Hölder continuity for sub-elliptic systems under the sub-quadratic controllable growth in Carnot groups

Jialin Wang; Dongni Liao; Zefeng Yu

Rendiconti del Seminario Matematico della Università di Padova (2013)

  • Volume: 130, page 169-202
  • ISSN: 0041-8994

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Wang, Jialin, Liao, Dongni, and Yu, Zefeng. "Hölder continuity for sub-elliptic systems under the sub-quadratic controllable growth in Carnot groups." Rendiconti del Seminario Matematico della Università di Padova 130 (2013): 169-202. <http://eudml.org/doc/275137>.

@article{Wang2013,
author = {Wang, Jialin, Liao, Dongni, Yu, Zefeng},
journal = {Rendiconti del Seminario Matematico della Università di Padova},
keywords = {optimal partial regularity; -harmonic approximation technique},
language = {eng},
pages = {169-202},
publisher = {Seminario Matematico of the University of Padua},
title = {Hölder continuity for sub-elliptic systems under the sub-quadratic controllable growth in Carnot groups},
url = {http://eudml.org/doc/275137},
volume = {130},
year = {2013},
}

TY - JOUR
AU - Wang, Jialin
AU - Liao, Dongni
AU - Yu, Zefeng
TI - Hölder continuity for sub-elliptic systems under the sub-quadratic controllable growth in Carnot groups
JO - Rendiconti del Seminario Matematico della Università di Padova
PY - 2013
PB - Seminario Matematico of the University of Padua
VL - 130
SP - 169
EP - 202
LA - eng
KW - optimal partial regularity; -harmonic approximation technique
UR - http://eudml.org/doc/275137
ER -

