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Regularity results and large time behavior for integro-differential equations with coercive Hamiltonians

  • Guy Barles
  • Shigeaki Koike
  • Olivier Ley
  • Erwin ToppEmail author
Article

Abstract

In this paper we obtain regularity results for elliptic integro-differential equations driven by the stronger effect of coercive gradient terms. This feature allows us to construct suitable strict supersolutions from which we conclude Hölder estimates for bounded subsolutions. In many interesting situations, this gives way to a priori estimates for subsolutions. We apply this regularity results to obtain the ergodic asymptotic behavior of the associated evolution problem in the case of superlinear equations. One of the surprising features in our proof is that it avoids the key ingredient which are usually necessary to use the strong maximum principle: linearization based on the Lipschitz regularity of the solution of the ergodic problem. The proof entirely relies on the Hölder regularity.

Mathematics Subject Classification

35R09 35B51 35B65 35D40 35B10 35B40 

Notes

Acknowledgments

G.B. and O.L. are partially supported by the ANR (Agence Nationale de la Recherche) through ANR WKBHJ (ANR-12-BS01-0020). S.K. is supported by Grant-in-Aid for Scientific Research (No. 23340028) of Japan Society for the Promotion of Science. E.T. was partially supported by CONICYT, Grants Capital Humano Avanzado, Cotutela en el Extranjero and Ayuda Realización Tesis Doctoral.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Guy Barles
    • 1
  • Shigeaki Koike
    • 2
  • Olivier Ley
    • 3
  • Erwin Topp
    • 1
    • 4
    Email author
  1. 1.Laboratoire de Mathématiques et Physique Théorique (UMR CNRS 7350), Fédération Denis Poisson (FR CNRS 2964)Université François Rabelais ToursToursFrance
  2. 2.Mathematical InstituteTohoku UniversitySendaiJapan
  3. 3.IRMARINSA de RennesRennesFrance
  4. 4.Departamento de Ingeniería Matemática (UMI 2807 CNRS)Universidad de ChileSantiagoChile

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