151 research outputs found
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 that in this case, Dirichlet eigenpairs are the only
accumulation points of the Robin eigenpairs with normalized eigenvectors. We
then provide a criteria to select accumulating sequences of eigenvalues and
eigenvectors and exhibit their full asymptotic with respect to the small
parameter
Boundary Integral Equations for the Transmission Eigenvalue Problem for Maxwell’s Equations
International audienceIn this paper we consider the transmission eigenvalue problem for Maxwell’s equations corresponding to non-magnetic inhomogeneities with contrast in electric permittivity that changes sign inside its support. We formulate the transmission eigenvalue problem as an equivalent homogeneous system of boundary integral equa- tion, and assuming that the contrast is constant near the boundary of the support of the inhomogeneity, we prove that the operator associated with this system is Fredholm of index zero and depends analytically on the wave number. Then we show the existence of wave numbers that are not transmission eigenvalues which by an application of the analytic Fredholm theory implies that the set of transmission eigenvalues is discrete with positive infinity as the only accumulation point
The linear sampling method for anisotropic media
AbstractWe consider the inverse scattering problem of determining the support of an anisotropic inhomogeneous medium from a knowledge of the incident and scattered time harmonic acoustic wave at fixed frequency. To this end, we extend the linear sampling method from the isotropic case to the case of anisotropic medium. In the case when the coefficients are real we also show that the set of transmission eigenvalues forms a discrete set
Analysis of topological derivative as a tool for qualitative identification
International audienceThe concept of topological derivative has proved effective as a qualitative inversion tool for a wave-based identification of finite-sized objects. Although for the most part, this approach remains based on a heuristic interpretation of the topological derivative, a first attempt toward its mathematical justification was done in Bellis et al. (Inverse Problems 29:075012, 2013) for the case of isotropic media with far field data and inhomogeneous refraction index. Our paper extends the analysis there to the case of anisotropic scatterers and background with near field data. Topological derivative-based imaging functional is analyzed using a suitable factorization of the near fields, which became achievable thanks to a new volume integral formulation recently obtained in Bonnet (J. Integral Equ. Appl. 29:271-295, 2017). Our results include justification of sign heuristics for the topological derivative in the isotropic case with jump in the main operator and for some cases of anisotropic media, as well as verifying its decaying property in the isotropic case with near field spherical measurements configuration situated far enough from the probing region
On the Regularity of Non-Scattering Anisotropic Inhomogeneities
In this paper we examine necessary conditions for an anisotropic
inhomogeneous medium to be non-scattering at a single wave number and for a
single incident field. These conditions are expressed in terms of the
regularity of the boundary of the inhomogeneity. We assume that the
coefficients, characterizing the constitutive material properties of the
medium, are sufficiently smooth, and the incident wave is appropriately
non-degenerate. Our analysis utilizes the Hodograph transform as well as
regularity results for nonlinear elliptic partial differential equations. Our
approach requires that the boundary a-priori is of class for
some
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