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6. Free Boundary Problems
and Phase Transitions

Initial and initial-boundary value problems for systems of partial differential
equations (PDEs) have functions or, more generally, distributions in the scalar
case and vector fields of functions or distributions in the vector-valued case as
solutions. Usually, the d-dimensional domain, on which the PDEs are posed,
is given and the problem formulation is based on a fixed geometry. Obviously,
there have to be compatibilities between the differential operator, particularly
its differential order and certain geometric properties, and the side (initial-
boundary) conditions and the geometry of the domain on which the problem is
posed in order to guarantee well-posedness of the problem under consideration.
In particular, for a given differential operator the number of initial-boundary
conditions and the geometry of the domain boundary are crucial for solvability,
uniqueness and continuous dependence on data.
Free boundary problems for PDEs have a totally different feature, namely that
geometric information is an inherent part of the solution. Typically, the solution
of afreeboundary problem consists of oneormorefunctions or distributions
d
ANDaset(thesocalledfreeboundary,subsetofR ),onwhichcertainconditions

Fig. 6.1. Layered iceberg, Lago Argentino