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6 Free Boundary Problems and Phase Transitions
91

Obviously, when F is a fixed hyper-surface in G, then one of the conditions
in (6.9) is redundant and the problem 6.8–6.10 is overdetermined.
In order to illustrate the difficulties of the obstacle problem, set

w = u − φ , x ∈ G .
Then, denoting h(x) = −Δφ(x), we can rewrite the Euler–Lagrange system (6.8)–
(6.10) as

(6.11)
Δw = h(x)1 {w>0}
w = ψ − φ on ∂G . (6.12)
Note that the minimisation of the Dirichlet functional over the set Y defined
in (6.2) leads to a simple linear problem while the minimisation over the con-
strained set X leads to a complicated nonlinear problem, as indicated by the
right hand side of (6.11)! Assuming a smooth obstacle we conclude that the right
hand side of the semilinear Poisson equation (6.11) is bounded in G,suchthat
by classical interior regularity results of linear uniformly elliptic equations we
conclude that the solution w is locally in the Sobolev space W 2,p for every p< ∞
(which is the space of locally p-integrable functions with locally p-integrable
weak second derivatives). By the Sobolev imbedding theorem we conclude that
w (and consequently u)islocally in thespace C 1,α (locally Hölder continuous
first derivatives) for every 0 < α < 1. More cannot be concluded from this simple
argument.
From the many results on optimal regularity of the solution of the obstacle
problem we cite the review [2], where optimal local regularity for u, i.e. u ∈ C 1,1
is shown, if the obstacle is sufficiently smooth. Note that the optimality of this
result follows trivially from the fact that Δu jumps from 0 to Δφ on the free
boundary F!Moreover, thefreeboundaryhas locallyfinite (n − 1)-dimensional
Hausdorf measure and is locally a C 1,α surface, for some α in the open interval
(0, 1), except at a ‘small’ set of singular points, contained in a smooth manifold.
Singularities can be excluded by assuming an exterior cone condition. Moreover,
if the free boundary is locally Lipschitz continuous, then it is locally as smooth as
thedata,inparticularitislocallyanalytic,ifthedataareanalytic.Weremarkthat
the proof of the optimal regularity of the free boundary requires deep insights
into elliptic theory, in particular the celebrated ‘monotonicity formula’ of Luis
1
Caffarelli .
A more complex application of free boundary problems arises in the theory
of phase transitions. A historically important example of a phase transition is
theformation of iceinthe polarsea,asoriginallyinvestigatedbythe Austrian
2
mathematician Josef Stefan (1835–1893). In the year 1891 Stefan published his

1
http://www.ma.utexas.edu/users/caffarel/
2
http://www-groups.dcs.st-and.ac.uk/∼history/Mathematicians/Stefan_Josef.html