98 research outputs found
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
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 in in \Om\times
(0,T) with \Om \subset M an open subset of a manifold with control
metric corresponding to and a measure 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
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 , 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 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 .Comment: We have corrected a few typos and added a few more details to the
proof of the Gaussian estimate
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 as \e\to 0. We
establish 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
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