Skip to main content
SpringerLink
Log in

Nonlinear Measure Data Problems

  • Published:
Milan Journal of Mathematics Aims and scope Submit manuscript

Abstract

We describe some basic results from regularity theory for solutions to elliptic quasilinear equations involving an assigned measure datum and we include some new integrability and differentiability results for sublinear problems.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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., Mingione G.: Gradient estimtes for the p(x)-Laplacean system. J. Reine Ang. Math. (Crelles J) 584, 117–148 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  2. Acerbi E., Mingione G.: Gradient estimates for a class of parabolic systems. Duke Math. J 136, 285–320 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  3. Adams D. R.: Traces of potentials arising from translation invariant operators. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (III) 25, 203–217 (1971)

    MATH  Google Scholar 

  4. Adams D.R.: A note on Riesz potentials. Duke Math. J 42, 765–778 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  5. Adams D.R., Hedberg L.I.: Function spaces and potential theory. Grundlehren der Mathematischen Wissenschaften 314. Springer-Verlag, Berlin (1996)

    Google Scholar 

  6. Adams D.R., Meyers N.G.: Thinnes and Wiener criteria for nonlinear potentials. Indiana Univ. Math. J 22, 169–197 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  7. Adams R.A.: Sobolev Spaces. Academic Press, New York (1975)

    MATH  Google Scholar 

  8. Alvino A., Ferone V., Trombetti G.: Estimates for the gradient of solutions of nonlinear elliptic equations with L 1-data. Ann. Mat. Pura Appl. (IV) 178, 129–142 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  9. Barenblatt G. I.: On self-similar motions of a compressible fluid in a porous medium. Akad. Nauk SSSR. Prikl. Mat. Meh 16, 679–698 (1952)

    MathSciNet  Google Scholar 

  10. Baroni P. & Habermann, J.: Calderón-Zygmund estimates for parabolic measure data problems. J. Differential Equations, doi:10.1016/j.jde.2011.08.016

  11. Baroni P. & Habermann, J.: New gradient estimates for parabolic equations. Houston J. Math., to appear.

  12. Benilan 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. (IV) 22, 241–273 (1995)

    MATH  Google Scholar 

  13. Boccardo L.: Problemi differenziali ellittici e parabolici con dati misure. Boll. Un. Mat. Ital. A 7(11), 439–461 (1997)

    MathSciNet  Google Scholar 

  14. Boccardo L.: Marcinkiewicz estimates for solutions of some elliptic problems with nonregular data. Ann. Mat. Pura Appl. (IV) 188, 591–601 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  15. Boccardo L., Dall’Aglio A., Gallouët T., Orsina L.: Nonlinear parabolic equations with measure data. J. Funct. Anal 147, 237–258 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  16. Boccardo L., Gallouët T.: Nonlinear elliptic and parabolic equations involving measure data. J. Funct. Anal 87, 149–169 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  17. Boccardo L., Boccardo L., Boccardo L.: Nonlinear elliptic equations with right-hand side measures. Comm. Partial Differential Equations 17, 641–655 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  18. Bojarski B., Iwaniec T.: Analytical foundations of the theory of quasiconformal mappings in \({\mathbb{R}_n}\). Ann. Acad. Sci. Fenn. Ser. A I Math 8, 257–324 (1983)

    MathSciNet  MATH  Google Scholar 

  19. Calderón A. P., Zygmund A.: On the existence of certain singular integrals. Acta Math 88, 85–139 (1952)

    Article  MathSciNet  MATH  Google Scholar 

  20. Calderón A.P., Zygmund A.: On singular integrals. Amer. J. Math 78, 289–309 (1956)

    Article  MathSciNet  MATH  Google Scholar 

  21. Dall’Aglio A.: Approximated solutions of equations with L 1 data. Application to the Hconvergence of quasi-linear parabolic equations. Ann. Mat. Pura Appl. (IV) 170, 207–240 (1996)

    MathSciNet  MATH  Google Scholar 

  22. 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. (IV) 28, 741–808 (1999)

    MathSciNet  MATH  Google Scholar 

  23. De Giorgi E.: Sulla differenziabilità e l’analiticit‘a delle estremali degli integrali multipli regolari. Mem. Accad. Sci. Torino Cl. Sci. Fis. Mat. Nat. (III) 125(3), 25–43 (1957)

