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Uniqueness of viscosity solutions for a class of integro-differential equations

  • Chenchen Mou
  • Andrzej Świe̜ch
Article

Abstract

We prove comparison theorems and uniqueness of viscosity solutions for a class of nonlocal equations. This class of equations includes Bellman–Isaacs equations containing operators of Lévy type with measures depending on x and control parameters, as well as elliptic nonlocal equations that are not strictly monotone in the u variable. The proofs use the knowledge about regularity of viscosity solutions of such equations.

Mathematics Subject Classification

35R09 35D40 35J60 47G20 45K05 93E20 

Keywords

Viscosity solution Integro-PDE Hamilton–Jacobi–Bellman–Isaacs equation Comparison theorem 

References

  1. 1.
    Alibaud N.: Existence, uniqueness and regularity for nonlinear parabolic equations with nonlocal terms. NoDEA Nonlinear Differ. Equ. Appl. 14(3–4), 259–289 (2007)zbMATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Alvarez O., Tourin A.: Viscosity solutions of nonlinear integro-differential equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 13(3), 293–317 (1996)zbMATHMathSciNetGoogle Scholar
  3. 3.
    Arisawa M.: A new definition of viscosity solutions for a class of second-order degenerate elliptic integro-differential equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 23(5), 695–711 (2006)zbMATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Barles G., Buckdahn R., Pardoux E.: Backward stochastic differential equations and integral-partial differential equations. Stoch. Stoch. Rep. 60(1-2), 57–83 (1997)zbMATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Barles G., Busca J.: Existence and comparison results for fully nonlinear degenerate elliptic equations without zeroth-order term. Commun. Partial Differ. Equ. 26(11–12), 2323–2337 (2001)zbMATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Barles G., Chasseigne E., Ciomaga A., Imbert C.: Lipschitz regularity of solutions for mixed integro-differential equations. J. Differ. Equ. 252, 6012–6060 (2012)zbMATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    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)zbMATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Barles G., Chasseigne E., Imbert C.: Hölder continuity of solutions of second-order non-linear elliptic integro-differential equations. J. Eur. Math. Soc. (JEMS) 13, 1–26 (2011)zbMATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    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)zbMATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Chasseigne E.: The Dirichlet problem for some non-local diffusion equations. Differ. Integral Equ. 20, 1389–1404 (2007)zbMATHMathSciNetGoogle Scholar
  11. 11.
    Crandall M.G., Ishii H., Lions P.L.: User’s guide to viscosity solutions of second order partial differential equations. Bull. Am. Math. Soc. 27(1), 1–67 (1991)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Caffarelli L.A., Silvestre L.: Regularity theory for fully nonlinear integro-differential equations. Commun. Pure Appl. Math. 62(5), 597–638 (2009)zbMATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    Caffarelli L.A., Silvestre L.: Regularity results for nonlocal equations by approximation. Arch. Ration. Mech. Anal. 200(1), 59–88 (2011)zbMATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    Caffarelli L.A., Silvestre L.: The Evans–Krylov theorem for nonlocal fully nonlinear equations. Ann. Math. (2) 174(2), 1163–1187 (2011)zbMATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    Guillen N., Schwab R.: Aleksandrov–Bakelman–Pucci type estimates for integro-differential equations. Arch. Ration. Mech. Anal. 206(1), 111–157 (2012)zbMATHMathSciNetCrossRefGoogle Scholar
  16. 16.
    Imbert C.: A non-local regularization of first order Hamilton–Jacobi equations. J. Differ. Equ. 211(1), 218–246 (2005)zbMATHMathSciNetCrossRefGoogle Scholar
  17. 17.
    Ishii H., Lions P.L.: Viscosity solutions of fully nonlinear second-order elliptic partial differential equations. J. Differ. Equ. 83(1), 26–78 (1990)zbMATHMathSciNetCrossRefGoogle Scholar
  18. 18.
    Jakobsen E.R., Karlsen K.H.: Continuous dependence estimates for viscosity solutions of integro-PDEs. J. Differ. Equ. 212(2), 278–318 (2005)zbMATHMathSciNetCrossRefGoogle Scholar
  19. 19.
    Jakobsen E.R., Karlsen K.H.: A “maximum principle for semicontinuous functions” applicable to integro-partial differential equations. NoDEA Nonlinear Differ. Equ. Appl. 13(2), 137–165 (2003)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Jensen R.: Uniqueness criteria for viscosity solutions of fully nonlinear elliptic partial differential equations. Indiana Univ. Math. J. 38(3), 629–667 (1989)zbMATHMathSciNetCrossRefGoogle Scholar
  21. 21.
    Jin, T., Xiong, J.: Schauder estimates for nonlocal fully nonlinear equations. Ann. Inst. H. Poincaré Anal. Non Linéaire. To appearGoogle Scholar
  22. 22.
    Koike S., Świech A.: Representation formulas for solutions of Isaacs integro-PDE. Indiana Univ. Math. J. 62(5), 1473–1502 (2013)zbMATHMathSciNetCrossRefGoogle Scholar
  23. 23.
    Kriventsov D.: C 1,α interior regularity for nonlinear nonlocal elliptic equations with rough kernels. Commun. Partial Differ. Equ. 38(12), 2081–2106 (2013)zbMATHMathSciNetCrossRefGoogle Scholar
  24. 24.
    Sayah A.: Équations d’Hamilton–Jacobi du premier ordre avec termes intgro-diffrentiels. I. Unicité des solutions de viscosité. II. Existence de solutions de viscosité. Commun. Partial Differ. Equ. 16(6–7), 1057–1093 (1991)zbMATHGoogle Scholar
  25. 25.
    Serra, J.: C σ+α regularity for concave nonlocal fully nonlinear elliptic equations with rough kernels. Calc. Var. Partial Differ. Equ. 54(1), 615–629 (2015)Google Scholar
  26. 26.
    Soner H.M.: Optimal control with state-space constraint. II. SIAM J. Control Optim. 24(6), 1110–1122 (1986)zbMATHMathSciNetCrossRefGoogle Scholar
  27. 27.
    Soner, H.M.: Optimal control of jump-Markov processes and viscosity solutions. In: Fleming, W.H., Lions, P.-L. (eds.) Stochastic differential systems, stochastic control theory and applications (Minneapolis, Minn., 1986), vol. 10, pp. 501–511. IMA Math. Appl. Springer, New York (1988)Google Scholar

Copyright information

© Springer Basel 2015

Authors and Affiliations

  1. 1.School of MathematicsGeorgia Institute of TechnologyAtlantaUSA

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