Abstract
AbstractThe 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.
Funder
Forschungszentrum Jülich GmbH
Publisher
Springer Science and Business Media LLC
Subject
Applied Mathematics,Computational Mathematics
Reference41 articles.
1. Abele, D., Kleefeld, A.: New Numerical Results for the Optimization of Neumann Eigenvalues. In: Constanda, C. (ed.) Computational and Analytic Methods in Science and Engineering, pp 1–20, Birkhäuser (2020)
2. Asante-Asamani, E.O., Kleefeld, A., Wade, B.A.: A second-order exponential time differencing scheme for non-linear reaction-diffusion systems with dimensional splitting. J. Comput. Phys. 415, 109490 (2020)
3. Atar, R.: Invariant wedges for a two-point reflecting Brownian motion and the “hot spots” problem. Electron. J. Probab. 6(18), 1–19 (2001)
4. Atar, R., Burdzy, K.: On Neumann eigenfunctions in lip domains. J. Am. Math. Soc. 17(2), 243–265 (2004)
5. Atkinson, K.E.: The numerical solution of integral equations of the second kind, Cambridge University Press (1997)
Cited by
6 articles.
订阅此论文施引文献
订阅此论文施引文献，注册后可以免费订阅5篇论文的施引文献，订阅后可以查看论文全部施引文献