Optimal boundary holes for the Sobolev trace constant

TL;DR

This paper investigates the minimization of the Sobolev trace Rayleigh quotient with boundary constraints, establishing existence of extremals, analyzing optimal boundary locations, and studying shape derivatives under boundary perturbations.

Contribution

It introduces a new boundary optimization problem for the Sobolev trace constant, proving extremal existence and analyzing boundary shape derivatives.

Findings

Existence of extremals for the boundary minimization problem.

Characterization of optimal boundary sets in specific cases.

Analysis of the shape derivative of the Sobolev trace constant.

Abstract

In this paper we study the problem of minimizing the Sobolev trace Rayleigh quotient $\|u\|_{W^{1,p}(\Omega)}^p / \|u\|_{L^q(\partial\Omega)}^p$ among functions that vanish in a set contained on the boundary $\partial\Omega$ of given boundary measure. We prove existence of extremals for this problem, and analyze some particular cases where information about the location of the optimal boundary set can be given. Moreover, we further study the shape derivative of the Sobolev trace constant under regular perturbations of the boundary set.