A hot spots theorem for the mixed eigenvalue problem with small Dirichlet region

TL;DR

This paper establishes that on convex domains, small connected Dirichlet regions prevent interior critical points of first mixed Laplace eigenfunctions, and provides new estimates for the eigenvalue to identify when this occurs.

Contribution

It introduces a new theorem linking the size of the Dirichlet region to the absence of interior critical points in mixed eigenfunctions, along with novel eigenvalue estimates.

Findings

No interior critical points when Dirichlet region is small and connected

New eigenvalue estimates for mixed Laplace problems

Explicit examples demonstrating the conditions

Abstract

We prove that on convex domains, first mixed Laplace eigenfunctions have no interior critical points if the Dirichlet region is connected and sufficiently small. We also find two seemingly new estimates on the first mixed eigenvalue to give explicit examples of when the Dirichlet region is sufficiently small.