Research project A new tool in spectral geometry

Spectral theory of partial differential equations is a classical topic in mathematical physics and has more than a century of history. Eigenvalues and eigenfunctions of PDEs appear, e.g., as fundamental tone and overtones of a vibrating membrane or as energy levels of a quantum system. They are very closely related to the geometry and topology of the underlying space, may it be a Euclidean domain, a manifold or a graph.

This research project is inspired by the famous hot spots conjecture: the hottest and coldest spots within an insulated, homogeneous medium should move away from each other and converge to the boundary for large time. Translated into mathematics, the first non-constant eigenfunction of the Laplacian with Neumann boundary conditions on a Euclidean domain without holes should attain its maximum and minimum only on the boundary. This conjecture, mentioned first in 1974, remains unsolved in full generality although physical intuition confirms it.

Based on a recent new approach to the hot spots conjecture introduced by the applicant, the planned project aims at studying critical points of eigenfunctions of the Laplacian and more general elliptic differential operators. In particular, new results on the hot spots conjecture and an increased understanding of the behavior of eigenfunctions and their relation to the geometry of the underlying domain shall be obtained. Moreover, it is planned to study isoperimetric problems for eigenvalues by means of the new tool.