Skip to main content
SpringerLink
Log in

A Harnack inequality for the quasi-minima of scalar integral functionals with general growth conditions

  • Published:
manuscripta mathematica Aims and scope Submit manuscript

Abstract

In this paper we proof a Harnack inequality and a regularity theorem for quasi-minima of scalar integral functionals with general growth conditions.

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. Adams, R.: Sobolev Spaces. Academic Press, New York (1975)

    MATH  Google Scholar 

  2. Ambrosio, L.: Lecture Notes on Partial Differential Equations. http://vcgmt.sns.it/papers/ambcarmas10

  3. Astarita, G., Marrucci, G.: Principles of Non-Newtonian Fluid Mechanics. McGraw-Hill, London (1974)

    MATH  Google Scholar 

  4. Bhattacharaya, T., Leonetti, F.: A new Poincaré inequality and its application to the regularity of minimizer of integral functionals with nonstandard growth. Nonlinear Anal. 17(9), 833–839 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bhattacharaya, T., Leonetti, F.: \(W^{2,2}\) regularity for weak solutions of elleptic systems with nonstandard growth. J. Math. Anal. Appl. 176, 224–234 (1993)

    Article  MathSciNet  Google Scholar 

  6. Breit, D., Fuchs, M.: The nonlinear Stokes problem with general potentials having superquadratic growth. J. Math. Fluid Mech. 13, 371–385 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  7. Breit, D., Stroffolini, B., Verde, A.: A general regularity theorem for functionals with \(\varphi \)-growth. J. Math. Anal. Appl. 383, 226–233 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  8. Cianchi, A., Fusco, N.: Gradient regularity for minimizers under general growth conditions. J. Reine Angew. Math. 507, 15–36 (1999)

    MathSciNet  MATH  Google Scholar 

  9. Dall’Aglio, A., Mascolo, E., Papi, G.: Regularity for local minima of functionala with nonstandard growth conditions. Rend. Mat. 18(VII), 305–326 (1998)

    MATH  Google Scholar 

  10. De Giorgi, E.: Sulla differenziabilità e l’analicità delle estremali degli integrali multipli. Mem. Accad. Sci. Torino. Cl. Sci. Fis. Mat. Nat. 3, 25–43 (1957)

    MathSciNet  MATH  Google Scholar 

  11. Di Benedetto, E., Trudinger, N.: Harnack inequalities for quasi-minima of variational integrals. Ann. Inst. H. Poincaré (Analyse Non Lineaire) 4, 295–308 (1984)

    MATH  Google Scholar 

  12. Diening, L., Ruzika, M.: Non-Newtonian fluids and function spaces. Nonlinear Anal. Funct. Spaces Appl. 8, 95–143 (2007)

    MATH  Google Scholar 

  13. Diening, L., Stroffolini, B., Verde, A.: Everywhere regulaity of functional with \(\varphi \)-growth. Manus. Math. 129, 440–481 (2009)

    Article  Google Scholar 

  14. Fuchs, M., Seregin, G.: A regularity theory for variational intgrals with LlnL-growth. Calc. Var. 6, 171–187 (1998)

    Article  MATH  Google Scholar 

  15. Fuchs, M., Mingione, G.: Full \(C^{1,\alpha }\)-regularity for free and constrained local minimizers of elliptic variational integrals with nearly linear growth. Manus. Math. 102, 227–250 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  16. Fuchs, M.: Local Lipschitz regularity of vector valued local minimizers of variational integrals with densities depending on the modulus of the gradient. Math. Nachr. 284, 266–272 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  17. Fusco, N., Sbordone, C.: Higher integrability of the gradient of minimizers of functionals with nonstandard growth conditions. Commun. Pure Appl. Math. 43, 673–683 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  18. Giaquinta, M., Giusti, E.: On the regularity of minima of variational integrals. Acta Math. 148, 31–46 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  19. Giaquinta, M., Giusti, E.: Quasi-minima. Ann. Inst. H. Poincarè (Analyse Non Lineaire) 1, 79–107 (1984)

    MathSciNet  MATH  Google Scholar 

  20. Giaquinta, M., Martinazzi, L.: An Introduction to the Regularity Theory for Elliptic Systems, Harmonic Maps and Minimal Graphs. S.N.S. Press, Pisa (2005)

    MATH  Google Scholar 

  21. Giusti, E.: Metodi Diretti nel Calcolo delle Variazioni. UMI, Bologna (1994)

    Google Scholar 

  22. Gosez, J.P.: Nonlinear elliptic problems for equations with rapidly (or slowly) increasing coefficents. Trans. Am. Math. Soc. 190, 163–205 (1974)

    Article  Google Scholar 

  23. Granucci, T.: Remarks on the continuity of the local minimizer of scalar integral functionals with nonstandard general growth conditions. J. Math. Res. 6(4), 1–17 (2014)

