Integral equation methods in electromagnetic scattering from anisotropic media

We investigate scattering of time‐harmonic electromagnetic waves by an anisotropic inhomogeneous medium. The problem is equivalently transformed into a system of strongly singular integral equations. The uniqueness and existence of a solution is shown and we examine the regularity of the solution by means of integral equations. We also prove the analyticity of the scattered field with respect to the refractive matrix and give a characterization of the derivatives in terms of solutions to anisotropic scattering problems. Copyright © 2000 John Wiley & Sons, Ltd.     

A new non-iterative singular sources method for the reconstruction of piecewise constant media.

Summary. We describe a novel non-iterative method for the reconstruction of a piecewise constant inhomogeneous medium in acoustic scattering, which we call the singular sources method. The basic idea of the method is to use the behaviour of the scattered field for singular incident fields (multipoles) to calculate the size of the refractive index n at some point z₀ on the boundary of the support of the scatterer and then eliminate this value from the data by subtracting a known piecewise constant background medium. The paper includes the theory for the singular sources method to locate the unknown support of an inhomogeneous medium for a known inhomogeneous background medium. Also, we give a new uniqueness proof for the reconstruction of a piecewise constant medium in two or three dimensions, using techniques that differ from those used to prove previous well-known results.Mathematics Subject Classification (2000): 35J05, 45Q05, 47A52, 78A46, 81U40Revised version received August 6, 2003     

Interior transmission eigenvalue problem with refractive index having C2-transition to the background medium

The interior transmission eigenvalue problem for scalar acoustics is studied for a new class of refractive index. Existence of an infinite discrete set of transmission eigenvalues in the case that the acoustic properties of a domain D???? ⁿ are allowed to have a C ²-transition to the homogeneous background medium is established. It is shown that the transmission problem has a weak formulation on certain weighted Sobolev spaces for this class of refractive index. The weak formulation and the discreteness of the spectrum is justified by using the Hardy inequality to prove compact imbedding theorems. Existence of transmission eigenvalues is demonstrated by investigating a generalized eigenvalue problem associated with the weak formulation.     

Stable methods for an inverse problem in acoustic scattering by an obstacle and an inhomogeneous medium

We consider (in two-dimensional Euclidean space) the scattering of a plane, time-harmonic acoustic wave by an inhomogeneous medium Ω with compact support and a bounded obstacle D lying completely outside of the inhomogeneous medium. We show that one may determine the shape of D and the local speed of sound in Ω from a knowledge of the asymptotic behavior of the scattered wave (i.e. the far field). This is done by considering a constrained optimization problem and employing integral equation and conformal mapping techniques. By assuming a priori that the functions which determine the shape of D and the local speed of sound in Ω lie in given compact sets, we show that the problem is stable, in the sense that the solution of the inverse scattering problem depends continuously on the far field data.     

Linearization Ill-Posedness for 2.5-D Ware Equation Inversion Model

Abstract For the weakly inhomogeneous acoustic medium in Ω={(x,y,z):z>0},we consider the inverse problemof determining the density function ρ(x,y).The inversion input for our inverse problem is the wave field givenon a line.We get an integral equation for the 2-D density perturbation from the linearization.By virtue of theintegral transform,we prove the uniqueness and the instability of the solution to the integral equation.Thedegree of ill-posedness for this problem is also given.     

Modelling Axi-symmetric Travelling Waves in a Dielectric with Nonlinear Refractive Index

We consider an isotropic dielectric with a nonlinear refractive index. The medium may be inhomogeneous but its spatial variation has an axial symmetry. We characterize all monochromatic axi-symmetric travelling waves as solution of a system of six second order differential equations on (0, ). Boundary conditions at 0 ensure the regularity of the fields on the axis. Guided waves satisfy additional conditions at . Special solutions of this system correspond to what are normally referred to as TE and TM modes.Lecture held in the Seminario Matematico e Fisico on June 13, 2003Received: February, 2004     

Optical Tomography for Variable Refractive Index with Angularly Averaged Measurements