References

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  1. [1] E. De Giorgi, Un esempio di estremali discontinue per un problema variazionale di tipo ellitico. Boll. Unione Mat. Italiana4 (1968), pp. 135–137. Zbl0155.17603MR227827
  2. [2] M. Giaquinta, Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems. Princeton Univ. Press, Princeton, 1983. Zbl0516.49003MR717034
  3. [3] M. Giaquinta, Introduction to regularity theory for nonlinear elliptic systems. Birkhäuser, Berlin, 1993. MR1239172
  4. [4] Y. Chen - L. Wu, Second order elliptic equations and elliptic systems. Science Press, Beijing, 2003. 
  5. [5] M. Giaquinta - G. Modica, Almost-everywhere regularity results for solutions of non linear elliptic systems. Manuscripta Math.28 (1979), pp. 109–158, Zbl0411.35018MR535699
  6. [6] E. Giusti - M. Miranda, Sulla regolarità delle soluzioni deboli di una classe di sistemi ellittici quasilineari. Arch. Rat. Mech. Anal.31 (1968), pp. 173–184. Zbl0167.10703MR235264
  7. [7] F. Duzaar - J. F. Grotowski, Partial regularity for nonlinear elliptic systems: The method of A-harmonic approximation, Manuscripta Math.103 (2000), pp. 267–298. Zbl0971.35025MR1802484
  8. [8] L. Simon, Lectures on Geometric Measure Theory. Australian National University Press, Canberra, 1983. Zbl0546.49019MR756417
  9. [9] F. Duzaar - J. F. Grotowski - M. Kronz, Regularity of almost minimizers of quasi-convex variational integrals with subquadratic growth. Annali Mat. Pura Appl. (4) 184 (2005), pp. 421–448. Zbl1223.49040MR2177809
  10. [10] F. Duzaar - G. Mingione, The p -harmonic approximation and the regularity of p -harmonic maps. Calc. Var. Partial Differential Equations20 (2004), pp. 235–256. Zbl1142.35433MR2062943
  11. [11] F. Duzaar - G. Mingione, Regularity for degenerate elliptic problems via p -harmonic approximation. Ann. Inst. H. Poincaré Anal. Non Linèaire21 (2004), pp. 735–766. Zbl1112.35078MR2086757
  12. [12] M. Carozza - N. Fusco - G. Mingione, Partial regularity of minimizers of quasiconvex integrals with subquadratic growth, Annali Mat. Pura Appl. (4) 175 (1998), pp. 141–164. Zbl0960.49025MR1748219
  13. [13] S. Chen - Z. Tan, The method of A-harmonic approximation and optimal interior partial regularity for nonlinear elliptic systems under the controllable growth condition. J. Math. Anal. Appl.335 (2007), pp. 20–42. Zbl05180310MR2340302
  14. [14] S. Chen - Z. Tan, Optimal interior partial regularity for nonlinear elliptic systems. Discrete Contin. Dyn. Syst.27 (2010), pp. 981–993. Zbl1191.35115MR2629569
  15. [15] L. Capogna - N. Garofalo, Regularity of minimizers of the calculus of variations in Carnot groups via hypoellipticity of systems of Hörmander type. J. European Math. Society5 (2003), pp. 1–40. Zbl1064.49026MR1961133
  16. [16] E. Shores, Hypoellipticity for linear degenerate elliptic systems in Carnot groups and applications, arXiv:math/0502569, pp. 27. 
  17. [17] A. Föglein, Partial regularity results for subelliptic systems in the Heisenberg group, Calc. Var. Partial Differential Equations32 (2008), pp. 25–51. Zbl1145.35059MR2377405
  18. [18] J. Wang - P. Niu, Optimal partial regularity for weak solutions of nonlinear sub-elliptic systems in Carnot groups. Nonlinear Analysis72 (2010), pp. 4162–4187. Zbl1191.35086MR2606775
  19. [19] L. Capogna - D. Danielli - N. Garofalo, An embedding theorem and the Harnak inequality for nonlinear subelliptic equations. Comm. Partial Differential Equations18 (1993), pp. 1765–1794. Zbl0802.35024MR1239930
  20. [20] G. Lu, Embedding theorems into Lipschitz and BMO spaces and applications to quasilinear subelliptic differential equations. Publ. Mat.40 (1996), pp. 301–329. Zbl0873.35006MR1425620
  21. [21] C. Xu, Regularity for quasi-linear second order subelliptic equations. Comm. Pure Appl. Math.45 (1992), pp. 77–96. Zbl0827.35023
  22. [22] L. Capogna, Regularity of quasi-linear equations in the Heisenberg group. Comm. Pure Appl. Math.50 (1997), pp. 867–889. Zbl0886.22006MR1459590
  23. [23] L. Capogna, Regularity for quasilinear equation and 1-quasiconformal maps in Carnot groups. Math. Ann.313 (1999), pp. 263–295. Zbl0927.35024MR1679786
  24. [24] S. Marchi, C 1 , α local regularity for the solutions of the p -Laplacian on the Heisenberg group for 2 p + 5 . Z. Anal. Anwendungen20 (2001), pp. 617–636. Zbl0988.35066MR1863937
  25. [25] S. Marchi, C 1 , α local regularity for the solutions of the p -Laplacian on the Heisenberg group for 1 + 1 5 p 2 . Comment. Math. Univ. Carolinae44 (2003), pp. 33–56. MR2045844
  26. [26] A. Domokos, Differentiability of solutions for the non-degenerate p -Laplacian in the Heisenberg group. J. Differential Equations.204 (2004), pp. 439–470. Zbl1065.35103MR2085543
  27. [27] A. Domokos, On the regularity of p -harmonic functions in the Heisenberg group. Ph. D. Thesis, University of Pittsburgh, 2004. Zbl1065.35103
  28. [28] J. Manfredi - G. Mingione, Regularity results for quasilinear elliptic equations in the Heisenberg group. Math. Ann.339 (2007), pp. 485–544. Zbl1128.35034MR2336058
  29. [29] G. Mingione - A. Zatorska-Goldstein - X. Zhong, Gradient regularity for elliptic equations in the Heisenberg group. Advances in Mathematics222 (2009), pp. 62–129. Zbl1175.35033MR2531368
  30. [30] N. Garofalo, Gradient bounds for the horizontal p -Laplacian on a Carnot group and some applications. Manuscripta Math.130 (2009), pp. 375–385. Zbl1179.35077MR2545524
  31. [31] G. Folland, Subelliptic estimates and function spaces on nilpotent Lie groups. Ark. Mat.13 (1975), pp. 161–207. Zbl0312.35026MR494315
  32. [32] E. Acerbi - N. Fusco, Regularity for minimizers of nonquadratic functionals: the case 1 p , J. Math. Anal. Appl.140 (1989), pp. 115–135. Zbl0686.49004MR997847
  33. [33] G. Lu, Embedding theorems on Campanato-Morrey space for vector fields on Hörmander type. Approx. Theory Appl.14 (1998), pp. 69–80. Zbl0916.46026MR1651473

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