Surface integral formulation of the interior transmission problem

Abstract

International audienceWe consider a surface integral formulation of the so-called interior transmission problem that appears in the study of inverse scattering problems from dielectric inclusions. In the case where the magnetic permeability contrast is zero, the main originality of our approach consists in still using classical potentials for the Helmholtz equation but in weaker trace space solutions. One major outcome of this study is to establish Fredholm properties of the problem for relaxed assumptions on the material coefficients. For instance we allow the contrast to change sign inside the medium. We also show how one can retrieve discreteness results for transmission eigenvalues in some particular situations

A boundary integral equation method for the transmission eigenvalue problem

“…From Theorem 3.2 in [2] we have that (S k n − S k ) is a pseudo-differential operator of order −7/2 which implies that the operator…”

“…More importantly, by a duality argument it is possible to extend the jump relations for single-layer potentials across ∂D to the case of potentials with weaker densities. More specifically, the following lemma is proven in Theorem 3.1 in [2] (see also [22]). …”

“…[6][7][8][9][10][11][12]). Various mathematical methods have been developed to study this eigenvalue problem, including variational methods, [6,10] boundary integral equation methods, [2,13] and semiclassical analysis. [9,11] On the other hand, it is shown in [14,15] that transmission eigenvalues can be determined from scattering data which together with the establishment of monotonicity properties of transmission eigenvalues in terms of the refractive index [6,16] opens the possibility to use transmission eigenvalues as target signature in the inverse medium problem.…”

“…In this paper we develop a boundary integral equation approach for the transmission eigenvalue problem in the case of a constant refractive index. Boundary integral equation methods were first used in the context of the transmission eigenvalue problem by Cossonnière and Haddar [1,2], who used Green's representation formula for v and w to derive a system of two linear boundary integral equations that are equivalent to the transmission eigenvalue problem. For the corresponding two by two matrix valued operator which depends non-linearly on the eigenvalue parameter k, Cossonnière and Haddar analyzed the kernel and also used it to numerically compute transmission eigenvalues.…”

“…Besides the reduction in the complexity of the integral equations with a reduction of the computational costs by about 50 %, an additional advantage of our approach lies in the fact that the integral equation based on the Dirichlet-to-Neumann operator (or on the Robin-to-Dirichlet operator) characterizes the interior transmission eigenvalues, i.e. the transmission eigenvalues for D, without any additional conditions whereas the integral equations from [1,2] simultaneously characterize both the interior and also the exterior transmission eigenvalues, i.e. the transmission eigenvalues for the complement of D. Hence, the latter integral equations require additional conditions to distinguish the interior and exterior eigenvalues within the numerical computation.…”