    MathSciNet  Google Scholar 

  24. DeVore, R.A. & Sharpley, R.C.: Maximal functions measuring smoothness. Mem. Amer. Math. Soc. 47 (1984), no. 293.

    Google Scholar 

  25. DiBenedetto E.: C 1+α local regularity of weak solutions of degenerate elliptic equations. Nonlinear Anal. TMA 7, 827–850 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  26. DiBenedetto E.: On the local behaviour of solutions of degenerate parabolic equations with measurable coefficients. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (IV) 13, 487–535 (1986)

    MathSciNet  MATH  Google Scholar 

  27. DiBenedetto E.: Degenerate parabolic equations Universitext. Springer-Verlag, New York (1993)

    Book  Google Scholar 

  28. DiBenedetto E., Friedman A.: Hölder estimates for nonlinear degenerate parabolic systems. J. reine ang. Math. (Crelles J.) 357, 1–22 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  29. DiBenedetto E., Gianazza U., Vespri V.: Alternative forms of the Harnack inequality for Non-Negative solutions to certain degenerate and singular parabolic equations. Rendiconti Lincei - Matematica e Applicazioni 20, 369–377 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  30. DiBenedetto E., Gianazza U., Vespri V.: Forward, backward and elliptic Harnack inequalities for non-negative solutions to certain singular parabolic partial differential equations. Ann. Scu. Norm. Sup. Pisa Cl. Sci. (V) 9, 385–422 (2010)

    MathSciNet  MATH  Google Scholar 

  31. DiBenedetto E., Manfredi J.J.: On the higher integrability of the gradient of weak solutions of certain degenerate elliptic systems. Amer. J. Math. 115, 1107–1134 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  32. Dolzmann G., Hungerbühler N., Müller S.: The p-harmonic system with measurevalued right hand side. Ann. Inst. H. Poincarè, Anal. Non Linèaire 14, 353–364 (1997)

    Article  MATH  Google Scholar 

  33. Dolzmann G., Hungerbühler N., Müller S.: Non-linear elliptic systems with measure-valued right hand side. Math. Z. 226, 545–574 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  34. 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. (Crelles J.) 520, 1–35 (2000)

    Article  MATH  Google Scholar 

  35. Duzaar F., Kristensen J., Mingione G.: The existence of regular boundary points for non-linear elliptic systems. J. Reine Ang. Math. (Crelles J.) 602, 17–58 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  36. Duzaar F., Mingione G.: Gradient estimates in nonlinear potential theory. Rendiconti Lincei - Matematica e Applicazioni 20, 179–190 (2009)

    MathSciNet  MATH  Google Scholar 

  37. Duzaar F., Mingione G.: Gradient estimates via non-linear potentials. Amer. J.Math 133, 1093–1149 (2011)

    MATH  Google Scholar 

  38. Duzaar F., Mingione G.: Gradient estimates via linear and nonlinear potentials. J. Funct. Anal 259, 2961–2998 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  39. Duzaar F., Mingione G.: Gradient continuity estimates. Calc. Var. & PDE 39, 379–418 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  40. Duzaar F., Mingione G.: Local Lipschitz regularity for degenerate elliptic systems. Ann. Inst. H. Poincaré, Anal. Non Linèaire 27, 1361–1396 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  41. Esposito L., Leonetti F., Mingione G.: Higher integrability for minimizers of integral functionals with (p, q) growth. J. Differential Equations 157, 414–438 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  42. Esposito L., Leonetti F., Mingione G.: Sharp regularity for functionals with (p, q) growth. J. Differential Equations 204, 5–55 (2004)

    MathSciNet  MATH  Google Scholar 

  43. Fuchs M., Reuling J.: Nonlinear elliptic systems involving measure data. Rend. Mat. Appl. (VII) 15, 311–319 (1995)

    MathSciNet  MATH  Google Scholar 

  44. Giusti E.: Direct methods in the calculus of variations. World Scientific Publishing Co., Inc., River Edge, NJ, 2003.

