Uniform bounds for Neumann heat kernels and their traces in convex sets
Analysis of PDEs
2026-01-13 v1 Spectral Theory
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.
Keywords
Cite
@article{arxiv.2601.07341,
title = {Uniform bounds for Neumann heat kernels and their traces in convex sets},
author = {Rupert L. Frank and Simon Larson},
journal= {arXiv preprint arXiv:2601.07341},
year = {2026}
}
Comments
The paper extends part of what was contained in an earlier version of arXiv:2407.11808