Singularities almost always scatter: Regularity results for non-scattering inhomogeneities

In this paper we examine necessary conditions for an inhomogeneity to be non-scattering, or equivalently, by negation, sufficient conditions for it to be scattering. These conditions are formulated in terms of the regularity of the boundary of the inhomogeneity. We examine broad classes of incident waves in both two and three dimensions. Our analysis is greatly influenced by the analysis carried out by Williams in order to establish that a domain, which does not possess the Pompeiu Property, has a real analytic boundary. That analysis, as well as ours, relies crucially on classical free boundary regularity results due to Kinderlehrer and Nirenberg, and Caffarelli.     

Domain sensitivity analysis of the acoustic far‐field pattern

We consider acoustic scattering problems described by the mixed boundary value problem for the scalar Helmholtz equation in the exterior of a 2D bounded domain or in the exterior of a crack. The boundary of the domain is assumed to have a finite set of corner points where the scattered wave may have singular behaviour. The paper is concerned with the sensitivity of the far‐field pattern with respect to small perturbations of the shape of the scatterer. Using a modification of the method of adjoint problems, we obtain an integral representation for the Gâteaux derivative which contains only boundary values of functions easily computable by standard BEM and which depends explicitly on the perturbation of the boundary. In some cases, we show the direct influence of the singularities of the solution on the sensitivity of the far‐field pattern. In this way, we generalize the domain sensitivity analysis developed earlier for smooth domains by Hettlich, Kirsch, Kress, Potthast and others. Finally, we show that the same approach can be applied to scattering from 3D domains with smooth edges. Copyright © 2002 John Wiley & Sons, Ltd.     

On the existence of transmission eigenvalues in an inhomogeneous medium

We prove the existence of transmission eigenvalues corresponding to the inverse scattering problem for isotropic and anisotropic media for both the scalar problem and Maxwell's equations. Considering a generalized abstract eigenvalue problem, we are able to extend the ideas of Päivärinta and Sylvester [Transmission eigenvalues, SIAM J. Math. Anal. 40, (2008) pp. 783–753] to prove the existence of transmission eigenvalues for a larger class of interior transmission problems. Our analysis includes both the case of a medium with positive contrast and of a medium with negative contrast provided that the contrasts are large enough.     

On the asymptotics of a Robin eigenvalue problem

The considered Robin problem can formally be seen as a small perturbation of a Dirichlet problem. However, due to the sign of the impedance value, its associated eigenvalues converge point-wise to ?∞ as the perturbation goes to zero. We prove in this case that Dirichlet eigenpairs are the only accumulation points of the Robin eigenpairs with normalized eigenvectors. We then provide a criterion to select accumulating sequences of eigenvalues and eigenvectors and exhibit their full asymptotic with respect to the small parameter.     

The inverse electromagnetic scattering problem for a partially coated dielectric

We use the linear sampling method to determine the shape and surface conductivity of a partially coated dielectric infinite cylinder from knowledge of the far field pattern of the scattered TM polarized electromagnetic wave at fixed frequency. A mathematical justification of the method is provided based on the use of a complete family of solutions. Numerical examples are given showing the efficiency of our method.     

Transmission Eigenvalues and the Riemann Zeta Function in Scattering Theory for Automorphic Forms on Fuchsian Groups of Type I

We introduce the concept of transmission eigenvalues in scattering theory for automorphic forms on fundamental domains generated by discrete groups acting on the hyperbolic upper half complex plane. In particular, we consider Fuchsian groups of type Ⅰ. Transmission eigenvalues are related to those eigen-parameters for which one can send an incident wave that produces no scattering. The notion of transmission eigenvalues, or non-scattering energies, is well studied in the Euclidean geometry, where in some cases these eigenvalues appear as zeros of the scattering matrix. As opposed to scattering poles, in hyperbolic geometry such a connection between zeros of the scattering matrix and non-scattering energies is not studied, and the goal of this paper is to do just this for particular arithmetic groups. For such groups, using existing deep results from analytic number theory, we reveal that the zeros of the scattering matrix, consequently non-scattering energies, are directly expressed in terms of the zeros of the Riemann zeta function. Weyl's asymptotic laws are provided for the eigenvalues in those cases along with estimates on their location in the complex plane.     

Boundary integral method for thermoelastic screen scattering problem in ℝ3

We investigate a three‐dimensional mathematical thermoelastic scattering problem from an open surface which will be referred to as a screen. Under the assumption of the local finite energy of the unified thermoelastic scattered field, we give a weak model on the appropriate Sobolev spaces and derive equivalent integral equations of the first kind for the jump of some trace operators on the open surface. Uniqueness and existence theorems are proved, the regularity and the singular behaviour of the solution near the edge are established with the help of the Wiener–Hopf method in the halfspace, the calculus of pseudodifferential operators on the basis of the strong ellipticity property and Gårding's inequality. An improved Galerkin scheme is provided by simulating the singular behaviour of the exact solution at the edge of the screen. Copyright © 2000 John Wiley & Sons, Ltd.     

The linear sampling method for the inverse electromagnetic scattering problem for screens

We consider the inverse electromagnetic scattering problem of determining the shape of a screen from a knowledge of the electric far field pattern of the scattered wave at fixed frequency. We adapt the linear sampling method invented by Colton and Kirsch (Inverse Problems 12 (1996) 383-393) for the case of obstacles with nonempty interior. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Domain sensitivity analysis of the elastic far-field patterns in scattering from nonsmooth obstacles

We consider elastic scattering problems described by the Dirichlet or the Neumann boundary value problem for the elastodynamic equation in the exterior of a 2D bounded domain or in the exterior of a crack. The boundary of the domain is assumed to have a finite set of corner points where the scattered wave may have singular behaviour. The paper is concerned with the sensitivity of the far scattered field with respect to small perturbations of the shape of the scatterer. Using a modification of the method of adjoint problems (K. Dems, Z. Mróz, Internat. J. Solids Structures 20 (1984) 527-552) we obtain a representation for the shape derivative which is well suited for a numerical realization with boundary element methods and which shows in some cases directly the influence of the singularities of the solution on the sensitivity of the far-field patterns.     

A boundary integral equation method for the transmission eigenvalue problem

We propose a new integral equation formulation to characterize and compute transmission eigenvalues for constant refractive index that play an important role in inverse scattering problems for penetrable media. As opposed to the recently developed approach by Cossonnière and Haddar [1,2] which relies on a two by two system of boundary integral equations our analysis is based on only one integral equation in terms of Dirichlet-to-Neumann or Robin-to-Dirichlet operators which results in a noticeable reduction of computational costs. We establish Fredholm properties of the integral operators and their analytic dependence on the wave number. Further we employ the numerical algorithm for analytic non-linear eigenvalue problems that was recently proposed by Beyn [3] for the numerical computation of transmission eigenvalues via this new integral equation.