Luca Capogna

An embedding theorem and the harnack inequality for nonlinear subelliptic equations

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CAPACITARY ESTIMATES AND THE LOCAL BEHAVIOR OF SOLUTIONS OF NONLINEAR SUBELLIPTIC EQUATIONS

We establish sharp capacitary estimates for Carnot-Carathéodory rings associated to a system of vector fields of Hörmander type. Such estimates are instrumental to the study of the local behavior of singular solutions of a wide class of nonlinear subelliptic equations. One of the main results is a generalization of fundamental estimates obtained independently by Sanchez-Calle and Nagel, Stein and Wainger.

Regularity of quasi-linear equations in the Heisenberg group

We prove the Cα regularity of the gradient of weak solutions of a class of quasi-linear equations in nilpotent stratified Lie groups of step two. As applications, we prove higher regularity theorems and a Liouville type theorem for 1-quasi-conformal mappings between domains of the Heisenberg group.

Uniform domains and quasiconformal mappings on the Heisenberg group

We prove that in the Heisenberg group the image of a uniform domain under a global quasiconformal homeomorphism is still a uniform domain. As a consequence, the class of NTA (non-tangentially accessible) domains in the Heisenberg group is also quasiconformally invariant. A large class of non-differentiable Lipschitz quasiconformal homeomorphisms is constructed. The images of smooth domains under these rough mappings give a class of non-smooth NTA domains in the Heisenberg group.

Subelliptic mollifiers and a basic pointwise estimate of Poincaré type

The aim of this paper is to prove a pointwise estimate which plays a fundamental role in the analysis of both linear and nonlinear partial differential equations arising from a system of non-commuting vector fields X = {X1, . . . ,Xm}. Although our result holds in a larger setting, for the unity of presentation we confine ourselves to the specific case of smooth vector fields X1, . . .Xm satisfying Hormander’s finite rank condition [H]: Rank (Lie[X1, . . . ,Xm ])(x ) = n for every x ∈ n . Let d : n × n → + be the Carnot-Caratheodory metric associated to X1, . . . ,Xm , and for x0 ∈ n and R > 0 set B = Bd (x0,R) = {y ∈ n |d (x0, y) 0 the symbol aB will denote the concentric ball Bd (x0, aR). For a given function u we let Xu = (X1u, . . . ,Xmu) and |Xu|2 = ∑m j =1(Xj u) 2. Throughout the paper we will use the standard notation uB = 1 |B | ∫ B u(y)dy . Our main result is Theorem 1.1. Let U ⊂⊂ n , and x0 ∈ U . There exist C , R0 > 0 and a > 1 depending only on U and the vector fields X1, . . . ,Xm such that for any u ∈ C 1(aB ), R ≤ R0, and x ∈ B one has |u(x ) − uB | ≤ C ∫

Properties of harmonic measures in the Dirichlet problem for nilpotent Lie groups of Heisenberg type

In groups of Heisenberg type we introduce a large class of domains, which we call ADP, admissible for the Dirichlet problem, and we prove that on the boundary of such domains, harmonic measure, ordinary surface measure, and the perimeter measure, are mutually absolutely continuous. We also establish the solvability of the Dirichlet problem when the boundary datum belongs to Lp, 1 < p ≤ ∞, with respect to the ordinary surface measure. Here, the harmonic measure is that relative to a sub-Laplacian associated with a basis of the first layer of the Lie algebra. A domain is called ADP if it is a nontangentially accessible domain and it satisfies an intrinsic outer ball condition.

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 in [4] and [12]. We establish two special cases of the comparison principle, existence, uniqueness and basic geometric properties of the flow.

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

AbstractWe consider the mixed problem,

Ahlfors type estimates for perimeter measures in Carnot-Carathéodory spaces

\left\{ \begin{array}{ll} \Delta u = 0 \quad & {\rm in }\, \Omega\\ \frac{\partial u }{\partial \nu} = f_N \quad & {\rm on }\, {\rm N} \\ u = f_D \quad & {\rm on}\,D \end{array} \right.