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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.

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  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)

    MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Article  Google Scholar 

  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)

    MathSciNet  MATH  Article  Google Scholar 

  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)

    MathSciNet  MATH  Article  Google Scholar 

  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)

    MathSciNet  ADS  MATH  Article  Google Scholar 

  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)

    MathSciNet  MATH  Article  Google Scholar 

  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. 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)

    MathSciNet  MATH  Article  Google Scholar 

  9. Bass R.F., Levin D.A.: Harnack inequalities for jump processes. Potential Anal. 17(4), 375–388 (2002)

    MathSciNet  MATH  Article  Google Scholar 

  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. Bjorland, C., Caffarelli, L., Figalli, A.: Non-local tug-of-war and the infinity fractional laplacian. Comm. Pure Appl. Math. (2012) (to appear)

  12. Brézis H., Kinderlehrer D.: The smoothness of solutions to nonlinear variational inequalities. Indiana Univ. Math. J. 23, 831–844 (1973/1974)

    Article  Google Scholar 

  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)

    MATH  Article  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  15. Caffarelli, L.A., Chan, C.H., Vasseur, A.: Regularity Theory for Nonlinear Integral operators. arXiv:1003.1699v1 [math.AP] (2010)

  16. Caffarelli, L.A., Silvestre, L.: The Evans–Krylov theorem for non local fully non linear equations. Ann. Math. (2) (2012) (to appear)

  17. Caffarelli, L.A., Silvestre, L.: Regularity results for nonlocal equations by approximation. Arch. Rational. Mech. Anal. (2012) (to appear)

  18. Caffarelli L., Silvestre L.: Regularity theory for fully nonlinear integro-differential equations. Commun. Pure Appl. Math. 62(5), 597–638 (2009)

    MathSciNet  MATH  Article  Google Scholar 

  19. Caffarelli, L., Vazquez, J.L.: Nonlinear Porous Medium Flow with Fractional Potential pressure.

  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. Caffarelli, L.A., Cabré, X.: Fully Nonlinear Elliptic Equations Vol. 43 of American Mathematical Society Colloquium Publications. American Mathematical Society, Providence, 1995

  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)

    MathSciNet  MATH  Article  Google Scholar 

  23. Cont, R., Tankov, P.: Financial Modelling with Jump Processes. Chapman & Hall/CRC Financial Mathematics Series. Chapman & Hall/CRC, Boca Raton, 2004

  24. Cont R., Voltchkova E.: Integro-differential equations for option prices in exponential Lévy models. Finance Stoch. 9(3), 299–325 (2005)

    MathSciNet  MATH  Article  Google Scholar 

  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)

    MathSciNet  MATH  Article  Google Scholar 

  26. Garding L.: An inequality for hyperbolic polynomials. J. Math. Mech. 8, 957–965 (1959)

    MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  ADS  MATH  Article  Google Scholar 

  28. Gilbarg D., Trudinger N.S.: Elliptic Partial Differential Equations of Second Order. Classics in Mathematics. Springer, Berlin, 2001 (Reprint of the 1998 edition)

  29. Gilboa G., Osher S.: Nonlocal operators with applications to image processing. Multiscale Model. Simul. 7(3), 1005–1028 (2008)

    MathSciNet  MATH  Article  Google Scholar 

  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)

    MathSciNet  MATH  Article  Google Scholar 

  31. Kassmann, M.: personal communication (2010)

  32. Kassmann, M., Mimica, A.: Analysis of jump processes with nondegenerate jumping kernels., arXiv:1109.3678v2 [math.PR] (2011)

  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, 1980

  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)

  35. Lasry J.-M., Lions P.-L.: A remark on regularization in Hilbert spaces. Israel J. Math. 55(3), 257–266 (1986)

    MathSciNet  MATH  Article  Google Scholar 

  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)

    MathSciNet  MATH  Article  Google Scholar 

  37. Lin F.-H.: Second derivative L p-estimates for elliptic equations of nondivergent type. Proc. Am. Math. Soc. 96(3), 447–451 (1986)

    MATH  Google Scholar 

  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)

  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. Schwab R.W.: Periodic homogenization for nonlinear integro-differential equations. SIAM J. Math. Anal. 42(6), 2652–2680 (2010)

    MathSciNet  MATH  Article  Google Scholar 

  41. Schwab, R.W.: Stochastic homogenization for some nonlinear integro-differential equations., arXiv:1101.6052v1 [math.AP] (2011)

  42. Silvestre L.: Hölder estimates for solutions of integro-differential equations like the fractional Laplace. Indiana Univ. Math. J. 55(3), 1155–1174 (2006)

    MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Article  Google Scholar 

  44. Soner H.M.: Optimal control with state-space constraint. II. SIAM J. Control Optim. 24(6), 1110–1122 (1986)

    MathSciNet  ADS  MATH  Article  Google Scholar 

  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, 1988

  46. Stein E.M.: Singular Integrals and Differentiability Properties of Functions. Princeton University Press, Princeton (1971)

    Google Scholar 

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Correspondence to Russell W. Schwab.

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Communicated by P.-L. Lions

Nestor Guillen was partially supported by NSF grant DMS-0654267 and Russell Schwab was supported by an NSF Postdoctoral Research Fellowship, grant DMS-0903064. This work was carried out during multiple visits of Nestor Guillen to The Center for Nonlinear Analysis at Carnegie Mellon University, for whose support the authors are also grateful. The authors would like to thank Takis Souganidis and Luis Silvestre for comments on the preliminary version of this manuscript, and the authors would like to extend a special thanks to the anonymous referee for helpful observations which led to some significant improvements in this work.

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Guillen, N., Schwab, R.W. Aleksandrov–Bakelman–Pucci Type Estimates for Integro-Differential Equations. Arch Rational Mech Anal 206, 111–157 (2012).

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  • Fractional Order
  • Viscosity Solution
  • Harnack Inequality
  • Obstacle Problem
  • Regularity Theory