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On the Regularity of p-Harmonic Functions in the Heisenberg Group

Giuseppe Mingione; Zatorska-Goldstein Anna; Xiao Zhong

Bollettino dell'Unione Matematica Italiana (2008)

  • Volume: 1, Issue: 1, page 243-253
  • ISSN: 0392-4041

Abstract

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We describe some recent results obtained in [29], where we prove regularity theorems for sub-elliptic equations in (horizontal) divergence form defined in the Heisenberg group, and exhibiting polynomial growth of order p. The main result tells that when p [ 2 , 4 ) solutions to possibly degenerate equations are locally Lipschitz continuous with respect to the intrinsic distance. In particular, such result applies to p-harmonic functions in the Heisenberg group. Explicit estimates are obtained, and eventually applied to obtain the suitable Calderón-Zygmund theory for the associated non-homogeneous problems.

How to cite

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Mingione, Giuseppe, Anna, Zatorska-Goldstein, and Zhong, Xiao. "On the Regularity of p-Harmonic Functions in the Heisenberg Group." Bollettino dell'Unione Matematica Italiana 1.1 (2008): 243-253. <http://eudml.org/doc/290476>.

@article{Mingione2008,
abstract = {We describe some recent results obtained in [29], where we prove regularity theorems for sub-elliptic equations in (horizontal) divergence form defined in the Heisenberg group, and exhibiting polynomial growth of order p. The main result tells that when $p \in [2,4)$ solutions to possibly degenerate equations are locally Lipschitz continuous with respect to the intrinsic distance. In particular, such result applies to p-harmonic functions in the Heisenberg group. Explicit estimates are obtained, and eventually applied to obtain the suitable Calderón-Zygmund theory for the associated non-homogeneous problems.},
author = {Mingione, Giuseppe, Anna, Zatorska-Goldstein, Zhong, Xiao},
journal = {Bollettino dell'Unione Matematica Italiana},
language = {eng},
month = {2},
number = {1},
pages = {243-253},
publisher = {Unione Matematica Italiana},
title = {On the Regularity of p-Harmonic Functions in the Heisenberg Group},
url = {http://eudml.org/doc/290476},
volume = {1},
year = {2008},
}

TY - JOUR
AU - Mingione, Giuseppe
AU - Anna, Zatorska-Goldstein
AU - Zhong, Xiao
TI - On the Regularity of p-Harmonic Functions in the Heisenberg Group
JO - Bollettino dell'Unione Matematica Italiana
DA - 2008/2//
PB - Unione Matematica Italiana
VL - 1
IS - 1
SP - 243
EP - 253
AB - We describe some recent results obtained in [29], where we prove regularity theorems for sub-elliptic equations in (horizontal) divergence form defined in the Heisenberg group, and exhibiting polynomial growth of order p. The main result tells that when $p \in [2,4)$ solutions to possibly degenerate equations are locally Lipschitz continuous with respect to the intrinsic distance. In particular, such result applies to p-harmonic functions in the Heisenberg group. Explicit estimates are obtained, and eventually applied to obtain the suitable Calderón-Zygmund theory for the associated non-homogeneous problems.
LA - eng
UR - http://eudml.org/doc/290476
ER -

