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

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Abstract

The aim of this work is to revisit viscosity solutions' theory for second-order elliptic integro-differential equations and to provide a general framework which takes into account solutions with arbitrary growth at infinity. Our main contribution is a new Jensen-Ishii's lemma for integro-differential equations, which is stated for solutions with no restriction on their growth at infinity. The proof of this result, which is of course a key ingredient to prove comparison principles, relies on a new definition of viscosity solution for integro-differential equation (equivalent to the two classical ones) which combines the approach with test-functions and sub-superjets. © 2007 Elsevier Masson SAS. All rights reserved.

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Barles, G., & Imbert, C. (2008). Second-order elliptic integro-differential equations: viscosity solutions’ theory revisited. Annales de l’Institut Henri Poincare (C) Analyse Non Lineaire, 25(3), 567–585. https://doi.org/10.1016/j.anihpc.2007.02.007

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