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


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