  We examine Serrin's classical overdetermined problem under a perturbation of
the Neumann boundary condition. The solution of the problem for a constant
Neumann boundary condition exists provided that the underlying domain is a
ball. The question arises whether for a perturbation of the constant there
still are domains admitting solutions to the problem. Furthermore, one may ask
whether a domain that admits a solution for the perturbed problem is unique up
to translation and whether it is close to the ball. We develop a new implicit
function theorem for a pair of Banach triplets that is applicable to nonlinear
problems with loss of derivatives except at the point under consideration.
Combined with a detailed analysis of the linearized operator, we prove the
existence and local uniqueness of a domain admitting a solution to the
perturbed overdetermined problem. Moreover, an optimal linear stability
estimate for the shape of a domain is established.
