Gradient regularity for elliptic equations in the Heisenberg Group
Abstract
We give dimension-free regularity conditions for a class of possibly degenerate sub-elliptic equations in the Heisenberg group exhibiting super-quadratic growth in the horizontal gradient; this solves an issue raised by Manfredi & Mingione (Math. Ann. 2007) where only dimension dependent bounds for the growth exponent are given. We also obtain explicit a priori local regularity estimates, and cover the case of the horizontal p-Laplacean operator, extending some regularity proven by Domokos & Manfredi (Cont. Math. 2005). In turn, the a priori estimates found are shown to imply the suitable local Calderon-Zygmund theory for the related class of non-homogeneous, possibly degenerate equations involving discontinuous coefficients. These last results extend to the sub-elliptic setting a few classical non-linear Euclidean results of Iwaniec and Dibenedetto & Manfredi, and to the non-linear case estimates of the same nature that were available in the sub-elliptic setting only for solutions to linear equations.
Cite
@article{arxiv.0708.3655,
title = {Gradient regularity for elliptic equations in the Heisenberg Group},
author = {Giuseppe Mingione and Anna Zatorska-Goldstein and Xiao Zhong},
journal= {arXiv preprint arXiv:0708.3655},
year = {2009}
}