Skip to main content
SpringerLink
Log in

Regularity for general functionals with double phase

  • Published:
Calculus of Variations and Partial Differential Equations Aims and scope Submit manuscript

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.

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.: 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)

  2. Acerbi, E., Mingione, G.: Regularity results for a class of functionals with non-standard growth. Arch. Ration. Mech. Anal. 156, 121–140 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  3. Acerbi, E., Mingione, G.: Regularity results for electrorheological fluids: the stationary case. C. R. Math. Acad. Sci. Paris 334, 817–822 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  4. Baroni, P.: Riesz potential estimates for a general class of quasilinear equations. Calc. Var. PDE 53, 803–846 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  5. Baroni, P., Colombo, M., Mingione, G.: Harnack inequalities for double phase functionals. Nonlinear Anal. 121, 206–222 (2015). (Special Issue for E. Mitidieri)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  7. Baroni, P., Colombo, M., Mingione, G.: Flow of double phase functionals (forthcoming)

  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)

    Article  MathSciNet  MATH  Google Scholar 

  9. Bildhauer, M., Fuchs, M.: \(C^{1,\alpha }\)-solutions to non-autonomous anisotropic variational problems. Calc. Var. PDE 24, 309–340 (2005)

    Article  MATH  Google Scholar 

  10. Bousquet, P., Brasco, L.: Global Lipschitz continuity for minima of degenerate problems. Math. Ann. 366, 1403–1450 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  11. Breit, D.: New regularity theorems for non-autonomous variational integrals with \((p, q)\)-growth. Calc. Var. PDE 44, 101–129 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  12. Byun, S.S., Oh, J.: Global gradient estimates for non-uniformly elliptic equations. Calc. Var. PDE 56(2), 36 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  15. Choe, H.J.: Interior behaviour of minimizers for certain functionals with nonstandard growth. Nonlinear Anal. 19, 933–945 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  16. Colombo, M., Mingione, G.: Regularity for double phase variational problems. Arch. Ration. Mech. Anal. 215, 443–496 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  17. Colombo, M., Mingione, G.: Bounded minimisers of double phase variational integrals. Arch. Ration. Mech. Anal. 218, 219–273 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  18. Colombo, M., Mingione, G.: Calderón–Zygmund estimates and non-uniformly elliptic operators. J. Funct. Anal. 270, 1416–1478 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  19. Cruz-Uribe, D., Hästö, P.: Extrapolation and interpolation in generalized Orlicz spaces. Trans. Am. Math. Soc. (to appear)

  20. Cupini, G., Marcellini, P., Mascolo, E.: Regularity of minimizers under limit growth conditions. Nonlinear Anal. 153, 294–310 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  21. De Giorgi, E.: Frontiere orientate di misura minima. Seminario di Matematica della Scuola Normale Superiore di Pisa, 1960–61

  22. Diening, L.: Maximal function on Musielak–Orlicz spaces and generalized Lebesgue spaces. Bull. Sci. Math. 129, 657–700 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  24. Duzaar, F., Mingione, G.: Harmonic type approximation lemmas. J. Math. Anal. Appl. 352, 301–335 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  26. Eleuteri, M., Marcellini, P., Mascolo, E.: Regularity for scalar integrals without structure conditions. Adv. Calc. Var. (to appear)

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

    Article  MathSciNet  MATH  Google Scholar 

  28. Fonseca, I., Malý, J., Mingione, G.: Scalar minimizers with fractal singular sets. Arch. Ration. Mech. Anal. 172, 295–307 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  29. Giaquinta, M., Giusti, E.: Differentiability of minima of nondifferentiable functionals. Invent. Math. 72, 285–298 (1983)

    Article  MathSciNet  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

  31. Harjulehto, P., Hästö, P.: Boundary regularity under generalized growth conditions. Preprint (2017)

  32. Harjulehto, P., Hästö, P., Toivanen, O.: Hölder regularity of quasiminimizers under generalized growth conditions. Calc. Var. PDE 56, 26 (2017)

