Eigenvalues of the Wentzell-Laplace Operator and of the Fourth Order Steklov Problems

TL;DR

This paper establishes bounds for eigenvalues of the Wentzell-Laplace operator and fourth order Steklov problems, providing isoperimetric inequalities and explicit eigenvalues for specific cases.

Contribution

It introduces sharp bounds and isoperimetric inequalities for eigenvalues of these operators, including explicit solutions on Euclidean balls.

Findings

Sharp upper and lower bounds for the first nonzero eigenvalue of Wentzell-Laplace operator.

Isoperimetric inequalities for eigenvalues in Euclidean domains.

Explicit eigenvalues and eigenfunctions for certain fourth order Steklov problems.

Abstract

We prove a sharp upper bound and a lower bound for the first nonzero eigenvalue of the Wentzell-Laplace operator on compact manifolds with boundary and an isoperimetric inequality for the same eigenvalue in the case where the manifold is a bounded domain in a Euclidean space. We study some fourth order Stekolv problems and obtain isoperimetric upper bound for the first eigenvalue of them. We also find all the eigenvalues and eigenfunctions for two kind of fourth order Stekolv problems on a Euclidean ball.