Geometry and topology of some overdetermined elliptic problems

We study necessary conditions on the geometry and the topology of domains in R2R2 that support a positive solution to a classical overdetermined elliptic problem{Δu+f(u)=0in Ωu=0on ∂Ω〈∇u,ν〉=constanton ∂Ω. The ideas and tools we use come from constant mean curvature surface theory. In particular, we obtain a partial answer to a question posed by H. Berestycki, L. Caffarelli and L. Nirenberg in 1997. We investigate also some boundedness properties of the solution u. Some of our results generalize to higher dimensions.