Existence of a weak solution for a n-dimensional elliptic-parabolic problem.

The paper deals with the existence of a weak solution for a problem with free boundaries separating a saturated, an unsaturated and a dry region. The idea is to follow the procedure by which a weak solution for the Stefan problem is defined; namely, to derive a system of equations, where the free boundaries are implicitly contained in the equations. To prove the existence of a weak solution an approximating sequence of functions is considered. A number of a priori estimates are derived, which allow the passage to the limit of the sequence. The convergence of the approximating sequence to a solution of the original problem is shown. A uniqueness result remains an open problem