On the hot spots conjecture in higher dimensions

TL;DR

This paper proves a strong form of the hot spots conjecture for certain symmetric and generalized lip domains in higher dimensions, using a variational approach to the vector-valued Laplace operator.

Contribution

It introduces a new variational proof for the hot spots conjecture in higher dimensions, extending previous results to broader classes of domains.

Findings

Proves the hot spots conjecture for a class of domains in $\

Provides a variational method that avoids stochastic analysis and deformation techniques.

Contains a new proof of Jerison and Nadirashvili's main result.

Abstract

We prove a strong form of the hot spots conjecture for a class of domains in $\mathbb{R}^d$ which are a natural generalization of the lip domains of Atar and Burdzy [J. Amer. Math. Soc. 17 (2004), 243-265] in dimension two, as well as for a class of symmetric domains in $\mathbb{R}^d$ generalizing the domains studied by Jerison and Nadirashvili [J. Amer. Math. Soc. 13 (2000), 741-772]. Our method of proof is based on studying a vector-valued Laplace operator whose spectrum contains the spectrum of the Neumann Laplacian. This proof is essentially variational and does not require tools from stochastic analysis, nor does it use deformation arguments. In particular, it contains a new proof of the main result of Jerison and Nadirashvili.