Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The hot spots conjecture for some non-convex polygons

Lawford Hatcher

TL;DR

This work establishes the hot spots conjecture for L-shaped domains via an elementary proof, and extends the analysis to Swiss cross translation surfaces and a broad family of L-tiled polygons. It introduces a strict mixed-Dirichlet–Neumann eigenvalue inequality and leverages nodal-derivative structure and boundary regularity near non-convex corners to control interior critical points. By exploiting symmetries and perturbation arguments, the authors show that second Neumann eigenfunctions on many non-convex tilings lack interior critical points and determine hot-spot locations on translation surfaces, unifying spectral behavior across several polygonal geometries. The results have implications for spectral geometry on flat surfaces and provide a robust framework for understanding eigenfunction extrema in non-convex domains.

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.

The hot spots conjecture for some non-convex polygons

TL;DR

This work establishes the hot spots conjecture for L-shaped domains via an elementary proof, and extends the analysis to Swiss cross translation surfaces and a broad family of L-tiled polygons. It introduces a strict mixed-Dirichlet–Neumann eigenvalue inequality and leverages nodal-derivative structure and boundary regularity near non-convex corners to control interior critical points. By exploiting symmetries and perturbation arguments, the authors show that second Neumann eigenfunctions on many non-convex tilings lack interior critical points and determine hot-spot locations on translation surfaces, unifying spectral behavior across several polygonal geometries. The results have implications for spectral geometry on flat surfaces and provide a robust framework for understanding eigenfunction extrema in non-convex domains.

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.
Paper Structure (12 sections, 38 theorems, 29 equations, 1 figure)

This paper contains 12 sections, 38 theorems, 29 equations, 1 figure.

Table of Contents

  1. Introduction
  2. L-shaped domains
  3. Swiss cross translation surfaces
  4. Hot spots of first mixed Dirichlet-Neumann eigenfunctions
  5. Hot spots in L-tiled domains
  6. Some preparatory results
  7. An eigenvalue inequality
  8. Proof of Theorem \ref{['mainthm']}
  9. Proof of Theorem \ref{['swissthm']}
  10. More eigenvalue inequalities
  11. First eigenfunctions with mixed boundary conditions
  12. Second Neumann eigenfunctions of L-tiled polygons

Key Result

Theorem 1.2

Let $L$ be an L-shaped domain, and let $u$ be a second Neumann eigenfunction of $L$. Then $u$ has no (non-vertex) critical points. In particular, the extrema of $u$ occur only at the diametric vertices of $L$. Moreover, if $L$ is canonically embedded, then $u$ may be chosen such that $\partial_xu>0$

Figures (1)

  • Figure 1: An example of an L-shaped domain and an example of each type of L-tiled domain. From left to right, the figure shows an L-shaped domain, a T-shaped domain, a Swiss cross domain, a U-shaped domain, an O-shaped domain, and an H-shaped domain. The dashed lines reveal the L-shaped domain by which each L-tiled domain is tiled.

Theorems & Definitions (84)

  • Definition 1.1: L-shaped domains
  • Theorem 1.2
  • Corollary 1.3
  • Corollary 1.4
  • Definition 1.6: T-shaped domains, Swiss cross domains, and Swiss cross surfaces
  • Theorem 1.7
  • Conjecture 1.8
  • Theorem 1.9
  • Theorem 1.10
  • Definition 1.11: U-shaped domains, O-shaped domains, and H-shaped domains
  • ...and 74 more