Partial data inverse problems and simultaneous recovery of boundary and coefficients for semilinear elliptic equations

We study various partial data inverse boundary value problems for the semilinear elliptic equation Δu + a(x, u) = 0 in a domain in Rn by using the higher order linearization technique introduced by Lassas– Liimatainen–Lin–Salo and Feizmohammadi–Oksanen. We show that the Dirichlet-to-Neumann map of the above equation determines the Taylor series of a(x, z) at z = 0 under general assumptions on a(x, z). The determination of the Taylor series can be done in parallel with the detection of an unknown cavity inside the domain or an unknown part of the boundary of the domain. The method relies on the solution of the linearized partial data Calder´on problem by Ferreira–Kenig–Sj¨ostrand–Uhlmann, and implies the solution of partial data problems for certain semilinear equations Δu + a(x, u) = 0 also proved by Krupchyk–Uhlmann. The results that we prove are in contrast to the analogous inverse problems for the linear Schr¨odinger equation. There recovering an unknown cavity (or part of the boundary) and the potential simultaneously are longstanding open problems, and the solution to the Calder´on problem with partial data is known only in special cases when n ≥ 3.