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 and Continuous Dynamical Systems, 2013, 33 (8) : 3391-3405. doi: 10.3934/dcds.2013.33.3391
References:
[1]

E.De Giorgi, "Frontiere orientate di misura minima," Seminaro Math. Scuola Norm. Sup. Pisa, 61(1960), Editrice Tecnico Scientifica, Pisa 57(1961).

[2]

E.De Giorgi, Un esempio di estremali discontinue per un problema variazionale di tipo ellittico, Boll.Un. Mat. Ital., 4 (1968), 135-137.

[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), 145-169.

[4]

P.A. Ivert, Regularit'dtsuntersuchungen von Lösungen ellipti-scher Sy steme von quasilinearen Differen-tialgleichungen zweiter Ordnung, Manus. Math., 30 (1979), 53-88. doi: 10.1007/BF01305990.

[5]

C. Hamburger, Partial boundary regularity of solutions of nonlinear superelliptic systems, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat., 8 (2007), 63-81.

[6]

L. Beck, Partial regularity for weak solutions of nonlinear elliptic systems: the subquadratic case, Manus. Math., 123 (2007), 453-491. doi: 10.1007/s00229-007-0100-8.

[7]

L.Simon, "Lectures on Geometric Measure Theory," Canberra: Australian National University Press, 1983.

[8]

W.K. Allard, On the first variation of a varifold, Annals of Math., 95 (1972), 417-491.

[9]

L.Simon, "Theorems on Regularity and Singularity of Energy Minimizing Maps,", Basel, ().  doi: 10.1007/978-3-0348-9193-6.

[10]

R.Schoen and K.Uhlenbeck, A regularity theorem for harmonic maps, J.Diff.Geom., 17 (1982), 307-335.

[11]

F.Duzaar and K.Steffen, Optimal interior and boundary regularity for almost minimizers to elliptic variational integrals, J.Reine Angew. Math., 546 (2002), 73-138. doi: 10.1515/crll.2002.046.

[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), 267-298. doi: 10.1007/s002290070007.

[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), 20-42. doi: 10.1016/j.jmaa.2007.01.042.

[14]

S. Chen and Z. Tan, Optimal interior partial regularity for nonlinear elliptic systems, Discrete Cont Dyn-A, 27 (2010), 981-993. doi: 10.3934/dcds.2010.27.981.

[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), 421-448. doi: 10.1007/s10231-004-0117-5.

[16]

F. Duzaar and G. Mingione, Regularity for degenerate elliptic problems via $p$-harmonic approximation, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, 21 (2004), 735-766. doi: 10.1016/j.anihpc.2003.09.003.

[17]

L. Capogna, Regularity for quasilinear equation and 1-quasiconformal maps in Carnot groups, Math. Ann., 313 (1999), 263-295. doi: 10.1007/s002080050261.

[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), 1-40. doi: 10.1007/s100970200043.

[19]

E. Shores, Hypoellipticity for linear degenerate elliptic systems in Carnot groups and applications, p27. arXiv: math. AP/ 0502569.

[20]

A. Föglein, Partial regularity results for subelliptic systems in the Heisenberg group, Cacl Var Partial Dif., 32 (2008), 25-51. doi: 10.1007/s00526-007-0127-4.

[21]

J. Wang and P. Niu, Optimal Partial regularity for weak solutions of nonlinear sub-elliptic systems in Carnot groups, Nonlinear Anal., 72 (2010), 4162-4187. doi: 10.1016/j.na.2010.01.048.

[22]

G. Folland, Subelliptic estimates and function spaces on nilpotent Lie groups, Ark. Mat., 13 (1975), 161-207.

[23]

D. Jerison, The poincaré inequality for vector fields satisfying Hörmander's condition, Duke Math. J., 53 (1986), 503-523. doi: 10.1215/S0012-7094-86-05329-9.

[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), 1081-1144. doi: 10.1002/(SICI)1097-0312(199610)49:10<1081::AID-CPA3>3.0.CO;2-A.

[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), 141-164. doi: 10.1007/BF01783679.

