Overdetermined boundary problems with nonconstant Dirichlet and Neumann data

Abstract: In this paper we consider the overdetermined boundary problem for a general second order semilinear elliptic equation on bounded domains of $\mathbf{R}^n$, where one prescribes both the Dirichlet and Neumann data of the solution. We are interested in the case where the data are not necessarily constant and where the coefficients of the equation can depend on the position, so that the overdetermined problem does not generally admit a radial solution. Our main result is that, nevertheless, under minor technical hypotheses nontrivial solutions to the overdetermined boundary problem always exist.