The hot spots conjecture for some non-convex polygons

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.