The hot spots conjecture can be false: some numerical examples

• Published in: Advances in computational mathematics 2021-12, Vol.47 (6), Article 85
• Main Author: Kleefeld, Andreas
• Format: Article
• Language: English
• Subjects: Advances in Computational Integral Equations; Algorithms; Boundary conditions; Boundary integral method; Collocation methods; Computational mathematics; Computational Mathematics and Numerical Analysis; Computational Science and Engineering; Domains; Eigenvalues; Eigenvectors; Integral equations; Laplace equation; Mathematical and Computational Biology; Mathematical Modeling and Industrial Mathematics; Mathematics; Mathematics and Statistics; Matrix; Visualization
• ISSN: 1019-7168; 1572-9044
• DOI: 10.1007/s10444-021-09911-5

The hot spots conjecture is only known to be true for special geometries. This paper shows 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.