The dam problem with leaky boundary conditions

The goal of this paper is to study the fluid flow through a two-dimensional porous medium when we impose a leaky boundary condition. We show in particular that the situation is quite different from the one with the usual Dirichlet boundary condition.This work was initiated at the Universidad Complutense in Madrid and at the University of Bonn. We would like to thank these institutions. We also thank the I.M.A. (University of Minnesota) for providing its support and a nice working atmosphere to the second author during the completion of this paper.