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6 Free Boundary Problems and Phase Transitions
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(see [9]). Now consider the degenerate parabolic PDE for the enthalpy:
e t = Δβ(e)+ f , x ∈ G , t> 0 (6.22)

subject to an initial condition

e(t = 0) = e 0 (6.23)
which is such that θ 0 = β(e 0 ). Note that the temperature can be calculated
uniquely from the enthalpy but not the other way around! Also we prescribe
appropriate boundary conditions of Dirichlet or Neumann type (in correspon-
dence to the boundary conditions (6.15)) on the fixed boundary ∂G:

on ∂G , t> 0 (6.24)
e = e 1
where, again, e 1 is such that θ 1 = β(e 1 ), or, resp.

on ∂G , t> 0 . (6.25)
grad e.ν = f 1
It is a simple exercise in distributional calculus to show that a smooth solu-
tion e of (6.22)–(6.25), which is such that its 0-level set is a smooth surface in G
for t> 0, gives a smooth solution θ of the Stefan problem (6.13)–(6.18), simply
by defining the temperature θ = β(e) and the free boundary Γ(t) as the 0-level
set of e(·, t). The nice feature of the nonlinear initial-boundary value problem
for the degenerate parabolic equation (6.22) is the fact that the phase transition
boundary Γ(t) does not appear explicitely. This allows for somewhat simpler
analytical and numerical approaches.
For a collection of analytical results and references on the Stefan problem
and its variants we refer to [8].
Comments on the Images 6.1–6.8 The Images 6.1–6.7 show icebergs in Patag-
onian lakes. Clearly, the free Stefan boundary is not visible itself, since it is the
ice-water phase transition under the water surface. In Image 6.5 and in Image 6.6
we getaglimpse of it, though …Whatwesee on theother images is –atleast
in part – the intersection of the free (Stefan) boundary with the fixed boundary
(water surface). Note that about 7/8 of the mass of a typical iceberg is under
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water !
Also the air-ice interface of icebergs, which is very well visible in most of the
Images 6.1–6.7 is determined by a free boundary problem, however, of much
more complicated nature than the Stefan Problem determining the ice-water
phase transition. Clearly, various mechanisms enter in the formation of the
above-water surface of an iceberg: the formation process of the iceberg itself
(mostly through calving from a glacier) giving the initial condition, the wind
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pattern, erosion by waves, ablation (through solar radiation), melting … .
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http://www.wordplay.com/tourism/icebergs
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http://www.wordplay.com/tourism/icebergs