On Generalized Ventcel's Type Boundary Conditions for Laplace Operator in a Bounded Domain

Abstract

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.

Artificial conditions for the linear elasticity equations

“…Now, applying Point 3 of Proposition 3.4, we consider a real R > 1 such that the system (2.10) admits a unique solution u on the domain B R \ B 1 as soon as γ ∈ S. Consider a domain ω close to B 1 and ask if system (2.10) admits a unique solution u on the domain B R \ ω for the same set of parameters S. Our aim is to prove that the perturbed Dirichlet-to-Neumann map depends continuously (as operator) on smooth perturbations of the domain. We adapt directly [9…”

“…The difference with the situation presented in [9] is that one deals with the elasticity system instead of the Laplace equation. Hence the transported weak formation for (3.3) is more complicated: one has to transport the symmetrized gradient instead of the usual gradient.…”

“…Then, one adapts straightforwardly step-by-step the proof of Theorem 3.1 in [9]. This is left to the reader.…”

“…Whatever the sign of the parameter ν ∈ (−1, 0.5), the obtained approximate boundary condition leads to a non-coercive weak formulation. A similar problem has been investigated in [9] for the scalar Laplace equation.…”

“…Finally, in the context of wall laws for rough boundary, Bresch and Milisic obtained in [12] a second order wall law which is written in the form 6) where β and γ are solutions of the two cells problem and have opposite signs. In a first work on the scalar case [9], we studied the existence and uniqueness for the non-coercive Ventcel problem. In this paper we give the first results, to the best of our knowledge, in the vectorial case.…”