Gradient regularity for elliptic equations in the Heisenberg group |
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Authors: | Giuseppe Mingione Anna Zatorska-Goldstein Xiao Zhong |
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Institution: | a Dipartimento di Matematica, Università di Parma, Viale G.P. Usberti 53/A, I-43100 Parma, Italy b Instytut Matematyki Stosowanej i Mechaniki, Uniwersytet Warszawski, Banacha 2, 02-097 Warszawa, Poland c Department of Mathematics and Statistics, PO Box 35 (MaD) FI-40014, University of Jyväskylä, Finland |
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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, 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. |
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Keywords: | primary 35H20 35J70 |
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