# American Institute of Mathematical Sciences

August  2013, 33(8): 3391-3405. doi: 10.3934/dcds.2013.33.3391

## Optimal partial regularity results for nonlinear elliptic systems in Carnot groups

 1 Department of Mathematics and Information Science, Zhangzhou Normal University, Zhangzhou 363000, Fujian, China 2 School of Mathematical Science, Xiamen University, Xiamen 361005, Fujian

Received  May 2012 Revised  July 2012 Published  January 2013

In this paper, we consider partial regularity for weak solutions of second-order nonlinear elliptic systems in Carnot groups. By the method of A-harmonic approximation, we establish optimal interior partial regularity of weak solutions to systems under controllable growth conditions with sub-quadratic growth in Carnot groups.
Citation: Shuhong Chen, Zhong Tan. Optimal partial regularity results for nonlinear elliptic systems in Carnot groups. Discrete & Continuous Dynamical Systems - A, 2013, 33 (8) : 3391-3405. doi: 10.3934/dcds.2013.33.3391
##### References:
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##### References:
 [1] E.De Giorgi, "Frontiere orientate di misura minima,", Seminaro Math. Scuola Norm. Sup. Pisa, (1960). Google Scholar [2] E.De Giorgi, Un esempio di estremali discontinue per un problema variazionale di tipo ellittico,, Boll.Un. Mat. Ital., 4 (1968), 135. Google Scholar [3] M. Giaquinta and G. Modica, Regularity results for some classes of higher order non linear elliptic systems,, J.Reine Angew. Math., 311-312 (1979), 311. Google Scholar [4] P.A. Ivert, Regularit'dtsuntersuchungen von Lösungen ellipti-scher Sy steme von quasilinearen Differen-tialgleichungen zweiter Ordnung,, Manus. Math., 30 (1979), 53. doi: 10.1007/BF01305990. Google Scholar [5] C. Hamburger, Partial boundary regularity of solutions of nonlinear superelliptic systems,, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat., 8 (2007), 63. Google Scholar [6] L. Beck, Partial regularity for weak solutions of nonlinear elliptic systems: the subquadratic case,, Manus. Math., 123 (2007), 453. doi: 10.1007/s00229-007-0100-8. Google Scholar [7] L.Simon, "Lectures on Geometric Measure Theory,", Canberra: Australian National University Press, (1983). Google Scholar [8] W.K. Allard, On the first variation of a varifold,, Annals of Math., 95 (1972), 417. Google Scholar [9] L.Simon, "Theorems on Regularity and Singularity of Energy Minimizing Maps,", Basel, (). doi: 10.1007/978-3-0348-9193-6. Google Scholar [10] R.Schoen and K.Uhlenbeck, A regularity theorem for harmonic maps,, J.Diff.Geom., 17 (1982), 307. Google Scholar [11] F.Duzaar and K.Steffen, Optimal interior and boundary regularity for almost minimizers to elliptic variational integrals,, J.Reine Angew. Math., 546 (2002), 73. doi: 10.1515/crll.2002.046. Google Scholar [12] F.Duzaar and J.F.Grotowski, Optimal interior partial regularity for nonlinear elliptic systems: The method of A-harmonic approximation,, Manus. Math., 103 (2000), 267. doi: 10.1007/s002290070007. Google Scholar [13] S. Chen and Z. Tan, The method of A-harmonic approximation and optimal interior interior partial regularity for nonlinear elliptic systems under the controllable growth condition,, J. Math. Anal. Appl., 335 (2007), 20. doi: 10.1016/j.jmaa.2007.01.042. Google Scholar [14] S. Chen and Z. Tan, Optimal interior partial regularity for nonlinear elliptic systems,, Discrete Cont Dyn-A, 27 (2010), 981. doi: 10.3934/dcds.2010.27.981. Google Scholar [15] F. Duzaar, J.F. Grotowski and M. Kronz, Regularity of almost minimizers of quasi-convex variational integrals with subquadratic growth,, Ann. Mat. Pura Appl., 184 (2005), 421. doi: 10.1007/s10231-004-0117-5. Google Scholar [16] F. Duzaar and G. Mingione, Regularity for degenerate elliptic problems via $p$-harmonic approximation,, Ann. Inst. Henri Poincaré, 21 (2004), 735. doi: 10.1016/j.anihpc.2003.09.003. Google Scholar [17] L. Capogna, Regularity for quasilinear equation and 1-quasiconformal maps in Carnot groups,, Math. Ann., 313 (1999), 263. doi: 10.1007/s002080050261. Google Scholar [18] L. Capogna and N.Garofalo, Regularity of minimizers of the calculus of variations in Carnot groups via hypoellipticity of systems of Hörmander type,, J. Eur. Math. Soc., 5 (2003), 1. doi: 10.1007/s100970200043. Google Scholar [19] E. Shores, Hypoellipticity for linear degenerate elliptic systems in Carnot groups and applications,, p27. arXiv: math. AP/ 0502569., (0502). Google Scholar [20] A. Föglein, Partial regularity results for subelliptic systems in the Heisenberg group,, Cacl Var Partial Dif., 32 (2008), 25. doi: 10.1007/s00526-007-0127-4. Google Scholar [21] J. Wang and P. Niu, Optimal Partial regularity for weak solutions of nonlinear sub-elliptic systems in Carnot groups,, Nonlinear Anal., 72 (2010), 4162. doi: 10.1016/j.na.2010.01.048. Google Scholar [22] G. Folland, Subelliptic estimates and function spaces on nilpotent Lie groups,, Ark. Mat., 13 (1975), 161. Google Scholar [23] D. Jerison, The poincaré inequality for vector fields satisfying Hörmander's condition,, Duke Math. J., 53 (1986), 503. doi: 10.1215/S0012-7094-86-05329-9. Google Scholar [24] N. Garofalo and D. Nhieu, Isoperimetric and Sobolev inequalities for Carnot-Carathéodory spaces and the existence of minimal surfaces,, Comm. Pure Appl. Math., 49 (1996), 1081. doi: 10.1002/(SICI)1097-0312(199610)49:10<1081::AID-CPA3>3.0.CO;2-A. Google Scholar [25] M. Carozza, N. Fusco and G. Mingione, Partial regularity of minimizers of quasiconvex integrals with subquadratic growth,, Ann. Mat. Pura Appl. IV. Ser., 175 (1998), 141. doi: 10.1007/BF01783679. Google Scholar [26] E. Acerbi and N. Fusco, Regularity for minimizers of non-quadratic functionals: the case $1 < p < 2$,, J. Math. Anal. Appl., 140 (1989), 115. doi: 10.1016/0022-247X(89)90098-X. Google Scholar
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