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 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.
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 .
@article{BUMI_2008_9_1_1_243_0, author = {Giuseppe Mingione and Zatorska-Goldstein Anna and Xiao Zhong}, title = {On the Regularity of p-Harmonic Functions in the Heisenberg Group}, journal = {Bollettino dell'Unione Matematica Italiana}, volume = {1}, year = {2008}, pages = {243-253}, zbl = {1164.35039}, mrnumber = {2388006}, language = {en}, url = {http://dml.mathdoc.fr/item/BUMI_2008_9_1_1_243_0} }
Mingione, Giuseppe; Anna, Zatorska-Goldstein; Zhong, Xiao. On the Regularity of p-Harmonic Functions in the Heisenberg Group. Bollettino dell'Unione Matematica Italiana, Tome 1 (2008) pp. 243-253. http://gdmltest.u-ga.fr/item/BUMI_2008_9_1_1_243_0/
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