We classify the solutions to an overdetermined elliptic problem in the plane in the finite connectivity case. This is achieved by establishing a one-to-one correspondence between the solutions to this problem and a certain type of minimal surfaces.
Laurent Hauswirth pointed out to me that Problem has been studied by D. Khavinson, E. Lundberg and R. Teodorescu in a recent paper @cite_5 . They obtain classification results in the @math -dimensional case under stronger topological hypotheses than ours. They prove that if an exceptional domain @math is the complement of a bounded domain, then it is the complement of a disk; if it is simply-connected and Smirnov, then it is a half-plane or the domain @math , up to similitude. In the simply-connected case, their results are stronger than ours because they do not assume that the boundary of @math has a finite number of components. They also prove that in higher dimension, an exceptional domain in @math whose complement is bounded, connected and has @math boundary, is the exterior of a sphere.
@cite_5
2116325468
We investigate a problem posed by L. Hauswirth, F. H 'elein, and F. Pacard, namely, to characterize all the domains in the plane that admit a "roof function", i.e., a positive harmonic function which solves simultaneously a Dirichlet problem with null boundary data, and a Neumann problem with constant boundary data. Under some a priori assumptions, we show that the only three examples are the exterior of a disk, a halfplane, and a nontrivial example. We show that in four dimensions the nontrivial simply connected example does not have any axially symmetric analog containing its own axis of symmetry.