The hot spots conjecture can be false: Some numerical examples

TL;DR

This paper demonstrates numerically that the hot spots conjecture, previously verified only for special geometries, can fail in simple bounded domains with holes, using boundary integral equations and high-accuracy eigenvalue computations.

Contribution

It introduces a numerical approach to show that the hot spots conjecture can be false in domains with holes, providing explicit counterexamples.

Findings

The hot spots conjecture can fail in domains with holes.

Boundary element collocation method yields highly accurate eigenvalues.

Counterexamples with up to five holes are constructed.

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.