2021/09/14 by Mayukh Mukherjee, Mukherjee, Mayukh, Soumyajit Saha +1
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #Graph theory and applications #Mathematical Approximation and Integration #Mathematical Physics (math-ph)
paper · pdf · doi:10.48550/arxiv.2109.06531
openalex publication_date 2021/09/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.