On generalized Ventcel's type boundary conditions for Laplace operator in a bounded domain

Virginie Bonnaillie-Noël (1) , Marc Dambrine (2) , Frédéric Hérau (3) , Grégory Vial (1)

1 IRMAR - Institut de Recherche Mathématique de Rennes
2 LMAP - Laboratoire de Mathématiques et de leurs Applications [Pau]
3 LMR - Laboratoire de Mathématiques de Reims

Ventcel boundary conditions are second order differential conditions that appear in asymptotic models. Like Robin boundary conditions, they lead to wellposed variational problems under a sign condition of a coefficient. Nevertheless situations where this condition is violated appeared in several works. The wellposedness of such problems was still open. This manuscript establishes, in the generic case, the existence and uniqueness of the solution for the Ventcel boundary value problem without the sign condition. Then, we consider perforated geometries and give conditions to remove the genericity restriction.

Mots clés: Ventcel boundary condition; pseudodifferential operators; asymptotic analysis; shape dependency

Domaines: Optimisation et contrôle [math.OC]

Virginie Bonnaillie-Noël, Marc Dambrine, Frédéric Hérau, Grégory Vial. On generalized Ventcel's type boundary conditions for Laplace operator in a bounded domain. SIAM Journal on Mathematical Analysis, 2010, 42 (2), pp.931-945. ⟨10.1137/090756521⟩. ⟨hal-00376786⟩