A perturbation theory of overdetermined boundary value problems

We study the shape of a bounded domain for which a parametrized overdetermined boundary value problem admits a solution. Unlike a typical nonlinear problem where the nondegeneracy of the linearized operator implies a local one-to-one correspondence between parameters and solutions, overdetermined problems generally fail to follow this scenario because of a loss of derivatives. We develop a perturbation theory of overdetermined problems based on a characterization of an evolving domain by a geometric evolution equation. We show that if the linearized operator satisfies some monotonicity condition in addition to the nondegeneracy, then there exists a monotonically increasing family of domains admitting solvability of the corresponding overdetermined problem under a small continuous deformation of parameters.