preprint
Inserted: 1 aug 2025
Year: 2025
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{fed1959} and widely developed in the
following years (see for instance the recent book by Rataj and Zh{\"a}le
\cite{ratzah2019}). 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{angmasmir2013}, but the technique's used here
are completely different and also the final result is slightly different.