Neumann to Steklov eigenvalues: asymptotic and monotonicity results

TL;DR

This paper investigates the relationship between Neumann and Steklov eigenvalues, analyzing their asymptotic behavior and monotonicity, and establishing that Steklov eigenvalues locally minimize Neumann eigenvalues.

Contribution

It provides explicit formulas for derivatives of eigenvalues at the limit and demonstrates monotonicity and local minimality properties.

Findings

Neumann eigenvalues approach Steklov eigenvalues asymptotically

Explicit formulas for derivatives at the limiting problem

Steklov eigenvalues locally minimize Neumann eigenvalues

Abstract

We consider the Steklov eigenvalues of the Laplace operator as limiting Neumann eigenvalues in a problem of mass concentration at the boundary of a ball. We discuss the asymptotic behavior of the Neumann eigenvalues and find explicit formulas for their derivatives at the limiting problem. We deduce that the Neumann eigenvalues have a monotone behavior in the limit and that Steklov eigenvalues locally minimize the Neumann eigenvalues.