Lipschitz regularity results for a class of obstacle problems with nearly linear growth


This paper deals with the Lipschitz continuity of solutions to variational obstacle problems with nearly linear growth. The main tool used here is a new higher differentiability result which reveals to be crucial because it allows us to perform the linearization procedure to transform the constrained problem in an unconstrained one and it permits us to deduce the equivalence between our minimization problem and its corresponding variational formulation. Our results hold true for a large class of example for which the Lavrentiev phenomenon does not occur, not necessarily for lagrangians dependent on the modulus of the gradient. We assume the same Sobolev regularity both for the gradient of the obstacle and for the coefficients.

This is a preview of subscription content, log in to check access.


  1. 1.

    Acerbi, E., Bouchitté, G., Fonseca, I.: Relaxation of convex functionals: the gap problem. Ann. IHP Anal. Non Linear 20, 359–390 (2003)

    MathSciNet  MATH  Google Scholar 

  2. 2.

    Acerbi, E., Fusco, N.: Regularity for minimizers of nonquadratic functionals: the case \(1 < p < 2\). J. Math. Anal. Appl. 140(1), 115–135 (1989)

    MathSciNet  MATH  Google Scholar 

  3. 3.

    Bella, P., Schäffner, M.: On the regularity of minimizers for scalar integral functionals with \((p,q)-\)growth. Anal. PDE. (2019) (to appear)

  4. 4.

    Benassi, C., Caselli, M.: Lipschitz continuity results for obstacle problems. Rend. Lincei Mat. Appl. 31(1), 191–210 (2020)

    MathSciNet  MATH  Google Scholar 

  5. 5.

    Bögelein, V., Duzaar, F., Marcellini, P.: Parabolic systems with \(p, q\)-growth: a variational approach. Arch. Ration. Mech. Anal. 210(1), 219–267 (2013)

    MathSciNet  MATH  Google Scholar 

  6. 6.

    Breit, D., De Maria, B., Passarelli di Napoli, A.: Regularity for non-autonomous functionals with almost linear growth. Manuscr. Math. 136(1–2), 83–114 (2011)

    MathSciNet  MATH  Google Scholar 

  7. 7.

    Buttazzo, G., Belloni, M.: A survey of old and recent results about the gap phenomenon in the calculus of variations. In: Lucchetti, R., Revalski, J. (eds.) Recent Developments in Well-posed Variational Problems, Mathematical Applications, vol. 331, pp. 1–27. Kluwer Academic Publishers, Dordrecht (1995)

    Google Scholar 

  8. 8.

    Buttazzo, G., Mizel, V.J.: Interpretation of the Lavrentiev phenomenon by relaxation. J. Funct. Anal. 110(2), 434–460 (1992)

    MathSciNet  MATH  Google Scholar 

  9. 9.

    Capone, C.: An higher integrability result for the second derivatives of the solutions to a class of elliptic PDE’s. Manuscr. Math. (2020).

    Article  Google Scholar 

  10. 10.

    Capone, C., Radice, T.: A regularity result for a class of elliptic equations with lower order terms. J. Ellipt. Parabol. Equ. (2020).

    Article  Google Scholar 

  11. 11.

    Carozza, M., Kristensen, J., Passarelli di Napoli, A.: Regularity of minimizers of autonomous convex variational integrals. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) XIII, 1065–1089 (2014)

    MathSciNet  MATH  Google Scholar 

  12. 12.

    Carozza, M., Kristensen, J., Passarelli di Napoli, A.: On the validity of the Euler Lagrange system. Commun. Pure Appl. Anal. 14(1), 51–62 (2018)

    MathSciNet  MATH  Google Scholar 

  13. 13.

    Caselli, M., Eleuteri, M., Passarelli di Napoli, A.: Regularity results for a class of obstacle problems with \(p,q\)-growth conditions. arXiv:1907.08527 (2019) (preprint)

  14. 14.

    Caselli, M., Gentile, A., Giova, R.: Regularity results for solutions to obstacle problems with Sobolev coefficients. J. Differ. Equ. 269(10), 8308–8330 (2020)

    MathSciNet  MATH  Google Scholar 

  15. 15.

    Colombo, M., Mingione, G.: Regularity for double phase variational problems. Arch. Ration. Mech. Anal. 215(2), 443–496 (2015)

    MathSciNet  MATH  Google Scholar 

  16. 16.

    Cupini, G., Giannetti, F., Giova, R., Passarelli di Napoli, A.: Regularity results for vectorial minimizers of a class of degenerate convex integrals. J. Differ. Equ. 265(9), 4375–4416 (2018)

    MathSciNet  MATH  Google Scholar 

  17. 17.

    De Filippis, C.: Regularity results for a class of nonautonomous obstacle problem with \((p, q)\)-growth. J. Math. Anal. Appl. (2019).

    Article  Google Scholar 

  18. 18.

