Qualitative properties of the heat content

TL;DR

This paper investigates the mathematical properties of heat content in various domains, establishing conditions for monotonicity and convexity, and providing counterexamples in Euclidean space.

Contribution

It introduces the concept of strictly decreasing temperature sets and explores their impact on heat content monotonicity and convexity in different geometries.

Findings

Strictly decreasing temperature sets ensure monotone heat content.

Counterexamples show heat content can be non-monotone in Euclidean space.

Heat content can be monotone but not convex under certain conditions.

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.