Journal of Fixed Point Theory and Applications

, Volume 19, Issue 4, pp 2697–2731 | Cite as

\({{\varvec{C}}}^\mathbf{1 }\)-regularity for minima of functionals with \({\varvec{p}}({\varvec{x}})\)-growth

Article
  • 81 Downloads

Abstract

We prove the \(C^1\)-regularity for local minima to the functionals with p(x)-growth such that \(w\mapsto \int _\Omega \left[ f(x,w,Dw)+gw\right] \, \mathrm{d}x\), where f satisfies a certain p(x)-growth condition. We assume that g belongs to the Lorentz space \(L^{n,1}\), and that \(p(\cdot )\) and \(f(\cdot ,z,\xi )\) satisfy Dini type continuity conditions. Our assumptions are sharp with respect to the type of regularity results obtained.

Keywords

Functional with variable growth Gradient continuity Dini continuity Lorentz space 

Mathematics Subject Classification

Primary 49N60 Secondary 35j20 

References

  1. 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)CrossRefMATHGoogle Scholar
  2. 2.
    Acerbi, E., Mingione, G.: Gradient estimates for the \(p(x)\)-Laplacean system. J. Reine Angew. Math. 584, 117–148 (2005)CrossRefMATHGoogle Scholar
  3. 3.
    Baroni, P., Colombo, M., Mingione, G.: Non-autonomous functionals, borderline cases and related function classes. St. Petersburg Math. J. 27(3), 347–379 (2016)CrossRefMATHGoogle Scholar
  4. 4.
    Baroni, P., Kuusi, T., Mingione, G.: Borderline gradient continuity of minima. J. Fixed Point Theory Appl. 15(2), 537–575 (2014)CrossRefMATHGoogle Scholar
  5. 5.
    Bögelein, V., Habermann, J.: Gradient estimates via non standard potentials and continuity. Ann. Acad. Sci. Fenn. Math. 35(2), 641–678 (2010)CrossRefMATHGoogle Scholar
  6. 6.
    Cianchi, A.: Maximizing the \(L^\infty \) norm of the gradient of solutions to the Poisson equation. J. Geom. Anal. 2(6), 499–515 (1992)CrossRefMATHGoogle Scholar
  7. 7.
    Cianchi, A., Maz’ya, V.: Global Lipschitz regularity for a class of quasilinear elliptic equations. Comm. Partial Differential Equations 36(1), 100–133 (2011)CrossRefMATHGoogle Scholar
  8. 8.
    Cianchi, A., Maz’ya, V.: Global boundedness of the gradient for a class of nonlinear elliptic systems. Arch. Ration. Mech. Anal. 212(1), 129–177 (2014)CrossRefMATHGoogle Scholar
  9. 9.
    Cianchi, A., Maz’ya, V.: Global gradient estimates in elliptic problems under minimal data and domain regularity. Commun. Pure Appl. Anal. 14(1), 285–311 (2015)CrossRefMATHGoogle Scholar
  10. 10.
    Colombo, M., Mingione, G.: Calderón-Zygmund estimates and non-uniformly elliptic operators. J. Funct. Anal. 270(4), 1416–1478 (2016)CrossRefMATHGoogle Scholar
  11. 11.
    Dacorogna, B.: Direct Methods in the Calculus of Variations, Appl. Math. Sci., vol. 78, 2nd edn. Springer, New York (2008)Google Scholar
  12. 12.
    DiBenedetto, E.: \(C^{1+\alpha }\) local regularity of weak solutions of degenerate elliptic equations. Nonlinear Anal. 7(8), 827–850 (1983)CrossRefMATHGoogle Scholar
  13. 13.
    Di Fazio, G.: \(L^p\) estimates for divergence form elliptic equations with discontinuous coefficients. Boll. Un. Mat. Ital. A (7) 10(2), 409–420 (1996)Google Scholar
  14. 14.
    Diening, L., Harjulehto, P., Hästö, P., Růžička, M.: Lebesgue and Sobolev Spaces with Variable Exponents. Lecture Notes in Math, vol. 2017. Springer, Berlin (2011)Google Scholar
  15. 15.
    Duzaar, F., Mingione, G.: Gradient continuity estimates. Calc. Var. Partial Differ. Equ. 39(3–4), 379–418 (2010)CrossRefMATHGoogle Scholar
  16. 16.