In optical tomography one seeks to use near-infrared light to determine the optical absorption and scattering properties of a medium X ? ? ⁿ . If the refractive index is constant throughout the medium, the steady-state case is modeled by the stationary linear transport equation in terms of the Euclidean metric. In this work we consider the case of variable refractive index where the dynamics are modeled by writing the transport equation in terms of a Riemannian metric; in the absence of interaction, photons follow the geodesics of this metric. In particular we study the problem where our measurements allow the application of an in-going flux depending on both position and direction, but we allow only a weighted average measurement of the out-going flux. We show that making measurements on all of ? X determines the extinction coefficient and that once this is known, under additional assumptions, measurements at a single point on ? X determine the scattering kernel.     

An inverse problem for Maxwell's equations in isotropic and non-stationary media

In this article, we consider Maxwell's equations in an isotropic, inhomogeneous and non-stationary medium. We discuss an inverse problem of determining the t-independent components of the coefficients ?, μ in the constitutive relations from a finite number of interior measurements. We prove a Lipschitz stability estimate for the inverse problem by applying the argument on the basis of Carleman estimate.     

A Steady Weak Solution of the Equations of Motion of a Viscous Incompressible Fluid through Porous Media in a Domain with a Non-Compact Boundary

We assume that Ω is a domain in ℝ² or in ℝ³ with a non-compact boundary, representing a generally inhomogeneous and anisotropic porous medium. We prove the weak solvability of the boundary-value problem, describing the steady motion of a viscous incompressible fluid in Ω. We impose no restriction on sizes of the velocity fluxes through unbounded components of the boundary of Ω. The proof is based on the construction of appropriate Galerkin approximations and study of their convergence. In Sect. 4, we provide several examples of concrete forms of Ω and prescribed velocity profiles on ∂Ω, when our main theorem can be applied.     

A Modified Linear Sampling Method Valid for all Frequencies and an Application to the Inverse Inhomogeneous Medium Problem

We consider the inverse inhomogeneous medium scattering problem in acoustics where one tries to recover the support of anomalies in a medium by interrogating the region of interest with plane waves at fixed frequency. We discuss the validity of the Linear Sampling Method (LSM) for solving this inverse problem, and its connection to an unusual eigenvalue problem. It turns out that the existence of non-trivial solutions of the so-called interior transmission eigenvalue problem results in the failure of the LSM. We then propose a modification of the LSM that avoids these shortcomings and is numerically sound. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The operator equations of Lippmann–Schwinger type for acoustic and electromagnetic scattering problems in L 2

This article is concerned with the scattering of acoustic and electromagnetic time harmonic plane waves by an inhomogeneous medium. These problems can be translated into volume integral equations of the second kind – the most prominent example is the Lippmann–Schwinger integral equation. In this work, we study a particular class of scattering problems where the integral operator in the corresponding operator equation of Lippmann–Schwinger type fails to be compact. Such integral equations typically arise if the modelling of the inhomogeneous medium necessitates space-dependent coefficients in the highest order terms of the underlying partial differential equation. The two examples treated here are acoustic scattering from a medium with a space-dependent material density and electromagnetic medium scattering where both the electric permittivity and the magnetic permeability vary. In these cases, Riesz theory is not applicable for the solution of the arising integral equations of Lippmann–Schwinger type. Therefore, we show that positivity assumptions on the relative material parameters allow to prove positivity of the arising volume potentials in tailor-made weighted spaces of square integrable functions. This result merely holds for imaginary wavenumber and we exploit a compactness argument to conclude that the arising integral equations are of Fredholm type, even if the integral operators themselves are not compact. Finally, we explain how the solution of the integral equations in L ² affects the notion of a solution of the scattering problem and illustrate why the order of convergence of a Galerkin scheme set up in L ² does not suffer from our L ² setting, compared to schemes in higher order Sobolev spaces.     

Time-periodic Stokes equations with inhomogeneous Dirichlet boundary conditions in a half-space

The time-periodic Stokes problem in a half-space with fully inhomogeneous right-hand side is investigated. Maximal regularity in a time-periodic L^p setting is established. A method based on Fourier multipliers is employed that leads to a decomposition of the solution into a steady-state and a purely oscillatory part in order to identify the suitable function spaces.     

