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The existence of an infinite discrete set of transmission eigenvalues
"... Abstract. We prove the existence of an infinite discrete set of transmission eigenvalues corresponding to the scattering problem for isotropic and anisotropic inhomogeneous media for both the Helmholtz and Maxwell’s equations. Our discussion includes the case of the interior transmission problem for ..."
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Abstract. We prove the existence of an infinite discrete set of transmission eigenvalues corresponding to the scattering problem for isotropic and anisotropic inhomogeneous media for both the Helmholtz and Maxwell’s equations. Our discussion includes the case of the interior transmission problem for an inhomogeneous medium with cavities, i.e. subregions with contrast zero. Key words. Interior transmission problem, transmission eigenvalues, inhomogeneous medium, inverse scattering. AMS subject classifications. 35R30, 35Q60, 35J40, 78A25. 1. Introduction. The
On the existence of transmission eigenvalues in an inhomogeneous medium, (will appear as an article
 in Applicable Analysis), Technical report, n o RR6779, INRIA, 2008, http://hal.inria.fr/ inria00347840/fr
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The inverse electromagnetic scattering problem for a partially coated dielectric
 J. Comp. Appl. Math
"... Abstract. The inverse electromagnetic scattering problem for anisotropic media plays a special role in inverse scattering theory due to the fact that the (matrix) index of refraction is not uniquely determined from the far field pattern of the scattered field even if multifrequency data is availabl ..."
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Cited by 20 (10 self)
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Abstract. The inverse electromagnetic scattering problem for anisotropic media plays a special role in inverse scattering theory due to the fact that the (matrix) index of refraction is not uniquely determined from the far field pattern of the scattered field even if multifrequency data is available. In this paper we describe how transmission eigenvalues can be determined from the far field pattern and be used to obtain upper and lower bounds on the norm of the index of refraction. Numerical examples will be given for the case when the scattering object is an infinite cylinder and the inhomogeneous medium is orthotropic. 1.
The computation of lower bounds for the norm of the index of refraction in an anisotropic media from far field data
 Jour. Integral Equations and Applications
"... Dedicated to Professor Dr. Rainer Kress on the occasion of his 65th birthday and the pleasure that knowing him has given to our lives! ABSTRACT. We consider the scattering of time harmonic electromagnetic plane waves by a bounded, inhomogeneous, anisotropic dielectric medium and show that under cert ..."
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Cited by 19 (8 self)
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Dedicated to Professor Dr. Rainer Kress on the occasion of his 65th birthday and the pleasure that knowing him has given to our lives! ABSTRACT. We consider the scattering of time harmonic electromagnetic plane waves by a bounded, inhomogeneous, anisotropic dielectric medium and show that under certain assumptions a lower bound on the norm of the (matrix) index of refraction can be obtained from a knowledge of the smallest transmission eigenvalue corresponding to the medium. Numerical examples are given showing the efficaciousness of our estimates. 1. Introduction. Anisotropic
Identification of partially coated anisotropic buried objects using electromagnetic Cauchy data
 in "J. Integral Equations and Applications
"... ABSTRACT. We consider the three dimensional electromagnetic inverse scattering problem of determining information about a target buried in a known inhomogeneous medium from a knowledge of the electric and magnetic fields corresponding to time harmonic electric dipoles as incident fields. The scatter ..."
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Cited by 5 (3 self)
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ABSTRACT. We consider the three dimensional electromagnetic inverse scattering problem of determining information about a target buried in a known inhomogeneous medium from a knowledge of the electric and magnetic fields corresponding to time harmonic electric dipoles as incident fields. The scattering object is assumed to be an anisotropic dielectric that is (possibly) partially coated by a thin layer of highly conducting material. The data is measured at a given surface containing the object in its interior. Our concern is to determine the shape of this scattering object and some information on the surface conductivity of the coating without any knowledge of the index of refraction of the inhomogeneity. No a priori assumption is made on the extent of the coating, i.e., the object can be fully coated, partially coated or not coated at all. Our method, introduced in [14, 17], is based on the linear sampling method and reciprocity gap functional for reconstructing the shape of the scattering object. The algorithm consists in solving a set of linear integral equations of the first kind for several sampling points and three linearly independent polarizations. The solution of these integral equations is also used to determine the surface conductivity. 1. Introduction. The
Nature of the transmission eigenvalue spectrum for elastic bodies
, 2010
"... This study develops a spectral theory of the interior transmission problem (ITP) for heterogeneous and anisotropic elastic solids. The importance of this subject stems from its central role in a certain class of inverse scattering theories (the socalled qualitative methods) involving penetrable sca ..."
