Advertisement

Mathematische Annalen

, Volume 369, Issue 3–4, pp 1211–1236 | Cite as

Continuous viscosity solutions for nonlocal Dirichlet problems with coercive gradient terms

  • Gonzalo Dávila
  • Alexander Quaas
  • Erwin Topp
Article

Abstract

In this paper we study existence of solutions of nonlocal Dirichlet problems that include a coercive gradient term, whose scaling strictly dominates the one of the integro-differential operator. For such problems the stronger effect of the gradient term may give rise to solutions not attaining the boundary data or discontinuous solutions on the boundary. Our main result states that under suitable conditions over the right-hand side and boundary data, there is a (unique) Hölder continuous viscosity solution attaining the boundary data in the classical sense. This result is accomplished by the construction of suitable barriers which, as a byproduct, lead to regularity results up to the boundary for the solution.

Notes

Acknowledgments

We would like to thank the referee for the careful reading of this manuscript, which lead to several improvements in the presentation of our results. G. Dávila was partially supported by Fondecyt Grant No. 11150880. A. Q. were partially supported by Fondecyt Grant No. 1151180, Programa Basal, CMM. U. de Chile and Millennium Nucleus Center for Analysis of PDE NC130017. E. T. was partially supported by Fondecyt Postdoctoral Grant No. 3150100 and Conicyt PIA Grant No. 79150056.

References

  1. 1.
    Alarcón, S., García-Melián, J., Quaas, A.: Existence and non-existence of solutions to elliptic equations with a general convection term. Proc. R. Soc. Edinburgh Sec A 144(2), 225–239 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Alvarez, O., Tourin, A.: Viscosity solutions of nonlinear integro-differential equations. Annales de L’I.H.P., section C 13(3):293–317 (1996)Google Scholar
  3. 3.
    Barles, G., Chasseigne, E., Imbert, C.: On the Dirichlet Problem for Second Order Elliptic Integro-Differential Equations Indiana U. Math, Journal (2008)Google Scholar
  4. 4.
    Barles, G., Da Lio, F.: On the generalized Dirichlet problem for viscous Hamilton-Jacobi Equations. J. Math. Pures et Appl. 83(1), 53–75 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Barles, G., Imbert, C.: Second-order elliptic integro-differential equations: viscosity solutions’ theory revisited. IHP Anal. Non Linéare 25(3), 567–585 (2008)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Barles, G., Koike, S., Ley, O. and Topp, E.: Regularity results and large time behavior for integro-differential equations with coercive Hamiltonians. Calc. Var. Partial Differ. Eq. (2014). doi: 10.1007/s00526-014-0794-x
  7. 7.
    Barles, G., Perthame, B.: Exit time problems in optimal control and vanishing viscosity method. SIAM J. Control Optim. 26, 1133–1148 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Barles, G., Topp, E.: Existence, uniqueness and asymptotic behavior for nonlocal parabolic problems with dominating gradient terms. SIAM J. Math. Anal. 48(2), 1512–1547 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Barrios, B., Del Pezzo, L., García-Melián, J., Quaas, A.: A priori bounds and existence of solutions for some nonlocal elliptic problems (Preprint)Google Scholar
  10. 10.
    Biswas, I.H.: On zero-sum stochastic differential games with jump-diffusion driven state: a viscosity solution framework. SIAM J. Control Optim 50(4):1823–1858Google Scholar
  11. 11.
    Bogdan, K., Burdzy, K., Chen, Z.-Q.: Censored stable processes. Prob. Theory Rel. Fields 127(1), 89–152 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Buckdahn, R., Hu, Y., Li, J.: Stochastic representation for solutions of Isaacs type integralpartial differential equations. Stoch. Process. Appl. 121(12), 2715–2750 (2011)CrossRefzbMATHGoogle Scholar
  13. 13.
    Caffarelli, L., Silvestre, L.: Regularity theory for nonlocal integro-differential equations. Comm. Pure Appl. Math 62(5), 597–638 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Capuzzo-Dolcetta, I., Leoni, F., Porretta, A.: Hölder estimates for degenerate elliptic equations with coercive Hamiltonians. Trans. Am. Math. Soc. 362(9), 4511–4536 (2010)CrossRefzbMATHGoogle Scholar
  15. 15.
    Chasseigne, E.: The Dirichlet problem for some nonlocal diffusion equations. Differ. Integral Equ. 20(12), 1389–1404 (2007)MathSciNetzbMATHGoogle Scholar
  16. 16.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Da Lio, F.: Comparison Results for quasilinear equations in annular domains and applications. Commun. Partial Differ. Equ. 27(1 & 2), 283–323 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Di Neza, E., Palatucci, G., Valdinoci, E.: Hitchhiker’s guide to the fractional Sobolev spaces. Bull. Sci. Math. 136(5), 521–573 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Felmer, P., Quaas, A.: Fundamental solutions and Liouville type theorems for nonlinear integral operators. Adv. Math. 226(3), 2712–2738 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Felmer, P., Quaas, A.: Fundamental solutions for a class of Isaacs integral operators. Discrete Contin. Dyn. Syst. 30(2), 493508 (2011)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Felmer, P., Topp, E.: Uniform equicontinuity for a family of zero order operators approaching the fractional Laplacian. Commun. Partial Differ. Equ. 40(9), 1591–1618 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Fleming, W., Soner, H.: Controlled Markov processes and viscosity solutions applications of mathematics. Springer-Verlag, New York (1993)zbMATHGoogle Scholar
  23. 23.
    Lasry, J.M., Lions, P.L.: Nonlinear elliptic equations with singular boundary conditions and stochastic control with state constraints. Math. Ann. 283, 583–630 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Gilbarg, D., Trudinger, N.S.: Elliptic partial differential equations of second order. Springer-Verlag, Berlin (2001)zbMATHGoogle Scholar
  25. 25.
    Kawohl, B., Kutev, N.: A study on gradient blow up for viscosity solutions of fully nonlinear, uniformly elliptic equations. Act. Math. Scientia 32 B (1):15–40 (2012)Google Scholar
  26. 26.
    Quaas, A., Salort, A.: Principal eigenvalue of a integro-differential elliptic equation with a drift term, in preparationGoogle Scholar
  27. 27.
    Ros-Oton, X., Serra, J.: The Dirichlet problem for the fractional Laplacian: Regularity up to the boundary. Journal de Mathèmatiques Pures et Appliquées 101(3), 275–302 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Soner, H.M.: Optimal: control problems with state-space constraints. SIAM J. on Control and Optimization 24, Part I: pp 552–562 II, 1110–1122 (1986)Google Scholar
  29. 29.
    Tchamba, T.T.: Large time behavior of solutions of viscous hamilton-jacobi equations with superquadratic Hamiltonian. Asymptot. Anal. 66, 161–186 (2010)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Topp, E.: Existence and uniqueness for integro-differential equations with dominating drift terms. Commun. Partial Differ. Equ. 39(8), 1523–1554 (2014)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Departamento de MatemáticaUniversidad Técnica Federico Santa MaríaValparaísoChile
  2. 2.Departamento de Matemática y C.C.Universidad de Santiago de ChileSantiagoChile

Personalised recommendations