On the Regularity of p-Harmonic Functions in the Heisenberg Group

Mingione, Giuseppe ; Anna, Zatorska-Goldstein ; Zhong, Xiao

Bollettino dell'Unione Matematica Italiana, Tome 1 (2008), p. 243-253 / Harvested from Biblioteca Digitale Italiana di Matematica

Abstract

We describe some recent results obtained in [29], where we prove regularity theorems for sub-elliptic equations in (horizontal) divergence form defined in the Heisenberg group, and exhibiting polynomial growth of order p. The main result tells that when $p \in [2,4)$ solutions to possibly degenerate equations are locally Lipschitz continuous with respect to the intrinsic distance. In particular, such result applies to p-harmonic functions in the Heisenberg group. Explicit estimates are obtained, and eventually applied to obtain the suitable Calderón-Zygmund theory for the associated non-homogeneous problems.

Résumé

Descriviamo alcuni recenti risultati ottenuti in [29], dove si dimostrano teoremi di regolarità per soluzioni di equazioni sub-ellittiche in forma di divergenza orizzontale, nel gruppo di Heisenberg. I risultati coprono il caso di operatori a crescita p, come il p-Laplaciano nel gruppo di Heisenberg, e sono ottenuti sotto l'ipotesi adimensionale $p \in [2,4)$.

Détail

Publié le : 2008-02-01

MR 2388006 | Zbl 1164.35039

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