Gradient regularity for elliptic equations in the Heisenberg Group

TL;DR

This paper establishes dimension-free regularity conditions for degenerate sub-elliptic equations in the Heisenberg group, providing explicit local estimates and extending classical Euclidean results to the sub-elliptic setting.

Contribution

It introduces new dimension-independent regularity criteria and local estimates for sub-elliptic equations, including the horizontal p-Laplacean, extending prior Euclidean and linear sub-elliptic results.

Findings

Dimension-free regularity conditions established

Explicit local a priori estimates derived

Extension of Euclidean non-linear results to sub-elliptic setting

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.