Hyperbolic Solutions to Bernoulli's Free Boundary Problem

Bernoulli's free boundary problem is an overdetermined problem in which one seeks an annular domain such that the capacitary potential satisfies an extra boundary condition. There exist two different types of solutions called elliptic and hyperbolic solutions. Elliptic solutions are "stable" solutions and tractable by the super and subsolution method, variational methods and the implicit function theorem of Nash-Moser, while hyperbolic solutions are "unstable" solutions of which the qualitative behavior is less known. We introduce a new implicit function theorem based on the parabolic maximal regularity, which is applicable to problems with loss of derivatives. In this approach, the existence of foliated hyperbolic solutions as well as elliptic solutions is reduced to the solvability of a non-local geometric flow, and the latter is established by clarifying the spectral structure of the linearized operator by harmonic analysis.