    Article  MathSciNet  Google Scholar 

  24. Granucci, T.: \(L^{\Phi }-L^{\infty }\) estimations and remarks on the the hőlder continuity of the quasi-minima of scalar integral functionals with general growths. Bol. Soc. Mat. Mex. 22(1), 165–212 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  25. Granucci, T.: An Harnack inequality for quasi-minima of scalar integral functionals with nearly linear growth conditions. Afr. Mat. (forthcoming)

  26. Granucci, T.: Remarks on the Harnack inequality of local-minima of scalar integral functionals with general growth conditions. J. Appl. Math. Phys. 2, 194–203 (2014)

    Article  Google Scholar 

  27. Granucci, T.: Hölder continuity for scalar minimizers of integals of the calculus of variations with general growths. Afr. Mat. 25(1), 197–212 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  28. Ladyženskaya, O.A., Ural’ceva, N.N.: Local estimates for the gradient of solutions of non-uniformly elliptic and parabolic equations. Commun. Pure Appl. Math. 23, 677–703 (1970)

    Article  MathSciNet  Google Scholar 

  29. Ladyženskaya, O.A., Ural’ceva, N.N.: Linear and Quasilinear Elliptic Equations. Accademic Press, New Work (1968)

    Google Scholar 

  30. Ladyženskaya, O.A., Ural’ceva, N.N.: Quasilinear elliptic equations and variational problems with many indipendet variables. Usp. Mat. Nauk. 16, 19–92 (1961) [English traslation in, Russ. Math. Surv. 16, 17–91 (1961)]

  31. Lieberman, G.M.: The natural generalization of the natural conditions of ladyzhenskaya and ural’tseva for elliptic equations. Commun. Part. Differ. Equ. 16, 331–361 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  32. Klimov, V.S.: Embedding theorems and continuity of generalized solutions of quasilinear elliptic equations. Differ. Equ. 36(6), 870–877 (2000) [translated from Differ. Uravn. 784–791 (2000)]

  33. Krasnosel’skij, M.A., Rutickii, Y.B.: Convex Function and Orlicz Spaces. Noordhoff, Groningen (1961)

    Google Scholar 

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

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  37. Marcellini, P.: Everywhere regularity for a class of elliptic systems without growth conditions. Ann. Sc. Norm. Super. Pisa 23, 1–25 (1996)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  39. Mascolo, E., Papi, G.: Local bounddeness of minimizers of integrals of the calculus of variations. Ann. Mat. Pura Appl. 167, 323–339 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  40. Mascolo, E., Papi, G.: Harnack inequality for minimizer of integral functionals with general growth conditions. Nonlinear Differ. Equ. Appl. 3, 231–244 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  41. Mingione, G., Siepe, F.: Full \(C^{1,\alpha }\) regularity for minimizers of integral functionals with L logL growth. Z. Anal. Anw. 18(4), 1083–1100 (1999)

    Article  MATH  Google Scholar 

  42. Moscariello, G., Nania, L.: Hölder continuity of minimizers of functionals with nonstandard growth conditions. Ricerche di Matematica 15(2), 259–273 (1991)

    MATH  Google Scholar 

  43. Moser, J.: A new proof of De Giorgi’s theorem concerning the regularity problem for elliptic differential equations. Commun. Pure Appl. Math. 13, 457–468 (1960)

    Article  MathSciNet  MATH  Google Scholar 

  44. Moser, J.: On Harnack’s theorem for elliptic differential equations. Commun. Pure Appl. Math. 14, 577–591 (1961)

    Article  MathSciNet  MATH  Google Scholar 

  45. Nash, J.: Continuity of solutions of parabolic and elliptic differential equations. Am. J. Math. 80, 931–953 (1958)

    Article  MATH  Google Scholar 

  46. Rao, M.M., Ren, Z.D.: Theory of Orlicz Spaces. Marcel Dekker, New York (1991)

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  48. Tilli, P.: Remarks on the Hölder continuity of solutions to elliptic equations in divergence form. Calc. Var. 25(3), 395–401 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  49. Trudinger, N.S.: On Harnack type inequalities and their application to quasilinear elliptic partial differential equations. Commun. Pure Appl. Math. 20, 721–747 (1967)

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Istituto A. Volta, via Roma 77, Bagno a Ripoli, Firenze, Italy

    Tiziano Granucci

  2. Istituto P. Calamandrei, via Ghibellina 77, Florence, Italy

    Tiziano Granucci

Authors
  1. Tiziano Granucci
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Tiziano Granucci.

Additional information

Dedicated to Caterina and Delia

This paper is in final form and no version of it will be submitted for publication elsewhere.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Granucci, T. A Harnack inequality for the quasi-minima of scalar integral functionals with general growth conditions. manuscripta math. 152, 345–380 (2017). https://doi.org/10.1007/s00229-016-0861-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00229-016-0861-z

Mathematics Subject Classification

Search

Navigation