Heat content asymptotics for sets with positive reach

TL;DR

This paper investigates the short-time asymptotic behavior of heat content for sets with positive reach, extending classical results to include certain non-smooth and singular sets with well-defined normal geometry.

Contribution

It provides new asymptotic formulas for heat content in sets with positive reach, using novel techniques differing from previous related works.

Findings

Derived asymptotic behavior of heat content for sets with positive reach

Extended analysis to non-smooth and singular sets with well-defined normal geometry

Introduced new techniques differing from prior methods

Abstract

In this paper we study the heat content for sets with positive reach. In details, we investigate the asymptotic behavior of the heat content of bounded subsets of the Euclidean space with positive reach. The concept of positive reach was introduced by Federer in \cite{fed_1959} and widely developed in the following years (see for instance the recent book by Rataj and Zh{\"a}le \cite{rat_zah_2019}). It extends the class of sets with smooth boundaries to include certain non-smooth and singular sets while still admitting a well-defined normal geometry. For such sets $E\subseteq\Rn$, we analyze the short-time asymptotics of the heat content $\|T_t\mathbbm{1}_E\|_2$, where $T_t\mathbbm{1}_E$ is the soluzion of the heat equation in $\Rn$ with initial condition $\mathbbm{1}_E$. The present paper is in the spirit of Angiuli, Massari and Miranda Jr.\cite{ang_mas_mir_2013}, but the technique's used here are completely different and also the final result is slightly different.