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.

伯努利的自由边界问题是一个过分确定的问题，在该问题中，人们寻求一个环形区域，以使电容势满足额外的边界条件。存在两种不同类型的解，称为椭圆解和双曲解。椭圆形解决方案是“稳定”的解决方案，可以通过Nash-Moser的上级解决方案和子解决方案，变分方法和隐函数定理进行处理，而双曲解是“不稳定”的解决方案，其定性行为鲜为人知。我们介绍了一个基于抛物线最大正则性的新隐函数定理，它适用于导数损失的问题。在这种方法中，叶状双曲线解和椭圆解的存在被减少到非局部几何流的可解性，