Heat profile, level sets and hot spots of Laplace eigenfunctions

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 correspondin...

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Bibliographic Details
Published in:arXiv.org 2022-03
Main Authors: Mukherjee, Mayukh, Saha, Soumyajit
Format: Article
Language:English
Subjects:
Boundary conditions
Brownian motion
Dirichlet problem
Eigenvalues
Eigenvectors
Ground state
Heat transmission
Online Access:Get full text
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Summary: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.
ISSN:2331-8422