Shape optimization problems involving nonlocal and nonlinear operators

TL;DR

This paper studies shape optimization problems involving nonlocal, nonlinear operators, proving existence of solutions and analyzing their asymptotic behavior as certain parameters approach limits, with extensions to anisotropic cases.

Contribution

It establishes the existence of solutions for a broad class of nonlocal, nonlinear shape optimization problems and examines their asymptotic and anisotropic behaviors.

Findings

Existence of a minimum point for the shape functional F.

Asymptotic behavior of solutions as s approaches 1.

Extension of results to anisotropic operators.

Abstract

In this research, we investigate a general shape optimization problem in which the state equation is expressed using a nonlocal and nonlinear operator. We prove the existence of a minimum point for a functional $F$ defined on the family of all 'quasi-open' subsets of a bounded open set $\Omega$ in $\mathbb{R}^n$. This is ensured under the condition that $F$ demonstrates decreasing behavior concerning set inclusion and is lower semicontinuous with respect to a suitable topology associated with the fractional $p$-Laplacian under Dirichlet boundary conditions. Moreover, we study the asymptotic behavior of the solutions when $s\to1$ and extend this result to the anisotropic case.