Continuous viscosity solutions for nonlocal Dirichlet problems with coercive gradient terms
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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.
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.
- 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.Barles, G., Chasseigne, E., Imbert, C.: On the Dirichlet Problem for Second Order Elliptic Integro-Differential Equations Indiana U. Math, Journal (2008)Google Scholar
- 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
- 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.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
- 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.Quaas, A., Salort, A.: Principal eigenvalue of a integro-differential elliptic equation with a drift term, in preparationGoogle Scholar
- 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