On the 𝐶^{1,𝛼} regularity of 𝑝-harmonic functions in the Heisenberg group

Ricciotti, Diego

doi:10.1090/proc/13961

On the $C^{1,\alpha}$ regularity of -harmonic functions in the Heisenberg group

Author: Diego Ricciotti
Journal: Proc. Amer. Math. Soc. 146 (2018), 2937-2952
MSC (2010): Primary 35H20, 35J70
DOI: https://doi.org/10.1090/proc/13961
Published electronically: February 8, 2018
MathSciNet review: 3787355
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We present a proof of the local Hölder regularity of the horizontal derivatives of weak solutions to the -Laplace equation in the Heisenberg group $\mathbb{H}^1$ for .

[1] A. Bonfiglioli, E. Lanconelli, and F. Uguzzoni, Stratified Lie groups and potential theory for their sub-Laplacians, Springer Monographs in Mathematics, Springer, Berlin, 2007. MR 2363343
[2] Luca Capogna, Regularity of quasi-linear equations in the Heisenberg group, Comm. Pure Appl. Math. 50 (1997), no. 9, 867–889. MR 1459590, https://doi.org/10.1002/(SICI)1097-0312(199709)50:9<867::AID-CPA3>3.0.CO;2-3
[3] Luca Capogna and Nicola Garofalo, Regularity of minimizers of the calculus of variations in Carnot groups via hypoellipticity of systems of Hörmander type, J. Eur. Math. Soc. (JEMS) 5 (2003), no. 1, 1–40. MR 1961133, https://doi.org/10.1007/s100970200043
[4] L. Capogna, G. Citti, E. Le Donne, and A. Ottazzi,
Conformality and Q-harmonicity in sub-Riemannian manifolds,
preprint, 2017.
[5] Ennio De Giorgi, Sulla differenziabilità e l’analiticità delle estremali degli integrali multipli regolari, Mem. Accad. Sci. Torino. Cl. Sci. Fis. Mat. Nat. (3) 3 (1957), 25–43 (Italian). MR 0093649
[6] E. DiBenedetto, 𝐶^{1+𝛼} local regularity of weak solutions of degenerate elliptic equations, Nonlinear Anal. 7 (1983), no. 8, 827–850. MR 709038, https://doi.org/10.1016/0362-546X(83)90061-5
[7] András Domokos, Differentiability of solutions for the non-degenerate 𝑝-Laplacian in the Heisenberg group, J. Differential Equations 204 (2004), no. 2, 439–470. MR 2085543, https://doi.org/10.1016/j.jde.2004.05.009
[8] András Domokos and Juan J. Manfredi, 𝐶^{1,𝛼}-regularity for 𝑝-harmonic functions in the Heisenberg group for 𝑝 near 2, The 𝑝-harmonic equation and recent advances in analysis, Contemp. Math., vol. 370, Amer. Math. Soc., Providence, RI, 2005, pp. 17–23. MR 2126699, https://doi.org/10.1090/conm/370/06827
[9] András Domokos and Juan J. Manfredi, Subelliptic Cordes estimates, Proc. Amer. Math. Soc. 133 (2005), no. 4, 1047–1056. MR 2117205, https://doi.org/10.1090/S0002-9939-04-07819-0
[10] Lawrence C. Evans, A new proof of local 𝐶^{1,𝛼} regularity for solutions of certain degenerate elliptic p.d.e, J. Differential Equations 45 (1982), no. 3, 356–373. MR 672713, https://doi.org/10.1016/0022-0396(82)90033-X
[11] B. Franchi, G. Lu, and R. L. Wheeden, Representation formulas and weighted Poincaré inequalities for Hörmander vector fields, Ann. Inst. Fourier (Grenoble) 45 (1995), no. 2, 577–604 (English, with English and French summaries). MR 1343563
[12] Enrico Giusti, Direct methods in the calculus of variations, World Scientific Publishing Co., Inc., River Edge, NJ, 2003. MR 1962933
