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Regularity for general functionals with double phase

  • Paolo Baroni
  • Maria Colombo
  • Giuseppe Mingione
Article

Abstract

We prove sharp regularity results for a general class of functionals of the type
$$\begin{aligned} w \mapsto \int F(x, w, Dw) \, dx, \end{aligned}$$
featuring non-standard growth conditions and non-uniform ellipticity properties. The model case is given by the double phase integral
$$\begin{aligned} w \mapsto \int b(x,w)(|Dw|^p+a(x)|Dw|^q) \, dx,\quad 1<p < q, \quad a(x)\ge 0, \end{aligned}$$
with \(0<\nu \le b(\cdot )\le L \). This changes its ellipticity rate according to the geometry of the level set \(\{a(x)=0\}\) of the modulating coefficient \(a(\cdot )\). We also present new methods and proofs that are suitable to build regularity theorems for larger classes of non-autonomous functionals. Finally, we disclose some new interpolation type effects that, as we conjecture, should draw a general phenomenon in the setting of non-uniformly elliptic problems. Such effects naturally connect with the Lavrentiev phenomenon.

Mathematics Subject Classification

49N60 35D10 

References

  1. 1.
    Acerbi, E., Fusco, N.: An Approximation Lemma for \(W^{1,p}\) Functions. Material Instabilities in Continuum Mechanics (Edinburgh, 1985–1986), pp. 1–5. Oxford Science Publications, Oxford University Press, New York (1988)Google Scholar
  2. 2.
    Acerbi, E., Mingione, G.: Regularity results for a class of functionals with non-standard growth. Arch. Ration. Mech. Anal. 156, 121–140 (2001)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Acerbi, E., Mingione, G.: Regularity results for electrorheological fluids: the stationary case. C. R. Math. Acad. Sci. Paris 334, 817–822 (2002)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Baroni, P.: Riesz potential estimates for a general class of quasilinear equations. Calc. Var. PDE 53, 803–846 (2015)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Baroni, P., Colombo, M., Mingione, G.: Harnack inequalities for double phase functionals. Nonlinear Anal. 121, 206–222 (2015). (Special Issue for E. Mitidieri) MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Baroni, P., Colombo, M., Mingione, G.: Non-autonomous functionals, borderline cases and related function classes. St. Petersb. Math. J. 27, 347–379 (2016). (Special Issue for N. Uraltseva) CrossRefMATHGoogle Scholar
  7. 7.
    Baroni, P., Colombo, M., Mingione, G.: Flow of double phase functionals (forthcoming)Google Scholar
  8. 8.
    Baroni, P., Lindfors, C.: The Cauchy–Dirichlet problem for a general class of parabolic equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 34, 593–624 (2017)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Bildhauer, M., Fuchs, M.: \(C^{1,\alpha }\)-solutions to non-autonomous anisotropic variational problems. Calc. Var. PDE 24, 309–340 (2005)CrossRefMATHGoogle Scholar
  10. 10.
    Bousquet, P., Brasco, L.: Global Lipschitz continuity for minima of degenerate problems. Math. Ann. 366, 1403–1450 (2016)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Breit, D.: New regularity theorems for non-autonomous variational integrals with \((p, q)\)-growth. Calc. Var. PDE 44, 101–129 (2012)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Byun, S.S., Oh, J.: Global gradient estimates for non-uniformly elliptic equations. Calc. Var. PDE 56(2), 36 (2017)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Byun, S.S., Oh, J.: Global gradient estimates for the borderline case of double phase problems with BMO coefficients in nonsmooth domains. J. Differ. Equ. 263, 1643–1693 (2017)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Byun, S.S., Ok, J., Ryu, S.: Global gradient estimates for elliptic equations of \(p(x)\)-Laplacian type with BMO nonlinearity. J. Reine Angew. Math. (Crelles J.) 715, 1–38 (2016)MathSciNetMATHGoogle Scholar
