No abstract

No abstract

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. © 1997 John Wiley & Sons, Inc.

No abstract

No abstract

No abstract

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 L p , 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.

Abstract. We prove the hypoellipticity for systems of Hörmander type with constant coefficients in Carnot groups of step 2. This result is used to implement blow-up methods and prove partial regularity for local minimizers of non-convex functionals, and for solutions of non-linear systems which appear in the study of non-isotropic metric structures with scalings. We also establish estimates of the Hausdorff dimension of the singular set.