Skip to main content
Log in

Nonlocal Equations with Measure Data

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

We develop an existence, regularity and potential theory for nonlinear integrodifferential equations involving measure data. The nonlocal elliptic operators considered are possibly degenerate and cover the case of the fractional p-Laplacean operator with measurable coefficients. We introduce a natural function class where we solve the Dirichlet problem, and prove basic and optimal nonlinear Wolff potential estimates for solutions. These are the exact analogs of the results valid in the case of local quasilinear degenerate equations established by Boccardo and Gallouët (J Funct Anal 87:149–169, 1989, Partial Differ Equ 17:641–655, 1992) and Kilpeläinen and Malý (Ann Scuola Norm Sup Pisa Cl Sci (IV) 19:591–613, 1992, Acta Math 172:137–161, 1994). As a consequence, we establish a number of results that can be considered as basic building blocks for a nonlocal, nonlinear potential theory: fine properties of solutions, Calderón–Zygmund estimates, continuity and boundedness criteria are established via Wolff potentials. A main tool is the introduction of a global excess functional that allows us to prove a nonlocal analog of the classical theory due to Campanato (Ann Mat Pura Appl (IV) 69:321–381, 1965). Our results cover the case of linear nonlocal equations with measurable coefficients, and the one of the fractional Laplacean, and are new already in such cases.

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

Access this article

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.

Similar content being viewed by others

References

  1. Alberti G., Bellettini G.: A nonlocal anisotropic model for phase transitions. I. The optimal profile problem. Math. Ann. 310, 527–560 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  2. Alibaud N., Andreianov B., Bendahmane M.: Renormalized solutions of the fractional Laplace equation. C. R. Math. Acad. Sci. Paris 348, 759–762 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  3. Barles G., Chasseigne E., Imbert C.: Hölder continuity of solutions of second-order non-linear elliptic integro-differential equations. J. Eur. Math. Soc. (JEMS) 13, 1–26 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  4. 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. (IV) 22, 241–273 (1995)

    MATH  Google Scholar 

  5. Bjorland C., Caffarelli L., Figalli A.: Non-local tug-of-war and the infinity fractional Laplacian. Adv. Math. 230, 1859–1894 (2012)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  7. Boccardo L., Gallouët T.: Nonlinear elliptic equations with right-hand side measures. Commun. Partial Differ. Equ. 17, 641–655 (1992)

    Article  MATH  Google Scholar 

  8. Boccardo L., Gallouët T.: W 1,1-solutions in some borderline cases of Calderón–Zygmund theory. J. Differ. Equ. 253, 2698–2714 (2012)

    Article  ADS  MATH  Google Scholar 

  9. Boccardo L., Gallouët T., Orsina L.: Existence and uniqueness of entropy solutions for nonlinear elliptic equations with measure data. Ann. Inst. H. Poincarè Anal. Non Linéaire 13, 539–551 (1996)

    MATH  Google Scholar 

  10. Bonforte M., Väzquez J.L.: Quantitative local and global a priori estimates for fractional nonlinear diffusion equations. Adv. Math. 250, 242–284 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  11. Cabré X., Sola-Morales X.: Layer solutions in a half-space for boundary reactions. Commun. Pure Appl. Math. 58, 1678–1732 (2005)

    Article  MATH  Google Scholar 

  12. Cacace S., Garroni A.: A multi-phase transition model for the dislocations with interfacial microstructure. Interfaces Free Bound 11, 291–316 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  13. Caffarelli L., Silvestre L.: An extension problem related to the fractional Laplacian. Commun. PDE 32, 1245–1260 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  14. Caffarelli L., Silvestre L.: Regularity results for nonlocal equations by approximation. Arch. Ration. Mech. Anal. 200(1), 59–88 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  15. Caffarelli L., Vasseur A.: Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation. Ann. Math. (II) 171, 1903–1930 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  16. Campanato S.: Equazioni ellittiche del II ordine e spazi \({\mathfrak{L}^{(2,\lambda)}}\). Ann. Mat. Pura Appl. (IV) 69, 321–381 (1965)

    Article  MATH  MathSciNet  Google Scholar 

  17. Chen H., Veron L.: Semilinear fractional elliptic equations involving measures. J. Differ. Equ. 25, 565–590 (1973)

    Google Scholar 

  18. Cianchi A.: Non-linear potentials, local solutions to elliptic equations and rearrangements. Ann. Sc. Norm. Super. Pisa Cl. Sci. (V) 10, 335–361 (2011)

