Fully anisotropic elliptic problems with minimally integrable data

  • Angela Alberico
  • Iwona Chlebicka
  • Andrea CianchiEmail author
  • Anna Zatorska-Goldstein


We investigate nonlinear elliptic Dirichlet problems whose growth is driven by a general anisotropic N-function, which is not necessarily of power-type and need not satisfy the \(\Delta _2\) nor the \(\nabla _2\)-condition. Fully anisotropic, non-reflexive Orlicz–Sobolev spaces provide a natural functional framework associated with these problems. Minimal integrability assumptions are detected on the datum on the right-hand side of the equation ensuring existence and uniqueness of weak solutions. When merely integrable, or even measure, data are allowed, existence of suitably further generalized solutions—in the approximable sense—is established. Their maximal regularity in Marcinkiewicz-type spaces is exhibited as well. Uniqueness of approximable solutions is also proved in case of \(L^1\)-data.

Mathematics Subject Classification

35J25 35J60 35B65 



We wish to thank the referee for his careful reading of the paper, for his valuable comments.


This research was partly funded by: (i) Italian Ministry of University and Research (MIUR), Research Project Prin 2015 “Partial differential equations and related analytic-geometric inequalities”, No. 2015HY8JCC; (ii) GNAMPA of the Italian INdAM—National Institute of High Mathematics (Grant No. not available); (iii) NCN Polish Grant No. 2016/23/D/ST1/01072.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.


