Skip to main content
SpringerLink
Log in

Regularity results for minimizers of (2, q)-growth functionals in the Heisenberg Group

  • Published:
manuscripta mathematica Aims and scope Submit manuscript

Abstract

We consider integral functionals in the Heisenberg group, whose convex C 2-integrand has quadratic growth from below, and growth of order q > 2 from above. We prove Hölder regularity for the full gradient of minimizers under the condition that q is less than an explicitly calculated dimension-dependent bound.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Acerbi E., Mingione G.: Regularity results for a class of functionals with non-standard growth. Arch. Ration. Mech. Anal. 156(2), 121–140 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  2. Acerbi E., Mingione G.: Regularity results for stationary electro-rheological fluids. Arch. Ration. Mech. Anal. 164(3), 213–259 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  3. Acerbi E., Mingione G.: Gradient estimates for the p(x)-Laplacian system. J. Reine Angew. Math. 584, 117–148 (2005)

    MATH  MathSciNet  Google Scholar 

  4. Capogna L.: Regularity of quasi-linear equations in the Heisenberg group. Commun. Pure Appl. Math. 50(9), 867–889 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  5. Capogna L.: Regularity for quasilinear equations and 1-quasiconformal maps in Carnot groups. Math. Ann. 313(2), 263–295 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  6. Capogna L., Danielli D., Garofalo N.: An embedding theorem and the Harnack inequality for nonlinear subelliptic equations. Commun. Partial Differ. Equ. 18(9–10), 1765–1794 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  7. Capogna L., Garofalo N.: Regularity of minimizers of the calculus of variations in Carnot groups via hypoellipticity of systems of Hörmander type. J. Eur. Math. Soc. 5(1), 1–40 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  8. Domokos A.: Differentiability of solutions for the non-degenerate p-Laplacian in the Heisenberg group. J. Differ. Equ. 204(2), 439–470 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  9. Domokos, A.: On the regularity of p-harmonic functions in the Heisenberg group. PhD thesis, University of Pittsburgh (2004)

  10. Domokos, A., Manfredi, J.J.: C 1,α-regularity for p-harmonic functions in the Heisenberg group for p near 2. In: Poggi-Corradini, P. (ed.), The p-Harmonic Equation and Recent Advances in Analysis. Proceedings of the 3rd prairie analysis seminar, Manhattan, KS, USA, October 17–18, 2003. Providence, RI: American Mathematical Society (AMS). Contemporary Mathematics 370, 17–23 (2005)

  11. Domokos A., Manfredi J.J.: Subelliptic Cordes estimates. Proc. Am. Math. Soc. 133(4), 1047–1056 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  12. Domokos A., Manfredi J.J.: Nonlinear subelliptic equations. Manuscr. Math. 130(2), 251–271 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  13. Esposito L., Leonetti F., Mingione G.: Higher integrability for minimizers of integral functionals with (p, q) growth. J. Differ. Equ. 157(2), 414–438 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  14. Esposito L., Leonetti F., Mingione G.: Regularity results for minimizers of irregular integrals with (p, q) growth. Forum Math. 14(2), 245–272 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  15. Esposito L., Leonetti F., Mingione G.: Sharp regularity for functionals with (p, q) growth. J. Differ. Equ. 204(1), 5–55 (2004)

    MATH  MathSciNet  Google Scholar 

  16. Föglein, A.: Regularität von Lösungen gewisser Systeme elliptischer partieller Differentialgleichungen in der Heisenberg-Gruppe. Diplomarbeit, Friedrich-Alexander-Universität Erlangen-Nürnberg (2005)

  17. Föglein A.: Partial regularity results for subelliptic systems in the Heisenberg group. Calc. Var. Partial Differ. Equ. 32(1), 25–51 (2008)

    Article  MATH  Google Scholar 

  18. Föglein, A.: Regularity results for minimizers of integrals with (2, q)-growth in the Heisenberg group. PhD thesis, Friedrich-Alexander-Universität Erlangen-Nürnberg (2009)

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

    Article  MATH  MathSciNet  Google Scholar 

  20. Garofalo N.: Gradient bounds for the horizontal p-Laplacian on a Carnot group and some applications. Manuscr. Math. 130(3), 375–385 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  21. Giaquinta M.: Growth conditions and regularity. A counterexample. Manuscr. Math. 59, 245–248 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  22. Giusti, E.: Direct Methods in the Calculus of Variations, vol. vii, 403 pp. World Scientific, Singapore (2003)

