The one-phase Bernoulli problem minimizes a Dirichlet energy together with the volume of the positive
phase. For a nonnegative Sobolev function 
on a domain 
,
the energy is
 |
A local minimizer is compared with every nonnegative Sobolev competitor of the same boundary trace on each
compactly contained ball. Its free boundary
is the boundary of 
inside the domain. The flat one-homogeneous example
is 
for a unit vector 
.