    De Filippis, C., Mingione, G.: On the regularity of minima of nonautonomous functionals. J. Geom. Anal. 30, 1584–1626 (2020)

    MathSciNet  MATH  Google Scholar 

  19. 19.

    De Filippis, C., Palatucci, G.: Hölder regularity for nonlocal double phase equations. J. Differ. Equ. 267(1), 547–586 (2018)

    MATH  Google Scholar 

  20. 20.

    Duzaar, F.: Variational inequalities for harmonic maps. J. Reine Angew. Math. 374, 39–60 (1987)

    MathSciNet  MATH  Google Scholar 

  21. 21.

    Duzaar, F., Grotowski, J.F., Kronz, M.: Regularity of almost minimizers of quasi-convex variational integrals with subquadratic growth. Ann. Mat. Pura Appl. 184, 421–448 (2005)

    MathSciNet  MATH  Google Scholar 

  22. 22.

    Eleuteri, M., Marcellini, P., Mascolo, E.: Lipschitz continuity for energy integrals with variable exponents. Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 27(1), 61–87 (2016)

    MathSciNet  MATH  Google Scholar 

  23. 23.

    Eleuteri, M., Marcellini, P., Mascolo, E.: Lipschitz estimates for systems with ellipticity conditions at infinity. Ann. Mat. Pura Appl. 195(4), 1575–1603 (2016)

    MathSciNet  MATH  Google Scholar 

  24. 24.

    Eleuteri, M., Marcellini, P., Mascolo, E.: Local Lipschitz continuity of minimizers with mild assumptions on the \(x-\)dependence. Discrete Contin. Dyn. Syst. Ser. S 12(2), 251–265 (2019)

    MathSciNet  MATH  Google Scholar 

  25. 25.

    Eleuteri, M., Marcellini, P., Mascolo, E.: Regularity for scalar integrals without structure conditions. Adv. Calc. Var. 13(2), 279–300 (2020)

    MathSciNet  MATH  Google Scholar 

  26. 26.

    Eleuteri, M., Passarelli di Napoli, A.: Higher differentiability for solutions to a class of obstacle problems. Calc. Var. 57(5), 115 (2018)

    MathSciNet  MATH  Google Scholar 

  27. 27.

    Eleuteri, M., Passarelli di Napoli, A.: Regularity results for a class of non-differentiable obstacle problems. Nonlinear Anal. 194, 111434 (2020)

    MathSciNet  MATH  Google Scholar 

  28. 28.

    Eleuteri, M., Passarelli di Napoli, A.: On the validity of variational inequalities for obstacle problems with non-standard growth (submitted)

  29. 29.

    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 

  30. 30.

    Esposito, A., Leonetti, F., Petricca, P.V.: Absence of Lavrentiev gap for non-autonomous functionals with \((p, q)\)-growth. Adv. Nonlinear Anal. 8(1), 73–78 (2019)

    MathSciNet  MATH  Google Scholar 

  31. 31.

    Fuchs, M.: Variational inequalities for vector valued functions with non convex obstacles. Analysis 5, 223–238 (1985)

    MathSciNet  MATH  Google Scholar 

  32. 32.

    Fuchs, M.: \(p\)-harmonic obstacle problems. Part I: partial regularity theory. Ann. Mat. Pura Appl. 156, 127–158 (1990)

    MathSciNet  MATH  Google Scholar 

  33. 33.

    Fuchs, M.: Topics in the Calculus of Variations, Advanced Lectures in Mathematics. Vieweg, Berlin (1994)

    Google Scholar 

  34. 34.

    Fuchs, M., Gongbao, L.: Variational inequalities for energy functionals with nonstandard growth conditions. Abstr. Appl. Anal. 3, 41–64 (1998)

    MathSciNet  MATH  Google Scholar 

  35. 35.

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

    MATH  Google Scholar 

  36. 36.

    Fuchs, M., Seregin, G.: A regularity theory for variational integrals with \(L \ln L\)-growth. Calc. Var. 6, 171–187 (1998)

    MathSciNet  MATH  Google Scholar 

  37. 37.

    Gavioli, C.: Higher differentiability for a class of obstacle problems with nonstandard growth conditions. Forum Mat. 31(6), 1501–1516 (2019)

    MATH  Google Scholar 

  38. 38.

    Gavioli, C.: A priori estimates for solutions to a class of obstacle problems under \(p, q\)-growth conditions. J. Ellipt. Parabol. Equ. 5(2), 325–437 (2019)

    MathSciNet  MATH  Google Scholar 

  39. 39.

    Gavioli, C.: Higher differentiability for a class of obstacle problems with nearly linear growth conditions (submitted)

  40. 40.

    Gentile, A.: Regularity for minimizers of a class of non-autonomous functionals with sub-quadratic growth. Adv. Calc. Var. (2020).

    Article  Google Scholar 

  41. 41.

