On a Bernoulli problem with geometric constraints

Antoine Laurain (1) , Yannick Privat (2)

1 HU Berlin - Humboldt-Universität zu Berlin = Humboldt University of Berlin = Université Humboldt de Berlin
2 IRMAR - Institut de Recherche Mathématique de Rennes

A Bernoulli free boundary problem with geometrical constraints is studied. The domain $\Om$ is constrained to lie in the half space determined by $x_1\geq 0$ and its boundary to contain a segment of the hyperplane $\{x_1=0\}$ where non-homogeneous Dirichlet conditions are imposed. We are then looking for the solution of a partial differential equation satisfying a Dirichlet and a Neumann boundary condition simultaneously on the free boundary. The existence and uniqueness of a solution have already been addressed and this paper is devoted first to the study of geometric and asymptotic properties of the solution and then to the numerical treatment of the problem using a shape optimization formulation. The major difficulty and originality of this paper lies in the treatment of the geometric constraints.

Mots clés: free boundary problem; Bernoulli condition; shape optimization; parameterization method

Domaines: Equations aux dérivées partielles [math.AP]

Antoine Laurain, Yannick Privat. On a Bernoulli problem with geometric constraints. ESAIM: Control, Optimisation and Calculus of Variations, 2012, 18 (1), pp.157-180. ⟨10.1051/cocv/2010049⟩. ⟨hal-00462494v2⟩