An upper bound on the Hot Spots constant

TL;DR

This paper establishes a universal upper bound on the maximum of the first nontrivial Neumann eigenfunction relative to its boundary maximum, demonstrating the Hot Spots Conjecture cannot be invalidated by an unbounded factor across smooth domains.

Contribution

It proves a uniform upper bound on the Hot Spots constant for all smooth, connected domains, advancing understanding of eigenfunction maxima in spectral geometry.

Findings

The maximum of the first nontrivial Neumann eigenfunction is at most 60 times its boundary maximum.

The constant 60 is uniform across all smooth, connected domains in any dimension.

The optimal constant is at least slightly greater than 1, as shown by Kleefeld's example.

Abstract

Let $D \subset \mathbb{R}^d$ be a bounded, connected domain with smooth boundary and let $-\Delta u = \mu_1 u$ be the first nontrivial eigenfunction of the Laplace operator with Neumann boundary conditions. We prove $$ \max_{x \in D} ~u(x) \leq 60 \cdot \max_{x \in \partial D} ~u(x)$$ and emphasize that this constant is uniform among all connected domains with smooth boundary in all dimensions. In particular, the Hot Spots Conjecture cannot fail by an arbitrary factor. The inequality also holds for other (Neumann-)eigenfunctions (possibly with a different constant) provided the eigenvalue is smaller than the first Dirichlet eigenvalue. An example of Kleefeld shows that the optimal constant is at least $1 + 10^{-3}$.