Uniform bounds for Neumann heat kernels and their traces in convex sets

TL;DR

This paper establishes uniform bounds for Neumann heat kernels and their traces in convex domains, providing insights into their small-time behavior and extending results to Lipschitz domains.

Contribution

It introduces a domain-dependent bound capturing the first two terms of the heat trace expansion valid for all times, with a focus on convex and Lipschitz domains.

Findings

Bound on the heat trace capturing first two terms in small-time expansion

Uniform on-diagonal heat kernel expansion near the boundary

Extension of results to Lipschitz domains

Abstract

We prove a bound on the heat trace of the Neumann Laplacian on a convex domain that captures the first two terms in its small-time expansion, but is valid for all times and depends on the underlying domain only through very simple geometric characteristics. This is proved via a precise and uniform expansion of the on-diagonal heat kernel close to the boundary. Most of our results are valid without the convexity assumption and we also consider two-term asymptotics for the heat trace for Lipschitz domains.