  1. 1.
    Adams, R.A., Fournier, J.J.F.: Sobolev Spaces, second edn. Academic Press, Amsterdam (2003)zbMATHGoogle Scholar
  2. 2.
    Alberico, A.: Boundedness of solutions to anisotropic variational problems. Commun. Partial Differ. Equ. 36, 470–486 (2011). et Erratum et Corrigendum to: “Boundedness of solutions to anisotropic variational problems”. Commun. Partial Differ. Equ. 36, 470–486 (2011); Commun. Partial Differ. Equ. 41(2016), 877–878Google Scholar
  3. 3.
    Alberico, A., di Blasio, G., Feo, F.: An eigenvalue problem for an anisotropic \({\Phi } \)–Laplacian. arXiv:1906.07593v2
  4. 4.
    Alberico, A., di Blasio, G., Feo, F.: A priori estimates for solutions to anisotropic elliptic problems via symmetrization. Math. Nachr. 290, 986–1003 (2017) MathSciNetCrossRefGoogle Scholar
  5. 5.
    Alberico, A., Cianchi, A.: Comparison estimates in anisotropic variational problems. Manuscr. Math. 126, 481–503 (2008)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Alvino, A., Ferone, V., Trombetti, G.: Estimates for the gradient of solutions of nonlinear elliptic equations with \(L^1\) data. Ann. Mat. Pura. Appl. 178, 129–142 (2000)Google Scholar
  7. 7.
    Alvino, A., Mercaldo, A.: Nonlinear elliptic problems with \(L^1\) data: an approach via symmetrization methods. Mediterr. J. Math. 5, 173–185 (2008)Google Scholar
  8. 8.
    Barletta, G., Cianchi, A.: Dirichlet problems for fully anisotropic elliptic equations. Proc. R. Soc. Edinb. Sect. A 147, 25–60 (2017)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Baroni, P.: Riesz potential estimates for a general class of quasilinear equations. Calc. Var. Partial Differ. Equ. 53, 803–846 (2015)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Beck, L., Mingione, G.: Lipschitz bounds and non-uniform ellipticity. Commun. Pure Appl. Math. (to appear)Google Scholar
  11. 11.
    Bénilan, P., Boccardo, L., Gallouët, T., Gariepy, R., Pierre, M., Vázquez, J.-L.: An \(L^1\)-theory of existence and uniqueness of solutions of nonlinear elliptic equations. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 22, 241–273 (1995)Google Scholar
  12. 12.
    Benkirane, A., Bennouna, J.: Existence of renormalized solutions for some elliptic problems involving derivatives of nonlinear terms in Orlicz spaces. In: Partial Differential Equations. Lecture Notes in Pure and Applied Mathematics, vol. 229, pp. 139–147. Dekker, New York (2002)Google Scholar
  13. 13.
    Bennett, C., Sharpley, R.: Interpolation of Operators. Academic Press, Boston (1988) zbMATHGoogle Scholar
  14. 14.
    Boccardo, L., Gallouët, T.: Nonlinear elliptic and parabolic equations involving measure data. J. Funct. Anal. 87, 149–169 (1989)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Boccardo, L., Gallouët, T., Marcellini, P.: Anisotropic equations in \(L^1\). Differ. Integral Equ. 9, 209–212 (1996)Google Scholar
  16. 16.
    Bogovskiĭ, M.E.: Solution of the first boundary value problem for an equation of continuity of an incompressible medium. Dokl. Akad. Nauk SSSR 248, 1037–1040 (1979)MathSciNetGoogle Scholar
  17. 17.
    Carozza, M., Kristensen, J., Passarelli di Napoli, A.: Regularity of minimizers of autonomous convex variational integrals. Ann. Sc. Norm. Super. Pisa Cl. Sci. 13, 1065–1089 (2014)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Carozza, M., Passarelli di Napoli, A.: Partial regularity for anisotropic functionals of higher order. ESAIM Control Optim. Calc. Var. 13, 692–706 (2007)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Chlebicka, I.: A pocket guide to nonlinear differential equations in Musielak–Orlicz spaces. Nonlinear Anal. 175, 1–27 (2018)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Chlebicka, I.: Gradient estimates for problems with Orlicz growth. Nonlinear Anal. (2018).
  21. 21.
    Chlebicka, I.: Regularizing effect of the lower-order terms in elliptic problems with Orlicz growth. Israel J. Math. (to appear)Google Scholar
  22. 22.
    Cianchi, A.: Boundedness of solutions to variational problems under general growth conditions. Commun. Partial Differ. Equ. 22, 1629–1646 (1997)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Cianchi, A.: A fully anisotropic Sobolev inequality. Pac. J. Math. 196, 283–295 (2000)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Cianchi, A.: Local boundedness of minimizers of anisotropic functionals. Ann. Inst. H. Poincaré Anal. Nonlinéaire 17, 147–168 (2000)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Cianchi, A.: Optimal Orlicz–Sobolev embeddings. Rev. Mat. Iberoam. 20, 427–474 (2004)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Cianchi, A.: Symmetrization in anisotropic elliptic problems. Commun. Partial Differ. Equ. 32, 693–717 (2007)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Cianchi, A., Maz’ya, V.: Quasilinear elliptic problems with general growth and merely integrable, or measure, data. Nonlinear Anal. 164, 189–215 (2017)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Dal Maso, A.: Approximated solutions of equations with \(L^1\) data. Application to the \(H\)-convergence of quasi-linear parabolic equations. Ann. Mat. Pura Appl. 170, 207–240 (1996)Google Scholar
  29. 29.
    Dal Maso, G., Murat, F., Orsina, L., Prignet, A.: Renormalized solutions of elliptic equations with general measure data. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 28, 741–808 (1999)Google Scholar
  30. 30.
    Dolzmann, G., Hungerbühler, N., Müller, S.: Uniqueness and maximal regularity for nonlinear elliptic systems of \(n\)-Laplace type with measure valued right hand side. J. Reine Angew. Math. 520, 1–35 (2000)Google Scholar
  31. 31.
