Sharp bounds on the failure of the hot spots conjecture

TL;DR

This paper establishes sharp bounds on the hot spots ratio for domains in high dimensions, showing it approaches e^{1/2} as dimension increases and providing insights into the conjecture's validity.

Contribution

It determines the maximal hot spots ratio in all dimensions, proves extremizing sets do not exist, and shows the conjecture holds asymptotically in measure as dimension grows.

Findings

Max hot spots ratio converges to e^{1/2} as dimension 

Extremizing sets do not exist for dimensions 

Hot spots conjecture is asymptotically true in measure as dimension 

Abstract

The hot spots ratio of a domain $\Omega\subset \mathbb{R}^d$ measures the degree of failure of Rauch's hot spots conjecture on that domain. We identify the largest possible value of this ratio over all connected Lipschitz domains $\Omega\subset \mathbb{R}^d$, for any dimension $d$. As $d\to \infty$, we show that this maximal ratio converges to $\sqrt{e}$, which asymptotically matches the previous best known upper bound by Mariano, Panzo and Wang. For $d\ge 2$, we show that sets extremizing the hot spots ratio do not exist, and extremizing sequences must converge to a ball at a quantitative rate. We then give a sharp bound on the measure of the set for which the first Neumann eigenfunction exceeds its maximal boundary value. From this we deduce that the hot spots conjecture is asymptotically true "in measure'' as $d\to \infty$.