拟线性次椭圆方程很弱解的正则性

数学物理学报 ›› 2010, Vol. 30 ›› Issue (2): 432-439.

拟线性次椭圆方程很弱解的正则性

郑神州, 王喜芬

北京交通大学理学院, 北京 100044

收稿日期:2008-12-29 修回日期:2009-11-08 出版日期:2010-04-25 发布日期:2010-04-25
基金资助:
国家自然科学基金(10671022)资助.

Regularity of Very Weak Solutions for a Class of Quasilinear Subelliptic Equations

ZHENG Shen-Zhou, WANG Xi-Fen

College of Sciences, Beijing Jiaotong University, Beijing 100044

Received:2008-12-29 Revised:2009-11-08 Online:2010-04-25 Published:2010-04-25
Supported by:
国家自然科学基金(10671022)资助.

摘要/Abstract

摘要：

该文利用扰动向量场的Hodge分解及估计构造关于弱解X 梯度场的反向Hölder 不等式, 从而建立了Carnot 群上的散度型拟线性次椭圆方程很弱解梯度的可积性指数的提高, 得到其很弱解是经典意义的弱解结论.

关键词: 次椭圆方程, 很弱解, Carnot-Caratheodory空间, Hodge分解, 反向Holder不等式

Abstract:

According to the construction of generalized reverse Hölder inequality by Hodge decomposition and estimates on disturbing vector fields, the authors obtain the improvement of integrability of very weak solutions of quasilinear subelliptic equations on Carnot group.

Key words: Subelliptic operators, Very weak solutions, Carnot-Caratheodorys spaces, Hodge decomposition, Reverse Holder inequality

中图分类号:

35J60

郑神州, 王喜芬. 拟线性次椭圆方程很弱解的正则性[J]. 数学物理学报, 2010, 30(2): 432-439.

ZHENG Shen-Zhou, WANG Xi-Fen. Regularity of Very Weak Solutions for a Class of Quasilinear Subelliptic Equations[J]. Acta mathematica scientia,Series A, 2010, 30(2): 432-439.

参考文献

[1] Anna Z. Very weak solutions of nonlinear subelliptic equations. Annales Academiae Scientiarum Fennicae Mathematica, 2005, 30: 407--436

[2] Garofalo N, Nhieu D M. Isoperimetric and Sobolev inequalities for Carnot-Carathéodory spaces and the
existence of minmal surfaces. Comm Pure Appl Math, 1996, 49: 1081--1144

[3] Gutiérrez C E, Lanconelli E. Maximum principle, non-homogeneous Harnack inequality, and Liouville theorems for X-elliptic operations. Comm Part Diff Eqs, 2003, 28(11/12): 1833--1862

[4] Iwaniec T. p-harmonic tensor and quasiregular mappings. Ann Math, 1992, 136(3): 589--624

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

[6] Iwaniec T, Lutoborski A. Integral estimates for null Lagrangians. Arch Rat Mech Anal, 1993, 125: 25--29

[7] Iwaniec T, Migliaccio L, Nania L, Sbordone C. Integrability and removability results for quasiregular mappings in high dimensions. Math Scand, 1994, 75(2): 263--279

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

[9] Carozza M, Antonia P Di N. On very weak solutions of a class of nonlinear elliptic systems. Comment Math Univ Carolinae, 2000, 41(3): 493--508

[10] Zheng S Z, Fang A N. Regularity of very weak solutions for a class of nonlinear elliptic systems. Acta Mathematica Sinica, 1998, 14: 733--740

[11] Zheng S Z. The L^p-integrability of Beltrami's equations with non-natural exponents. Acta Mathematica Scientia, 1998, 18: 13--20

[12] Zheng S Z. Regualrity results for the generalized Beltrami system. Acta Math Sin, 2004, 20(2): 193--204

[13] Zheng S Z. Removable singularities of solutions of A-harmonic type equation. Acta Math Appl Sin, 2004, 20(1): 115--122

[14] Hömander L. Hypoellptic second order differential equations. Acta Math, 1967, 119: 147--171

[15] Rothschild L P, Stein E M. Hypoelliptic differential operators and nilpotent groups. Acta Math, 1976, 138: 247--320

[16] Folland G B. Subelliptic estimates and function spaces on nilpotent Lie groups. Ark Mat, 1975, 13(2): 161--207

[17] Stein E M. Singular Integrals and Differentiability Properties of Functions. Princeton: Princeton Univ Press, 1970

[18] Di Fazio G, Zamboni P. Hölder continuity for quasilinear subelliptic equations in Carnot Carathéodory spaces.
Math Nachr, 2004, 272: 3--10