    MATH  MathSciNet  Google Scholar 

  19. 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)

    MATH  MathSciNet  Google Scholar 

  20. Di Castro, A., Kuusi, T., Palatucci, G.: Local behavior of fractional p-minimizers. (submitted paper)

  21. Di Castro A., Kuusi T., Palatucci G.: Nonlocal Harnack inequalities. J. Funct. Anal. 267, 1807–1836 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  22. Di Nezza E., Palatucci G., Valdinoci E.: Hitchhiker’s guide to the fractional Sobolev spaces. Bull. Sci. Math. 136, 521–573 (2012)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  24. Gilboa G., Osher S.: Nonlocal operators with applications to image processing. Multiscale Model. Simul. 7, 1005–1028 (2008)

    Article  MATH  MathSciNet  Google Scholar 

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

    Book  MATH  Google Scholar 

  26. Havin M., Maz’ja V.G.: Non-linear potential theory. Russ. Math. Surv. 27, 71–148 (1972)

    MathSciNet  Google Scholar 

  27. Heinonen J., Kilpeläinen T., Martio O.: Nonlinear Potential Theory of Degenerate Elliptic Equations. Oxford Mathematical Monographs, New York (1993)

    MATH  Google Scholar 

  28. Jaye B., Verbitsky I.: Local and global behaviour of solutions to nonlinear equations with natural growth terms. Arch. Ration. Mech. Anal. 204, 627–681 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  29. Karlsen K.H., Petitta F., Ulusoy S.: A duality approach to the fractional Laplacian with measure data. Publ. Mat. 55, 151–161 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  30. Kilpeläinen T.: Hölder continuity of solutions to quasilinear elliptic equations involving measures. Potential Anal. 3, 265–272 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  31. Kilpeläinen T., Kuusi T., Tuhola-Kujanpää A.: Superharmonic functions are locally renormalized solutions. Ann. Inst. H. Poincaré, Anal. Non Linèaire 28, 775–795 (2011)

    Article  ADS  MATH  Google Scholar 

  32. Kilpeläinen T., Malý J.: Degenerate elliptic equations with measure data and nonlinear potentials. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (IV) 19, 591–613 (1992)

    MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  34. Klimsiak T., Rozkosz A.: Dirichlet forms and semilinear elliptic equations with measure data. J. Funct. Anal. 265, 890–925 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  35. 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  MATH  MathSciNet  Google Scholar 

  36. Kuusi T., Mingione G.: Universal potential estimates. J. Funct. Anal. 262, 4205–4269 (2012)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  38. Lieberman G.M.: Sharp forms of estimates for subsolutions and supersolutions of quasilinear elliptic equations involving measures. Commun. Partial Differ. Equ. 18, 1191–1212 (1993)

    Article  MATH  MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  40. Maz’ya V.: The continuity at a boundary point of the solutions of quasi-linear elliptic equations. (Russian) Vestn. Leningr. Univ. 25, 42–55 (1970)

    MATH  Google Scholar 

  41. Mingione G.: Nonlinear measure data problems. Milan J. Math. 79, 429–496 (2011)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  43. Phuc N.C., Verbitsky I.E.: Singular quasilinear and Hessian equations and inequalities. J. Funct. Anal. 256, 1875–1906 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  44. Stein E.M., Weiss G.: Introduction to Fourier Analysis on Euclidean Spaces. Princeton Mathematics Series, vol. 32. Princeton Univ. Press, Princeton (1971)

    Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  46. Vázquez J.L.: Barenblatt solutions and asymptotic behaviour for a nonlinear fractional heat equation of porous medium type. J. Eur. Math. Soc. (JEMS) 16, 769–803 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  47. Vázquez, J.L.: Recent progress in the theory of nonlinear diffusion with fractional Laplacian operators. DCDS (to appear)

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Yannick Sire.

Additional information

Communicated by L. Caffarelli

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kuusi, T., Mingione, G. & Sire, Y. Nonlocal Equations with Measure Data. Commun. Math. Phys. 337, 1317–1368 (2015). https://doi.org/10.1007/s00220-015-2356-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-015-2356-2

Keywords

Navigation