On the eigenvalues of a biharmonic Steklov problem

TL;DR

This paper investigates the spectral properties of a biharmonic Steklov eigenvalue problem, demonstrating domain dependence, analyticity of eigenvalue functions under perturbations, and the optimality of balls for the first positive eigenvalue.

Contribution

It introduces a new analysis of the biharmonic Steklov problem, including boundary limit processes, eigenvalue sensitivity to domain changes, and optimality results for spherical domains.

Findings

Eigenvalues depend analytically on domain perturbations.

Balls are critical points for eigenvalue functions under measure constraints.

Balls maximize the first positive eigenvalue among fixed-measure domains.

Abstract

We consider an eigenvalue problem for the biharmonic operator with Steklov-type boundary conditions. We obtain it as a limiting Neumann problem for the biharmonic operator in a process of mass concentration at the boundary. We study the dependence of the spectrum upon the domain. We show analyticity of the symmetric functions of the eigenvalues under isovolumetric perturbations and prove that balls are critical points for such functions under measure constraint. Moreover, we show that the ball is a maximizer for the first positive eigenvalue among those domains with a prescribed fixed measure.