A Subelliptic Analogue of Aronson-Serrin's Harnack Inequality

We show that the Harnack inequality for a class of degenerate parabolic quasilinear PDE \p_t u=-X_i^* A_i(x,t,u,Xu)+ B(x,t,u,Xu), associated to a system of Lipschitz continuous vector fields $X=(X_1,...,X_m)$ in in \Om\times (0,T) with \Om \subset M an open subset of a manifold $M$ with control metric $d$ corresponding to $X$ and a measure $d\sigma$ follows from the basic hypothesis of doubling condition and a weak Poincar\'e inequality. We also show that such hypothesis hold for a class of Riemannian metrics g_\e collapsing to a sub-Riemannian metric \lim_{\e\to 0} g_\e=g_0 uniformly in the parameter \e\ge 0

Regularity of mean curvature flow of graphs on Lie groups free up to step 2

We consider (smooth) solutions of the mean curvature flow of graphs over bounded domains in a Lie group free up to step two (and not necessarily nilpotent), endowed with a one parameter family of Riemannian metrics \sigma_\e collapsing to a subRiemannian metric $\sigma_0$ as \e\to 0. We establish $C^{k,\alpha}$ estimates for this flow, that are uniform as \e\to 0 and as a consequence prove long time existence for the subRiemannian mean curvature flow of the graph. Our proof extend to the setting of every step two Carnot group (not necessarily free) and can be adapted following our previous work in \cite{CCM3} to the total variation flow.Comment: arXiv admin note: text overlap with arXiv:1212.666

Regularity for Subelliptic PDE Through Uniform Estimates in Multi-Scale Geometries

We aim at reviewing and extending a number of recent results addressing stability of certain geometric and analytic estimates in the Riemannian approximation of subRiemannian structures. In particular we extend the recent work of the the authors with Rea [19] and Manfredini [17] concerning stability of doubling properties, Poincar\'e inequalities, Gaussian estimates on heat kernels and Schauder estimates from the Carnot group setting to the general case of H\"ormander vector fields

Harnack estimates for degenerate parabolic equations modeled on the subelliptic p-Laplacian

We establish a Harnack inequality for a class of quasi-linear PDE modeled on the prototype {equation*} \partial_tu= -\sum_{i=1}^{m}X_i^\ast (|\X u|^{p-2} X_i u){equation*} where $p\ge 2$, \ \X = (X_1,..., X_m) is a system of Lipschitz vector fields defined on a smooth manifold \M endowed with a Borel measure $\mu$, and $X_i^*$ denotes the adjoint of $X_i$ with respect to $\mu$. Our estimates are derived assuming that (i) the control distance $d$ generated by \X induces the same topology on \M; (ii) a doubling condition for the $\mu$-measure of $d-$metric balls and (iii) the validity of a Poincar\'e inequality involving \X and $\mu$. Our results extend the recent work in \cite{DiBenedettoGianazzaVespri1}, \cite{K}, to a more general setting including the model cases of (1) metrics generated by H\"ormander vector fields and Lebesgue measure; (2) Riemannian manifolds with non-negative Ricci curvature and Riemannian volume forms; and (3) metrics generated by non-smooth Baouendi-Grushin type vector fields and Lebesgue measure. In all cases the Harnack inequality continues to hold when the Lebesgue measure is substituted by any smooth volume form or by measures with densities corresponding to Muckenhoupt type weights

Uniform Gaussian bounds for subelliptic heat kernels and an application to the total variation flow of graphs over Carnot groups

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 $t\to \infty$.Comment: We have corrected a few typos and added a few more details to the proof of the Gaussian estimate