Archive for Rational Mechanics and Analysis

, Volume 206, Issue 1, pp 111–157 | Cite as

Aleksandrov–Bakelman–Pucci Type Estimates for Integro-Differential Equations



In this work we provide an Aleksandrov–Bakelman–Pucci type estimate for a certain class of fully nonlinear elliptic integro-differential equations, the proof of which relies on an appropriate generalization of the convex envelope to a nonlocal, fractional-order setting and on the use of Riesz potentials to interpret second derivatives as fractional order operators. This result applies to a family of equations involving some nondegenerate kernels and, as a consequence, provides some new regularity results for previously untreated equations. Furthermore, this result also gives a new comparison theorem for viscosity solutions of such equations which depends only on the L and L n norms of the right-hand side, in contrast to previous comparison results which utilize the continuity of the right-hand side for their conclusions. These results appear to be new, even for the linear case of the relevant equations.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Abels H., Kassmann M.: The Cauchy problem and the martingale problem for integro-differential operators with non-smooth kernels. Osaka J. Math. 46(3), 661–683 (2009)MathSciNetMATHGoogle Scholar
  2. 2.
    Awatif S.: Équations d’Hamilton–Jacobi du premier ordre avec termes intégro-différentiels. I. Unicité des solutions de viscosité. Commum. Partial Differ. Equ. 16(6–7), 1057–1074 (1991)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Barles G., Chasseigne E., Imbert C.: On the dirichlet problem for second-order elliptic integro-differential equations. Indiana Univ. Math. J. 57(1), 213–246 (2008)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Barles G., Chasseigne E., Imbert C.: Hölder continuity of solutions of second-order elliptic integro-differential equations. J. Eur. Math. Soc. 13(1), 1–26 (2011)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Barles G., Imbert C.: Second-order elliptic integro-differential equations: viscosity solutions’ theory revisited. Ann. Inst. H. Poincaré Anal. Non Linéaire 25(3), 567–585 (2008)MathSciNetADSMATHCrossRefGoogle Scholar
  6. 6.
    Barlow M.T., Bass R.F., Chen Z.-Q., Kassmann M.: Non-local Dirichlet forms and symmetric jump processes. Trans. Am. Math. Soc. 361(4), 1963–1999 (2009)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Bass, R.F., Kassmann, M.: Harnack inequalities for non-local operators of variable order. Trans. Am. Math. Soc. 357(2), 837–850 (electronic) (2005)Google Scholar
  8. 8.
    Bass R.F., Kassmann M.: Hölder continuity of harmonic functions with respect to operators of variable order. Commun. Partial Differ. Equ. 30(7–9), 1249–1259 (2005)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Bass R.F., Levin D.A.: Harnack inequalities for jump processes. Potential Anal. 17(4), 375–388 (2002)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Bass, R.F., Levin, D.A.: Transition probabilities for symmetric jump processes. Trans. Am. Math. Soc. 354(7), 2933–2953 (electronic) (2002)Google Scholar
  11. 11.
    Bjorland, C., Caffarelli, L., Figalli, A.: Non-local tug-of-war and the infinity fractional laplacian. Comm. Pure Appl. Math. (2012) (to appear)Google Scholar
  12. 12.
    Brézis H., Kinderlehrer D.: The smoothness of solutions to nonlinear variational inequalities. Indiana Univ. Math. J. 23, 831–844 (1973/1974)CrossRefGoogle Scholar
  13. 13.
    Caffarelli L., Crandall M.G., Kocan M., Swįch A.: On viscosity solutions of fully nonlinear equations with measurable ingredients. Commun. Pure Appl. Math. 49(4), 365–397 (1996)MATHCrossRefGoogle Scholar
  14. 14.
    Caffarelli L., Nirenberg L., Spruck J.: The Dirichlet problem for nonlinear second-order elliptic equations. III. Functions of the eigenvalues of the Hessian. Acta Math. 155(3–4), 261–301 (1985)MathSciNetMATHGoogle Scholar
  15. 15.
    Caffarelli, L.A., Chan, C.H., Vasseur, A.: Regularity Theory for Nonlinear Integral operators. arXiv:1003.1699v1 [math.AP] (2010)Google Scholar
  16. 16.
    Caffarelli, L.A., Silvestre, L.: The Evans–Krylov theorem for non local fully non linear equations. Ann. Math. (2) (2012) (to appear)Google Scholar
  17. 17.
    Caffarelli, L.A., Silvestre, L.: Regularity results for nonlocal equations by approximation. Arch. Rational. Mech. Anal. (2012) (to appear)Google Scholar
  18. 18.
    Caffarelli L., Silvestre L.: Regularity theory for fully nonlinear integro-differential equations. Commun. Pure Appl. Math. 62(5), 597–638 (2009)MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Caffarelli, L., Vazquez, J.L.: Nonlinear Porous Medium Flow with Fractional Potential pressure.
  20. 20.
    Caffarelli, L.A.: Interior a priori estimates for solutions of fully nonlinear equations. Ann. Math. (2) 130(1), 189–213 (1989)Google Scholar
  21. 21.
