Skip to main content
SpringerLink
Log in

Comparison estimates in anisotropic variational problems

  • Published:
manuscripta mathematica Aims and scope Submit manuscript

Abstract

A pointwise inequality between the radially decreasing symmetrals of minimizers of (possibly) anisotropic variational problems and the minimizers of suitably symmetrized problems is established. As a consequence, a priori sharp estimates for norms of the relevant minimizers are derived.

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., Fusco, N.: Partial regularity under anisotropic (p, q) conditions. J. Differ. Equ. 107, 46–67 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  2. Alvino, A., Lions, P.L., Trombetti, G.: On optimization problems with prescribed rearrangements. Nonlinear Anal. 13, 185–220 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  3. Alvino, A., Trombetti, G.: Sulle migliori costanti di maggiorazione per una classe di equazioni ellittiche degeneri. Ricerche Mat. 27, 413–428 (1978)

    MathSciNet  MATH  Google Scholar 

  4. Bennett, C., Sharpley, R.: Interpolation of Operators. Academic Press, Boston (1988)

    MATH  Google Scholar 

  5. Boccardo, L., Gallouët, T., Marcellini, P.: Anisotropic equations in L 1. Differ. Int. Equ. 9, 209–212 (1996)

    MATH  Google Scholar 

  6. Boccardo, L., Marcellini, P., Sbordone, C.: L ∞-regularity for variational problems with sharp nonstandard growth conditions. Boll. Un. Mat. Ital. A (7) 4, 219–225 (1990)

    MathSciNet  MATH  Google Scholar 

  7. Brothers, J.E., Ziemer, W.P.: Minimal rearrangements of Sobolev functions. J. Reine Angew Math. 384, 153–179 (1988)

    MathSciNet  MATH  Google Scholar 

  8. Cianchi, A.: Boundedness of solutions to variational problems under general growth conditions. Comm. Partial Differ. Equ. 22, 1629–1646 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  9. Cianchi, A.: A fully anisotropic Sobolev inequality. Pac. J. Math. 196, 283– (2000)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  11. Cianchi, A.: Optimal Orlicz–Sobolev embeddings. Rev. Math. Iberoam. 20, 427–474 (2004)

    MathSciNet  MATH  Google Scholar 

  12. Cianchi, A.: Symmetrization in anisotropic elliptic problems. Comm. Partial Differ. Equ. 32, 693–717 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  13. Cianchi, A., Schianchi, R.: A priori sharp estimates for minimizers. Boll. Un. Mat. Ital. B 7, 821–831 (1993)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  15. Fragalà, I., Gazzola, F., Kawohl, B.: Existence and nonexistence results for anisotropic quasilinear elliptic equations. Ann. Inst. Henri Poincaré, Anal. non linéaire 21, 715–734 (2004)

    Article  MATH  Google Scholar 

  16. Fragalà, I., Gazzola, F., Liebermann, G.: Regularity and nonexistence results for anisotropic quasilinear elliptic equations in convex domains. Disc. Cont. Dynam. Syst. Suppl. Vol. pp. 280–286 (2005)

  17. 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 

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

    Article  MathSciNet  MATH  Google Scholar 

  19. Hong, M.-C.: Some remarks on minimizers of variational integrals with non-standard growth conditions. Boll. Un. Mat. Ital. A 6, 91–101 (1992)

    MathSciNet  MATH  Google Scholar 

  20. Kawohl, B.: Rearrangements and convexity of level sets in PDE. Lecture Notes in Mathematics, vol. 1150, Springer, Berlin (1985)

  21. Kesavan, S.: Symmetrization & Applications. World Scientific, Hackensack, HJ (2006)

    MATH  Google Scholar 

  22. Klimov, V.S.: Isoperimetric inequalities and embedding theorems, Dokl. Akad. Nauk. SSSR 217 (1974) (Russian); english translation in Soviet Math. Dokl. 15 (1974)

  23. Klimov, V.S.: Imbedding theorems and geometric inequalities, Izv. Akad. Nauk SSSR, Ser. Mat. 40 (1976) (Russian); english translation in Math. USSR Izvestiya 10, 615–638 (1976)

  24. Leonetti, F.: Weak differentiability for solutions to nonlinear elliptic systems with (p, q) growth conditions. Ann. Mat. Pura. Appl. 162, 349–366 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  25. Lieberman, G.: 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 of the calculus of variations with non standard growth conditions. Arch. Rat. Mech. Anal. 105, 267–284 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  27. Maz’ya, V.G.: On weak solutions of the Dirichlet and Neumann problems. Trans. Moscow Math. Soc. 20, 135–172 (1969)

    Google Scholar 

  28. Maz’ya, V.G.: Sobolev Spaces. Springer, Berlin (1985)

    Google Scholar 

  29. Schianchi, R.: An estimate for the minima of the functionals of the calculus of variations. Differ. Integr. Equ. 2, 326–332 (1989)

    MathSciNet  MATH  Google Scholar 

  30. Skaff, M.S.: Vector-valued Orlicz spaces, generalized N-functions I. Pac. J. Math. 28, 193–206 (1969)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  32. Stroffolini, B.: Some remarks on the regularity of anisotropic variational problems. Rend. Accad. Naz. Sci. XV Mem. Mat. 17, 229–239 (1993)

    MathSciNet  MATH  Google Scholar 

  33. Talenti, G.: Elliptic equations and rearrangements. Ann. Sc. Norm. Sup. Pisa IV 3, 697–718 (1976)

    MathSciNet  MATH  Google Scholar 

  34. Talenti, G.: Best constant in Sobolev inequality. Ann. Mat. Pura Appl. 110, 353–372 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  35. Talenti, G.: Nonlinear elliptic equations, rearrangements of functions and Orlicz spaces. Ann. Mat. Pura Appl. 120, 159–184 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  36. Tersenov, Al.S., Tersenov, Ak.S.: The problem of Dirichlet for anisotropic quasilinear degenerate elliptic equations. J. Differ. Equ. 235, 376–396 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  37. Trombetti, G.: Symmetrization methods for partial differential equations. Boll. Un. Mat. Ital. B 3, 601–634 (2000)

    MathSciNet  MATH  Google Scholar 

  38. Trudinger, N.S.: An imbedding theorem for H 0 (G , Ω) spaces. Studia Math. 50, 17–30 (1974)

    MathSciNet  Google Scholar 

  39. Vazquez, J.L.: Symmetrization and mass comparison for degenerate nonlinear parabolic and related elliptic equations. Adv. Nonlinear Stud. 5, 87–131 (2005)

    MathSciNet  MATH  Google Scholar 

  40. Ziemer, W.P.: Weakly Differentiable Functions. Springer, New York (1989)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Istituto per le Applicazioni del Calcolo “M. Picone”, Sez. Napoli-C.N.R., Via P. Castellino 111, 80131, Napoli, Italy

    Angela Alberico

  2. Dipartimento di Matematica e Applicazioni per l’Architettura, Università di Firenze, Piazza Ghiberti 27, 50122, Firenze, Italy

    Andrea Cianchi

Authors
  1. Angela Alberico
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Andrea Cianchi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Angela Alberico.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Alberico, A., Cianchi, A. Comparison estimates in anisotropic variational problems. manuscripta math. 126, 481–503 (2008). https://doi.org/10.1007/s00229-008-0183-x

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00229-008-0183-x

Mathematics Subject Classification (2000)

Search

Navigation