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.

Inverse transport and diffusion problems in photoacoustic imaging with nonlinear absorption

“…Our main strategy is to use the linearization technique of Isakov and others [28,29,30,31,32] in dealing with nonlinear equations to decompose the inverse problem to the semilinear radiative transport equation ( 1) into an inverse coefficient problem for the linear transport equation where we reconstruct σ a and σ s by the result of Bal-Jollivet-Jungon [6], and an inverse source problem for the linear transport equation where we reconstruct the two-photon absorption coefficient σ b . This is the same type of strategy that have been successfully employed to solve many inverse problems for nonlinear PDEs recently; see for instance [4,11,13,14,15,18,20,25,26,33,35,36,38,39,40,41,43,44,45,46,47,50,58,61,62,63] and reference therein.…”

A remark on inverse problems for nonlinear magnetic Schrödinger equations on complex manifolds

“…Multiplying the difference of two equations in (2.4) by a harmonic function v (3) ∈ C ∞ (M), integrating over M and using Green's formula, we obtain that…”

“…(2.6) Subtracting (2.6) from (2.5) and letting v (3) = 1, we get M A, dv (1) g v (2) dV g = 0, (2.7) for all harmonic functions v (1) , v (2) ∈ C ∞ (M). Applying Proposition A.1 to (2.7), we conclude that A| ∂M = 0.…”

“…and therefore, A, dv (3) g = 0, (2.12) for every harmonic function v (3) ∈ C ∞ (M). Let p ∈ M int and by assumption (b), there exist f 1 , .…”

“…Theorem 1.1 is a direct consequence of the main result of [19], combined with some boundary determination results of [38] and of Appendix A, as well as the higher order linearization procedure introduced in [28] in the hyperbolic case, and in [15], [31] in the elliptic case. We refer to [20] where the method of a first order linearization was pioneered in the study of inverse problems for nonlinear PDE, and to [3], [10], [44], and [45] where a second order linearization was successfully exploited. The crucial fact used in the proof of the main result of [19], indispensable for our Theorem 1.1, is that both holomorphic and antiholomorphic functions are harmonic on Kähler manifolds.…”