Optimization problem for extremals of the trace inequality in domains with holes

TL;DR

This paper investigates how the Sobolev trace constant changes with respect to holes in a domain, deriving a formula for the first variation and analyzing optimality conditions in symmetric cases.

Contribution

It provides a formula for the first variation of the Sobolev trace constant with respect to holes and analyzes the criticality and optimality of symmetric holes in specific domains.

Findings

Symmetric holes are critical under volume-preserving deformations in a centered ball.

Symmetric holes are not always optimal for the Sobolev trace constant.

The first variation formula helps identify critical configurations for the trace inequality.

Abstract

We study the Sobolev trace constant for functions defined in a bounded domain $\O$ that vanish in the subset $A.$ We find a formula for the first variation of the Sobolev trace with respect to hole. As a consequence of this formula, we prove that when $\O$ is a centered ball, the symmetric hole is critical when we consider deformation that preserve volume but is not optimal for some case.