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

Rupert L. Frank; Simon Larson

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arXiv · 2601.07341

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

Rupert L. Frank·Simon Larson

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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.

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Rupert L. Frank, Simon Larson. 2026-01-12. Uniform bounds for Neumann heat kernels and their traces in convex sets. https://arxiv.org/abs/2601.07341

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