Some nonlocal optimal design problems

TL;DR

This paper explores two optimal design problems related to fractional Sobolev spaces, establishing their connection and analyzing the behavior as the fractional parameter approaches 1.

Contribution

It introduces a relationship between two fractional Sobolev space-based design problems and studies their convergence as the fractional parameter tends to 1.

Findings

Established a link between the two optimal design problems.

Analyzed the convergence of solutions as s approaches 1.

Provided insights into the behavior of fractional Sobolev space problems.

Abstract

In this paper we study two optimal design problems associated to fractional Sobolev spaces $W^{s,p}(\Omega)$. Then we find a relationship between these two problems and finally we investigate the convergence when $s\uparrow 1$.