Persistency of wellposedness of Ventcel’s boundary value problem under shape deformations

Abstract

Ventcel boundary conditions are second order differential conditions that appear in asymptotic models. Like Robin boundary conditions, they lead to well-posed variational problems under a sign condition of the coefficient. This is achieved when physical situations are considered. Nevertheless, situations where this condition is violated appeared in several recent works where absorbing boundary conditions or equivalent boundary conditions on rough surfaces are sought for numerical purposes. The well-posedness of such problems was recently investigated: up to a countable set of parameters, existence and uniqueness of the solution for the Ventcel boundary value problem holds without the sign condition. However, the values to be avoided depend on the domain where the boundary value problem is set. In this work, we address the question of the persistency of the solvability of the boundary value problem under domain deformation.