Skip to main content
SpringerLink
Log in

Local boundedness of solutions to quasilinear elliptic systems

  • Published:
Manuscripta Mathematica Aims and scope Submit manuscript

Abstract

The mathematical analysis to achieve everywhere regularity in the interior of weak solutions to nonlinear elliptic systems usually starts from their local boundedness. Having in mind De Giorgi’s counterexamples, some structure conditions must be imposed to treat systems of partial differential equations. On the contrary, in the scalar case of a general elliptic single equation a well established theory of regularity exists. In this paper we propose a unified approach to local boundedness of weak solutions to a class of quasilinear elliptic systems, with a structure condition inspired by Ladyzhenskaya–Ural’tseva’s work for linear systems, as well as valid for the general scalar case. Our growth assumptions on the nonlinear quantities involved are new and general enough to include anisotropic systems with sharp exponents and the p, q-growth case.

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 stationary electro-rheological fluids. Arch. Ration. Mech. Anal. 164, 213–259 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bildhauer M., Fuchs M.: Higher integrability of the gradient for vectorial minimizers of decomposable variational integrals. Manuscr. Math. 123, 269–283 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  3. Boccardo L., Marcellini P., Sbordone C.: L ∞-regularity for variational problems with sharp nonstandard growth conditions. Boll. Un. Mat. Ital. A 4, 219–225 (1990)

    MathSciNet  MATH  Google Scholar 

  4. Carozza M., Leonetti F., Passarelli di Napoli A.: Vector valued minimizers of anisotropic functionals: fractional differentiability and estimate for the singular set. Manuscr. Math. 128, 51–68 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  5. Cianchi A.: Local boundedness of minimizers of anisotropic functionals. Ann. Inst. Henri Poincaré Anal. Non Lin. 17, 147–168 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  6. Cupini G., Marcellini P., Mascolo E.: Regularity under sharp anisotropic general growth conditions. Discr. Contin. Dyn. Syst. Ser. B 11, 66–86 (2009)

    MathSciNet  Google Scholar 

  7. Dall’Aglio A., Mascolo E.: L ∞ estimates for a class of nonlinear elliptic systems with nonstandard growth. Atti Sem. Mat. Fis. Univ. Modena 50, 65–83 (2002)

    MathSciNet  MATH  Google Scholar 

  8. De Giorgi E.: Un esempio di estremali discontinue per un problema variazionale di tipo ellittico. Boll. Un. Mat. Ital. (4) 1, 135–137 (1968)

    MathSciNet  MATH  Google Scholar 

  9. De Maria B., Passarelli di Napoli A.: Partial regularity for non autonomous functionals with non standard growth conditions. Calc. Var. Partial Differ. Equ. 38, 417–439 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  10. Frehse J.: Una generalizzazione di un controesempio di De Giorgi nella teoria delle equazioni ellittiche. Boll. Un. Mat. Ital. (4) 3, 998–1002 (1970)

    MathSciNet  MATH  Google Scholar 

  11. Frehse J.: A discontinuous solution of a mildly nonlinear elliptic system. Math. Z. 134, 229–230 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  12. Fusco N., Sbordone C.: Local boundedness of minimizers in a limit case. Manuscr. Math. 69, 19–25 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  13. Fusco N., Sbordone C.: Some remarks on the regularity of minima of anisotropic integrals. Comm. Partial Differ. Equ. 18, 153–167 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  14. Giaquinta M.: Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems, Annals of Mathematics Studies, 105. Princeton University Press, Princeton (1983)

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  16. Giusti E.: Direct Methods in the Calculus of Variations. World Scientific Publishing Co., Inc, River Edge (2003)

    Book  MATH  Google Scholar 

  17. Giusti E., Miranda M.: Un esempio di soluzioni discontinue per un problema di minimo relativo ad un integrale regolare del calcolo delle variazioni. Boll. Un. Mat. Ital. (4) 1, 219–226 (1968)

    MathSciNet  MATH  Google Scholar 

  18. Hildebrandt S., Widman K.: Some regularity results for quasilinear elliptic systems of second order. Math. Z. 142, 67–86 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  19. Hildebrandt S., Widman K.: On the Hölder continuity of weak solutions of quasilinear elliptic systems of second order. Ann. Sci. Norm. Super. Pisa Cl. Sci. (4) 4, 145–178 (1977)

