Abstract: Our main interest lies in the shape of a bounded domain for which a parametrized overdetermined boundary value problem admits a solution. Unlike a typical nonlinear problem where the non-degeneracy 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. In this talk, I will explain a general perturbation result in overdetermined problems based on a characterization of an evolving domain by a geometric evolution equation. It turns out that the non-degeneracy and an additional monotonicity condition of the linearized operator are the properties inherited by the original problem, i.e., these linear properties imply the existence of a monotonically increasing family of domains admitting solvability of the corresponding overdetermined problem under a small continuous deformation of parameters.