References

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  1. ACERBI, E. - MINGIONE, G., Gradient estimates for the p ( x ) -Laplacean system, J. reine ang. Math. (Crelles)584 (2005), 117-148 Zbl1093.76003MR2155087DOI10.1515/crll.2005.2005.584.117
  2. BILDHAUER, M. - FUCHS, M. - ZHONG, X., A regularity theory for scalar local minimizers of splitting-type variational integrals, Ann. Scu. Norm. Sup. Pisa, Cl. Sci. (5), to appear Zbl1141.49032MR2370266
  3. BRAMANTI, M. - BRANDOLINI, L., L p estimates for nonvariational hypoelliptic operators with VMO coefficients, Trans. Amer. Math. Soc.352 (2000), 781-822. Zbl0935.35037MR1608289DOI10.1090/S0002-9947-99-02318-1
  4. CAFFARELLI, L. - PERAL, I., On W 1 , p ) estimates for elliptic equations in divergence form, Comm. Pure Appl. Math.51 (1998), 1-21. MR1486629DOI10.1002/(SICI)1097-0312(199801)51:1<1::AID-CPA1>3.3.CO;2-N
  5. CAPOGNA, L., Regularity of quasi-linear equations in the Heisenberg group, Comm. Pure Appl. Math.50 (1997), 867-889. Zbl0886.22006MR1459590DOI10.1002/(SICI)1097-0312(199709)50:9<867::AID-CPA3>3.0.CO;2-3
  6. CAPOGNA, L. - DANIELLI, D. - GAROFALO, N., An embedding theorem and the Harnack inequality for nonlinear subelliptic equations, Comm. P.D.E18 (1993), 1765-1794. Zbl0802.35024MR1239930DOI10.1080/03605309308820992
  7. CAPOGNA, L. - GAROFALO, N., Regularity of minimizers of the calculus of variations in Carnot groups via hypoellipticity of systems of Hormander type, J. Eur. Math. Soc. (JEMS) 5 (2003), 1-40. Zbl1064.49026MR1961133DOI10.1007/s100970200043
  8. CHANILLO, S. - MANFREDI, J. J., Sharp global bounds for the Hessian on pseudo-Hermitian manifolds Preprint 2006. Zbl1205.32027MR2603179DOI10.1007/978-0-8176-4588-5_8
  9. CHIARENZA, F. - FRASCA, M. - LONGO, P., W 2 , p -solvability of the Dirichlet problem for nondivergence elliptic equations with VMO coefficients, Trans. Amer. Math. Soc.336 (1993), 841-853. Zbl0818.35023MR1088476DOI10.2307/2154379
  10. DIBENEDETTO, E. - MANFREDI, J. J., On the higher integrability of the gradient of weak solutions of certain degenerate elliptic systems, Amer. J. Math.115 (1993), 1107-1134. Zbl0805.35037MR1246185DOI10.2307/2375066
  11. DOMOKOS, A., Differentiability of solutions for the non-degenerate p-Laplacian in the Heisenberg group, J. Differential Equations, 204 (2004), 439-470. Zbl1065.35103MR2085543DOI10.1016/j.jde.2004.05.009
  12. DOMOKOS, A. - MANFREDI, J. J., Subelliptic Cordes estimates, Proc. Amer. Math. Soc.133 (2005), 1047-1056. Zbl1081.35015MR2117205DOI10.1090/S0002-9939-04-07819-0
  13. DOMOKOS, A. - MANFREDI, J. J., C 1 , α -regularity for p -harmonic functions in the Heisenberg group for p near 2, Contemp. Math.370 (2005), 17-23. Zbl1073.22004MR2126699DOI10.1090/conm/370/06827
  14. ESPOSITO, L. - LEONETTI, F. - MINGIONE, G., Regularity for minimizers of irregular integrals with ( p , q ) -growth, Forum Mathematicum14 (2002), 245-272. Zbl0999.49022MR1880913DOI10.1515/form.2002.011
  15. ESPOSITO, L. - LEONETTI, F. - MINGIONE, G., Sharp regularity for functionals with ( p , q ) growth, J. Differential Equations204 (2004), 5-55. Zbl1072.49024MR2076158DOI10.1016/j.jde.2003.11.007
  16. FÖGLEIN, A., Partial regularity results for sub-elliptic systems in the Heisenberg group, Calc. Var. - PDE, to appear. MR2377405DOI10.1007/s00526-007-0127-4
  17. FOLLAND, G. B., Applications of analysis on nilpotent groups to partial differential equations, Bulletin of AMS83 (1977), 912-930. Zbl0371.35008MR457928DOI10.1090/S0002-9904-1977-14326-7
  18. FOLLAND, G. B., Subelliptic estimates and function spaces on nilpotent Lie groups, Ark. Mat.13 (1975), 161-207. Zbl0312.35026MR494315DOI10.1007/BF02386204
  19. HÖRMANDER, L., Hypoelliptic second order differential equations, Acta Mathematica119 (1967), 147-171. MR222474DOI10.1007/BF02392081
  20. IWANIEC, T., Projections onto gradient fields and L p -estimates for degenerated elliptic operators, Studia Math.75 (1983), 293-312. Zbl0552.35034MR722254DOI10.4064/sm-75-3-293-312
  21. IWANIEC, T. - SBORDONE, C., Weak minima of variational integrals. J. reine angew. math. (Crelle J.)454 (1994), 143-161. Zbl0802.35016MR1288682DOI10.1515/crll.1994.454.143
  22. KOHN, J. J., Pseudo-differential operators and hypoellipticity. Partial differential equations (Proc. Sympos. Pure Math., Vol. XXIII, Univ. California, Berkeley, Calif., 1971), 61-69. MR338592
  23. KRISTENSEN, J. - MINGIONE, G., The singular set of minima of integral functionals, Arch. Ration. Mech. Anal.180 (2006), 331-398. Zbl1116.49010MR2214961DOI10.1007/s00205-005-0402-5
  24. LADYZHENSKAYA, O. A. - URAL'TSEVA, N. N., Linear and quasilinear elliptic equations, Academic Press, New York-London1968. MR244627
  25. LU, G., Embedding theorems into Lipschitz and BMO spaces and applications to quasilinear subelliptic differential equations, Publ. Mat.40 (1996), 301-329 Zbl0873.35006MR1425620DOI10.5565/PUBLMAT_40296_04
  26. MANFREDI, J. J. - MINGIONE, G., Regularity results for quasilinear elliptic equations in the Heisenberg Group, Mathematische Annalen, 339 (2007) Zbl1128.35034MR2336058DOI10.1007/s00208-007-0121-3
  27. MARCELLINI, P., Everywhere regularity for a class of elliptic systems without growth conditions. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 23 (1996), 1-25. Zbl0922.35031MR1401415
  28. MARCHI, S., C 1 , α local regularity for the solutions of the p -Laplacian on the Heisenberg group for 2 p 5 , Z. Anal. Anwendungen, 20 (2001), 617-636. Erratum: Z. Anal. Anwendungen, 22 (2003), 471-472. Zbl0988.35066MR2000279DOI10.4171/ZAA/1157
  29. MINGIONE, G. - ZATORSKA-GOLDSTEIN, A. - ZHONG, X., Gradient regularity for elliptic equations in the Heisenberg Group. Arxiv Preprint, 2007. MR2531368DOI10.1016/j.aim.2009.03.016

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