Advertisement

Archive for Rational Mechanics and Analysis

, Volume 227, Issue 2, pp 663–714 | Cite as

Nonlinear Calderón–Zygmund Theory in the Limiting Case

  • Benny Avelin
  • Tuomo Kuusi
  • Giuseppe Mingione
Article

Abstract

We prove a maximal differentiability and regularity result for solutions to nonlinear measure data problems. Specifically, we deal with the limiting case of the classical theory of Calderón and Zygmund in the setting of nonlinear, possibly degenerate equations and we show a complete linearization effect with respect to the differentiability of solutions. A prototype of the results obtained here tells for instance that if
$$-{\rm div} \, (|Du|^{p-2}Du)=\mu \quad \mbox{in} \ \Omega\subset\mathbb{R}^n,$$
with \({\mu}\) being a Borel measure with locally finite mass on the open subset \({\Omega\subset \mathbb{R}^n}\) and \({p > 2-1/n}\), then
$$|Du|^{p-2}Du \in W^{\sigma, 1}_{\rm{loc}}(\Omega)\quad \mbox{for \, every} \ \sigma \in (0,1).$$
The case \({\sigma=1}\) is obviously forbidden already in the classical linear case of the Poisson equation \({-\triangle u=\mu}\).

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Adams, R.A., Fournier, J.J.F.: Sobolev Spaces, 2nd edn. Pure and Appl. Math., vol. 140, Elsevier/Academic Press, Amsterdam, 2003Google Scholar
  2. 2.
    Baroni P.: Riesz potential estimates for a general class of quasilinear equations. Calc. Var. PDE 53, 803–846 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Boccardo L., Gallouët T.: Nonlinear elliptic and parabolic equations involving measure data. J. Funct. Anal. 87, 149–169 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Boccardo L., Gallouët T.: Nonlinear elliptic equations with right-hand side measures. Commun. Partial Differ. Equ. 17, 641–655 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    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)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Boccardo L., Gallouët T., Vázquez J.L.: Nonlinear elliptic equations in \({\mathbb{R}^N}\) without growth restrictions on the data. J. Differ. Equ. 105, 334–363 (1993)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Brezis H., Strauss W. A.: Semi-linear second-order elliptic equations in L 1. J. Math. Soc. Jpn. 25, 565–590 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Brasco, L., Santambrogio, F.: A sharp estimate à la Calderón–Zygmund for the p-Laplacian. Commun. Cont. Math., to appearGoogle Scholar
  9. 9.
    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)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Damascelli L., Sciunzi B.: Regularity, monotonicity and symmetry of positive solutions of m-Laplace equations. J. Diff. Equ. 206, 483–515 (2004)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    DiBenedetto E.: Degenerate Parabolic Equations. Universitext. Springer, New York (1993)CrossRefzbMATHGoogle Scholar
  12. 12.
    DiBenedetto E., Manfredi J.: On the higher integrability of the gradient of weak solutions of certain degenerate elliptic systems. Am. J. Math. 115, 1107–1134 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Diening L., Kaplický P., Schwarzacher S.: BMO estimates for the p-Laplacian. Nonlinear Anal. 75, 637–650 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Di Nezza E., Palatucci G., Valdinoci E.: Hitchhiker’s guide to the fractional Sobolev spaces. Bull. Sci. Math., 136, 521–573 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Duzaar F., Mingione G.: Gradient estimates via non-linear potentials. Am. J. Math. 133, 1093–1149 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Duzaar F., Mingione G.: Gradient estimates via linear and nonlinear potentials. J. Funct. Anal. 259, 2961–2998 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Fonseca I., Fusco N.: Regularity results for anisotropic image segmentation models. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (IV). 24, 463–499 (1997)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Giusti E.: Direct Methods in the Calculus of Variations. World Scientific Publishing Co., Inc., River Edge, NJ (2003)CrossRefzbMATHGoogle Scholar
  19. 19.
    Hamburger C.: Regularity of differential forms minimizing degenerate elliptic functionals. J. Reine Angew. Math. (Crelles J.) 431, 7–64 (1992)MathSciNetzbMATHGoogle Scholar
  20. 20.
