Hot Spots Conjecture for General Simply Connected Planar Domains and Higher Dimensions

Determine whether the Hot Spots Conjecture holds for all bounded, simply connected domains in the plane and for domains in higher-dimensional Euclidean spaces; specifically, establish whether every eigenfunction corresponding to the smallest positive Neumann eigenvalue of the Laplacian attains its maximum and minimum only on the boundary of the domain.

Background

The Hot Spots Conjecture, first formulated by Rauch, asserts that the eigenfunction corresponding to the smallest positive Neumann eigenvalue of the Laplacian on a bounded domain attains its maximum and minimum only on the boundary. The conjecture is verified for simple shapes and certain classes of planar domains, while multiply-connected counterexamples exist.

Despite progress and several partial results, the status of the conjecture remains unresolved in broad settings. The author notes that it remains open for general simply connected domains in the plane and in higher dimensions, and it is believed to be true at least for convex domains. This paper presents a variational approach that yields some known results but does not resolve the general case.