The hot spots conjecture can be false: Some numerical examples

2021/01/04 by Andreas Kleefeld, Kleefeld, Andreas · 2 citations
Engineering · #Advanced Numerical Methods in Computational Mathematics #Analysis of PDEs (math.AP) #Electromagnetic Simulation and Numerical Methods #FOS: Mathematics #Numerical Analysis (math.NA) #Numerical methods in engineering

paper · pdf · doi:10.48550/arxiv.2101.01210

openalex publication_date 2021/01/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The hot spots conjecture is only known to be true for special geometries. It can be shown numerically that the hot spots conjecture can fail to be true for easy to construct bounded domains with one hole. The underlying eigenvalue problem for the Laplace equation with Neumann boundary condition is solved with boundary integral equations yielding a non-linear eigenvalue problem. Its discretization via the boundary element collocation method in combination with the algorithm by Beyn yields highly accurate results both for the first non-zero eigenvalue and its corresponding eigenfunction which is due to superconvergence. Additionally, it can be shown numerically that the ratio between the maximal/minimal value inside the domain and its maximal/minimal value on the boundary can be larger than 1+10^-3. Finally, numerical examples for easy to construct domains with up to five holes are provided which fail the hot spots conjecture as well.

Cited by

• Analysis of the transmission eigenvalue problem for biharmonic scattering considering penetrable scatterers
• Sharp bounds on the failure of the hot spots conjecture

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