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

TL;DR

The paper derives uniform two-term bounds for the Neumann heat trace on bounded convex domains, refining Brown's classical small-time expansion by providing an explicit remainder that scales algebraically with the inradius r_in(Ω). The authors develop a rigorous framework to approximate the Neumann heat kernel: in the bulk by the whole-space kernel, and near the boundary by a half-space kernel, using a quantitative notion of boundary flatness and a large, well-controlled set of near-boundary points. A semi-abstract trace estimate is established that translates pointwise kernel bounds into trace bounds with an error expressed in terms of θ_Ω and related geometric data, and this is specialized to obtain a convex-domain two-term bound valid for all times t>0. The work also includes refined bulk estimates and detailed analysis of the Neumann boundary-value problem, along with explicit constructions of large good boundary sets for convex domains, enabling precise, geometry-dependent control of the heat-kernel and trace behavior with potential applications to spectral-shape optimization and related problems.

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.

Theorems & Definitions (71)

• Theorem 1.1: Brown93
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Proposition 2.1
• proof
• Proposition 2.2
• proof