Journal of University of Chinese Academy of Sciences >
Interior Morrey estimates for divergence degenerate elliptic systems in Carnot groups
Received date: 2012-12-05
Revised date: 2013-03-29
Online published: 2014-01-15
Supported by
Supported by National Natural Science Foundation of China(11271299, 11001221) and Northewstern Polytechnical University jichu yanjiu jijin tansuo xiangmu(JC201124)
Let X1,…,Xq be the basis of the space of horizontal vector fields on a homogeneous Carnot group G =(R n,o) (q<n). The interior Morrey estimates for weak solutions to the following divergence degenerate elliptic system
10???
are obtained when the coefficients aαβij satisfy the strong Legendre condition in Ω⊂ G and belong to the space VMOloc(Ω).
Key words： Carnot group; subelliptic systems; Morrey estimates
ZHU Maochun , FENG Xiaojing . Interior Morrey estimates for divergence degenerate elliptic systems in Carnot groups[J]. Journal of University of Chinese Academy of Sciences, 2014 , 31(1) : 10 -16 . DOI: 10.7523/j.issn.2095-6134.2014.01.003
[1] Capogna L, Garofalo N. Regularity of minimizers of the calculus of variations in Carnot groups via hypoellipticity of systems of Hormander type[J]. J European Math Soc, 2003, 5: 1-40.
[2] Shores E. Hypoellipticity for linear degenerate elliptic systems in Carnot groups and applications[R]. Fayetteville:University of Arkansas, arXiv: mathAP/050256, 2005.
[3] Wang J, Niu P. Optimal partial regularity for weak solutions of nonlinear sub-elliptic systems in Carnot groups[J]. Nonlinear Analysis, 2010, 72: 4162-4187.
[4] Wang J, Liao D. Optimal partial regularity for sub-elliptic systems with sub-quadratic growth in Carnot groups[J]. Nonlinear Analysis, 2012, 75:2499-2519.
[5] Zhu M, Bramanti M, Niu P. Interior HW1, p estimates for divergence degenerate elliptic systems in Carnot groups[J]. J Math Anal Appl, 2013, 399: 442-458.
[6] Wang L. A geometric approach to the Calderon-Zygmund estimates[J]. Acta Mathematica Sinica, 2003, 19: 381-396.
[7] Byun S, Wang L. Elliptic equations with BMO coefficients in Reifenberg domains[J]. Comm Pure and Appl Math, 2004, 57: 1283-1310.
[8] Byun S, Wang L. Gradient estimates for elliptic systems in non-smooth domains[J]. Math Ann, 2008, 241: 629-650.
[9] Mengesha T, Phuc N C. Weighted and regularity estimates for nonlinear equations on Reifenberg flat domains[J]. J Diff Eqs, 2011, 250: 2485-2507.
[10] Folland G B. Subelliptic estimates and function spaces on nilpotent Lie groups[J]. Ark Mat, 1975, 13: 161-207.
[11] Bonfiglioli A, Lanconelli E, Uguzzoni F. Stratified Lie groups and potential theory for their sub-Laplacians[M]. Springer Monographs in Mathematics. Berlin:Springer, 2007.
[12] Stein E M. Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals[M]. Princeton Mathematical Series, 43. Monographs in Harmonic Analysis, Ⅲ. Princeton, NJ: Princeton University Press, 1993.
[13] Bramanti M, Brandolini L. Lp-estimates for nonvariational hypoelliptic operators with VMO coefficients[J]. Trans Amer Math Soc, 2000, 352: 781-822.
[14] Bramanti M, Brandolini L. Estimates of BMO type for singular integrals on spaces of homogeneous type and applications to hypoelliptic PDES[J]. Revista Matematica Iberoamericana, 2005, 21: 511-556.
[15] Coifman R, Weiss G. Analyse harmonique non-commutative sur certains espaces homogènes[M]. Lecture Notes in Mathematics, 242. Springer-Verlag, Berlin-Heidelberg-New York, 1971.
[16] Maclas R, Segovia C. A well behaved quasi-distance for spaces of homogeneous type[J]. Trabajos de Matematica, Inst Argentino Mat, 1981, 32: 1-18.
[17] Grafakos L. Classical and modern Fourier analysis[M]. Upper Saddle River NJ: Pearson Education Inc/Prentice Hall, 2004.
/
| 〈 |
|
〉 |