On the lowest eigenvalue of Laplace operators with mixed boundary conditions

TL;DR

This paper investigates how the lowest eigenvalue of a Robin-type Laplace operator varies with boundary conditions and domain size, providing estimates for convex domains based on geometric and boundary parameters.

Contribution

It offers new two-sided estimates for the lowest eigenvalue depending on boundary conditions and domain inradius for convex domains, advancing understanding of spectral properties.

Findings

Derived two-sided bounds for eigenvalues in convex domains

Analyzed asymptotic behavior in shrinking and expanding domains

Connected eigenvalue estimates to geometric and boundary parameters

Abstract

In this paper we consider a Robin-type Laplace operator on bounded domains. We study the dependence of its lowest eigenvalue on the boundary conditions and its asymptotic behavior in shrinking and expanding domains. For convex domains we establish two-sided estimates on the lowest eigenvalues in terms of the inradius and of the boundary conditions.