| 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.
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 non-degeneracy, then there exists a monotonically increasing family of domains admitting solvability of the corresponding overdetermined problem under a small continuous deformation of parameters. |
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