次椭圆拟线性方程在外区域上的Liouville定理

数学物理学报 ›› 2012, Vol. 32 ›› Issue (4): 644-653.

次椭圆拟线性方程在外区域上的Liouville定理

郑神州|卢寒芳

北京交通大学理学院北京 100044

收稿日期:2010-10-08 修回日期:2012-02-06 出版日期:2012-08-25 发布日期:2012-08-25
基金资助:
国家自然科学基金(11071012)资助

Liouville Theorems on Subelliptic Quasilinear Equations in Unbounded Exterior Domain

ZHENG Shen-Zhou, LU Han-Fang

School of Science, Beijing Jiaotong University, Beijing 100044

Received:2010-10-08 Revised:2012-02-06 Online:2012-08-25 Published:2012-08-25
Supported by:
国家自然科学基金(11071012)资助

摘要/Abstract

摘要：

研究一类次椭圆拟线性方程在具有Carnot-Caratheodory空间度量下的外区域的分析性质, 得到了在各种情况下无界外区域上的Liouville型定理结论.

关键词: Carnot-Caratheodory空间, 次椭圆, 外区域, Liouville定理

Abstract:

In this paper, the analytic properties is considered for a class of subelliptic quasilinear equations in unbounded exterior domian. we establish various Liuoville theorems in unbounded exterior domain under the different settings with n≥3.

Key words: Carnot-Caratheodory spaces, Sub-elliptic, Exterior domain, Liouville theorem

中图分类号:

35C15

郑神州, 卢寒芳. 次椭圆拟线性方程在外区域上的Liouville定理[J]. 数学物理学报, 2012, 32(4): 644-653.

ZHENG Shen-Zhou, LU Han-Fang. Liouville Theorems on Subelliptic Quasilinear Equations in Unbounded Exterior Domain[J]. Acta mathematica scientia,Series A, 2012, 32(4): 644-653.

参考文献

[1]} H\"omander L. Hypoellptic second order differential equations. Acta Math, 1967, 119: 147--171

[2] Folland G. Subelliptic estimates and function spaces on nilpotent Lie groups. Arkiv Mat, 1975, 13: 161--207

[3] Capogna L, Danielli D, Garofalo N. An embedding theorem and the Harnack inequality for nonlinear subelliptic equations. Comm Part Diff Eqs, 1993, 18: 1765--1794

[4] Capogna L, Danielli D, Garofalo N. The geometric Sobolev embedding for vector fields and the isoperimetric inequality. Comm
Anal Geom, 1994, 2: 203--215

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

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

[7] Mazzoni G. Green function for X-elliptic operators. Manuscripta Math, 2004, 115: 207--238

[8] Xu C J, Zuily C. Higher interior regularity for quasilinear subelliptic systems. Calc Var, 1997, 5: 323--343

[9] Trudinger N S, Wang X J. On the weak continuity of elliptic operators and applications to potential theory. Amer J Math, 2002, 124: 369--410

[10] Capogna L. Regularity for quasilinear equations and 1-quasiconformal maps in Carnot groups. Math Ann, 1999, 313: 263--295

[11] Di Fazio G, Zamboni P. H\"older continuity for quasilinear subelliptic equations in Carnot Carathéodory spaces. Math Nachr, 2004, 272: 3--10

[12] Damascelli L, Sciunzi B. Harnack inequalities, maximum and comparison principles and regularity of positive solutions of m-Laplase equations. Calc Vari, 2005, 25(2): 139--159

[13] Manfredi J J, Mingione G. Regularity results for quasilinear elliptic equations in the Heisenberg group. Math Ann, 2007, 339: 485--544

[14] Zheng S Z, Feng Z S. Green functions for a class of nonlinear degenerate operators with X-ellipticity. To appear in Transactions of AMS,
2011, 38Pages

[15] Bellaiche A, Risler J J. Sub-Riemannian Geometry. Boston: Birkhäser Verlag, 1996

[16] Bidaut-Veron M -F, Pohozaev S. Nonexistence results and estimates for some nonlinear elliptic problems. J Anal Math, 2001, 84: 1--49

[17] Serrin J, Zou H H. Cauchy-Liouville and universal boundedness theorems for quasilinear elliptic equations and inequalities. Acta Math, 2002, 189: 79--142

[18] Heinonen J, Kilpel\"ainen T, Martio O. Nonlinear potential theory of degenerate elliptic equations. Oxford: Clarendon Press, 1993

[19] Astarita G, Marrucci G. Principles of Non-Newtonian Fluid Mechanics. New York: McGrawHill, 1974

[20] Nagel A, Stein E, Wainger S. Balls and metrics defined by vector fields I: basic properties. Acta Math, 1985, 155: 103--147

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