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6 Free Boundary Problems and Phase Transitions
88

on theunknown function(s)are prescribed. Ifweassumefor amomentthatthe
free boundary is fixed, then, typically, the problem would be over-determined.
So, in fact, the additional conditions are needed to determine the free boundary
itself.
Regularity (smoothness) is an important issue in PDE theory. Usually, a cer-
tain degree of regularity (differentiability in the classical or weak sense) of the
solutions of initial-boundary value problems is necessary to prove their unique-
ness, their continuous dependence on the data, to carry out certain scaling
limits, as done in singular perturbation theory, and to devise efficient numerical
discretisation techniques. For free boundary problems the situation is definitely
more complex. Not only the regularity of the unknown functions is important,
but also the regularity of the unknown set, the free boundary. Typical questions,
which arise, are: does the free boundary have empty topological interior? What
are its measure theoretical properties? Is it a (finite union of) smooth manifolds?
What is the optimal regularity of the free boundary? In many cases the study
of the optimal regularity of the free boundary is of paramount importance for
understanding the solution of the free boundary problem under consideration,
to prove uniqueness, stability etc.
Obviously, themathematicalliteratureoffreeboundaryproblemsisvast, at
this point we refer to the book [4] for a review of basic analytical tools and for
further references.
To start a more concrete discussion, we consider the maybe best-studied free
boundary problem, the so called obstacle problem for the Laplace operator. Let
us consider the classical Dirichlet functional
1
D(v):= | grad v| dx , (6.1)
2
2
G
d
where G is a bounded domain in R with a sufficiently smooth boundary ∂G.
Also, let us fix the boundary values of v and, for the moment, the considered
class of functions
1
Y := v ∈ H (G) | v = ψ on ∂G , (6.2)
1
where ψ is a prescribed function in the Sobolev space H (G), which consists of
those square integrable functions, defined almost everywhere in G with values
in R, which also have a square integrable distributional gradient.
It is an easy exercise to show that the minimum u of the functional D over the
set of functions Y is the unique harmonic function on G assuming the boundary
values ψ,i.e. u uniquely solves the boundary value problem:

Δu = 0in G (6.3)
u = ψ on ∂G . (6.4)

The obstacle problem is obtained by a modification of this minimizing pro-
1
cedure. Letφ ∈ H (G) be another given function, the so called obstacle, and look