Abstract
The purpose of this chapter is to discuss some of the highlights of the mathematical theory of direct and inverse scattering and inverse source scattering problem for acoustic, elastic and electromagnetic waves. We also briefly explain the uniqueness of the external source for acoustic, elastic and electromagnetic waves equation. However, we must first issue a caveat to the reader. We will also present the recent results for inverse source problems. The resents results including a logarithmic estimate consists of two parts: the Lipschitz part data discrepancy and the high frequency tail of the source function. In general, it is known that due to the existence of non-radiation source, there is no uniqueness for the inverse source problems at a fixed frequency.
Highlights
This chapter tries to provide some results and materials on inverse scattering, direct scattering theory and inverse source scattering problems
We will focus on inverse source problems for acoustic, elastics and electromagnetic waves
(A is the elliptic operator A 1⁄4 À∇ða∇Þ þ b:∇ þ c with Reb 1⁄4 0, ∇b 1⁄4 0, and Imc ≤ 0, which coincides with the Laplace operator outside a ball B and which possesses the uniqueness of continuation property) or to the Helmholtz equation
Summary
This chapter tries to provide some results and materials on inverse scattering, direct scattering theory and inverse source scattering problems. Speaking, scattering theory is concerned with the effect an inhomogeneous medium has on an incident particle or wave. If the total field is viewed as the sum of an incident field ui and a scattered field us the direct scattering problem is to determine us form a knowledge of ui and the differential equation governing. There are even more in the inverse scattering problem of determining the nature of the inhomogeneity from a knowledge of the asymptotic behavior of us, i.e., to reconstruct the differential equation or its domain or source functions of definition from the behavior of solutions of the direct problems. We are following this notation; C denote generic constants depending on the domain Ω or domain D, which is different in different results, and kukðlÞðΩÞ denotes the standard norm in Sobolev space HlðΩÞ
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