Optimizing the ground of a Robin Laplacian: asymptotic behavior

TL;DR

This paper investigates how to optimize the principal eigenvalue of a Robin Laplacian on a bounded domain by adjusting the Robin parameter function, revealing connections to the domain's heat content and deriving asymptotic expansions.

Contribution

It introduces a novel approach linking eigenvalue optimization to the asymptotic behavior of the Dirichlet heat content, providing new insights and applications.

Findings

Derived a two-term asymptotic expansion of the principal eigenvalue.

Established a relation between eigenvalue optimization and heat content asymptotics.

Discussed applications of the asymptotic results in related problems.

Abstract

In this note we consider achieving the largest principle eigenvalue of a Robin Laplacian on a bounded domain $\Omega$ by optimizing the Robin parameter function under an integral constraint. The main novelty of our approach lies in establishing a close relation between the problem under consideration and the asymptotic behavior of the Dirichlet heat content of $\Omega$. By using this relation we deduce a two-term asymptotic expansion of the principle eigenvalue and discuss several applications.