  In this volume a theory for models of transport in the presence of a free
boundary is developed. Macroscopic laws of transport are described by PDEs.
When the system is open, there are several mechanisms to couple the system with
the external forces. Here a class of systems where the interaction with the
exterior takes place in correspondence of a free boundary is considered. Both
continuous and discrete models sharing the same structure are analyzed. In Part
I a free boundary problem related to the Stefan Problem is worked out in all
details. For this model a new notion of relaxed solution is proposed for which
global existence and uniqueness is proven. It is also shown that this is the
hydrodynamic limit of the empirical mass density of the associated particle
system. In Part II several other models are discussed. The expectation is that
the results proved for the basic model extend to these other cases. All the
models discussed in this volume have an interest in problems arising in several
research fields such as heat conduction, queuing theory, propagation of fire,
interface dynamics, population dynamics, evolution of biological systems with
selection mechanisms.In general researchers interested in the relations between
PDEs and stochastic processes can find in this volume an extension of this
correspondence to modern mathematical physics.