  45. Grafakos, L.: Classical and modern Fourier analysis. Pearson Edu. Inc., Upper Saddle River, NJ, 2004.

  46. Hamburger C.: Regularity of differential forms minimizing degenerate elliptic functionals. J. Reine Angew. Math. (Crelles J.) 431, 7–64 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  47. Havin M., Maz’ya V.G.: A nonlinear potential theory. Russ. Math. Surveys 27, 71–148 (1972)

    Google Scholar 

  48. Hedberg L.I., Wolff T.: Thin sets in nonlinear potential theory. Ann. Inst. Fourier (Grenoble) 33, 161–187 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  49. Heinonen, J. & Kilpeläinen, T. & Martio, O.: Nonlinear potential theory of degenerate elliptic equations. Oxford Mathematical Monographs., New York, 1993.

  50. Iwaniec T.: Projections onto gradient fields and L p-estimates for degenerated elliptic operators. Studia Math 75, 293–312 (1983)

    MathSciNet  MATH  Google Scholar 

  51. Jin T., Mazya V., Van Schaftingen J.: Pathological solutions to elliptic problems in divergence form with continuous coefficients. Comptes Rendus Mathematique 347, 773–778 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  52. Kilpeläinen T., Li G.: Estimates for p-Poisson equations. Diff. Int. Equ 13, 791–800 (2000)

    MATH  Google Scholar 

  53. Kilpeläinen T., Malý J.: The Wiener test and potential estimates for quasilinear elliptic equations. Acta Math 172, 137–161 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  54. Kilpeläinen, T. & Kuusi, T. & Tuhola-Kujanpää, A.: Superharmonic functions are locally renormalized solutions. Ann. Inst. H. Poincarè, Anal. Non Linèaire, doi:10.1016/j.anihpc.2011.03.004

  55. 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 

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

    Article  MathSciNet  MATH  Google Scholar 

  57. Korte R., Kuusi T.: A note on the Wolff potential estimate for solutions to elliptic equations involving measures. Adv. Calc. Var 3, 99–113 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  58. Kuusi T.: Harnack estimates for weak supersolutions to nonlinear degenerate parabolic equations. Ann. Sc. Norm. Super. Pisa Cl. Sci. (V) 7, 673–716 (2008)

    MathSciNet  MATH  Google Scholar 

  59. Kuusi, T. & Mingione, G.: Potential estimates and gradient boundedness for nonlinear parabolic systems. Rev. Mat. Iberoamericana, to appear.

  60. Kuusi, T. & Mingione, G.: Universal potential estimates. Preprint 2010.

  61. Kuusi T., Mingione G.: Endpoint and intermediate potential estimates for nonlinear equations. Boll. UMI (Ser. IX) 4, 149–157 (2011)

    MathSciNet  Google Scholar 

  62. Kuusi T., Mingione G.: Nonlinear potential estimates in parabolic problems. Rendiconti Lincei - Matematica e Applicazioni 22, 161–174 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  63. Kuusi, T. & Mingione, G.: TheWolff gradient bound for degenerate parabolic equations. Preprint 2010.

  64. Kuusi, T. & Mingione, G.: Gradient regularity for nonlinear parabolic equations. Preprint 2011.

  65. Kuusi, T. & Mingione, G.: New perturbation methods in nonlinear parabolic problems. Preprint 2011.

  66. Kuusi, T. & Mingione, G.: Interpolative intrinsic geometries and the regularity of nonlinear parabolic equations. Preprint 2011.

  67. Kuusi T., Mingione G.: A surprising linear type estimate for nonlinear elliptic equations. C. R. Acad. Sci. Paris Ser. I 349, 889–892 (2011)

    MATH  Google Scholar 

  68. Kuusi, T. & Mingione, G.: Linear potentials in nonlinear potential theory. Preprint 2011.

  69. Kuusi T., Parviainen M.: Existence for a degenerate Cauchy problem. Manuscripta Math 128, 213–249 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  70. Leonardi S., Stará J.: Regularity results for the gradient of solutions of linear elliptic systems with VMO-coefficients and L 1,λ data. Forum Math. 22, 913–940 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  71. Leray J., Lions J.-L.: Quelques résulatats de Višik sur les problèmes elliptiques nonlinéaires par les méthodes de Minty-Browder. Bull. Soc. Math. France 93, 97–107 (1965)