  15. 15.
    Choe, H.J.: Interior behaviour of minimizers for certain functionals with nonstandard growth. Nonlinear Anal. 19, 933–945 (1992)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Colombo, M., Mingione, G.: Regularity for double phase variational problems. Arch. Ration. Mech. Anal. 215, 443–496 (2015)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Colombo, M., Mingione, G.: Bounded minimisers of double phase variational integrals. Arch. Ration. Mech. Anal. 218, 219–273 (2015)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Colombo, M., Mingione, G.: Calderón–Zygmund estimates and non-uniformly elliptic operators. J. Funct. Anal. 270, 1416–1478 (2016)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Cruz-Uribe, D., Hästö, P.: Extrapolation and interpolation in generalized Orlicz spaces. Trans. Am. Math. Soc. (to appear)Google Scholar
  20. 20.
    Cupini, G., Marcellini, P., Mascolo, E.: Regularity of minimizers under limit growth conditions. Nonlinear Anal. 153, 294–310 (2017)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    De Giorgi, E.: Frontiere orientate di misura minima. Seminario di Matematica della Scuola Normale Superiore di Pisa, 1960–61Google Scholar
  22. 22.
    Diening, L.: Maximal function on Musielak–Orlicz spaces and generalized Lebesgue spaces. Bull. Sci. Math. 129, 657–700 (2005)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Diening, L., Stroffolini, B., Verde, A.: The \(\varphi \)-harmonic approximation and the regularity of \(\varphi \)-harmonic maps. J. Differ. Equ. 253, 1943–1958 (2012)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Duzaar, F., Mingione, G.: Harmonic type approximation lemmas. J. Math. Anal. Appl. 352, 301–335 (2009)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Eleuteri, M., Marcellini, P., Mascolo, E.: Lipschitz estimates for systems with ellipticity conditions at infinity. Ann. Mat. Pura Appl. (IV) 195, 1575–1603 (2016)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Eleuteri, M., Marcellini, P., Mascolo, E.: Regularity for scalar integrals without structure conditions. Adv. Calc. Var. (to appear)Google Scholar
  27. 27.
    Esposito, L., Leonetti, F., Mingione, G.: Sharp regularity for functionals with \((p, q)\)-growth. J. Differ. Equ. 204, 5–55 (2004)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Fonseca, I., Malý, J., Mingione, G.: Scalar minimizers with fractal singular sets. Arch. Ration. Mech. Anal. 172, 295–307 (2004)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Giaquinta, M., Giusti, E.: Differentiability of minima of nondifferentiable functionals. Invent. Math. 72, 285–298 (1983)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Giusti, E.: Direct Methods in the Calculus of Variations. World Scientific Publishing Co., Inc, River Edge (2003)CrossRefMATHGoogle Scholar
  31. 31.
    Harjulehto, P., Hästö, P.: Boundary regularity under generalized growth conditions. Preprint (2017)Google Scholar
  32. 32.
    Harjulehto, P., Hästö, P., Toivanen, O.: Hölder regularity of quasiminimizers under generalized growth conditions. Calc. Var. PDE 56, 26 (2017)CrossRefMATHGoogle Scholar
  33. 33.
    Hästö, P.: The maximal operator on generalized Orlicz spaces. J. Funct. Anal. 269, 4038–4048 (2015)MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Kristensen, J., Mingione, G.: The singular set of minima of integral functionals. Arch. Ration. Mech. Anal. 180, 331–398 (2006)MathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    Kristensen, J., Mingione, G.: Boundary regularity in variational problems. Arch. Ration. Mech. Anal. 198, 369–455 (2010)MathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    Kuusi, T., Mingione, G.: Guide to nonlinear potential estimates. Bull. Math. Sci. 4, 1–82 (2014)MathSciNetCrossRefMATHGoogle Scholar
  37. 37.
