Boundary behavior p-harmonic functions in the Heisenberg group
Abstract
We study the boundary behavior of nonnegative p-harmonic functions which vanish on a portion of the boundary of a domain in the Heisenberg group H^n. Our main results are: 1) An estimate from above which shows that, under suitable geometric assumptions on the relevant domain, such a p-harmonic function vanishes at most linearly with respect to the sub-Riemannian distance to the boundary. 2) An estimate from below which shows that for a (Euclidean) C^{1,1} domain, away from the characteristic set, such a p-harmonic function vanishes exactly like the distance to the boundary. By combining 1) and 2) we obtain a comparison theorem stating that, at least away from the characteristic set, any two such p-harmonic functions must vanish at the same rate.
Cite
@article{arxiv.1005.2602,
title = {Boundary behavior p-harmonic functions in the Heisenberg group},
author = {Nicola Garofalo and Nguyen Cong Phuc},
journal= {arXiv preprint arXiv:1005.2602},
year = {2010}
}