Eigenvalues of the Laplacian with moving mixed boundary conditions: the case of disappearing Neumann region

TL;DR

This paper investigates how the eigenvalues of the Laplacian change when the boundary conditions vary, especially when a Neumann boundary region shrinks to zero, providing bounds and sharp estimates.

Contribution

It introduces a novel analysis of eigenvalue variation under moving mixed boundary conditions, including sharp estimates for star-shaped Neumann regions.

Findings

Derived upper and lower bounds for eigenvalue variation.

Provided sharp estimates for strictly star-shaped Neumann regions.

Analyzed eigenvalue behavior with vanishing Neumann boundary portions.

Abstract

We deal with eigenvalue problems for the Laplacian with varying mixed boundary conditions, consisting in homogeneous Neumann conditions on a vanishing portion of the boundary and Dirichlet conditions on the complement. By the study of an Almgren type frequency function, we derive upper and lower bounds of the eigenvalue variation and sharp estimates in the case of a strictly star-shaped Neumann region.