Eigenvalue inequalities for the Laplacian with mixed boundary conditions

TL;DR

This paper establishes inequalities for Laplacian eigenvalues under mixed boundary conditions on complex domains, extending classical bounds between Dirichlet and Neumann eigenvalues.

Contribution

It introduces new inequalities relating mixed boundary eigenvalues to pure boundary eigenvalues, broadening the understanding of spectral bounds for Laplacians.

Findings

Eigenvalue bounds for mixed boundary conditions are derived.

Results extend classical inequalities between Dirichlet and Neumann eigenvalues.

The inequalities apply to polyhedral and general bounded domains.

Abstract

Inequalities for the eigenvalues of the (negative) Laplacian subject to mixed boundary conditions on polyhedral and more general bounded domains are established. The eigenvalues subject to a Dirichlet boundary condition on a part of the boundary and a Neumann boundary condition on the remainder of the boundary are estimated in terms of either Dirichlet or Neumann eigenvalues. The results complement several classical inequalities between Dirichlet and Neumann eigenvalues due to P\'{o}lya, Payne, Levine and Weinberger, Friedlander, and others.