  23. Hajlasz P., Koskela P.: Sobolev met Poincaré. Mem. Am. Math. Soc. 688, 101 (2000)

    MathSciNet  Google Scholar 

  24. Hörmander L.: Hypoelliptic second order differential equations. Acta Math. 119, 147–171 (1967)

    Article  MATH  MathSciNet  Google Scholar 

  25. Jerison D.: The Poincaré inequality for vector fields satisfying Hörmander’s condition. Duke Math. J. 53, 503–523 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  26. Kohn J.: Pseudo-differential operators and hypoellipticity. Partial diff. Equ., Berkeley 1971, Proc. Sympos. Pure Math. 23, 61–69 (1973)

    MathSciNet  Google Scholar 

  27. Korányi, Á.: Geometric aspects of analysis on the Heisenberg group. In: Topics in modern harmonic analysis, Proc. Semin., Torino and Milano 1982, vol. I, 209–258 (1983)

  28. Leonetti F.: Higher differentiability for weak solutions of elliptic systems with nonstandard growth conditions. Ric. Mat. 42(1), 101–122 (1993)

    MATH  MathSciNet  Google Scholar 

  29. Leonetti F.: Higher integrability for minimizers of integral functionals with nonstandard growth. J. Differ. Equ. 112(2), 308–324 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  30. Lu G.: Weighted Poincaré and Sobolev inequalities for vector fields satisfying Hörmander’s condition and applications. Rev. Mat. Iberoam. 8(3), 367–439 (1992)

    MATH  Google Scholar 

  31. Lu G.: Embedding theorems into Lipschitz and BMO spaces and applications to quasilinear subelliptic differential equations. Publ. Mat. Barc. 40(2), 301–329 (1996)

    MATH  Google Scholar 

  32. Manfredi J.J.: Regularity for minima of functionals with p-growth. J. Differ. Equ. 76(2), 203–212 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  33. Manfredi J.J., Mingione G.: Regularity results for quasilinear elliptic equations in the Heisenberg group. Math. Ann. 339, 485–544 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  34. Marcellini P.: On the definition and the lower semicontinuity of certain quasiconvex integrals. Ann. Inst. H. Poincaré Anal. Non Linéaire 3(5), 391–409 (1986)

    MATH  MathSciNet  Google Scholar 

  35. Marcellini P.: Regularity of minimizers of integrals of the calculus of variations with non-standard growth conditions. Arch. Ration. Mech. Anal. 105(3), 267–284 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  36. Marcellini P.: Regularity and existence of solutions of elliptic equations with p,q-growth conditions. J. Differ. Equ. 90(1), 1–30 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  37. Marcellini P.: Regularity for elliptic equations with general growth conditions. J. Differ. Equ. 105(2), 296–333 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  38. Marcellini P.: Regularity for some scalar variational problems under general growth conditions. J. Optim. Theory Appl. 90(1), 161–181 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  39. Marchi S.: C 1,α local regularity for the solutions of the p-Laplacian on the Heisenberg group for \({2 \le p < 1 + \sqrt{5}}\) . Z. Anal. Anwend. 20(3), 617–636 (2001)

    MATH  MathSciNet  Google Scholar 

  40. Marchi S.: L p regularity of the derivatives in the second commutator’s direction for nonlinear elliptic equations on the Heisenberg group. Rend. Accad. Naz. Sci. XL Mem. Math. Appl. 5, 1–15 (2002)

    MathSciNet  Google Scholar 

  41. Mingione G.: Regularity of minima: an invitation to the dark side of the calculus of variations. Appl. Math. 51(4), 355–426 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  42. Mingione G., Zatorska-Goldstein A., Zhong X.: On the regularity of p-harmonic functions in the Heisenberg group. Adv. Math. 222, 62–129 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  43. Moscariello G.: Local boundedness of minimizers of certain degenerate functionals of the calculus of variations. Nonlinear Anal. Theory Methods Appl. 23(12), 1587–1593 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  44. Moscariello G., Nania L.: Hölder continuity of minimizers of functionals with non standard growth conditions. Ric. Mat. 40(2), 259–273 (1991)

    MATH  MathSciNet  Google Scholar 

  45. Schmidt T.: Regularity of relaxed minimizers of quasiconvex variational integrals with (p, q)-growth. Arch. Ration. Mech. Anal. 193(2), 311–337 (2009)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. NWF-I Mathematik, Universität Regensburg, 93040, Regensburg, Germany

    Anna Föglein

Authors
  1. Anna Föglein
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Anna Föglein.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Föglein, A. Regularity results for minimizers of (2, q)-growth functionals in the Heisenberg Group. manuscripta math. 133, 131–172 (2010). https://doi.org/10.1007/s00229-010-0366-0

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00229-010-0366-0

Mathematics Subject Classification (2000)

Search

Navigation