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

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


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 



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.


  1. 1.
    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
  2. 2.
    Bardi, M., Da Lio, F.: On the strong maximum principle for fully nonlinear degenerate elliptic equations. Arch. Math. (Basel) 73(4), 276–285 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Barles, G.: Solutions de Viscosite des Equations de Hamilton–Jacobi Collection “Mathematiques et Applications” de la SIAM, no 17. Springer (1994)Google Scholar
  4. 4.
    Barles, G.: A short proof of the \(C^{0,\alpha }-\) regularity of viscosity subsolutions for superquadratic viscous Hamilton–Jacobi equations and applications. Nonlinear Anal. 73, 31–47 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Barles, G., Chasseigne, E., Ciomaga, A., Imbert, C.: Lipschitz regularity of solutions for mixed integro-differential equations. J. Differ. Equ. 252, 6012–6060 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Barles, G., Chasseigne, E., Ciomaga, A., Imbert, C.: Time behavior of periodic viscosity solutions for uniformly parabolic integro-differential equations. Calc. Var. Partial Differ. Equ. 50(1–2), 283–304 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Barles, G., Chasseigne, E., Georgelin, C., Jakobsen, E.: On Neumann type problems for nonlocal equations set in a half space. Trans. Am. Math. Soc. 366(9), 4873–4917 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    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)Google Scholar
  9. 9.
    Barles, G., Imbert, C.: Second-order eliptic integro-differential equations: viscosity solutions’ theory revisited. IHP Anal. Non Linéare 25(3), 567–585 (2008)zbMATHGoogle Scholar
  10. 10.
    Barles, G., Souganidis, P.E.: Space-time periodic solutions and long-time behavior of solutions of quasilinear parabolic equations. SIAM J. Math. Anal. 32, 1311–1323 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Bogdan, K., Burdzy, K., Chen, Z.-Q.: Censored stable processes. Probab. Theory Rel. Fields 127(1), 89–152 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Caffarelli, L., Silvestre, L.: Regularity theory for nonlocal integro-differential equations. Commun. Pure Appl. Math. 62(5), 597–638 (2009)CrossRefzbMATHGoogle Scholar
  13. 13.
    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
  14. 14.
    Cardaliaguet, P., Cannarsa, P.: Hölder estimates in space-time for viscosity solutions of Hamilton–Jacobi equations. CPAM 63(5), 590–629 (2010)zbMATHGoogle Scholar
  15. 15.
    Cardaliaguet, P., Rainer, C.: Höder regularity for viscosity solutions of fully nonlinear, local or nonlocal, Hamilton–Jacobi equations with superquadratic growth in the gradient. SIAM J. Control Optim. 49(2), 555–573 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Chasseigne, E.: The Dirichlet problem for some nonlocal diffusion equations. Differ. Integr. Equ. 20(12), 1389–1404 (2007)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Ciomaga, A.: On the strong maximum principle for second order nonlinear parabolic integro-differential equations. Adv. Differ. Equ. 17, 635–671 (2012)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Coville, J.: Maximum principles, sliding techniques and applications to nonlocal equations. Electron. J. Differ. Equ. 2007(68), 1–23 (2007)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Coville, J.: Remarks on the strong maximum principle for nonlocal operators. Electron. J. Differ. Equ. 66, 1–10 (2008)MathSciNetzbMATHGoogle Scholar
  20. 20.
    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
  21. 21.
    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
  22. 22.
    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
  23. 23.
    Fukushima, M., Oshima, Y., Takeda, M.: Dirichlet forms and symmetric Markov processes. de Grueter Stud. Math. 19 (1994)Google Scholar
  24. 24.
    Guan, Q.Y.: The integration by part of the regional fractional Laplacian. Commun. Math. Phys. 266, 289–329 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Guan, Q.Y., Ma, Z.M.: Reflected symmetric \(\alpha \) stable process and regional fractional Laplacian. Probab. Theory Relat. Fields 134, 649–694 (2006)CrossRefzbMATHGoogle Scholar
  26. 26.
    Ishii, H.: Existence and uniqueness of solutions of Hamilton–Jacobi equations. Funkcialaj Ekvacioj 29, 167–188 (1986)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Ishii, H., Nakamura, G.: A class of integral equations and approximation of p-Laplace equations. Calc. Var. Partial Differ. Equ. 37(3–4), 485–522 (2010)Google Scholar
  28. 28.
    Jacob, N.: Pseudo Differential Operators and Markov Process. Markov Process and Applications, vol. III. Imperial College Press, Princeton (2005)CrossRefGoogle Scholar
  29. 29.
    Kim, P.: Weak convergence of censored and reflected stable processes. Stoch. Process. Appl. 116(12), 1792–1814 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Koike, S.: A beginner’s guide to the theory of viscosity solutions. MSJ Memoirs, vol. 13. Mathematical Society of Japan, Tokyo (2004)Google Scholar
  31. 31.
    Sayah, A.: Équations d’Hamilton–Jacobi du emier Ordre Avec Termes Intégro-Différentiels. I. Unicité des solutions de viscosité. Commun. Partial Differ. Equ. 16, 1057–1074 (1991)CrossRefzbMATHGoogle Scholar
  32. 32.
    Sayah, A.: Équations d’Hamilton–Jacobi du emier Ordre Avec Termes Intégro-Différentiels. II. Existence de solutions de viscosité. Commun. Partial Differ. Equ. 16, 1075–1093 (1991)CrossRefzbMATHGoogle Scholar
  33. 33.
    Tchamba, T.T.: Large time behavior of solutions of viscous Hamilton–Jacobi equations with superquadratic Hamiltonian. Asymptot. Anal. 66, 161–186 (2010)MathSciNetzbMATHGoogle Scholar
  34. 34.
    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 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

Personalised recommendations