Ergodic problems and periodic homogenization for fully nonlinear equations in half-space type domains with Neumann boundary conditions

Guy Barles (1) , Francesca da Lio (2) , Pierre Louis Lions (3, 4) , Panagiotis E. Souganidis (5)

1 LMPT - Laboratoire de Mathématiques et Physique Théorique
2 Dipartimento di Matematica Pura e Applicata [Padova]
3 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
4 Collège de France - Chaire Équations aux dérivées partielles et applications
5 Department of Mathematics

We study periodic homogenization problems for second-order pde in half-space type domains with Neumann boundary conditions. In particular, we are interested in ``singular problems'' for which it is necessary to determine both the homogenized equation and boundary conditions. We provide new results for fully nonlinear equations and boundary conditions. Our results extend previous work of Tanaka in the linear, periodic setting in half-spaces parallel to the axes of the periodicity, and of Arisawa in a rather restrictive nonlinear periodic framework. The key step in our analysis is the study of associated ergodic problems in domains with similar structure.

Mots clés: Homogenization; nonlinear equations; Neumann boundary conditions

Domaines: Equations aux dérivées partielles [math.AP]; Mathématiques générales [math.GM]

Guy Barles, Francesca da Lio, Pierre Louis Lions, Panagiotis E. Souganidis. Ergodic problems and periodic homogenization for fully nonlinear equations in half-space type domains with Neumann boundary conditions. Indiana University Mathematics Journal, 2008, 57 (5), pp.2355--2376. ⟨hal-00426354⟩