2024/05/29 by Lawford Hatcher, Hatcher, Lawford · 2 citations
Computer Science · Mathematics · #35B38 #35J05 #35J25 #35P05 #58J50 #Analysis of PDEs (math.AP) #Computational Geometry and Mesh Generation #FOS: Mathematics #Mathematics and Applications #Point processes and geometric inequalities #Spectral Theory (math.SP)
paper · pdf · doi:10.48550/arxiv.2405.19508
openalex publication_date 2024/05/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.