Fractional heat content asymptotics for Carnot groups

Abstract

We propose a novel approach for studying small-time asymptotics of the fractional heat content of $C^2$ non-characteristic domains in Carnot groups. Denoting the sub-Laplacian operator by $\mathcal{L}$, the fractional heat content of a bounded domain $Ω$ is defined as $Q^{(α)}_Ω(t)=\int_Ωu_α(x,t) dx$, where $u_α$ is the solution to the heat equation corresponding to the fractional sub-Laplacian $\mathcal{L}_α:=\mathcal{L}^{α/2}$ with Dirichlet boundary condition on $Ω$. We prove that for $1\le α\le 2$, there exists explicit rate function $μ_α: (0,\infty)\to (0,\infty)$ such that \begin{align*} \lim_{t\to 0}\frac{|Ω|-Q^{(α)}_Ω(t)}{μ_α(t)}=|\partial Ω|_H, \end{align*} where $|Ω|$, $|\partial Ω|_H$ are the volume and horizontal perimeter of $Ω$ respectively. Moreover, the rate function $μ_α$ coincides with the same for the Euclidean case.