Gradient regularity for elliptic equations in the Heisenberg Group

Giuseppe Mingione; Anna Zatorska-Goldstein; Xiao Zhong

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arXiv · 0708.3655

Gradient regularity for elliptic equations in the Heisenberg Group

Giuseppe Mingione·Anna Zatorska-Goldstein·Xiao Zhong

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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.

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Giuseppe Mingione, Anna Zatorska-Goldstein, Xiao Zhong. 2007-08-27. Gradient regularity for elliptic equations in the Heisenberg Group. https://arxiv.org/abs/0708.3655

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