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6 Free Boundary Problems and Phase Transitions
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seminal paper [7], investigating the ice layer formation in a water-ice phase tran-
sition.Interestingly enough,Stefancomparedhis data,obtainedbymathemati-
cal modeling, to measurements taken in the quest of the search of a north-west
passage [11] through the northern polar sea. In his paper [7] Stefan investigated
the non-stationary transport of heat in the ice and formulated a free boundary
problem, which is now known as the classical Stefan problem and which has
given rise to the modern research area of phase transition modeling by free
boundary problems. As a basic reference we refer to [10].
Some of the photographs associated to this chapter show icebergs in lakes
of Patagonia. The evolution of their water-ice phase transition free boundary is
modeled by the 3-dimensional Stefan problem formulated below.
d
Therefore, consider a domain G ∈ R (of course d = 1, 2or3for physical
reasons but there is no mathematical reason to exclude larger dimensions here),
in which the ice-water ensemble is contained. At time t> 0 assume that the
domainGisdividedinto2subdomains,G 1 (t)containingthesolidphase(ice)and
G 2 (t) containing the liquid phase (water). These subdomains shall be separated
by asmoothsurface Γ(t), where the phase transition occurs. Γ(t)isthe free
boundary, an unknown of the Stefan problem. Heat transport is modeled by the
linear heat equation:
θ t = Δθ + f , x ∈ G 1 (t)and x ∈ G 2 (t), t> 0 , (6.13)

where f is a given function describing external heat sources/sinks. Here we
assumed that the local mass density, the heat conductivity and the heat capacity
at constant volume are equal and constant 1 in both phases. More realistically,
piecewise constants can be used for modeling purposes. The parabolic PDE
(6.13) has to be supplemented by an initial condition

θ(t = 0) = θ 0 in G (6.14)

and appropriate boundary conditions at the fixed boundary ∂G. Usually, the
temperatureisfixedthere

θ = θ 1 on ∂G (6.15)
or the heat flux through the boundary is given:

gradθ .ν = f 1 on ∂G , t> 0 . (6.16)
Here ν denotes the exterior unit normal to ∂G. Also, mixed Neumann–Dirichlet
boundary conditions can be prescribed, corresponding to different types of
boundary segments.
Disregarding the phase transition the problem (6.13), (6.14), (6.15) or (6.16)
is well-posed. Thus, additional conditions are needed to determine the free
boundary. The physically intuitive condition says that the temperature at the free
boundary is the constant melting temperature θ m of the solid phase. Obviously