    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)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Kilpeläinen T., Malý J.: The Wiener test and potential estimates for quasilinear elliptic equations. Acta Math. 172, 137–161 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Kilpeläinen T., Kuusi T., Tuhola-Kujanpää A.: Superharmonic functions are locally renormalized solutions. Ann. Inst. H. Poincaré, Anal. Non Linèaire 28, 775–795 (2011)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Kristensen J., Mingione G.: The singular set of minima of integral functionals. Arch. Ration. Mech. Anal. 180, 331–398 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Kristensen J., Mingione G.: Boundary regularity in variational problems. Arch. Ration. Mech. Anal. 198, 369–455 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Kuusi T., Mingione G.: Universal potential estimates. J. Funct. Anal. 262, 4205–4269 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Kuusi T., Mingione G.: Linear potentials in nonlinear potential theory. Arch. Ration. Mech. Anal. 207, 215–246 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Kuusi T., Mingione G.: The Wolff gradient bound for degenerate parabolic equations. J. Eur. Math. Soc. 16, 835–892 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Kuusi T., Mingione G.: Guide to nonlinear potential estimates. Bull. Math. Sci. 4, 1–82 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Kuusi, T., Mingione, G.: Vectorial nonlinear potential theory. J. Eur. Math. Soc., to appearGoogle Scholar
  30. 30.
    Littman W., Stampacchia G., Weinberger H. F.: Regular points for elliptic equations with discontinuous coefficients. Ann. Scu. Norm. Sup. Pisa (III). 17, 43–77 (1963)MathSciNetzbMATHGoogle Scholar
  31. 31.
    Lou H.: On singular sets of local solutions to p-Laplace equations. Chin. Ann. Math. Ser. B 29, 521–530 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Manfredi J.J.: Regularity for minima of functionals with p-growth. J. Differ. Equ. 76, 203–212 (1988)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Manfredi, J.J.: Regularity of the Gradient for a Class of Nonlinear Possibly Degenerate Elliptic Equations, Ph.D. Thesis. University of Washington, St. LouisGoogle Scholar
  34. 34.
    Manfredi J.J.: p-harmonic functions in the plane. Proc. Am. Math. Soc. 103, 473–479 (1988)MathSciNetzbMATHGoogle Scholar
  35. 35.
    Mingione G.: The singular set of solutions to non-differentiable elliptic systems. Arch. Ration. Mech. Anal. 166, 287–301 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Mingione G.: The Calderón–Zygmund theory for elliptic problems with measure data. Ann. Scu. Norm. Sup. Pisa Cl. Sci. (V). 6, 195–261 (2007)zbMATHGoogle Scholar
  37. 37.
    Mingione G.: Gradient estimates below the duality exponent. Math. Ann. 346, 571–627 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Mingione G.: Gradient potential estimates. J. Eur. Math. Soc. 13, 459–486 (2011)MathSciNetzbMATHGoogle Scholar
  39. 39.
    Mingione, G.: Nonlinear Calderón–Zygmund theory. EMS, ETH Zürich series, to appearGoogle Scholar
  40. 40.
    Rahman, Q.I., Schmeisser, G.: Analytic Theory of Polynomials. London Mathematical Society Monographs. New Series, vol. 26. Oxford University Press, Oxford, 2002Google Scholar
  41. 41.
    Schur I.: About the characteristic roots of a linear substitution with an application to the theory of integral equations (German). Math. Ann. 66, 488–510 (1909)MathSciNetCrossRefGoogle Scholar
  42. 42.
    Simon, J.: Régularité de la solution d’une équation non linaire dans RN. (French). In: Journées d’Analyse Non Linéaire (Proc. Conf., Besanon, 1977), Lecture Notes in Math., vol. 665, pp. 205–227. Springer, Berlin, 1978Google Scholar
  43. 43.
    Uhlenbeck K.: Regularity for a class of non-linear elliptic systems. Acta Math. 138, 219–240 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Ural’tseva N.N.: Degenerate quasilinear elliptic systems. Zap. Na. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI). 7, 184–222 (1968)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.Department of MathematicsUppsala UniversityUppsalaSweden
  2. 2.Department of Mathematics and Systems AnalysisAalto UniversityAaltoFinland
  3. 3.Dipartimento di Matematica e InformaticaUniversità di ParmaParmaItaly

Personalised recommendations