    Gentile, A.: Higher differentiability results for solutions to a class of non-autonomous obstacle problems with sub-quadratic growth conditions. arXiv:2007.04064(preprint)

  42. 42.

    Giova, R.: Higher differentiability for \(n-\)harmonic systems with Sobolev coefficients. J. Differ. Equ. 259(11), 5667–5687 (2015)

    MathSciNet  MATH  Google Scholar 

  43. 43.

    Giova, R., Passarelli di Napoli, A.: Regularity results for a priori bounded minimizers of non-autonomous functionals with discontinuous coefficients. Adv. Calc. Var. 12(1), 85–110 (2019)

    MathSciNet  MATH  Google Scholar 

  44. 44.

    Giusti, E.: Direct Methods in the Calculus of Variations. World Scientific, Singapore (2003)

    Google Scholar 

  45. 45.

    Greco, L., Iwaniec, T., Sbordone, C.: Variational integrals of nearly linear growth. Differ. Integr. Equ. 10(4), 687–716 (1997)

    MathSciNet  MATH  Google Scholar 

  46. 46.

    Hariulehto, P., Hästö, P.: Double phase image restoration. arXiv:1906.09837 (2019) (preprint)

  47. 47.

    Leone, C., Passarelli di Napoli, A., Verde, A.: Lipschitz regularity for some asymptotically subquadratic problems. Nonlinear Anal. 67, 1532–1539 (2007)

    MathSciNet  MATH  Google Scholar 

  48. 48.

    Marcellini, P.: On the definition and the lower semicontinuity of certain quasiconvex integrals. Ann. I. Poincaré Sect. C Tome 3(5), 391–409 (1986)

    MathSciNet  MATH  Google Scholar 

  49. 49.

    Marcellini, P.: Un example de solution discontinue d’un problème variationnel dans le cas scalaire, Preprint 11, Istituto Matematico “U. Dini”, Università di Firenze (1987)

  50. 50.

    Marcellini, P.: Regularity of minimizers of integrals of the calculus of variations with nonstandard growth conditions. Arch. Ration. Mech. Anal. 105(3), 267–284 (1989)

    MathSciNet  MATH  Google Scholar 

  51. 51.

    Marcellini, P.: Regularity and existence of solutions of elliptic equations with \(p, q\)-growth conditions. J. Differ. Equ. 90(1), 1–30 (1991)

    MathSciNet  MATH  Google Scholar 

  52. 52.

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

    MathSciNet  MATH  Google Scholar 

  53. 53.

    Marcellini, P.: A variational approach to parabolic equations under general and \(p, q\)-growth conditions. Nonlinear Anal. 194, 111456 (2019)

    MathSciNet  MATH  Google Scholar 

  54. 54.

    Marcellini, P.: Regularity under general and \(p, q\)-growth conditions. Discrete Contin. Dyn. Syst. S Ser. 13, 2009–2031 (2020)

    MathSciNet  MATH  Google Scholar 

  55. 55.

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

    MathSciNet  MATH  Google Scholar 

  56. 56.

    Passarelli di Napoli, A.: Existence and regularity results for a class of equations with logarithmic growth. Nonlinear Anal. 125, 290–309 (2015)

    MathSciNet  MATH  Google Scholar 

  57. 57.

    Passarelli di Napoli, A.: Higher differentiability of minimizers of variational integrals with Sobolev coefficients. Adv. Calc. Var. 7(1), 59–89 (2014)

    MathSciNet  MATH  Google Scholar 

  58. 58.

    Passarelli di Napoli, A.: Higher differentiability of solutions of elliptic systems with Sobolev coefficients: the case \(p = n = 2\). Potent. Anal. 41(3), 715–735 (2014)

    MathSciNet  MATH  Google Scholar 

  59. 59.

    Passarelli di Napoli, A.: Regularity results for non-autonomous variational integrals with discontinuous coefficients. Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 26(4), 475–496 (2015)

    MathSciNet  MATH  Google Scholar 

  60. 60.

    Ragusa, M.A., Tachikawa, A.: Regularity for minimizers for functionals of double phase with variable exponents. Adv. Nonlinear Anal. 9(1), 710–728 (2019)

    MathSciNet  MATH  Google Scholar 

  61. 61.

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

    MathSciNet  MATH  Google Scholar 

Download references

Author information



Corresponding author

Correspondence to Samuele Riccò.

Ethics declarations

Conflicts of interest

The authors declare that they have no conflict of interest.

Ethical approval

This article does not contain any studies with human participants or animals performed by the authors.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Bertazzoni, G., Riccò, S. Lipschitz regularity results for a class of obstacle problems with nearly linear growth. J Elliptic Parabol Equ 6, 883–918 (2020).

Download citation


  • Regularity of solutions
  • Obstacle problems
  • A priori estimates
  • Nearly linear growth

Mathematics Subject Classification

  • Primary 49N60
  • 35J85
  • Secondary 35B65
  • 35B45
  • 49J40