Second-order elliptic integro-differential equations : viscosity solutions' theory revisited

Barles, Guy; Imbert, Cyril

doi:10.1016/j.anihpc.2007.02.007

Barles, Guy ; Imbert, Cyril

Annales de l'I.H.P. Analyse non linéaire, Tome 25 (2008) no. 3, pp. 567-585

MR Zbl | 5 citations dans Numdam

DOI : 10.1016/j.anihpc.2007.02.007

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     author = {Barles, Guy and Imbert, Cyril},
     title = {Second-order elliptic integro-differential equations : viscosity solutions' theory revisited},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {567--585},
     year = {2008},
     publisher = {Elsevier},
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     doi = {10.1016/j.anihpc.2007.02.007},
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     zbl = {1155.45004},
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     url = {https://www.numdam.org/articles/10.1016/j.anihpc.2007.02.007/}
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Barles, Guy; Imbert, Cyril. Second-order elliptic integro-differential equations : viscosity solutions' theory revisited. Annales de l'I.H.P. Analyse non linéaire, Tome 25 (2008) no. 3, pp. 567-585. doi: 10.1016/j.anihpc.2007.02.007

Bibliographie
Cité par

[1] N. Alibaud, C. Imbert, A non-local perturbation of first order Hamilton-Jacobi equations with unbounded data, submitted for publication, 2007.

[2] Alvarez O., Tourin A., Viscosity solutions of nonlinear integro-differential equations, Ann. Inst. H. Poincaré Anal. Non Linéaire 13 (3) (1996) 293-317. | Zbl | MR | Numdam

[3] Arisawa M., A new definition of viscosity solutions for a class of second-order degenerate elliptic integro-differential equations, Ann. Inst. H. Poincaré 23 (5) (2006) 695-711. | Zbl | MR | Numdam

[4] Arisawa M., Corrigendum in the comparison theorems in “a new definition of viscosity solutions for a class of second-order degenerate elliptic integro-differential equations”, Ann. Inst. H. Poincaré 24 (1) (2006) 167-169. | Zbl | MR | Numdam

[5] Barles G., Buckdahn R., Pardoux E., Backward stochastic differential equations and integral-partial differential equations, Stochastics Stochastics Rep. 60 (1-2) (1997) 57-83. | Zbl | MR

[6] Bensaoud I., Sayah A., Stability results for Hamilton-Jacobi equations with integro-differential terms and discontinuous Hamiltonians, Arch. Math. (Basel) 79 (5) (2002) 392-395. | Zbl | MR

[7] Bertoin J., Lévy processes, Cambridge Tracts in Mathematics, vol. 121, Cambridge University Press, Cambridge, 1996. | Zbl | MR

[8] Crandall M.G., Ishii H., Lions P.-L., User's guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. (N.S.) 27 (1) (1992) 1-67. | Zbl | MR

[9] Garroni M.G., Menaldi J.L., Second Order Elliptic Integro-Differential Problems, Chapman & Hall/CRC Research Notes in Mathematics, vol. 430, Chapman & Hall/CRC, Boca Raton, FL, 2002. | Zbl | MR

[10] Imbert C., A non-local regularization of first order Hamilton-Jacobi equations, J. Differential Equations 211 (1) (2005) 218-246. | Zbl | MR

[11] C. Imbert, R. Monneau, E. Rouy, Homogenization of first order equations with $u / ϵ$ -periodic Hamiltonians. Part II: application to dislocation dynamics, Commun. Partial Differential Equations, submitted for publication. | Zbl | MR

[12] Ishii H., On uniqueness and existence of viscosity solutions of fully nonlinear second-order elliptic PDEs, Comm. Pure Appl. Math. 42 (1) (1989) 15-45. | Zbl | MR

[13] Jakobsen E.R., Karlsen K.H., A “maximum principle for semicontinuous functions” applicable to integro-partial differential equations, NoDEA Nonlinear Differential Equations Appl. 13 (2) (2006) 137-165. | Zbl | MR

[14] Jensen R., The maximum principle for viscosity solutions of fully nonlinear second order partial differential equations, Arch. Rational Mech. Anal. 101 (1) (1988) 1-27. | Zbl | MR

[15] Øksendal B., Sulem A., Applied Stochastic Control of Jump Diffusions, Universitext, Springer-Verlag, Berlin, 2005. | Zbl | MR

[16] Pham H., Optimal stopping of controlled jump diffusion processes: a viscosity solution approach, J. Math. Systems Estim. Control 8 (1) (1998), 27 pp. (electronic). | Zbl | MR

[17] Sayah A., Équations d'Hamilton-Jacobi du premier ordre avec termes intégro-différentiels. I. Unicité des solutions de viscosité. II. Existence de solutions de viscosité, Comm. Partial Differential Equations 16 (6-7) (1991) 1057-1093. | Zbl | MR

[18] Silvestre L., Hölder estimates for solutions of integro-differential equations like the fractional Laplace, Indiana Univ. Math. J. 55 (3) (2006) 1155-1174. | Zbl | MR

[19] Soner H.M., Optimal control with state-space constraint. II, SIAM J. Control Optim. 24 (6) (1986) 1110-1122. | Zbl | MR

[20] Woyczyński W.A., Lévy processes in the physical sciences, in: Lévy processes, Birkhäuser Boston, Boston, MA, 2001, pp. 241-266. | Zbl | MR

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