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6 Free Boundary Problems and Phase Transitions
98

Fig. 6.6. Complicated structure of the free boundary and its intersection with the fixed boundary

we can normalize θ m = 0 and regard θ from now on as the difference between
the local temperature and the melting temperature:
θ = 0at Γ(t) . (6.17)

Note that the condition (6.17) cannot suffice to determine the free boundary.
Fixing Γ(t) arbitrarily (in a non-degenerate way) leaves us with two decoupled
linear boundary valueproblemsfor theheatequation, oneineachphasewith
Dirichlet boundary data on the interface. Both of these problems are uniquely
solvable!
The second interface condition, derived from local energy balance [9], reads:

Lv n = [gradθ .n] , (6.18)

where n denotes the unit normal to the interface, [.] stands for the jump across
the interface and v n for the interface velocity in orthogonal direction. L is the
latent heat parameter representing the energy needed for a phase change.
When the interface is a regular surface, given by the equation H(x, t)=0,we
have

v n = −H t grad H · n .