Functional Analysis and Applications Seminar 

23/09/2005, 14:00 — 15:00 — Room P3.10, Mathematics Building
Eugene Shargorodsky, King's College, London, England

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.