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A note on hot-spots free subregions of convex domains

Jonathan Rohleder

TL;DR

This work addresses where interior critical points of the second Neumann eigenfunction $\psi_2$ can occur on convex planar domains. It combines a diameter-based spectral bound $\mu_2 \le \frac{4 j_0^2}{\operatorname{diam}(\Omega)^2}$ with a Miyamoto-inspired Bessel-function comparison to derive an explicit interior exclusion region, namely dist$(\mathbf{x_0},\partial\Omega) > \frac{j_1}{2 j_0}\operatorname{diam}(\Omega)$. Consequently, interior hot spots cannot lie near the center, and a corollary states that if $\mu_2 \le \frac{j_1^2}{\operatorname{diam}(\Omega)^2}$, then $\psi_2$ has no interior critical points, confirming the hot spots conjecture in this regime (in view of the Payne–Weinberger bound $\mu_2 \ge \frac{\pi^2}{\operatorname{diam}(\Omega)^2}$). The results illuminate how domain geometry (via diameter) controls the possible localization of extrema of $\psi_2$, particularly for domains close to a disk or elongated shapes.

Abstract

The second eigenfunction of the Neumann Laplacian on convex, planar domains is considered. Inspired by the famous hot spots conjecture and a related result of Steinerberger, we show that potential critical points of this eigenfunction (and, in particular, interior ''hot spots'') cannot be located ''near the center'' of the domain. The region in which critical points are excluded is described explicitly.

A note on hot-spots free subregions of convex domains

TL;DR

This work addresses where interior critical points of the second Neumann eigenfunction can occur on convex planar domains. It combines a diameter-based spectral bound with a Miyamoto-inspired Bessel-function comparison to derive an explicit interior exclusion region, namely dist. Consequently, interior hot spots cannot lie near the center, and a corollary states that if , then has no interior critical points, confirming the hot spots conjecture in this regime (in view of the Payne–Weinberger bound ). The results illuminate how domain geometry (via diameter) controls the possible localization of extrema of , particularly for domains close to a disk or elongated shapes.

Abstract

The second eigenfunction of the Neumann Laplacian on convex, planar domains is considered. Inspired by the famous hot spots conjecture and a related result of Steinerberger, we show that potential critical points of this eigenfunction (and, in particular, interior ''hot spots'') cannot be located ''near the center'' of the domain. The region in which critical points are excluded is described explicitly.

Paper Structure

This paper contains 2 sections, 3 theorems, 19 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['thm:main']}

Key Result

Theorem 1

There exists a universal constant $c > 0$ such that for any convex bounded domain $\Omega \subset \mathop{\mathrm{\mathbb{R}}}\nolimits^2$, if $\mathbf{x, y} \in \partial \Omega$ are at maximal distance, i.e., $|\mathbf{x} - \mathbf{y}| = \mathop{\mathrm{diam}}\nolimits (\Omega)$, then $\psi_2$ atta

Figures (2)

  • Figure 1: A circle and an ellipse; according to Theorem \ref{['thm:main']}, critical points of $\psi_2$ may only be located in the respective gray region.
  • Figure 2: Typical structure of the nodal lines of $w$ in $\Omega$; $w$ is positive on both $\Omega_1^+$ and $\Omega_2^+$.

Theorems & Definitions (6)

  • Theorem : Steinerberger S20
  • Theorem 1.1
  • Lemma 2.1
  • proof
  • proof : Proof of Theorem \ref{['thm:main']}
  • Remark 2.2