    MathSciNet  Google Scholar 

  20. Krömer S.: A priori estimates in L ∞ for non-diagonal perturbed quasilinear systems. Ann. Sci. Norm. Super. Pisa Cl. Sci. (5) 8, 417–428 (2009)

    MATH  Google Scholar 

  21. Ladyzhenskaya O., Ural’tseva N.: Linear and Quasilinear Elliptic Equations. Academic Press, New York (1968)

    MATH  Google Scholar 

  22. Landes R.: Some remarks on bounded and unbounded weak solutions of elliptic systems. Manuscr. Math. 64, 227–234 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  23. Landes R.: On the regularity of weak solutions of certain elliptic systems. Calc. Var. Partial Differ. Equ. 25, 247–255 (2005)

    Article  MathSciNet  Google Scholar 

  24. Lieberman G.M.: The natural generalization of the natural conditions of Ladyzhenskaya and Ural’tseva for elliptic equations. Comm. Partial Differ. Equ. 16, 311–361 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  25. Lieberman G.M.: Gradient estimates for anisotropic elliptic equations. Adv. Differ. Equ. 10, 767–812 (2005)

    MathSciNet  MATH  Google Scholar 

  26. Marcellini P.: Regularity of minimizers of integrals in the calculus of variations with non standard growth conditions. Arch. Rational Mech. Anal. 105, 267–284 (1989)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  29. Marcellini P.: Everywhere regularity for a class of elliptic systems without growth conditions. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 23, 1–25 (1996)

    MathSciNet  MATH  Google Scholar 

  30. Marcellini P., Papi G.: Nonlinear elliptic systems with general growth. J. Differ. Equ. 221, 412–443 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  31. Meier M.: Boundedness and integrability properties of weak solutions of quasilinear elliptic systems. J. Reine Angew. Math. 333, 191–220 (1982)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  33. Mingione G.: Singularities of minima: a walk on the wild side of the calculus of variations. J. Glob. Optim. 40, 209–223 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  34. Nečas, J.: Example of an irregular solution to a nonlinear elliptic system with analytic coefficients and conditions for regularity. In: Kluge, R. (ed.) Theory of Nonlinear Operators (Proceedings of 4th International Summer School. Academy of Science, Berlin, 1975), pp. 197–206. Akademie-Verlag, Berlin (1977)

  35. Serrin J.: Local behavior of solutions of quasi-linear equation. Acta Math. 111, 247–302 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  36. Serrin J.: Isolated singularities of solutions of quasi-linear equations. Acta Math. 113, 219–240 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  37. Stroffolini B.: Global boundedness of solutions of anisotropic variational problems. Boll. Un. Mat. Ital. A 5, 345–352 (1991)

    MathSciNet  MATH  Google Scholar 

  38. Šverák V., Yan X.: A singular minimizer of a smooth strongly convex functional in three dimensions. Calc. Var. Partial Differ. Equ. 10, 213–221 (2000)

    Article  MATH  Google Scholar 

  39. Talenti G.: Boundedness of minimizers. Hokkaido Math. J. 19, 259–279 (1990)

    MathSciNet  MATH  Google Scholar 

  40. Troisi M.: Teoremi di inclusione per spazi di Sobolev non isotropi. Ricerche di Mat. 18, 3–24 (1969)

    MathSciNet  MATH  Google Scholar 

  41. Uhlenbeck K.: Regularity for a class of non-linear elliptic systems. Acta Math. 138, 219–240 (1977)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dipartimento di Matematica, Università di Bologna, Piazza di Porta S. Donato 5, 40126, Bologna, Italy

    Giovanni Cupini

  2. Dipartimento di Matematica “U. Dini”, Università di Firenze, Viale Morgagni 67/A, 50134, Firenze, Italy

    Paolo Marcellini & Elvira Mascolo

Authors
  1. Giovanni Cupini
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Paolo Marcellini
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Elvira Mascolo
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Paolo Marcellini.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Cupini, G., Marcellini, P. & Mascolo, E. Local boundedness of solutions to quasilinear elliptic systems. manuscripta math. 137, 287–315 (2012). https://doi.org/10.1007/s00229-011-0464-7

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00229-011-0464-7

Mathematics Subject Classification (2000)

Search

Navigation