Nonlinear elliptic-parabolic problems

We introduce a notion of viscosity solutions for a general class of elliptic-parabolic phase transition problems. These include the Richards equation, which is a classical model in filtration theory. Existence and uniqueness results are proved via the comparison principle. In particular, we show existence and stability properties of maximal and minimal viscosity solutions for a general class of initial data. These results are new even in the linear case, where we also show that viscosity solutions coincide with the regular weak solutions introduced in [Alt&Luckhaus 1983].