A free boundary problem involving a cusp

We consider a stationary free boundary problem describing the stationary flow of fresh and salt water in a porous medium. The salt water is supposed to be stagnant, while the fresh water on top of it is drawn into wells. In a previous work it has been shown, that for pumping rates $Q < Q_{cr$ a solution with smooth interface exists. In this part we study the case $Q=Q_{cr$ in two dimensions. We show that the interface has isolated singularities. At each singularity the free boundary develops a cusp or becomes vertical. By means of local analysis techniques we obtain the asymptotic behaviour of the free boundary at these singularities.