    Duzaar, F., Mingione, G.: Local Lipschitz regularity for degenerate elliptic systems. Ann. Inst. H. Poincaré Anal. Non Linéaire 27(6), 1361–1396 (2010)CrossRefMATHGoogle Scholar
  17. 17.
    Fan, X., Zhao, D.: A class of De Giorgi type and Hölder continuity. Nonlinear Anal. 36(3), 295–318 (1999)CrossRefMATHGoogle Scholar
  18. 18.
    Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order, Grundlehren Math. Wiss., vol. 224. Springer, Berlin (1977)Google Scholar
  19. 19.
    Giusti, E.: Direct methods in the calculus of variations. World Scientific Publishing Co., Inc., River Edge (2003)CrossRefMATHGoogle Scholar
  20. 20.
    Iwaniec, T., Verde, A.: On the operator \(\fancyscript {L}(f)=f\log |f|\). J. Funct. Anal. 169(2), 391–420 (1999)CrossRefMATHGoogle Scholar
  21. 21.
    Jin, T., Maz’ya, V., Van Schaftingen, J.: Pathological solutions to elliptic problems in divergence form with continuous coefficients. C. R. Math. Acad. Sci. Paris 347(12–14), 773–778 (2009)CrossRefMATHGoogle Scholar
  22. 22.
    Kilpeläinen, T., Koskela, P.: Global integrability of the gradients of solutions to partial differential equations. Nonlinear Anal. 23(7), 899–909 (1994)CrossRefMATHGoogle Scholar
  23. 23.
    Kristensen, J., Mingione, G.: The singular set of minima of integral functionals. Arch. Ration. Mech. Anal. 180(3), 331–398 (2006)CrossRefMATHGoogle Scholar
  24. 24.
    Kuusi, T., Mingione, G.: Linear potentials in nonlinear potential theory. Arch. Ration. Mech. Anal. 207(1), 215–246 (2013)CrossRefMATHGoogle Scholar
  25. 25.
    Kuusi, T., Mingione, G.: Universal potential estimates. J. Funct. Anal. 262(10), 4205–4269 (2012)CrossRefMATHGoogle Scholar
  26. 26.
    Kuusi, T., Mingione, G.: Guide to nonlinear potential estimates. Bull. Math. Sci. 4(1), 1–82 (2014)CrossRefMATHGoogle Scholar
  27. 27.
    Kuusi, T., Mingione, G.: A nonlinear Stein theorem. Calc. Var. Partial Differ. Equ. 51(1–2), 45–86 (2014)CrossRefMATHGoogle Scholar
  28. 28.
    Kuusi, T., Mingione, G.: The Wolff gradient bound for degenerate parabolic equations. Arch. Ration. Mech. Anal. 212(3), 727–780 (2014)CrossRefMATHGoogle Scholar
  29. 29.
    Manfredi, J.: Regularity for minima of functionals with \(p\)- growth. J. Differ. Equ. 76(2), 203–212 (1988)CrossRefMATHGoogle Scholar
  30. 30.
    Marcellini, P.: Regularity of minima of integrals of the calculus of variations with non standard growth conditions. Arch. Ration. Mech. Anal. 105, 267–284 (1989)CrossRefMATHGoogle Scholar
  31. 31.
    Marcellini, P.: Regularity and existence of solutions of elliptic equations with \(p, q\)-growth conditions. J. Differ. Equ. 90, 1–30 (1991)CrossRefMATHGoogle Scholar
  32. 32.
    Marcellini, P.: Regularity for elliptic equations with general growth conditions. J. Differ. Equ. 105, 296–333 (1993)CrossRefMATHGoogle Scholar
  33. 33.
    Ok, J.: Gradient continuity for \(p(\cdot )\)-Laplace systems. Nonlinear Anal. 141, 139–166 (2016)CrossRefMATHGoogle Scholar
  34. 34.
    Ok, J.: Gradient continuity for nonlinear obstacle problems. Mediterr. J. Math. 14(1), 24 , Art. 16(2017)Google Scholar
  35. 35.
    Ok, J.: Harnack inequality for a class of functionals with non-standard growth via De Giorgi’s method. Adv. Nonlinear Anal. doi: 10.1515/anona-2016-0083
  36. 36.
    Stein, E.M.: Editor’s note: the differentiability of functions in \(R^n\). Ann. Math. (2) 113(2), 383–385 (1981)Google Scholar

Copyright information

© Springer International Publishing 2017

Authors and Affiliations

  1. 1.Department of Applied MathematicsKyung Hee UniversityYonginKorea

Personalised recommendations