Improved upper bounds for the Hot Spots constant of Lipschitz domains

TL;DR

This paper extends the Hot Spots constant to Lipschitz domains, providing new upper bounds that improve previous results and relate to the weak Hot Spots conjecture across various dimensions.

Contribution

It generalizes the Hot Spots constant to Lipschitz domains and derives a new dimension-dependent upper bound formula for these domains.

Findings

Derived a new upper bound formula for the Hot Spots constant.

Improved bounds for Lipschitz domains in low and high dimensions.

Established an if and only if condition for the weak Hot Spots conjecture.

Abstract

The Hot Spots constant for bounded smooth domains was recently introduced by Steinerberger (2021) as a means to control the global extrema of the first nontrivial eigenfunction of the Neumann Laplacian by its boundary extrema. We generalize the Hot Spots constant to bounded Lipschitz domains and show that it leads to an if and only if condition for the weak Hot Spots conjecture HS2 from Ba\~{n}uelos and Burdzy (1999). We also derive a new general formula for a dimension-dependent upper bound that can be tailored to any specific class of domains. This formula is then used to compute upper bounds for the Hot Spots constant of the class of all bounded Lipschitz domains in $\mathbb{R}^d$ for both small $d$ and for asymptotically large $d$ that significantly improve upon the existing results.