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Cited by 3 (1 self)
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This study develops a spectral theory of the interior transmission problem (ITP) for heterogeneous and anisotropic elastic solids. The importance of this subject stems from its central role in a certain class of inverse scattering theories (the socalled qualitative methods) involving penetrable scatterers. Although simply stated as a coupled pair of elastodynamic wave equations, the ITP for elastic bodies is neither selfadjoint nor elliptic. To help deal with such impediments, earlier studies have established the wellposedness of an elastodynamic ITP under notably restrictive assumptions on the contrast in elastic parameters between the scatterer and the background solid. Due to the lack of problem selfadjointness, however, these studies were successful in substantiating only the discreteness of the relevant eigenvalue spectrum, but not its existence. The aim of this work is to provide a systematic treatment of the ITP for heterogeneous and anisotropic elastic bodies that transcends the limitations of earlier treatments. Considering a broad range of materialcontrast configurations (both in terms of elastic tensors and mass densities), this paper investigates the questions of the solvability of the ITP, the discreteness of its eigenvalues and, for the first time, of the actual existence of such eigenvalue spectrum. Necessitated by the breadth of material configurations studied, the relevant claims are established through the development of a suite of variational formulations, each customized to meet the needs of a particular subclass of eigenvalue problems. As a secondary result, the lower and upper bounds on the first transmission eigenvalue are obtained in terms of the elasticity and mass density contrasts between the obstacle and the background. Given the fact that the transmission eigenvalues are computable from experimental observations of the scattered field, such estimates may have significant potential toward exposing the nature (e.g. compliance) of penetrable scatterers in elasticity. 1
Inequalities in Inverse Scattering Theory
 J. of Inverse and Illposed Problems
"... Abstract. We consider the scattering of time harmonic electromagnetic plane waves by a bounded, inhomogeneous dielectric medium that is partially coated by a thin metallic layer in R 2. We use far field pattern of the scattered waves at a fixed frequency as data to determine the support D of the inh ..."
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Abstract. We consider the scattering of time harmonic electromagnetic plane waves by a bounded, inhomogeneous dielectric medium that is partially coated by a thin metallic layer in R 2. We use far field pattern of the scattered waves at a fixed frequency as data to determine the support D of the inhomogeneous obstacle, the surface conductivity that characterizes the coating and the relative permittivity. No a prior information on the material properties of the scatterer is needed. The support D is determined by the linear sampling method which is based on the approximate solution of the far field equation. This solution is also used to obtain lower bounds for the surface conductivity and relative permittivity. The techniques for solving this inverse scattering problem rely on the analysis of a non standard boundary value problem known as the interior transmission problem. Key words. Inverse scattering problem, inhomogeneous medium, interior transmission problem, electromagnetic waves, mixed boundary value problems, qualitative approaches in inverse scattering. AMS classification. 35P25, 35R30, 78A45. 1.
A NONCONFORMING FINITE ELEMENT METHOD FOR FOURTH ORDER CURL EQUATIONS IN R 3
"... Abstract. In this paper we present a nonconforming finite element method for solving fourth order curl equations in three dimensions arising from magnetohydrodynamics models. We show that the method has an optimal error estimate for a model problem involving both (∇×) 2 and (∇×) 4 operators. The ele ..."
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Abstract. In this paper we present a nonconforming finite element method for solving fourth order curl equations in three dimensions arising from magnetohydrodynamics models. We show that the method has an optimal error estimate for a model problem involving both (∇×) 2 and (∇×) 4 operators. The element has a very small number of degrees of freedom, and it imposes the interelement continuity along the tangential direction which is appropriate for the approximation of magnetic fields. We also provide explicit formulae of basis functions for this element. 1.
Author manuscript, published in "Journal of Elasticity (2010) in press" Journal of Elasticity manuscript No.
, 2010
"... (will be inserted by the editor) On the existence and uniqueness of a solution to the interior transmission problem for piecewisehomogeneous solids ..."
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(will be inserted by the editor) On the existence and uniqueness of a solution to the interior transmission problem for piecewisehomogeneous solids