Direct and inverse problems for the nonlinear time-harmonic Maxwell equations in Kerr-type media

Abstract

In the current paper we consider an inverse boundary value problem of electromagnetism in a nonlinear Kerr medium. We show the unique determination of the electromagnetic material parameters and the nonlinear susceptibility parameters of the medium by making electromagnetic measurements on the boundary. We are interested in the case of the time-harmonic Maxwell equations.

The Dirichlet-to-Neumann map for a semilinear wave equation on Lorentzian manifolds

“…We are now in a position to prove Proposition 4.1. Notice that a fourth order linearization of the DN maps Λ g,q,e β a (f ) = Λ g,0,a (f ), with f = 1 f (1) + 2 f (2) + 3 f (3) + 4 f (3) (notice the repetition of f (3) ), leads to the integral identity…”

“…Since the work [31], rapid progress has been made on the study of inverse problems for nonlinear equations. See [37,30,36,14,47,13,44,11,20,23,12,4,35,19,32] for results on hyperbolic equations and [3,10,8,34,33,21,29,27,2,28,26,9] for elliptic equations. For hyperbolic equations, the recovery of time-dependent coefficients is possible for some nonlinear equations, whereas the corresponding problems for linear equations are still largely open.…”

On an inverse boundary value problem for a nonlinear elastic wave equation

“…The well-definedness of Λ for small f is guaranteed by the well-posedness of (1) with small boundary data: Theorem 2]). Assume f ∈ C m ([0, T ] × ∂Ω), m ≥ 3 is supported away from t = 0.…”

“…The linearization of Λ itself has already been used in [8]. Higher order linearization of Dirichlet-to-Neumann map and the resulted integral identities for semilinear and quasilinear elliptic equations are used [32,17,1,5,23,24,12,19]. Assume u solves (1) with Dirichlet boundary value…”

“…Applying ∂ 2 ∂ǫ 1 ∂ǫ 2 to (1), we obtain the equation for 1) , u (2) ), (t, x) ∈ (0, T ) × Ω, U (12) (t, x) = 0, (t, x) ∈ (0, T ) × ∂Ω, U (12) (0, x) = ∂ ∂t U (12) (0, x) = 0, x ∈ Ω.…”