We introduce a new approach for studying the uniqueness and stability of a domain admitting the solvability of an overdetermined problem.
One of the key observations is that the deformation of a continuously varying domain for a parametrized overdetermined problem forms an analytic semiflow.
This allows us to obtain the uniqueness of a domain for the original ``stationary'' overdetermined problem together with a quantitative estimate of its shape by clarifying the dynamical structure of the semiflow.
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