%0 Journal Article
%T Uniform Gaussian bounds for subelliptic heat kernels and an application to the total variation flow of graphs over Carnot groups
%A Luca Capogna
%A Giovanna Citti
%A Maria Manfredini
%J Mathematics
%D 2012
%I arXiv
%R 10.2478/agms-2013-0006
%X In this paper we study heat kernels associated to a Carnot group $G$, endowed with a family of collapsing left-invariant Riemannian metrics $\sigma_\e$ which converge in the Gromov-Hausdorff sense to a sub-Riemannian structure on $G$ as $\e\to 0$. The main new contribution are Gaussian-type bounds on the heat kernel for the $\sigma_\e$ metrics which are stable as $\e\to 0$ and extend the previous time-independent estimates in \cite{CiMa-F}. As an application we study well posedness of the total variation flow of graph surfaces over a bounded domain in $(G,\s_\e)$. We establish interior and boundary gradient estimates, and develop a Schauder theory which are stable as $\e\to 0$. As a consequence we obtain long time existence of smooth solutions of the sub-Riemannian flow ($\e=0$), which in turn yield sub-Riemannian minimal surfaces as $t\to \infty$.
%U http://arxiv.org/abs/1212.6665v2