Gradient regularity for elliptic equations in the Heisenberg group

51Citations
Citations of this article
11Readers
Mendeley users who have this article in their library.

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 in [J.J. Manfredi, G. Mingione, Regularity results for quasilinear elliptic equations in the Heisenberg group, Math. Ann. 339 (2007) 485-544], 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 in [A. Domokos, J.J. Manfredi, C1, α-regularity for p-harmonic functions in the Heisenberg group for p near 2, in: Contemp. Math., vol. 370, 2005, pp. 17-23]. In turn, using some recent techniques of Caffarelli and Peral [L. Caffarelli, I. Peral, On W1, p estimates for elliptic equations in divergence form, Comm. Pure Appl. Math. 51 (1998) 1-21], the a priori estimates found are shown to imply the suitable local Calderón-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 [T. Iwaniec, Projections onto gradient fields and Lp-estimates for degenerated elliptic operators, Studia Math. 75 (1983) 293-312; E. DiBenedetto, J.J. Manfredi, On the higher integrability of the gradient of weak solutions of certain degenerate elliptic systems, Amer. J. Math. 115 (1993) 1107-1134], 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. © 2009 Elsevier Inc. All rights reserved.

Cite

CITATION STYLE

APA

Mingione, G., Zatorska-Goldstein, A., & Zhong, X. (2009). Gradient regularity for elliptic equations in the Heisenberg group. Advances in Mathematics, 222(1), 62–129. https://doi.org/10.1016/j.aim.2009.03.016

Register to see more suggestions

Mendeley helps you to discover research relevant for your work.

Already have an account?

Save time finding and organizing research with Mendeley

Sign up for free