    Donaldson, T.K.: Nonlinear elliptic boundary value problems in Orlicz–Sobolev spaces. J. Differ. Equ. 10, 507–528 (1971)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Donaldson, T.K., Trudinger, N.S.: Orlicz–Sobolev spaces and imbedding theorems. J. Funct. Anal. 8, 52–75 (1971)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Dong, G., Fang, X.: Existence results for some nonlinear elliptic equations with measure data in Orlicz–Sobolev spaces. Bound. Value Probl. (2015).
  34. 34.
    Elmahi, A., Meskine, D.: Elliptic inequalities with lower order terms and \(L^1\) data in Orlicz spaces. J. Math. Anal. Appl. 328, 1417–1434 (2007)Google Scholar
  35. 35.
    Esposito, L., Leonetti, F., Mingione, G.: Sharp regularity for functionals with \((p, q)\) growth. J. Differ. Equ. 204, 5–55 (2004)Google Scholar
  36. 36.
    Fiorenza, A., Sbordone, C.: Existence and uniqueness results for solutions of nonlinear equations with right-hand side in \(L^1\). Stud. Math. 127, 223–231 (1998)Google Scholar
  37. 37.
    Gossez, J.-P.: Nonlinear elliptic boundary value problems for equations with rapidly (or slowly) increasing coefficients. Trans. Am. Math. Soc. 190, 163–205 (1974)MathSciNetCrossRefGoogle Scholar
  38. 38.
    Gossez, J.-P.: Some approximation properties in Orlicz–Sobolev spaces. Stud. Math. 74, 17–24 (1982)MathSciNetCrossRefGoogle Scholar
  39. 39.
    Gwiazda, P., Skrzypczak, I., Zatorska-Goldstein, A.: Existence of renormalized solutions to elliptic equation in Musielak–Orlicz space. J. Differ. Equ. 264, 341–377 (2018)MathSciNetCrossRefGoogle Scholar
  40. 40.
    Gwiazda, P., Wittbold, P., Wróblewska, A., Zimmermann, A.: Renormalized solutions of nonlinear elliptic problems in generalized Orlicz spaces. J. Differ. Equ. 253, 635–666 (2012)MathSciNetCrossRefGoogle Scholar
  41. 41.
    Innamorati, A., Leonetti, F.: Global integrability for weak solutions to some anisotropic elliptic equations. Nonlinear Anal. 113, 430–434 (2015)MathSciNetCrossRefGoogle Scholar
  42. 42.
    Klimov, V.S.: Imbedding theorems and geometric inequalities. Izv. Akad. Nauk SSSR Ser. Mat. 40, 645–671 (in Russian); English translation: Math. USSR Izvestiya 10(1976), 615–638 (1976)Google Scholar
  43. 43.
    Korolev, A.G.: On the boundedness of generalized solutions of elliptic differential equations with nonpower nonlinearities. Mat. Sb. 180, 78–100 (1989) (in Russian); English translation: Math. USSR-Sb. 66(1990), 83–106 (1989)Google Scholar
  44. 44.
    Lieberman, G.: The natural generalization of the natural conditions of Ladyzhenskaya and Ural’tseva for elliptic equations. Commun. Partial Differ. Equ. 16, 311–361 (1991)CrossRefGoogle Scholar
  45. 45.
    Lions, P.-L., Murat, F.: Sur les solutions renormalisées d’équations elliptiques non linéaires. ManuscriptGoogle Scholar
  46. 46.
    Marcellini, P.: Regularity and existence of solutions of elliptic equations with \(p, q\)-growth conditions. J. Differ. Equ. 90, 1–30 (1991)Google Scholar
  47. 47.
    Mingione, G.: The Calderón–Zygmund theory for elliptic problems with measure data. Ann. Sc. Norm. Super. Pisa Cl. Sci. 6, 195–261 (2007)MathSciNetzbMATHGoogle Scholar
  48. 48.
    Mingione, G.: Gradient estimates below the duality exponent. Math. Ann. 346, 571–627 (2010)MathSciNetCrossRefGoogle Scholar
  49. 49.
    Mustonen, V., Tienari, M.: On monotone-like mappings in Orlicz–Sobolev spaces. Math. Bohem. 124, 255–271 (1999)MathSciNetzbMATHGoogle Scholar
  50. 50.
    Rakotoson, J.-M.: Uniqueness of renormalized solutions in a \(T\)-set for the \(L^1\)-data problem and the link between various formulations. Indiana Univ. Math. J. 43, 685–702 (1994)Google Scholar
  51. 51.
    Rao, M.M., Ren, Z.D.: Theory of Orlicz Spaces. Marcel Dekker, New York (1991)zbMATHGoogle Scholar
  52. 52.
    Rao, M.M., Ren, Z.D.: Applications of Orlicz Spaces. Marcel Dekker, New York (2002)zbMATHGoogle Scholar
  53. 53.
    Serrin, J.: Pathological solutions of elliptic differential equations. Ann. Scuola Norm. Sup. Pisa 18, 385–387 (1964)MathSciNetzbMATHGoogle Scholar
  54. 54.
    Schappacher, G.: A notion of Orlicz spaces for vector valued functions. Appl. Math. 50, 355–386 (2005)MathSciNetCrossRefGoogle Scholar
  55. 55.
    Skaff, M.S.: Vector valued Orlicz spaces generalized \(N\)-functions, I. Pac. J. Math. 28, 193–206 (1969)Google Scholar
  56. 56.
    Skaff, M.S.: Vector valued Orlicz spaces, II. Pac. J. Math. 28, 413–430 (1969)MathSciNetCrossRefGoogle Scholar
  57. 57.
    Stroffolini, B.: Global boundedness of solutions of anisotropic variational problems. Boll. Un. Mat. Ital. A 5, 345–352 (1991)MathSciNetzbMATHGoogle Scholar
  58. 58.
    Talenti, G.: Nonlinear elliptic equations, rearrangements of functions an Orlicz spaces. Ann. Mat. Pura Appl. 120, 159–184 (1979)MathSciNetCrossRefGoogle Scholar
  59. 59.
    Talenti, G.: Boundedness of minimizers. Hokkaido Math. J. 19, 259–279 (1990)MathSciNetCrossRefGoogle Scholar
  60. 60.
    Trudinger, N.S.: An imbedding theorem for \(H_{0}(G,\,\Omega )\) spaces. Stud. Math. 50, 17–30 (1974)Google Scholar
  61. 61.
    Vétois, J.: Existence and regularity for critical anisotropic equations with critical directions. Adv. Differ. Equ. 16, 61–83 (2011)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • Angela Alberico
    • 3
  • Iwona Chlebicka
    • 2
  • Andrea Cianchi
    • 1
    • 3
    Email author
  • Anna Zatorska-Goldstein
    • 2
  1. 1.Dipartimento di Matematica e Informatica “U. Dini”Università di FirenzeFirenzeItaly
  2. 2.Institute of Applied Mathematics and MechanicsUniversity of WarsawWarsawPoland
  3. 3.Istituto per le Applicazioni del Calcolo “M. Picone”Consiglio Nazionale delle RicercheNaplesItaly

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