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6 Free Boundary Problems and Phase Transitions
90

for a mimimizer of the energy functional D,which is in Y AND which nowhere
(in the almost everywhere sense) in G stays below the obstacle φ. More formally,
consider the convex set of functions:

X := Y ∩ {v | v ≥ φ} (6.5)
and find:

u := argmin v∈X D(v) . (6.6)

Clearly, the obstacle φ cannot stay above the function ψ on the boundary of
G,i.e.weassume:

φ ≤ ψ on ∂G . (6.7)

In thetwo 2-dimensionalcasethe solution of theobstacleproblem canbe
seen as the (small amplitude) displacement of an elastic membrane, fixed at the
boundary, minimizing its total energy under the constraint of having to stay
above a solid obstacle.
It is actually easy to show that the obstacle problem (6.6) has a unique
solution (minimizer) u ∈ X, all the technical mathematical analysis goes into
the investigation of the regularity properties of its solution u and of the free
boundary defined below.
We define the non-coincidence set N as

N := {x ∈ G | u(x) > φ(x)}

and the coincidence set C
C := {x ∈ G | u(x) = φ(x)} .

Then the free boundary F is defined as that part of the topological boundary of
N, which lies in G,i.e.
F := ∂N ∩ G ,

in other words it is the interface between the sets C and N.
It is a simple exercise to derive the Euler–Lagrange equations of the minimi-
sation problem (6.6). We find, assuming sufficient regularity of the minimizer u
and of the free boundary F:

Δu = 0in N , u = φ in C and − Δu ≥ 0in G (6.8)
u = φ and grad u · n = gradφ · n on F , (6.9)

where n denotes a unit normal vector to F, and finally:

u = ψ on ∂G . (6.10)