    MathSciNet  MATH  Google Scholar 

  72. Lieberman G.M.: Sharp forms of estimates for subsolutions and supersolutions of quasilinear elliptic equations involving measures. Comm. PDE 18, 1191–1212 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  73. Lindqvist P.: On the definition and properties of p-superharmonic functions. J. Reine Angew. Math. (Crelles J.) 365, 67–79 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  74. Lukkari T., Maeda F.Y., Marola N.: Wolff potential estimates for elliptic equations with nonstandard growth and applications. Forum Math. 22, 1061–1087 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  75. Mingione G.: The singular set of solutions to non-differentiable elliptic systems. Arch. Ration. Mech. Anal 166, 287–301 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  76. Mingione G.: Regularity of minima: an invitation to the Dark Side of the Calculus of Variations. Applications of Mathematics 51, 355–425 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  77. Mingione G.: The Calderón-Zygmund theory for elliptic problems with measure data. Ann Scu. Norm. Sup. Pisa Cl. Sci. (V) 6, 195–261 (2007)

    MathSciNet  MATH  Google Scholar 

  78. Mingione G.: Gradient estimates below the duality exponent. Math. Ann. 346, 571–627 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  79. Mingione G.: Gradient potential estimates. J. Europ. Math. Soc 13, 459–486 (2011)

    MathSciNet  MATH  Google Scholar 

  80. Mingione G.: Nonlinear aspects of Calderón-Zygmund theory. Jahresbericht der DMV 112, 159–191 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  81. Phuc N.C., Verbitsky I.E.: Quasilinear and Hessian equations of Lane-Emden type. Ann. of Math. (II) 168, 859–914 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  82. Phuc N.C., Verbitsky I.E.: Quasilinear and Hessian equations of Lane-Emden type. J. Funct. Anal 256, 1875–1906 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  83. Scheven, C.: Elliptic obstacle problems with measure data: Potentials and low order regularity. Preprint 2011.

  84. Scheven, C.: Gradient potential estimates in non-linear elliptic obstacle problems with measure data. Preprint 2011.

  85. Serrin J.: Pathological solutions of elliptic differential equations. Ann. Scuola Norm. Sup. Pisa (III) 18, 385–387 (1964)

    MathSciNet  MATH  Google Scholar 

  86. Serrin J.: Local behavior of solutions of quasi-linear equations. Acta Math 111, 247–302 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  87. Stein, E. M.: Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals. Princeton Math. Series, 43. Princeton University Press, Princeton, NJ, 1993.

  88. Stein, E. M. & Weiss, G.: Introduction to Fourier analysis on Euclidean spaces. Princeton Math. Ser., 32. Princeton Univ. Press, Princeton, N.J. 1971.

  89. Talenti G.: Elliptic equations and rearrangements. Ann Scu. Norm. Sup. Pisa Cl. Sci. (IV) 3, 697–717 (1976)

    MathSciNet  MATH  Google Scholar 

  90. Trudinger N.S., Wang X.J.: On the weak continuity of elliptic operators and applications to potential theory. Amer. J. Math 124, 369–410 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  91. Trudinger N.S., Wang X.J.: Quasilinear elliptic equations with signed measure data. Disc. Cont. Dyn. Systems A 23, 477–494 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  92. Vázquez, J.L.: Smoothing and decay estimates for nonlinear diffusion equations. Equations of porous medium type. Oxford Lecture Series in Math. Appl., 33. Oxford University Press, Oxford, 2006. xiv+234 pp.

  93. Zhong X.: On nonhomogeneous quasilinear elliptic equations. Dissertation, University of Jyväskylä, 1998. Ann. Acad. Sci. Fenn. Math. Diss. 117 (1998).

Download references

Author information

Authors and Affiliations

  1. Dipartimento di Matematica, Università di Parma, Viale G. P. Usberti 53/a, Campus, I-43100, Parma, Italy

    Giuseppe Mingione

Authors
  1. Giuseppe Mingione
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Giuseppe Mingione.

Additional information

To Lucio Boccardo on his mth birthday, $${m \in}$$ (60, 65)

This work has been supported by the ERC grant 207573 “Vectorial problems”.

Lecture held in the Seminario Matematico e Fisico di Milano on March 4, 2010

Rights and permissions

Reprints and permissions

About this article

Cite this article

Mingione, G. Nonlinear Measure Data Problems. Milan J. Math. 79, 429–496 (2011). https://doi.org/10.1007/s00032-011-0168-1

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00032-011-0168-1

Mathematics Subject Classification (2010)

Keywords

Search

Navigation