On Neumann and oblique derivatives boundary conditions for nonlocal elliptic equations

Guy Barles (1, 2) , Christine Georgelin (1, 2) , Espen R. Jakobsen (3)

1 FDP - Fédération de recherche Denis Poisson
2 LMPT - Laboratoire de Mathématiques et Physique Théorique
3 Department of Mathematical Sciences

Inspired by the penalization of the domain approach of Lions \& Sznitman, we give a sense to Neumann and oblique derivatives boundary value problems for nonlocal, possibly degenerate elliptic equations. Two different cases are considered: (i) homogeneous Neumann boundary conditions in convex, possibly non-smooth and unbounded domains, and (ii) general oblique derivatives boundary conditions in smooth, bounded, and possibly non-convex domains. In each case we give apropriate definitions of viscosity solutions and prove uniqueness of solutions of the corresponding boundary value problems. We prove that these boundary value problems arise in the penalization of the domain limit from whole space problems and obtain as a corollary the existence of solutions of these problems.

Mots clés: Nonlocal Elliptic equation; Neumann-type boundary conditions; general nonlocal operators; reflection; viscosity solutions; Lévy process

Domaines: Equations aux dérivées partielles [math.AP]

Guy Barles, Christine Georgelin, Espen R. Jakobsen. On Neumann and oblique derivatives boundary conditions for nonlocal elliptic equations. 2013. ⟨hal-00793393v1⟩