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Second-order elliptic integro-differential equations: viscosity solutions’ theory revisited. (English) Zbl 1155.45004
Three types of viscosity solution of second order nonlinear elliptic integrodifferential equations are defined to account for solutions with arbitrary growth at infinity. Stability results for these viscosity solutions are derived. The Jensen-Ishii lemma is generalized and applied to prove comparison theorems.

MSC:
 45K05 Integro-partial differential equations 45M10 Stability theory for integral equations 49L25 Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games 45G10 Other nonlinear integral equations
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References:
 [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, 293-317, (1996) · Zbl 0870.45002 [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, 695-711, (2006) · Zbl 1105.45004 [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, 167-169, (2006) · Zbl 1125.45008 [5] Barles, G.; Buckdahn, R.; Pardoux, E., Backward stochastic differential equations and integral-partial differential equations, Stochastics stochastics rep., 60, 1-2, 57-83, (1997) · Zbl 0878.60036 [6] Bensaoud, I.; Sayah, A., Stability results for hamilton – jacobi equations with integro-differential terms and discontinuous Hamiltonians, Arch. math. (basel), 79, 5, 392-395, (2002) · Zbl 1022.35004 [7] Bertoin, J., Lévy processes, Cambridge tracts in mathematics, vol. 121, (1996), Cambridge University Press Cambridge · Zbl 0861.60003 [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, 1-67, (1992) · Zbl 0755.35015 [9] Garroni, M.G.; Menaldi, J.L., Second order elliptic integro-differential problems, Chapman & Hall/CRC research notes in mathematics, vol. 430, (2002), Chapman & Hall/CRC Boca Raton, FL · Zbl 0806.45007 [10] Imbert, C., A non-local regularization of first order hamilton – jacobi equations, J. differential equations, 211, 1, 218-246, (2005) · Zbl 1073.35059 [11] C. Imbert, R. Monneau, E. Rouy, Homogenization of first order equations with $$u / \operatorname{\&z.epsiv;}$$-periodic Hamiltonians. Part II: application to dislocation dynamics, Commun. Partial Differential Equations, submitted for publication · Zbl 1143.35005 [12] Ishii, H., On uniqueness and existence of viscosity solutions of fully nonlinear second-order elliptic pdes, Comm. pure appl. math., 42, 1, 15-45, (1989) · Zbl 0645.35025 [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, 137-165, (2006) · Zbl 1105.45006 [14] Jensen, R., The maximum principle for viscosity solutions of fully nonlinear second order partial differential equations, Arch. rational mech. anal., 101, 1, 1-27, (1988) · Zbl 0708.35019 [15] Øksendal, B.; Sulem, A., Applied stochastic control of jump diffusions, Universitext, (2005), Springer-Verlag Berlin · Zbl 1074.93009 [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 0899.60039 [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, 1057-1093, (1991) · Zbl 0742.45004 [18] Silvestre, L., Hölder estimates for solutions of integro-differential equations like the fractional Laplace, Indiana univ. math. J., 55, 3, 1155-1174, (2006) · Zbl 1101.45004 [19] Soner, H.M., Optimal control with state-space constraint. II, SIAM J. control optim., 24, 6, 1110-1122, (1986) · Zbl 0619.49013 [20] Woyczyński, W.A., Lévy processes in the physical sciences, (), 241-266 · Zbl 0982.60043
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