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A Boundary value problem for laplace's equation

We consider the boundary value problem Δu=0 in Ω, ∂2 n/∂n 2on a smooth bounded domain . It is shown that if Ω is simply connected, or if Ω is doubly connected with at least one of the boundaries star-shaped, then the only solutions are linear functions. We also show that this is not true for doubly connected regions in general.