Gradient regularity for elliptic equations in the Heisenberg group
Gradient regularity for elliptic equations in the Heisenberg group
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DOI：
10.1016/j.aim.2009.03.016
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发表时间：
2007-08
影响因子：
1.7
通讯作者：
G. Mingione;Anna Zatorska-Goldstein;X. Zhong
中科院分区：
文献类型：
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作者：
G. Mingione;Anna Zatorska-Goldstein;X. Zhong
本文给出了海森堡群中一类可能退化的次椭圆方程在水平梯度上具有超二次增长的无量纲正则性条件，解决了[J. J. Manfredi，G. Mingione，Regularity results for quasilinear elliptical equations in the Heisenberg group，Math.Ann.339（2007）485-544]，其中仅给出了增长指数的维度相关边界。我们还得到了显式的先验局部正则性估计，并覆盖了水平p-Laplacean算子的情况，推广了[A. Domokos，J. J. Manfredi，C1，海森堡群中p-调和函数的α-正则性，p接近2，在：Contemp。数学、第370卷，2005年，第370页。17-23]。反过来，利用Caffarelli和Peral [L.卡法雷利岛Peral，On W1，pestimates for elliptical equations in divergence form，Comm. Pure Appl. Math. 51（1998）1-21]，所发现的先验估计表明对于相关类的非齐次、可能退化的涉及不连续系数的方程，存在合适的局部Calderón-Zygmund理论.这些结果推广到了亚椭圆情形，一些经典的非线性欧几里德结果[T。张文龙，梯度场的投影与退化椭圆算子的Lp-估计，《数学研究》，75（1983）293-312; DiBenedetto，J. J. Manfredi，On the higher integrability of the gradient of weak solutions of certain degenerative elliptical systems，Amer. J. Math. 115（1993）1107-1134]，以及仅在次椭圆设置中可用于线性方程解的相同性质的非线性情况估计。
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,pestimates 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.