Heat profile, level sets and hot spots of Laplace eigenfunctions

TL;DR

This paper explores heat distribution, level sets, and hot spots of Laplace eigenfunctions using probabilistic methods, providing insights into eigenfunction behavior and bounds in various domains.

Contribution

It introduces probabilistic tools like Brownian motion and Feynman-Kac formulae to analyze eigenfunctions, including bounds and level set properties, especially in complex domains.

Findings

Bounds on supremum norms for Dirichlet eigenfunctions

Comparison of maximum interior and boundary temperatures for Neumann eigenfunctions

Analysis of level set proximity and domain geometry effects

Abstract

We use probabilistic tools based on Brownian motion and Feynman-Kac formulae to investigate the heat profile for the ground state Dirichlet and second Neumann eigenfunctions. Among other topics, we comment on supremum norm bounds for ground state Dirichlet eigenfunctions and look at the corresponding Neumann problem, namely the comparison of maximum temperatures on the interior and the boundary, the latter being partially motivated by the hot spots problem. We also investigate the proximity/distance of level sets of ground state Dirichlet eigenfunctions, some with analogous statements for Neumann eigenfunctions. Domains with bottlenecks make occasional appearances as an illuminating example as well as testing ground for our theory.