Shape optimization problems for nonlocal operators

TL;DR

This paper investigates shape optimization problems involving nonlocal operators, including fractional eigenvalues, and explores the transition from nonlocal to local equations, providing insights into their properties and solutions.

Contribution

It introduces a general framework for shape optimization with nonlocal operators and analyzes the nonlocal to local transition, which is a novel aspect in this context.

Findings

Characterization of shape optimization problems with nonlocal operators

Analysis of the transition from nonlocal to local state equations

Insights into fractional eigenvalues in shape optimization

Abstract

In this work we study a general shape optimization problem where the state equation is given in terms of a nonlocal operator. Examples of the problems considered are monotone combinations of fractional eigenvalues. Moreover, we also analyze the transition from nonlocal to local state equations.