一类A -调和方程的障碍问题的很弱解的全局正则性

数学物理学报 ›› 2014, Vol. 34 ›› Issue (1): 27-38.

一类A -调和方程的障碍问题的很弱解的全局正则性

周树清^1,2*|胡振华³|彭冬云¹

1.湖南师范大学数学与计算机科学学院长沙 410081;
2.高性能计算与随机信息处理省部共建教育部重点实验室长沙 410081;
3.湖南城市学院数学系湖南益阳 413000

收稿日期:2012-04-08 修回日期:2013-07-04 出版日期:2014-02-25 发布日期:2014-02-25
通讯作者: 周树清,zhoushuqing87@163.com E-mail:zhoushuqing87@163.com
基金资助:
国家自然科学基金(11271120, 10971061)、湖南省自科基金(11JJ6005)、湖南省重点学科建设项目和湖南师范大学青优培养计划(080640)资助.

Global Regularity for Very Weak Solutions to Obstacle Promlems Corresponding to a Class of A-Harmonic Equations

ZHOU Shu-Qing^1,2*, HU Zhen-Hua³, PENG Dong-Yun¹

1.School of Mathematics and Computer Science of Hunan Normal University, Changsha 410081;
2.Key Laboratory of High Performance Computing and Stochastic Information Processing, Changsha 410081;
3.Department of Mathematics, Hunan City University, Hunan |Yiyang 413000

Received:2012-04-08 Revised:2013-07-04 Online:2014-02-25 Published:2014-02-25
Contact: ZHOU Shu-Qing,zhoushuqing87@163.com E-mail:zhoushuqing87@163.com
Supported by:
国家自然科学基金(11271120, 10971061)、湖南省自科基金(11JJ6005)、湖南省重点学科建设项目和湖南师范大学青优培养计划(080640)资助.

摘要/Abstract

摘要：

应用Hodge分解定理, 得到了非齐次A -调和方程

-div(A(x, Du(x)))=f(x, u(x))
对应的障碍问题很弱解的局部和全局的W^{1, q}(Ω) -正则性, 其中, A(x, Du(x)), f(x, u(x))满足文中所给的条件, 从而推广了相关文献中的有关结果. 该结果在优化控制问题中有着广泛的应用.

关键词: 非齐次A -调和方程, 障碍问题, 优化控制, Hodge分解, 全局W^1, ^q(Ω) -正则性

Abstract:

Using Hodge decomposition theorem, the local and the global W^1, ^q(Ω)-regularity results for very weak solutions to the obstacle problems associated with the following non-homogeneous A-harmonic equations

-div(A(x, Du(x)))=f(x, u(x))
are obtained under certain conditions on A(x, Du(x)), f(x, u(x)) listed in the context. The results generalize the corresponding results in related literatures. The results can be widely applied to optimal control problems.

Key words: Non-homogeneous A-harmonic equations, Obstacle problems, Optimal control, Hodge decomposition, Global W^1, ^q(Ω)-regularity

中图分类号:

35J60

周树清, 胡振华, 彭冬云. 一类A -调和方程的障碍问题的很弱解的全局正则性[J]. 数学物理学报, 2014, 34(1): 27-38.

ZHOU Shu-Qing, HU Zhen-Hua, PENG Dong-Yun. Global Regularity for Very Weak Solutions to Obstacle Promlems Corresponding to a Class of A-Harmonic Equations[J]. Acta mathematica scientia,Series A, 2014, 34(1): 27-38.

参考文献

[1] Iwaniec T, Sbordone C. Weak minima of variational integrals. J Reine Angew Math, 1994, 454: 143--161

[2] Li Juan, Gao Hongya. Local regularity result for very weak solutions of obstacle problems. Radovi Matematicki, 2003, 12: 19--26

[3] Li Gongbao, Martio O. Local and global integrability of gradients in obstacle problems. Ann Acad Sci Fenn Ser A I Math, 1994, 19: 25--34

[4] Granlund S. An L^p-estimate for the gradient of extremals. Math Scand, 1982, 50: 66--72

[5] Giaquinta M. Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems. Ann Math Stud 105,
New Jersey: Princeton University Press, 1983

[6] Serrin J. Pathological solutions of elliptic equations. Ann Scor Norm Sup Pisa, 1964, 18: 385--387

[7] Elcrat A, Meyers N. Some results on regularity for non-linear elliptic systems and quasi-regular functions. Duke Math J, 1975, 42: 121--136

[8] Lewis L. On very weak solutions of certain elliptic systems. Comm Part Diff Equ, 1993, 18(9/10): 1515--1537

[9] Daniea G, Francesco L, Rosanna S. On the regularity of very weak minima. Proc Roy Soc of Edin, 1996, 126A: 287--296

[10] Daniea G, Rosanna S. Boundary higher integrability for the gradient of distributional solutions of nonlinear systems. Studia Math, 1997, 123(2): 175--184

[11] Zhou Shuqing, Gao Hongya, Zhu Huanran. Uniqueness of very weak solutions to a class of nonlinear elliptic equations. Chinese J of Contem Math, 2007, 28(1): 99--108

[12] 周树清. 一类非齐次A -调和方程组很弱解的正则性. 数学年刊, 2002, 23A(3): 283--288

[13] 周树清, 文海英, 方华强. 一类非齐次A -调和方程组很弱解的的性质. 数学物理学报, 2003, 23A(2): 135--144

[14] Bergounioux M. Optimal control of an obstacle problem. Appl Math Optim, 1997, 36: 47--172

[15] Caffarelli1 L A. The Obstacle problem revisited. J of Fourier Ana and Appl, 1998, 4(4/5): 383--402

[16] Caffarelli1 L A, Salsa S, Silvestre L. Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian. Invent math, 2008, 171: 425--461

[17] Lin Fanghua. On regularity and singularity of free boundaries in obstacle Problems. Chinese Ann Math, 2009, 30B(5): 645--652

[18] 高红亚, 何茜,牛红玲等. A -调和方程障碍问题很弱解的局部正则性. 数学物理学报, 2009, 29A(5): 1291--1297

[19] Gao Hongya, Guo Jing, Zuo Yali, et al. Local regularity result in obstacle. Acta Mathematica Scietia, 2010, 30B: 208--214

[20] Iwaniec T. p-Harmonic tensor and quasi-regular maps. Ann Math, 1992, 136(3): 589--624

[21] Iwaniec T, Migliaccio L, Sbordone C. Integrability and removability results for quasi-regular mappings in high dimension. Math Scand, 1994, 75: 263--279

[22] Gilbarg D, Trudinger N S. Elliptic Partial Differential Equations of Second Order. Berlin, Heidelberg: Springer-Verlag, 1977

[23] Morrey C B. Multiple Integrals in the Calculus of Variations. Berlin, Heidelberg: Springer-Verlag, 1966

[24] 郑神州. 拟正则映照的一些问题[D]. 上海: 上海交通大学, 1997

[25] Gehring F W. L^P-integrability of the partial derivatives of a quasi-conformal mapping. Acta Math, 1973, 130(3/4): 265--277