Spectral heat content for α-stable processes in C1,1 open sets

Abstract

In this paper we study the asymptotic behavior, as $t\downarrow 0$, of the spectral heat content $Q_D^{(\alpha )}(t)$ for isotropic $\alpha$-stable processes, $\alpha \in [1,2)$, in bounded $C^{1,1}$ open sets $D\subset {\mathbb{R}}^d$, $d\ge 2$. Together with the results from \cite{Val2017} for $d=1$ and \cite{GPS19} for $\alpha \in (0,1)$, the main theorem of this paper establishes the asymptotic behavior of the spectral heat content up to the second term for all $\alpha \in (0,2)$ and $d\ge 1$, and resolves the conjecture raised in \cite{Val2017}.