Inverse problems for nonlinear Helmholtz Schr\"odinger equations and time-harmonic Maxwell's equations with partial data

TL;DR

This paper develops methods for solving inverse boundary value problems for nonlinear Helmholtz Schrödinger and Maxwell's equations, using higher-order linearization and unique continuation to recover unknown coefficients and cavities from partial data.

Contribution

It introduces a higher-order linearization approach for nonlinear inverse problems and extends local uniqueness results to global cases with partial boundary data.

Findings

Established local uniqueness for partial data inverse problems.

Extended results to global uniqueness using Runge approximation and unique continuation.

Achieved simultaneous recovery of unknown cavities and coefficients.

Abstract

We consider Calder\'{o}n's inverse boundary value problems for a class of nonlinear Helmholtz Schr\"{o}dinger equations and Maxwell's equations in a bounded domain in $\R^n$. The main method is the higher-order linearization of the Dirichlet-to-Neumann map of the corresponding equations. The local uniqueness of the linearized partial data Calder\'{o}n's inverse problem is obtained following \cite{DKSU}. The Runge approximation properties and unique continuation principle allow us to extend to global situations. Simultaneous recovery of some unknown cavity$/$boundary and coefficients are given as some applications.