Gradient regularity for elliptic equations in the Heisenberg group

By Giuseppe Mingione, Anna Zatorska-Goldstein, Xiao Zhong. 2007

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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 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, C 1 , 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 W 1 , 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 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 [T. Iwaniec, Projections onto gradient fields and L p -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.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/1a807c4e01354ecb8f16968dc2d3a34bbcfa55e3

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Citations in the corpus

On certain anisotropic elliptic equations arising in congested optimal transport: Local gradient bounds (2014, background): “variational integrals similar to those considered here, in less degenerate situations) and(2) [16].”
Elliptic equations with measurable coefficients in Reifenberg domains (2010, methodology): “In fact, this approach and methods presented in [7,8] have been deployed and adjusted successfully to different settings for such gradient estimates in very recent papers, see [21,22,26].”