Minimizers for the fractional Sobolev inequality on domains

TL;DR

This paper investigates the existence of minimizers for the fractional Sobolev inequality within domains, revealing that such minimizers exist for certain domains like half-spaces and bounded regions, unlike in classical cases.

Contribution

The study demonstrates the existence of minimizers for the fractional Sobolev inequality in specific domains, contrasting with classical Sobolev inequality behavior.

Findings

Minimizers exist for the fractional Sobolev inequality in half-spaces.

Minimizers also exist for a large class of bounded domains.

This contrasts with the classical Sobolev inequalities where minimizers may not exist.

Abstract

We consider a version of the fractional Sobolev inequality in domains and study whether the best constant in this inequality is attained. For the half-space and a large class of bounded domains we show that a minimizer exists, which is in contrast to the classical Sobolev inequalities in domains.