  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 variational methods and maximum principles, 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.
Clarifying the spectral structure of the corresponding linearized operator by
harmonic analysis, we prove the existence of foliated hyperbolic solutions as
well as elliptic solutions in the same regularity class.