    Kuusi, T., Mingione, G.: Vectorial nonlinear potential theory. J. Eur. Math. Soc. (2018).  https://doi.org/10.4171/JEMS/780 MATHGoogle Scholar
  38. 38.
    Lavrentiev, M.: Sur quelques problèmes du calcul des variations. Ann. Mat. Pura Appl. 4, 7–28 (1926)CrossRefGoogle Scholar
  39. 39.
    Lieberman, G.M.: The natural generalization of the natural conditions of Ladyzhenskaya and Ural’tseva for elliptic equations. Commun. PDE 16, 311–361 (1991)CrossRefMATHGoogle Scholar
  40. 40.
    Manfredi, J.J.: Regularity for minima of functionals with \(p\)-growth. J. Differ. Equ. 76, 203–212 (1988)MathSciNetCrossRefMATHGoogle Scholar
  41. 41.
    Manfredi, J.J.: Regularity of the gradient for a class of nonlinear possibly degenerate elliptic equations. Ph.D. Thesis. University of Washington, St. Louis (1986)Google Scholar
  42. 42.
    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)CrossRefMATHGoogle Scholar
  43. 43.
    Marcellini, P.: Regularity and existence of solutions of elliptic equations with \(p, q\)-growth conditions. J. Differ. Equ. 90, 1–30 (1991)MathSciNetCrossRefMATHGoogle Scholar
  44. 44.
    Marcellini, P.: Everywhere regularity for a class of elliptic systems without growth conditions. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (IV) 23, 1–25 (1996)MathSciNetMATHGoogle Scholar
  45. 45.
    Mingione, G.: Regularity of minima: an invitation to the dark side of the calculus of variations. Appl. Math. 51, 355–425 (2006)MathSciNetCrossRefMATHGoogle Scholar
  46. 46.
    Ok, J.: Gradient estimates for elliptic equations with \(L^{p(\cdot )}\log L\) growth. Calc. Var. PDE 55(2), 30 (2016)MathSciNetCrossRefMATHGoogle Scholar
  47. 47.
    Ok, J.: Regularity of \(\omega \)-minimizers for a class of functionals with non-standard growth. Calc. Var. PDE 56(2), 31 (2017)MathSciNetCrossRefMATHGoogle Scholar
  48. 48.
    Perera, K., Squassina, M.: Existence results for double phase problems via Morse theory. Commun. Contemp. Math. 20, Article no. 1750023 (2018)Google Scholar
  49. 49.
    Ragusa, M.A., Tachikawa, A.: Boundary regularity of minimizers of \(p(x)\)-energy functionals. Ann. IHP Anal. non Linéare 33, 451–476 (2017)MathSciNetMATHGoogle Scholar
  50. 50.
    Ragusa, M.A., Tachikawa, A.: Partial regularity of \(p(x)\)-harmonic maps. Trans. Am. Math. Soc. 365, 3329–3353 (2013)MathSciNetCrossRefMATHGoogle Scholar
  51. 51.
    Tachikawa, A., Usuba, K.: Regularity results up to the boundary for minimizers of \(p(x)\)-energy with \(p(x)>1\). manus. math. 152, 127–151 (2017)MathSciNetCrossRefMATHGoogle Scholar
  52. 52.
    Uhlenbeck, K.: Regularity for a class of non-linear elliptic systems. Acta Math. 138, 219–240 (1977)MathSciNetCrossRefMATHGoogle Scholar
  53. 53.
    Ural’tseva, N.N.: Degenerate quasilinear elliptic systems. Zap. Na. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 7, 184–222 (1968)MathSciNetMATHGoogle Scholar
  54. 54.
    Ural’tseva, N.N., Urdaletova, A.B.: The boundedness of the gradients of generalized solutions of degenerate quasilinear non-uniformly elliptic equations. Vestnik Leningrad University Mathematics 19 (1983) (Russian) English. tran.: 16 (1984), 263–270Google Scholar
  55. 55.
    Zhikov, V.V.: Averaging of functionals of the calculus of variations and elasticity theory. Izv. Akad. Nauk SSSR Ser. Mat. 50, 675–710 (1986)MathSciNetGoogle Scholar
  56. 56.
    Zhikov, V.V.: On Lavrentiev’s phenomenon. Russ. J. Math. Phys. 3, 249–269 (1995)MathSciNetMATHGoogle Scholar
  57. 57.
    Zhikov, V.V.: On some variational problems. Russ. J. Math. Phys. 5, 105–116 (1997)MathSciNetMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Paolo Baroni
    • 1
  • Maria Colombo
    • 2
  • Giuseppe Mingione
    • 1
  1. 1.Dipartimento SMFIUniversità di ParmaParmaItaly
  2. 2.Institute for Theoretical StudiesETH ZürichZurichSwitzerland

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