[13] Piotr Hajłasz and Pekka Koskela, Sobolev met Poincaré, Mem. Amer. Math. Soc. 145 (2000), no. 688, x+101. MR 1683160, https://doi.org/10.1090/memo/0688
[14] Lars Hörmander, Hypoelliptic second order differential equations, Acta Math. 119 (1967), 147–171. MR 222474, https://doi.org/10.1007/BF02392081
[15] Juha Kinnunen, Niko Marola, Michele Miranda Jr., and Fabio Paronetto, Harnack’s inequality for parabolic De Giorgi classes in metric spaces, Adv. Differential Equations 17 (2012), no. 9-10, 801–832. MR 2985675
[16] Olga A. Ladyzhenskaya and Nina N. Ural’tseva, Linear and quasilinear elliptic equations, Translated from the Russian by Scripta Technica, Inc. Translation editor: Leon Ehrenpreis, Academic Press, New York-London, 1968. MR 0244627
[17] John L. Lewis, Regularity of the derivatives of solutions to certain degenerate elliptic equations, Indiana Univ. Math. J. 32 (1983), no. 6, 849–858. MR 721568, https://doi.org/10.1512/iumj.1983.32.32058
[18] Guozhen Lu, Embedding theorems into Lipschitz and BMO spaces and applications to quasilinear subelliptic differential equations, Publ. Mat. 40 (1996), no. 2, 301–329. MR 1425620, https://doi.org/10.5565/PUBLMAT_40296_04
[19] Juan Jose Manfredi, REGULARITY OF THE GRADIENT FOR A CLASS OF NONLINEAR POSSIBLY DEGENERATE ELLIPTIC EQUATIONS, ProQuest LLC, Ann Arbor, MI, 1986. Thesis (Ph.D.)–Washington University in St. Louis. MR 2635642
[20] Juan J. Manfredi and Giuseppe Mingione, Regularity results for quasilinear elliptic equations in the Heisenberg group, Math. Ann. 339 (2007), no. 3, 485–544. MR 2336058, https://doi.org/10.1007/s00208-007-0121-3
[21] S. Marchi, 𝐶^{1,𝛼} local regularity for the solutions of the 𝑝-Laplacian on the Heisenberg group for 2≤𝑝<1+√5, Z. Anal. Anwendungen 20 (2001), no. 3, 617–636. MR 1863937, https://doi.org/10.4171/ZAA/1035
S. Marchi, Erratum to: “𝐶^{1,𝛼} local regularity for the solutions of the 𝑝-Laplacian on the Heisenberg group for 2≤𝑝<1+√5” [Z. Anal. Anwendungen 20 (2001), no. 3, 617–636; MR1863937 (2002i:35037)], Z. Anal. Anwendungen 22 (2003), no. 2, 471–472. MR 2000279, https://doi.org/10.4171/ZAA/1157
[22] Silvana Marchi, 𝐶^{1,𝛼} local regularity for the solutions of the 𝑝-Laplacian on the Heisenberg group. The case 1+\frac1√5<𝑝≤2, Comment. Math. Univ. Carolin. 44 (2003), no. 1, 33–56. MR 2045844
[23] Giuseppe Mingione, Anna Zatorska-Goldstein, and Xiao Zhong, Gradient regularity for elliptic equations in the Heisenberg group, Adv. Math. 222 (2009), no. 1, 62–129. MR 2531368, https://doi.org/10.1016/j.aim.2009.03.016
[24] Diego Ricciotti, 𝑝-Laplace equation in the Heisenberg group, SpringerBriefs in Mathematics, Springer, [Cham]; BCAM Basque Center for Applied Mathematics, Bilbao, 2015. Regularity of solutions; BCAM SpringerBriefs. MR 3444525
[25] N. N. Ural′ceva, Degenerate quasilinear elliptic systems, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 7 (1968), 184–222 (Russian). MR 0244628
[26] X. Zhong,
Regularity for variational problems in the Heisenberg group,
preprint, 2017.
[27] S. Mukherjee and X. Zhong, $C^{1,\alpha }$ -regularity for variational problems in the Heisenberg group, preprint, 2017.