Universal Bounds for Traces of the Dirichlet Laplace Operator

TL;DR

This paper establishes universal upper bounds for the heat kernel trace of the Dirichlet Laplacian in various domains, improving existing inequalities and providing sharp asymptotic estimates for both finite and infinite volume cases.

Contribution

It introduces new universal bounds for the heat kernel trace of the Dirichlet Laplacian, refining previous inequalities and extending results to infinite volume domains.

Findings

Bounds decay exponentially for large t in finite volume domains

Results improve Kac's inequality for finite volume domains

Bounds incorporate sharp short-time asymptotics

Abstract

We derive upper bounds for the trace of the heat kernel $Z(t)$ of the Dirichlet Laplace operator in an open set $\Omega \subset \R^d$, $d \geq 2$. In domains of finite volume the result improves an inequality of Kac. Using the same methods we give bounds on $Z(t)$ in domains of infinite volume. For domains of finite volume the bound on $Z(t)$ decays exponentially as $t$ tends to infinity and it contains the sharp first term and a correction term reflecting the properties of the short time asymptotics of $Z(t)$. To prove the result we employ refined Berezin-Li-Yau inequalities for eigenvalue means.