非线性次椭圆方程障碍问题很弱解的高阶可积性

数学物理学报 ›› 2017, Vol. 37 ›› Issue (1): 122-145.

非线性次椭圆方程障碍问题很弱解的高阶可积性

杜广伟, 钮鹏程

西北工业大学应用数学系西安 710129

收稿日期:2016-06-21 修回日期:2016-11-20 出版日期:2017-02-26 发布日期:2017-02-26
通讯作者: 钮鹏程 E-mail:pengchengniu@nwpu.edu.cn
作者简介:杜广伟,E-mail:guangwei87@mail.nwpu.edu.cn
基金资助:
国家自然科学基金（11271299）

Higher Integrability for Very Weak Solutions of Obstacle Problems to Nonlinear Subelliptic Equations

Du Guangwei, Niu Pengcheng

Department of Applied Mathematics, Northwestern Polytechnical University, Xi'an 710129

Received:2016-06-21 Revised:2016-11-20 Online:2017-02-26 Published:2017-02-26
Supported by:
Supported by the NSFC (11271299)

摘要/Abstract

摘要：

该文首先证明了一类由满足Hörmander条件的向量场构成的次椭圆方程K_ψ，u₀^r-障碍问题很弱解的局部高阶可积性，进而说明了其很弱解即为经典意义下的弱解.作为其应用，得到了障碍问题很弱解的紧性结果.此外，在当区域Ω满足某容度条件假设时，证明了上述障碍问题很弱解的全局高阶可积性.

关键词: 非线性次椭圆方程, 障碍问题, 很弱解, 高阶可积性, 紧性

Abstract:

In this paper we first establish the local higher integrability for very weak solutions of the K_{ψ, u₀}^r-obstacle problems to the nonlinear subelliptic equations constructed by Hörmander vector fields which implies the very weak solutions are classical weak solutions. As an application, the compactness results are obtained. We also derive the global higher integrability for very weak solutions of the K_{ψ, u₀}^r-obstacle problems with a capacitary condition on Ω.

Key words: Nonlinear subelliptic equation, Obstacle problem, Very weak solutions, Higher integrability, Compactness result

中图分类号:

O175.2

杜广伟, 钮鹏程. 非线性次椭圆方程障碍问题很弱解的高阶可积性[J]. 数学物理学报, 2017, 37(1): 122-145.

Du Guangwei, Niu Pengcheng. Higher Integrability for Very Weak Solutions of Obstacle Problems to Nonlinear Subelliptic Equations[J]. Acta mathematica scientia,Series A, 2017, 37(1): 122-145.

参考文献

[1] Heinonen J, Kilpeläinen T, Martio O. Nonlinear Potential Theory of Degenerate Elliptic Equations. New York:Clarendon Press, 1993
[2] Caffarelli L A. The obstacle problem revisited. J Fourier Anal Appl, 1998, 4:383-402
[3] Kinderlehrer D, Stampacchia G. An Introduction to Variational Inequalities and Their Applications. Philadelphia, PA:SIAM, 2000
[4] Friedman A. Variational Principles and Free Boundary Problems. New York:Dover, 2010
[5] Iwaniec T, Sbordone C. Weak minima of variational integrals. J Reine Angew Math, 1994, 454:143-161
[6] Lewis J L. On very weak solutions to certain elliptic systems. Comm Partial Differential Equations, 1993, 18:1515-1537
[7] Li G B, Martio O. Local and global integrability of gradients in obstacle problems. Ann Acad Sci Fenn Ser A I Math, 1994, 19:25-34
[8] Kilpeläinen T, Koskela P. Global integrability of the gradients of solutions to partial differential equations. Nonlinear Anal, 1994, 23:899-909
[9] Bao G J, Wang T T, Li G F. On very weak solutions to a class of double obstacle problems. J Math Anal Appl, 2013, 402:702-709
[10] Gianazza U, Marchi S. Interior regularity for solutions to some degenerate quasilinear obstacle problems. Nonlinear Anal, 1999, 36:923-942
[11] Marchi S. Regularity for the solutions of double obstacle problems involving nonlinear elliptic operators on the Heisenberg group. Matematiche, 2001, 56:109-127
[12] Danielli D, Garofalo N, Petrosyan A. The sub-elliptic obstacle problem:C^1,α-regularity of the free boundary in Carnot groups of step two. Adv Math, 2007, 211:485-516
[13] Bigolin F. Regularity result for a class of obstacle problems in Heisenberg groups. Appl Math, 2013, 58:531-554
[14] Zatorska-Goldstein A. Very weak solutions of nonlinear subelliptic equations. Ann Acad Sci Fenn Math, 2005, 30:407-436
[15] Danielli D, Garofalo N, Phuc N C. Inequalities of Hardy-Sobolev type in Carnot-Carathéodory spaces//Maz'ya V. Sobolev Spaces in Mathematics I. Novosibirsk:Tamara Rozhkovskaya Publisher, 2009:117-151
[16] Hörmander L. Hypoelliptic second order differential equations. Acta Math, 1967, 119:147-171
[17] Chow W L. Über systeme von linearen partiellen differentialgleichungen erster ordnung. Math Ann, 1939, 117:98-105
[18] Nagel A, Stein E M, Wainger S. Balls and metrics defined by vector fields I:Basic properties. Acta Math, 1985, 155:103-147
[19] Garofalo N, Nhieu D M. Lipschitz continuity, global smooth approximations and extension theorems for Sobolev functions in Carnot-Carathéodory spaces. J Anal Math, 1998, 74:67-97
[20] Hajlasz P, Koskela P. Sobolev Meets Poincaré. Providence, RI:American Mathematical Society, 2000
[21] Franchi B, Serapioni R, Serra Cassano F. Approximation and imbedding theorems for weighted Sobolev spaces associated with Lipschitz continuous vector fields. Boll Un Mat Ital B, 1997, 11(7):83-117
[22] Lu G Z. Weighted Poincaré and Sobolev inequalities for vector fields satisfying Hörmander's condition and applications. Rev Mat Iberoamericana, 1992, 8:367-439
[23] Danielli D. Regularity at the boundary for solutions of nonlinear subelliptic equations. Indiana Univ Math J, 1995, 44:269-285
[24] Lewis J L. Uniformly fat sets. Trans Am Math Soc, 1988, 308:177-196
[25] Federer H. Geometric Measure Theory. Berlin:Springer-Verlag, 1969