Viewing the Steklov eigenvalues of the Laplace operator as critical Neumann eigenvalues

TL;DR

This paper explores the relationship between Steklov and Neumann eigenvalues of the Laplace operator, analyzing their asymptotic behavior, optimization properties, and effects of boundary mass perturbations.

Contribution

It demonstrates that Steklov eigenvalues serve as limits of Neumann eigenvalues and investigates their optimization and perturbation characteristics.

Findings

Steklov eigenvalues minimize Neumann eigenvalues in certain limits

Steklov eigenvalues depend continuously on boundary mass density

Steklov eigenvalues violate maximum principles in spectral optimization

Abstract

We consider the Steklov eigenvalues of the Laplace operator as limiting Neumann eigenvalues in a problem of boundary mass concentration. We discuss the asymptotic behavior of the Neumann eigenvalues in a ball and we deduce that the Steklov eigenvalues minimize the Neumann eigenvalues. Moreover, we study the dependence of the eigenvalues of the Steklov problem upon perturbation of the mass density and show that the Steklov eigenvalues violates a maximum principle in spectral optimization problems.