Inverse backscattering problem for Maxwell's equations

In this paper we consider the inverse backscattering problem for Maxwell's equations in a non-magnetic inhomogeneous medium, i.e. the magnetic permeability is a fixed constant. We show that the electric permittivity ε is uniquely determined by the trace of the backscattering kernel S(s, −θ, θ) for all s∈ℝ, θ∈ S ² provided that it is a priori close to a constant. © 1998 B. G. Teubner Stuttgart—John Wiley & Sons, Ltd.     

Optimization of the stressed state by a distribution of elastic parameters in elastic bodies

For designing an isotropic inhomogeneous body having a variable elastic modulus and a prescribed shape under the influence of an external force load we give a formulation and solution of the problem of determining the design that is optimal with respect to stress. The problem of optimal design reduces to a certain problem in the theory of elasticity for a non-linearelastic material. As an example we consider the problem of optimal design of an inhomogeneous cylinder. Four figures. Bibliography: 9 titles.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 30, 1989, pp. 78–82.     

Singular quasilinear elliptic problems with indefinite weights and critical potential

The Dirichlet problem for a quasilinear sub-critical inhomogeneous elliptic equation with critical potential and singular coefficients, which has indefinite weights in RN , is studied in this paper. We discuss the corresponding eigenvalue problems by the variational techniques and Picone’s identity, and obtain the existence of non-trivial solutions for the inhomogeneous Dirichlet problem by using Hardy inequality, Mountain Pass Lemma in conjunction with the property of eigenvalues.     

Inverse problems in underwater acoustics and approximation of the velocity field for signal propagation

A new approach is suggested for solving the inverse problem in underwater acoustics — the determination of the velocity of signal propagation in an inhomogeneous medium. The unknown velocity u(x(t), t) is assumed to be independent of the time t and is sought in the form of a polynomial with respect to the degree of the space coordinate x ₁, x₂, x₃.Translated fromVychislitel'naya i Prikladnaya Matematika, No. 69, pp. 102–105, 1989.     

The Additive Cousin Problem and Related Problems for Regular Functions with Parameter Taking Values in a Clifford Algebra

This article deals with the Runge type approximations problem, the solution of inhomogeneous system and the additive Cousin problem for Clifford-algebra-valued functions ?(x, y) which are regular with respect to x and real-analytic in y. Some main results concerning these three fundamental problems from complex analysis are proved for the class of functions mentioned above.     

Two-Scale Convergence Method and Scattering Problems

In this paper, we deal with the steady-state acoustic wave equation in the space ℝ³ diffracted by an obstacle made by an inhomogeneous medium and located in a bounded domain. The inhomogeneity of the medium depends on a parameter ε > 0. If the solution u _ε converges to a solution u ₀ of the limit problem as ε → 0, as in the homogenization process, then we can use the two-scale convergence method to study the convergence of the gradient.__________Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 10, Suzdal Conference-4, 2003.     

Asymptotic Solution of the Problem on an Accelerated Source of High-Frequency Oscillations

The problem of the wave field of a source moving in an inhomogeneous medium is considered. It is assumed that the velocity of the source is less than the velocity of sound for t < 0 and is greater than this velocity for t > 0 (subsonicsupersonic transition). An asymptotic expansion for the wave field in a neighborhood of the source is constructed on the basis of the well-known Hadamard ansatz. The expansion derived is uniform with respect to the velocity of the source and contains new special functions. These functions are generalizations of the Hankel and Bessel functions and possess some remarkable properties. Bibliography: 7 titles.     

Stationary solution of the Navier-Stokes equations in a 2d bounded domain for incompressible flow with discontinuous density

We consider the boundary value problem for the stationary Navier-Stokes equations describing an inhomogeneous incompressible fluid in a two dimensional bounded domain. We show the existence of a weak solution with boundary values for the density prescribed in L^￥L^{\infty}.