\(C^{\sigma +\alpha }\) regularity for concave nonlocal fully nonlinear elliptic equations with rough kernels

  • Joaquim SerraEmail author


We establish \(C^{\sigma +\alpha }\) interior estimates for concave nonlocal fully nonlinear equations of order \(\sigma \in (0,2)\) with rough kernels. Namely, we prove that if \(u\in C^{\alpha }(\mathbb {R}^n)\) solves in \(B_1\) a concave translation invariant equation with kernels in \(\mathcal L_0(\sigma )\), then u belongs to \(C^{\sigma +\alpha }(\overline{B_{1/2}})\), with an estimate. More generally, our results allow the equation to depend on x in a \(C^\alpha \) fashion. Our method of proof combines a Liouville theorem and a blow-up (compactness) procedure. Due to its flexibility, the same method can be useful in different regularity proofs for nonlocal equations.

Mathematics Subject Classification

35J60 45K05 



The author is indebted to Xavier Cabré, Hector Chang Lara, and Luis Silvestre for their interesting comments on the paper.


  1. 1.
    Caffarelli, L.A.: Interior a priori estimates for solutions of fully nonlinear equations. Ann. Math. 130, 189–213 (1989)zbMATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Caffarelli, L., Cabré, X.: Fully nonlinear elliptic equations. In: American Mathematical Society Colloquium Publications, vol. 43. American Mathematical Society, Providence (1995)Google Scholar
  3. 3.
    Caffarelli, L., Silvestre, L.: Regularity theory for fully nonlinear integro-differential equations. Commun. Pure Appl. Math. 62, 597–638 (2009)zbMATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Caffarelli, L., Silvestre, L.: Regularity results for nonlocal equations by approximation. Arch. Ration. Mech. Anal. 200, 59–88 (2011)zbMATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Caffarelli, L., Silvestre, L.: The Evans–Krylov theorem for nonlocal fully nonlinear equations. Ann. Math. 174, 1163–1187 (2011)zbMATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Evans, L.C.: Classical solutions of fully nonlinear, convex, second-order elliptic equations. Commun. Pure Appl. Math. 35, 333–363 (1982)zbMATHCrossRefGoogle Scholar
  7. 7.
    Gilbarg, D., Trudinger, N.S.: Elliptic partial differential equations of second order. In: Grundlehren der Mathematischen Wissenschaften, vol. 224. Springer, Berlin (1977)Google Scholar
  8. 8.
    Jin, T., Xiong, J.: Schauder estimates for nonlocal fully nonlinear equations. Ann. de l’Institut Henri Poincare (C) Non Linear Anal. (2014) (in press)Google Scholar
  9. 9.
    Krylov, N.V.: Boundedly inhomogeneous elliptic and parabolic equations. Izv. Akad. Nauk SSSR Ser. Mat. 46, 487–523 (1982)zbMATHMathSciNetGoogle Scholar
  10. 10.
    Kriventsov, D.: \(C^{1,\alpha }\) interior regularity for nonlocal elliptic equations with rough kernels. Commun. Partial Differ. Equ. 38, 2081–2106 (2013)zbMATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    Ros-Oton, X., Serra, J.: Boundary regularity for fully nonlinear integro-differential equations (2014).arXiv:1404.1197
  12. 12.
    Ros-Oton, X., Serra, J.: Regularity theory for general stable operators (2014).arXiv:1412.3892
  13. 13.
    Serra, J.: Regularity for fully nonlinear nonlocal parabolic equation with rought kerenls (2012). doi: 10.1007/s00526-014-0798-6
  14. 14.
    Silvestre, L.: Lecture notes on nonlocal equations (2014) (published online)Google Scholar
  15. 15.
    L. Silvestre and others, Nonlocal Equations Wiki.

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Departament de Matemàtica Aplicada IUniversitat Politècnica de CatalunyaBarcelonaSpain

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