ESAIM: Control, Optimisation and Calculus of Variations

Table of Contents - January 2012 - Volume 18, Issue 01  

Research Article

On a Bernoulli problem with geometric constraints

Antoine Lauraina1 and Yannick Privata2

a1 Karl-Franzens-University of Graz, Department of Mathematics and Scientific Computing, Heinrichstrasse 36, 8010 Graz, Austria. Antoine.Laurain@uni-graz.at

a2 IRMAR, ENS Cachan Bretagne, Univ. Rennes 1, CNRS, UEB, av. Robert Schuman, 35170 Bruz, France; Yannick.Privat@bretagne.ens-cachan.fr

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.

(Received March 11 2010)

(Revised July 18 2010)

(Revised September 16 2010)

(Online publication December 2 2010)

Key Words:

  • Free boundary problem;
  • Bernoulli condition;
  • shape optimization

Mathematics Subject Classification:

  • 49J10;
  • 35J25;
  • 35N05;
  • 65P05