Free Boundary Problems: Geometric Theory

The aim of this text is two-fold. The first is to introduce the reader to the theory of some classes of nonlinear evolution-type partial differential equations so that an engineer, physicist or other practitioner could form a clear conception of even the most recent results in the field. The author's main aim is to help the reader imagine (in space-time) the different processes (the solutions) with special regard to free boundaries and their dependence on the parameters of the model. The models considered are nonlinear degenerate parabolic equations and systems, and first and second order nonlinear hyperbolic equations. The equations considered arise in a great variety of practical situations. This is the reason why no special derivations of the models and applications of the results are given. The formal mathematical style is avoided. The author concentrates on the questions that are most important from the engineering point of view. These are illustrated with several explicit examples and figures.