An inverse problem for a semilinear elliptic equation on conformally transversally anisotropic manifolds

Abstract

Given a conformally transversally anisotropic manifold $(M,g)$, we consider the semilinear elliptic equation $$(-{\Delta }_g+V)u+q{u}^2=0\text{on M}.$$ We show that an a priori unknown smooth function $q$ can be uniquely determined from the knowledge of the Dirichlet-to-Neumann map associated to the semilinear elliptic equation. This extends the previously known results of the works [FO20, LLLS21a]. Our proof is based on analyzing higher order linearizations of the semilinear equation with non-vanishing boundary traces and also the study of interactions of two or more products of the so-called Gaussian quasimode solutions to the linearized equation.

39 pages. All comments are welcome

References in corpus (2)

• Reconstruction for the coefficients of a quasilinear elliptic partial differential equation
• Simultaneous recovery of piecewise analytic coefficients in a semilinear elliptic equation

Cited by in corpus (2)

• Stability estimates for inverse problems for semi-linear wave equations on Lorentzian manifolds
• Increasing stability of the first order linearized inverse Schrödinger potential problem with integer power type nonlinearities