Uniform bounds for the heat content of open sets in Euclidean space

TL;DR

This paper establishes bounds and geometric conditions for the heat content and heat loss of open sets in Euclidean space, providing insights into their thermal properties based on boundary smoothness and measure.

Contribution

It introduces new bounds and a geometric criterion for the finiteness of heat content in open sets with smooth boundaries in Euclidean space.

Findings

Derived bounds for heat content of open sets with smooth boundaries

Established a necessary and sufficient geometric condition for finiteness of heat content

Provided bounds for heat loss in open sets with finite measure

Abstract

We obtain (i) lower and upper bounds for the heat content of an open set in $\mathbb{R}^m$ with $R$-smooth boundary and finite Lebesgue measure, (ii) a necessary and sufficient geometric condition for finiteness of the heat content in $\mathbb{R}^m$, and corresponding lower and upper bounds, (iii) lower and upper bounds for the heat loss of an open set in $\mathbb{R}^m$ with finite Lebesgue measure.