JournalsaihpcVol. 25, No. 3pp. 567–585

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

  • Guy Barles

    Laboratoire de Mathématiques et Physique Théorique CNRS UMR 6083, Fédération Denis Poisson, Université François Rabelais, Parc de Grandmont, 37200 Tours, France

  • Cyril Imbert

    Polytech'Montpellier & Institut de mathématiques et de modélisation de Montpellier, UMR CNRS 5149, Université Montpellier II, CC 051, Place E. Bataillon, 34 095 Montpellier cedex 5, France
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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.

Cite this article

Guy Barles, Cyril Imbert, Second-order elliptic integro-differential equations: viscosity solutions' theory revisited. Ann. Inst. H. Poincaré Anal. Non Linéaire 25 (2008), no. 3, pp. 567–585

DOI 10.1016/J.ANIHPC.2007.02.007