Archive for Rational Mechanics and Analysis

, Volume 200, Issue 1, pp 59–88 | Cite as

Regularity Results for Nonlocal Equations by Approximation

  • Luis Caffarelli
  • Luis Silvestre


We obtain C 1,α regularity estimates for nonlocal elliptic equations that are not necessarily translation-invariant using compactness and perturbative methods and our previous regularity results for the translation-invariant case.


Viscosity Solution Elliptic Operator Regularity Result Nonlocal Operator Nonlocal Equation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Barles, G., Chasseigne, E., Imbert, C.: Hölder continuity of solutions of second-order non-linear elliptic integro-differential equations. J. Eur. Math. Soc. (to appear)Google Scholar
  2. 2.
    Caffarelli L., Silvestre L.: Regularity theory for fully nonlinear integro-differential equations. Commun. Pure Appl. Math. 62(5), 597–638 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Caffarelli L.A.: Interior a priori estimates for solutions of fully nonlinear equations. Ann. of Math. (2) 130(1), 189–213 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Caffarelli, L.A., Cabré, X.: Fully Nonlinear Elliptic Equations. American Mathematical Society Colloquium Publications, Vol. 43. American Mathematical Society, Providence, RI, 1995Google Scholar
  5. 5.
    Cordes H.O.: Über die erste Randwertaufgabe bei quasilinearen Differentialgleichungen zweiter Ordnung in mehr als zwei Variablen. Math. Ann. 131, 278–312 (1956)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Kassmann M.: A priori estimates for integro-differential operators with measurable kernels. Calc. Var. Partial Differ. Equ. 34(1), 1–21 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Mikulyavichyus R., Pragarauskas G.: Nonlinear potentials of the Cauchy–Dirichlet problem for the Bellman integro-differential equation. Liet. Mat. Rink. 36(2), 178–218 (1996)MathSciNetGoogle Scholar
  8. 8.
    Nirenberg, L.: On a generalization of quasi-conformal mappings and its application to elliptic partial differential equations. In Contributions to the Theory of Partial Differential Equations. Annals of Mathematics Studies, Vol. 33. Princeton University Press, Princeton, 95–100, 1954Google Scholar
  9. 9.
    Silvestre L.: Hölder estimates for solutions of integro-differential equations like the fractional laplace. Indiana Univ. Math. J. 55(3), 1155–1174 (2006)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Texas at AustinAustinUSA
  2. 2.Department of MathematicsUniversity of ChicagoChicagoUSA

Personalised recommendations