    Article  MATH  Google Scholar 

  33. Hästö, P.: The maximal operator on generalized Orlicz spaces. J. Funct. Anal. 269, 4038–4048 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  34. Kristensen, J., Mingione, G.: The singular set of minima of integral functionals. Arch. Ration. Mech. Anal. 180, 331–398 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  35. Kristensen, J., Mingione, G.: Boundary regularity in variational problems. Arch. Ration. Mech. Anal. 198, 369–455 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  36. Kuusi, T., Mingione, G.: Guide to nonlinear potential estimates. Bull. Math. Sci. 4, 1–82 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  37. Kuusi, T., Mingione, G.: Vectorial nonlinear potential theory. J. Eur. Math. Soc. (2018). https://doi.org/10.4171/JEMS/780

    MATH  Google Scholar 

  38. Lavrentiev, M.: Sur quelques problèmes du calcul des variations. Ann. Mat. Pura Appl. 4, 7–28 (1926)

    Article  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  40. Manfredi, J.J.: Regularity for minima of functionals with \(p\)-growth. J. Differ. Equ. 76, 203–212 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  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)

    Article  MATH  Google Scholar 

  43. 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 

  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)

    MathSciNet  MATH  Google Scholar 

  45. Mingione, G.: Regularity of minima: an invitation to the dark side of the calculus of variations. Appl. Math. 51, 355–425 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  46. Ok, J.: Gradient estimates for elliptic equations with \(L^{p(\cdot )}\log L\) growth. Calc. Var. PDE 55(2), 30 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  47. Ok, J.: Regularity of \(\omega \)-minimizers for a class of functionals with non-standard growth. Calc. Var. PDE 56(2), 31 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  48. Perera, K., Squassina, M.: Existence results for double phase problems via Morse theory. Commun. Contemp. Math. 20, Article no. 1750023 (2018)

  49. Ragusa, M.A., Tachikawa, A.: Boundary regularity of minimizers of \(p(x)\)-energy functionals. Ann. IHP Anal. non Linéare 33, 451–476 (2017)

    MathSciNet  MATH  Google Scholar 

  50. Ragusa, M.A., Tachikawa, A.: Partial regularity of \(p(x)\)-harmonic maps. Trans. Am. Math. Soc. 365, 3329–3353 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  53. Ural’tseva, N.N.: Degenerate quasilinear elliptic systems. Zap. Na. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 7, 184–222 (1968)

    MathSciNet  MATH  Google Scholar 

  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), 263–270

  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)

    MathSciNet  Google Scholar 

  56. Zhikov, V.V.: On Lavrentiev’s phenomenon. Russ. J. Math. Phys. 3, 249–269 (1995)

    MathSciNet  MATH  Google Scholar 

  57. Zhikov, V.V.: On some variational problems. Russ. J. Math. Phys. 5, 105–116 (1997)

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dipartimento SMFI, Università di Parma, Viale delle Scienze 53/a, Campus, 43124, Parma, Italy

    Paolo Baroni & Giuseppe Mingione

  2. Institute for Theoretical Studies, ETH Zürich, Clausiusstrasse 47, 8092, Zurich, Switzerland

    Maria Colombo

Authors
  1. Paolo Baroni
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Maria Colombo
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Giuseppe Mingione
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Giuseppe Mingione.

Additional information

Communicated by L. Ambrosio.

To Paolo Marcellini on his 70th birthday, with admiration for his pioneering work in the Calculus of Variations.

The authors thank the referee, for his/her careful reading of the original manuscript and for providing several remarks that eventually led to a better final version.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Baroni, P., Colombo, M. & Mingione, G. Regularity for general functionals with double phase. Calc. Var. 57, 62 (2018). https://doi.org/10.1007/s00526-018-1332-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00526-018-1332-z

Mathematics Subject Classification

Search

Navigation