Complex methods for Bernoulli free-boundary problems
A Bernoulli free-boundary problem is one of finding domains in the
plane on which a harmonic function simultaneously satisfies
homogeneous linear Dirichlet and inhomogeneous linear Neumann
boundary conditions. The boundary of such a domain (called the free
boundary because it is not prescribed a priori) is the essential
ingredient of a solution. The classical Stokes waves provide an
important example of a Bernoulli free-boundary problem. Existence,
multiplicity or uniqueness, and smoothness of boundaries are
important questions and, despite appearances, the problem of
determining free boundaries is nonlinear. The talk, based on a
joint work with J.F. Toland, will examine an equivalence between
these free-boundary problems and a set of nonlinear
pseudo-differential equations, for one real-valued function of one
real variable, which have the gradient structure of an
Euler-Lagrange equation and can be formulated in terms of
Riemann-Hilbert theory. The equivalence is global in the sense that
it involves no restriction on the amplitudes of solutions, nor on
their smoothness. Non-existence and regularity results will be
described and some important unresolved questions about precisely
how irregular a Bernoulli free boundary can be will be formulated.
