Free or moving boundary problems appear in many areas of mathemat-
ics and science in general. Typical examples are shape optimization (least
area for fixed volume, optimal insulation, minimal capacity potential at pre-
scribed volume), phase transitions (melting of a solid, Cahn-Hilliard), fluid
dynamics (incompressible or compressible flow in porous media, cavitation,
flame propagation), probability and statistics (optimal stopping time, hy-
pothesis testing, financial mathematics), among other areas.
They are also an important mathematical tool for proving the existence
of solutions in nonlinear problems, homogenization limits in random and
periodic media, etc.
A typical example of a free boundary problem is the evolution in time
of a solid-liquid configuration: suppose that we have a container D filled
with a material that is in solid state in some region QQ C D and liquid in
A0 = D\n0.
We know its initial temperature distribution TQ(X) and we can control
what happens on dD at all times (perfect insulation, constant temperature,
etc.). Then from this knowledge we should be able to reconstruct the solid-
liquid configuration, £^, A*, and the temperature distribution T(x,t) for all
times t 0.
Roughly, on Qt, At the temperature should satisfy some type of diffusion
equation, while across the transition surface, we should have some "balance"
conditions that express the dynamics of the melting process.
The separation surface dftt between solid and liquid is thus determined
implicitly by these "balance conditions". In attempting to construct solu-
tions to such a problem, one is thus confronted with a choice. We could try