Qualitative properties of the heat content

TL;DR

This work analyzes qualitative heat-flow properties without imposing boundary conditions on open sets in $\mathbb{R}^m$ and on Riemannian manifolds, focusing on the monotonicity and convexity of the heat content $H_{\Omega,\psi}(t)$. It introduces the concept of a strictly decreasing temperature set and proves that, whenever $H_{\Omega}(t)<\infty$ for all $t>0$, the function $t\mapsto H_{\Omega}(t)$ is decreasing and convex, with strict decrease under appropriate limiting conditions and geometric assumptions. In Euclidean space, the authors construct a strictly decreasing temperature set from two small balls and provide derivative bounds for $H_{\Omega}(t)$, while showing that nonconstant initial data $\psi$ can destroy monotonicity or convexity; they also derive explicit differential inequalities and extend results to general nonnegative initial data. The paper employs heat-kernel semigroup methods, spectral expansions, and Neumann kernels to obtain both global and time-local estimates, and presents sharp counterexamples that delineate the boundary between monotonicity/convexity and more complex heat-content behavior.

Abstract

We obtain monotonicity and convexity results for the heat content of domains in Riemannian manifolds and in Euclidean space subject to various initial temperature conditions. We introduce the notion of a strictly decreasing temperature set, and show that it is a sufficient condition to ensure monotone heat content. In addition, in Euclidean space, we construct a domain and an initial condition for which the heat content is not monotone, as well as a domain and an initial condition for which the heat content is monotone but not convex.

Theorems & Definitions (12)

• Definition 1
• Theorem 1
• Theorem 2
• Theorem 3
• Corollary 4
• Theorem 5
• Theorem 6
• proof
• proof
• proof