Inverse problems for semilinear elliptic PDE with a general nonlinearity a(x,u)

This article studies the inverse problem of recovering a nonlinearity in an elliptic equation from boundary measurements of solutions. Previous results based on first‐order linearization achieve this under a sign condition on , and results based on higher order linearization recover the Taylor series of with respect to . We improve these results and show that a general nonlinearity, and not just its Taylor series, is uniquely determined up to gauge near a fixed solution. Our method is based on constructing a good solution map that locally parametrizes solutions of the nonlinear equation by solutions of the linearized equation.