J depends on Ω and uΩ solution to a partial differential equation

Summary

Introduction

We are interested for shape optimization problems using fractional laplacian problems. We consider a functional J(Ω) depending on Ω and uΩ solution to the fractional Laplacian. The shape derivative of the function (2) is given by the following. Malick Fall et al.: On Shape Optimization Theory with Fractional Laplacian result. 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 uωsolution to (22).

Sobolev Inequalities

Shape Derivative of the Functional

Existence of Optimal Shape

Shape Derivative

Optimal Conditions

Conclusion