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Luca Capogna

Publications and source records attributed to Luca Capogna.

28 records · Page 2Linked to original sources
arXiv2013

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 $μ$, and $X_i^*$ denotes the adjoint of $X_i$ with respect to $μ$. 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 $μ$-measure of $d-$metric balls and (iii) the validity of a Poincaré inequality involving $\X$ and $μ$. 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 Hörmander 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.

math.AP↗
arXiv2013

Sub-Riemannian heat kernels and mean curvature flow of graphs

We introduce a sub-Riemannian analogue of the Bence-Merriman-Osher diffusion driven algorithm and show that it leads to weak solutions of the horizontal mean curvature flow of graphs over sub-Riemannian Carnot groups. The proof follows the nonlinear semi-group theory approach originally introduced by L. C. Evans in the Euclidean setting and is based on new results on the relation between sub-Riemannian heat flows of characteristic functions of subgraphs and the horizontal mean curvature of the corresponding graphs.

math.AP↗
arXiv2013

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 $σ_\e$ 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 $σ_\e$ 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 $t\to \infty$.

math.AP↗
arXiv2012

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 $dσ$ follows from the basic hypothesis of doubling condition and a weak Poincaré 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 $\e\ge 0$.

math.AP↗
arXiv2010

An Aronsson type approach to extremal quasiconformal mappings

We study $C^2$ extremal quasiconformal mappings in space and establish necessary and sufficient conditions for a `localized' form of extremality in the spirit of the work of G. Aronsson on absolutely minimizing Lipschitz extensions. We also prove short time existence for smooth solutions of a gradient flow of QC diffeomorphisms associated to the extremal problem.

math.AP↗
arXiv2008

Generalized mean curvature flow in Carnot groups

In this paper we study the generalized mean curvature flow of sets in the sub-Riemannian geometry of Carnot groups. We extend to our context the level sets method and the weak (viscosity) solutions introduced in the Euclidean setting by Evans-Spruck and Chen-Giga-Goto. We establish two special cases of the comparison principle, existence, uniqueness and basic geometric properties of the flow.

math.AP↗
arXiv2008

Regularity of non-characteristic minimal graphs in the Heisenberg group $H^1$

Minimal surfaces in the sub-Riemannian Heisenberg group can be constructed by means of a Riemannian approximation scheme, as limit of Riemannian minimal surfaces. We study the regularity of Lipschitz, non-characteristic minimal surfaces which arise as such limits. Our main results are a-priori estimates on the solutions of the approximating Riemannian PDE and the ensuing $C^{\infty}$ regularity of the sub-Riemannian minimal surface along its Legendrian foliation.

math.AP↗
arXiv2008

Mutual Absolute Continuity of Harmonic and Surface Measures for Hormander Type Operators

In this paper, we consider the Sub-Laplacian L which consists of sum of squares of smooth vector fields that satisfy Hormander's finite rank condition. We study the Dirichlet problem for this operator on domains that satisfy certain geometric conditions. For such domains, several key results are established. These results consist of 1) A reversed Holder inequality for the Poisson kernel 2) Harmonic measure (corresponding to L) and surface measure (as well as the H-Perimeter measure) are mutually absolutely continuous 3) A representation (hence solvability of the Dirichlet problem) for solutions to the Dirichlet problem.

math.AP↗
arXiv2005

The mixed problem in L^p for some two-dimensional Lipschitz domains

We consider the mixed problem for the Laplace operator in a class of Lipschitz graph domains in two dimensions with Lipschitz constant at most 1. The boundary of the domain is decomposed into two disjoint sets D and N. We suppose the Dirichlet data, f_D has one derivative in L^p(D) of the boundary and the Neumann data is in L^p(N). We find conditions on the domain and the sets D and N so that there is a p_0>1 so that for p in the interval (1,p_0), we may find a unique solution to the mixed problem and the gradient of the solution lies in L^p.

math.AP↗