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, FranceCyril 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
![Second-order elliptic integro-differential equations: viscosity solutions' theory revisited cover](/_next/image?url=https%3A%2F%2Fcontent.ems.press%2Fassets%2Fpublic%2Fimages%2Fserial-issues%2Fcover-aihpc-volume-25-issue-3.png&w=3840&q=90)
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