When a domain in the plane is specified by the
requirement that there exists a harmonic function which is zero on its
boundary and additionally satisfies a prescribed Neumann condition
there, the boundary is called a Bernoulli free boundary. (The boundary
is “free” because the domain is not known a
priori and the name Bernoulli was originally associated with such
problems in hydrodynamics.) Questions of existence, multiplicity or
uniqueness, and regularity of free boundaries for prescribed data need
to be addressed and their solutions lead to nonlinear problems.
In this paper an equivalence is established between Bernoulli
free-boundary problems and a class of equations for real-valued
functions of one real variable. The authors impose no restriction on
the amplitudes or shapes of free boundaries, nor on their
smoothness. Therefore the equivalence is global, and valid even for
very weak solutions.
An essential observation here is that the equivalent equations can
be written as nonlinear Riemann-Hilbert problems and the theory of
complex Hardy spaces in the unit disc has a central role. An
additional useful fact is that they have gradient structure, their
solutions being critical points of a natural Lagrangian. This means
that a canonical Morse index can be assigned to free boundaries and
the Calculus of Variations becomes available as a tool for the
study.
Some rather natural conjectures about the regularity of free
boundaries remain unresolved.