# Hot Spots in Convex Domains are in the Tips (up to an Inradius)

## TL;DR

This paper demonstrates that in convex domains, the extrema of the first nontrivial Neumann eigenfunction are located near the domain's tips, specifically close to points of maximum distance, confirming a refined aspect of the Hot Spots conjecture.

## Contribution

It establishes that the maxima and minima of the eigenfunction are confined within a universal constant times the inradius near the domain's extremal points, advancing understanding of eigenfunction localization.

## Key findings

- Maxima and minima are near points of maximum distance in the domain.
- The location of extrema is within a constant times the inradius of the tips.
- The result confirms a refined aspect of the Hot Spots conjecture.

## Abstract

Let $\Omega \subset \mathbb{R}^2$ be a bounded, convex domain and let $-\Delta \phi_1 = \mu_1 \phi_1$ be the first nontrivial Laplacian eigenfunction with Neumann boundary conditions. The Hot Spots conjecture claims that the maximum and minimum are attained at the boundary. We show that they are attained far away from one another: if $x_1, x_2 \in \Omega$ satisfy $\|x_1 - x_2\| = \mbox{diam}(\Omega)$, then every maximum and minimum is assumed within distance $c\cdot \mbox{inrad}(\Omega)$ of $x_1$ and $x_2$, where $c$ is a universal constant (which is the optimal scaling up to the value of $c$).