The hot spots conjecture in higher dimensions
By: James Kennedy
From: Departamento de Matemática, Faculdade de Ciências, Universidade de Lisboa, Portugal
At: Building C1, room 1.4.14
[2024-11-25]
The hot spots conjecture, first articulated by Jeffrey Rauch in the 1970s, asserts that if heat is diffusing within a perfectly insulated body, the hottest and coldest points in the body will migrate to the boundary as time passes. Mathematically, this reduces to the assertion that the eigenfunctions associated with the second eigenvalue of the Laplacian with Neumann (zero flux) boundary conditions on a domain, attain their maximum and minimum only at the boundary of the domain. To date only very partial results have been obtained, and almost only in dimension two: there are known counterexamples, while the conjecture is known to be true for a few special classes of domains, such as sufficiently symmetric domains, long thin domains, and triangles.
We will give a brief introduction to the conjecture, recalling the mathematical model behind it, and discussing the (relatively few) known results, both positive and negative. We will then introduce a new method based on a vector-valued Laplacian related to Maxwell's equations, which allows us to prove the conjecture for some classes of domains in three or more dimensions.
This talk is largely based on work with Jonathan Rohleder (Stockholm), supported by the FCT via grants UIDB/00208/2020 (DOI 10.54499/UIDB/00208/2020) and UIDP/00208/2020 (DOI 10.54499/UIDP/00208/2020), and project PTDC/MAT-PUR/1788/2020.
Project UIDB/00618/2020 https://doi.org/10.54499/UIDB/00618/2020