Nonlinear elliptic-parabolic problems

TL;DR

This paper develops a new framework for viscosity solutions to nonlinear elliptic-parabolic phase transition problems, including the Richards equation, establishing existence, uniqueness, and stability results that unify and extend previous theories.

Contribution

It introduces a general notion of viscosity solutions for elliptic-parabolic problems, proving foundational existence and uniqueness results that are novel even in linear cases.

Findings

Existence and uniqueness of viscosity solutions established

Maximal and minimal solutions exhibit stability

Viscosity solutions coincide with classical weak solutions in linear cases

Abstract

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].