    Caffarelli, L.A., Cabré, X.: Fully Nonlinear Elliptic Equations Vol. 43 of American Mathematical Society Colloquium Publications. American Mathematical Society, Providence, 1995Google Scholar
  22. 22.
    Caffarelli L.A., Souganidis P.E., Wang L.: Homogenization of fully nonlinear, uniformly elliptic and parabolic partial differential equations in stationary ergodic media. Commun. Pure Appl. Math. 58(3), 319–361 (2005)MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    Cont, R., Tankov, P.: Financial Modelling with Jump Processes. Chapman & Hall/CRC Financial Mathematics Series. Chapman & Hall/CRC, Boca Raton, 2004Google Scholar
  24. 24.
    Cont R., Voltchkova E.: Integro-differential equations for option prices in exponential Lévy models. Finance Stoch. 9(3), 299–325 (2005)MathSciNetMATHCrossRefGoogle Scholar
  25. 25.
    Crandall M.G., Ishii H., Lions P.-L.: User’s guide to viscosity solutions of second order partial differential equations. Bull. Am. Math. Soc. (N.S.) 27(1), 1–67 (1992)MathSciNetMATHCrossRefGoogle Scholar
  26. 26.
    Garding L.: An inequality for hyperbolic polynomials. J. Math. Mech. 8, 957–965 (1959)MathSciNetMATHGoogle Scholar
  27. 27.
    Giacomin G., Lebowitz J.L.: Phase segregation dynamics in particle systems with long range interactions. I. Macroscopic limits. J. Stat. Phys. 87(1–2), 37–61 (1997)MathSciNetADSMATHCrossRefGoogle Scholar
  28. 28.
    Gilbarg D., Trudinger N.S.: Elliptic Partial Differential Equations of Second Order. Classics in Mathematics. Springer, Berlin, 2001 (Reprint of the 1998 edition)Google Scholar
  29. 29.
    Gilboa G., Osher S.: Nonlocal operators with applications to image processing. Multiscale Model. Simul. 7(3), 1005–1028 (2008)MathSciNetMATHCrossRefGoogle Scholar
  30. 30.
    Jensen R., Lions P.-L., Souganidis P.E.: A uniqueness result for viscosity solutions of second order fully nonlinear partial differential equations. Proc. Am. Math. Soc. 102(4), 975–978 (1988)MathSciNetMATHCrossRefGoogle Scholar
  31. 31.
    Kassmann, M.: personal communication (2010)Google Scholar
  32. 32.
    Kassmann, M., Mimica, A.: Analysis of jump processes with nondegenerate jumping kernels., arXiv:1109.3678v2 [math.PR] (2011)Google Scholar
  33. 33.
    Kinderlehrer, D., Stampacchia, G.: An Introduction to Variational Inequalities And their Applications Vol. 88 Pure and Applied Mathematics. Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York, 1980Google Scholar
  34. 34.
    Landkof, N.S.: Foundations of Modern Potential Theory. Springer, New York, 1972 (Translated from the Russian by A. P. Doohovskoy, Die Grundlehren der mathematischen Wissenschaften, Band 180)Google Scholar
  35. 35.
    Lasry J.-M., Lions P.-L.: A remark on regularization in Hilbert spaces. Israel J. Math. 55(3), 257–266 (1986)MathSciNetMATHCrossRefGoogle Scholar
  36. 36.
    Lewy H., Stampacchia G.: On existence and smoothness of solutions of some non-coercive variational inequalities. Arch. Rational Mech. Anal. 41, 241–253 (1971)MathSciNetMATHCrossRefGoogle Scholar
  37. 37.
    Lin F.-H.: Second derivative L p-estimates for elliptic equations of nondivergent type. Proc. Am. Math. Soc. 96(3), 447–451 (1986)MATHGoogle Scholar
  38. 38.
    Oberman A.M.: The convex envelope is the solution of a nonlinear obstacle problem. Proc. Am. Math. Soc. 135(6), 1689–1694 (electronic)(2007)Google Scholar
  39. 39.
    Pham H.: Optimal stopping of controlled jump diffusion processes: a viscosity solution approach. J. Math. Syst. Estim. Control 8(1), 27 (electronic) (1998)Google Scholar
  40. 40.
    Schwab R.W.: Periodic homogenization for nonlinear integro-differential equations. SIAM J. Math. Anal. 42(6), 2652–2680 (2010)MathSciNetMATHCrossRefGoogle Scholar
  41. 41.
    Schwab, R.W.: Stochastic homogenization for some nonlinear integro-differential equations., arXiv:1101.6052v1 [math.AP] (2011)Google Scholar
  42. 42.
    Silvestre L.: Hölder estimates for solutions of integro-differential equations like the fractional Laplace. Indiana Univ. Math. J. 55(3), 1155–1174 (2006)MathSciNetMATHGoogle Scholar
  43. 43.
    Silvestre L.: Regularity of the obstacle problem for a fractional power of the Laplace operator. Commun. Pure Appl. Math. 60(1), 67–112 (2007)MathSciNetMATHCrossRefGoogle Scholar
  44. 44.
    Soner H.M.: Optimal control with state-space constraint. II. SIAM J. Control Optim. 24(6), 1110–1122 (1986)MathSciNetADSMATHCrossRefGoogle Scholar
  45. 45.
    Soner, H.M.: Optimal control of jump-Markov processes and viscosity solutions. In: Stochastic differential systems, stochastic control theory and applications (Minneapolis, MN, 1986), Vol. 10 IMA Vol. Math. Appl., pp. 501–511. Springer, New York, 1988Google Scholar
  46. 46.
    Stein E.M.: Singular Integrals and Differentiability Properties of Functions. Princeton University Press, Princeton (1971)Google Scholar

Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of TexasAustinUSA
  2. 2.Department of Mathematical SciencesCarnegie Mellon UniversityPittsburghUSA

Personalised recommendations