On a Bernoulli problem with geometric constraints

¹ Karl-Franzens-University of Graz, Department of Mathematics and Scientific Computing, Heinrichstrasse 36, 8010 Graz, Austria
Antoine.Laurain@uni-graz.at
² IRMAR, ENS Cachan Bretagne, Univ. Rennes 1, CNRS, UEB, av. Robert Schuman, 35170 Bruz, France
Yannick.Privat@bretagne.ens-cachan.fr

Received: 11 March 2010
Revised: 18 July 2010
Revised: 16 September 2010

Abstract

A Bernoulli free boundary problem with geometrical constraints is studied. The domain Ω is constrained to lie in the half space determined by x₁ ≥ 0 and its boundary to contain a segment of the hyperplane  {x₁ = 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.