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Continuous viscosity solutions for nonlocal Dirichlet problems with coercive gradient terms

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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.

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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.

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Authors and Affiliations

  1. Departamento de Matemática, Universidad Técnica Federico Santa María, Casilla: v-110, Avda. España 1680, Valparaíso, Chile

    Gonzalo Dávila & Alexander Quaas

  2. Departamento de Matemática y C.C., Universidad de Santiago de Chile, Casilla 307, Santiago, Chile

    Erwin Topp

Authors
  1. Gonzalo Dávila
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  2. Alexander Quaas
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  3. Erwin Topp
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Correspondence to Gonzalo Dávila.

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Dávila, G., Quaas, A. & Topp, E. Continuous viscosity solutions for nonlocal Dirichlet problems with coercive gradient terms. Math. Ann. 369, 1211–1236 (2017). https://doi.org/10.1007/s00208-016-1481-3

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  • DOI: https://doi.org/10.1007/s00208-016-1481-3

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