On the heat content of a polygon

TL;DR

This paper analyzes the initial heat content of polygonal and fractal polyhedral domains in Euclidean space, deriving asymptotic behaviors as time approaches zero, with implications for understanding heat distribution in complex shapes.

Contribution

It provides the first detailed asymptotic analysis of heat content for polygons and applies these results to fractal polyhedra, extending classical heat content theory.

Findings

Asymptotic expansion of heat content for polygons as t → 0

Application of results to fractal polyhedra

Insights into heat distribution in complex geometries

Abstract

Let $D$ be a bounded, connected, open set in Euclidean space $\mathbb{R}^{2}$ with polygonal boundary. Suppose $D$ has initial temperature $1$ and the complement of $D$ has initial temperature $0$. We obtain the asymptotic behaviour of the heat content of $D$ as time $t \downarrow 0$. We then apply this result to compute the heat content of a particular fractal polyhedron as $t \downarrow 0$.