[26]

E. Acerbi and N. Fusco, Regularity for minimizers of non-quadratic functionals: the case $1 < p < 2$, J. Math. Anal. Appl., 140 (1989), 115-135. doi: 10.1016/0022-247X(89)90098-X.

show all references

References:
[1]

E.De Giorgi, "Frontiere orientate di misura minima," Seminaro Math. Scuola Norm. Sup. Pisa, 61(1960), Editrice Tecnico Scientifica, Pisa 57(1961).

[2]

E.De Giorgi, Un esempio di estremali discontinue per un problema variazionale di tipo ellittico, Boll.Un. Mat. Ital., 4 (1968), 135-137.

[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), 145-169.

[4]

P.A. Ivert, Regularit'dtsuntersuchungen von Lösungen ellipti-scher Sy steme von quasilinearen Differen-tialgleichungen zweiter Ordnung, Manus. Math., 30 (1979), 53-88. doi: 10.1007/BF01305990.

[5]

C. Hamburger, Partial boundary regularity of solutions of nonlinear superelliptic systems, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat., 8 (2007), 63-81.

[6]

L. Beck, Partial regularity for weak solutions of nonlinear elliptic systems: the subquadratic case, Manus. Math., 123 (2007), 453-491. doi: 10.1007/s00229-007-0100-8.

[7]

L.Simon, "Lectures on Geometric Measure Theory," Canberra: Australian National University Press, 1983.

[8]

W.K. Allard, On the first variation of a varifold, Annals of Math., 95 (1972), 417-491.

[9]

L.Simon, "Theorems on Regularity and Singularity of Energy Minimizing Maps,", Basel, ().  doi: 10.1007/978-3-0348-9193-6.

[10]

R.Schoen and K.Uhlenbeck, A regularity theorem for harmonic maps, J.Diff.Geom., 17 (1982), 307-335.

[11]

F.Duzaar and K.Steffen, Optimal interior and boundary regularity for almost minimizers to elliptic variational integrals, J.Reine Angew. Math., 546 (2002), 73-138. doi: 10.1515/crll.2002.046.

[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), 267-298. doi: 10.1007/s002290070007.

[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), 20-42. doi: 10.1016/j.jmaa.2007.01.042.

[14]

S. Chen and Z. Tan, Optimal interior partial regularity for nonlinear elliptic systems, Discrete Cont Dyn-A, 27 (2010), 981-993. doi: 10.3934/dcds.2010.27.981.

[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), 421-448. doi: 10.1007/s10231-004-0117-5.

[16]

F. Duzaar and G. Mingione, Regularity for degenerate elliptic problems via $p$-harmonic approximation, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, 21 (2004), 735-766. doi: 10.1016/j.anihpc.2003.09.003.

[17]

L. Capogna, Regularity for quasilinear equation and 1-quasiconformal maps in Carnot groups, Math. Ann., 313 (1999), 263-295. doi: 10.1007/s002080050261.

[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), 1-40. doi: 10.1007/s100970200043.

[19]

E. Shores, Hypoellipticity for linear degenerate elliptic systems in Carnot groups and applications, p27. arXiv: math. AP/ 0502569.

[20]

A. Föglein, Partial regularity results for subelliptic systems in the Heisenberg group, Cacl Var Partial Dif., 32 (2008), 25-51. doi: 10.1007/s00526-007-0127-4.

[21]

J. Wang and P. Niu, Optimal Partial regularity for weak solutions of nonlinear sub-elliptic systems in Carnot groups, Nonlinear Anal., 72 (2010), 4162-4187. doi: 10.1016/j.na.2010.01.048.

[22]

G. Folland, Subelliptic estimates and function spaces on nilpotent Lie groups, Ark. Mat., 13 (1975), 161-207.

[23]

D. Jerison, The poincaré inequality for vector fields satisfying Hörmander's condition, Duke Math. J., 53 (1986), 503-523. doi: 10.1215/S0012-7094-86-05329-9.

[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), 1081-1144. doi: 10.1002/(SICI)1097-0312(199610)49:10<1081::AID-CPA3>3.0.CO;2-A.

[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), 141-164. doi: 10.1007/BF01783679.

[26]

E. Acerbi and N. Fusco, Regularity for minimizers of non-quadratic functionals: the case $1 < p < 2$, J. Math. Anal. Appl., 140 (1989), 115-135. doi: 10.1016/0022-247X(89)90098-X.

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