The hot spots conjecture for some non-convex polygons

TL;DR

This paper provides new proofs and results confirming the hot spots conjecture for specific non-convex polygons, including L-shaped domains and related tiled domains, using eigenvalue inequalities and eigenfunction analysis.

Contribution

It introduces an elementary proof for the hot spots conjecture on L-shaped domains and extends the results to certain non-convex tiled domains, utilizing eigenfunction regularity properties.

Findings

Proved the hot spots conjecture for L-shaped domains.

Established eigenvalue inequalities for non-convex polygons.

Identified locations of hot spots in Swiss cross translation surfaces.

Abstract

We give an elementary new proof of the hot spots conjecture for L-shaped domains. This result, in addition to a new eigenvalue inequality, allows us to locate the hot spots in Swiss cross translation surfaces. We then prove, in several cases, that first mixed Dirichlet-Neumann eigenfunctions of the Laplacian on L-shaped domains also have no interior critical points. As a combination of these results, we prove the hot spots conjecture for five classes of domains tiled by L-shaped domains, including a class of non-simply connected domains. An interesting feature of the proofs is that we make positive use of the lack of regularity of eigenfunctions on non-convex polygons.