Hölder continuity for sub-elliptic systems under the sub-quadratic controllable growth in Carnot groups
Rendiconti del Seminario Matematico della Università di Padova, Volume 130  (2013), p. 169-202
@article{RSMUP_2013__130__169_0,
author = {Wang, Jialin and Liao, Dongni and Yu, Zefeng},
title = {H\"older continuity for sub-elliptic systems under the sub-quadratic controllable growth in Carnot groups},
journal = {Rendiconti del Seminario Matematico della Universit\a di Padova},
publisher = {Seminario Matematico of the University of Padua},
volume = {130},
year = {2013},
pages = {169-202},
mrnumber = {3148637},
language = {en},
url = {http://www.numdam.org/item/RSMUP_2013__130__169_0}
}

Wang, Jialin; Liao, Dongni; 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, Volume 130 (2013) , pp. 169-202. http://www.numdam.org/item/RSMUP_2013__130__169_0/`

[1] E. De Giorgi, Un esempio di estremali discontinue per un problema variazionale di tipo ellitico. Boll. Unione Mat. Italiana 4 (1968), pp. 135-137. | MR 227827

[2] M. Giaquinta, Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems. Princeton Univ. Press, Princeton, 1983. | MR 717034

[3] M. Giaquinta, Introduction to regularity theory for nonlinear elliptic systems. Birkhäuser, Berlin, 1993. | MR 1239172

[4] Y. Chen - L. Wu, Second order elliptic equations and elliptic systems. Science Press, Beijing, 2003.

[5] M. Giaquinta - G. Modica, Almost-everywhere regularity results for solutions of non linear elliptic systems. Manuscripta Math. 28 (1979), pp. 109-158, | MR 535699

[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. | MR 235264

[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. | MR 1802484

[8] L. Simon, Lectures on Geometric Measure Theory. Australian National University Press, Canberra, 1983. | MR 756417

[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. | MR 2177809

[10] F. Duzaar - G. Mingione, The $p$ -harmonic approximation and the regularity of $p$ -harmonic maps. Calc. Var. Partial Differential Equations 20 (2004), pp. 235-256. | MR 2062943

[11] F. Duzaar - G. Mingione, Regularity for degenerate elliptic problems via $p$ -harmonic approximation. Ann. Inst. H. Poincaré Anal. Non Linèaire 21 (2004), pp. 735-766. | MR 2086757

[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. | MR 1748219

[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. | MR 2340302

[14] S. Chen - Z. Tan, Optimal interior partial regularity for nonlinear elliptic systems. Discrete Contin. Dyn. Syst. 27 (2010), pp. 981-993. | MR 2629569

[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. Society 5 (2003), pp. 1-40. | MR 1961133

[16] E. Shores, Hypoellipticity for linear degenerate elliptic systems in Carnot groups and applications, arXiv:math/0502569, pp. 27.

[17] A. Föglein, Partial regularity results for subelliptic systems in the Heisenberg group, Calc. Var. Partial Differential Equations 32 (2008), pp. 25-51. | MR 2377405

[18] J. Wang - P. Niu, Optimal partial regularity for weak solutions of nonlinear sub-elliptic systems in Carnot groups. Nonlinear Analysis 72 (2010), pp. 4162-4187. | MR 2606775

[19] L. Capogna - D. Danielli - N. Garofalo, An embedding theorem and the Harnak inequality for nonlinear subelliptic equations. Comm. Partial Differential Equations 18 (1993), pp. 1765-1794. | MR 1239930

[20] G. Lu, Embedding theorems into Lipschitz and BMO spaces and applications to quasilinear subelliptic differential equations. Publ. Mat. 40 (1996), pp. 301-329. | MR 1425620

[21] C. Xu, Regularity for quasi-linear second order subelliptic equations. Comm. Pure Appl. Math. 45 (1992), pp. 77-96.

[22] L. Capogna, Regularity of quasi-linear equations in the Heisenberg group. Comm. Pure Appl. Math. 50 (1997), pp. 867-889. | MR 1459590

[23] L. Capogna, Regularity for quasilinear equation and 1-quasiconformal maps in Carnot groups. Math. Ann. 313 (1999), pp. 263-295. | MR 1679786

[24] S. Marchi, ${C}^{1,\alpha }$ local regularity for the solutions of the $p$ -Laplacian on the Heisenberg group for $2p+\sqrt{5}$ . Z. Anal. Anwendungen 20 (2001), pp. 617-636. | MR 1863937

[25] S. Marchi, ${C}^{1,\alpha }$ local regularity for the solutions of the $p$ -Laplacian on the Heisenberg group for $1+\frac{1}{\sqrt{5}}p\le 2$ . Comment. Math. Univ. Carolinae 44 (2003), pp. 33-56. | MR 2045844

[26] A. Domokos, Differentiability of solutions for the non-degenerate $p$ -Laplacian in the Heisenberg group. J. Differential Equations. 204 (2004), pp. 439-470. | MR 2085543

[27] A. Domokos, On the regularity of $p$ -harmonic functions in the Heisenberg group. Ph. D. Thesis, University of Pittsburgh, 2004.

[28] J. Manfredi - G. Mingione, Regularity results for quasilinear elliptic equations in the Heisenberg group. Math. Ann. 339 (2007), pp. 485-544. | MR 2336058

[29] G. Mingione - A. Zatorska-Goldstein - X. Zhong, Gradient regularity for elliptic equations in the Heisenberg group. Advances in Mathematics 222 (2009), pp. 62-129. | MR 2531368

[30] N. Garofalo, Gradient bounds for the horizontal $p$ -Laplacian on a Carnot group and some applications. Manuscripta Math. 130 (2009), pp. 375-385. | MR 2545524

[31] G. Folland, Subelliptic estimates and function spaces on nilpotent Lie groups. Ark. Mat. 13 (1975), pp. 161-207. | MR 494315

[32] E. Acerbi - N. Fusco, Regularity for minimizers of nonquadratic functionals: the case $1p$ , J. Math. Anal. Appl. 140 (1989), pp. 115-135. | MR 997847

[33] G. Lu, Embedding theorems on Campanato-Morrey space for vector fields on Hörmander type. Approx. Theory Appl. 14 (1998), pp. 69-80. | MR 1651473