The Impedance Boundary Value Problem for the Helmholtz Equation in a Half-Plane

Scattering of Electromagnetic Waves by Rough Interfaces and Inhomogeneous Layers

“…This present paper is intended, in part, as a contribution to the mathematical analysis of rough surface scattering problems and of the well-posedness of their formulation as integral equations. It is related, in terms of results and methods of argument, to recent studies of scattering of a wave incident from a homogeneous half-space onto an inhomogeneous impedance plane [5]; of electromagnetic waves by a one-dimensional perfectly conducting rough surface [6,8]; of electromagnetic waves by an inhomogeneous conducting or dielectric layer on a perfectly conducting plate [7]; and of acoustic waves by an inhomogeneous layer on a rigid plate [31]. In particular, the present study is closest to this last paper [31] in that the existence proof depends on the same novel results on the solvability of systems of weakly singular second-kind integral equations on unbounded domains.…”

Scattering by the local perturbation of an open periodic waveguide in the half plane

“…Therefore, a l,j = 0 for all l, j, which yields u (2) = 0. Theorem 3.1 has been shown, and in next section we will show the uniqueness of u (1) .…”

“…(iii) u vanishes for x 2 = 0, (2) = 0 for x 2 > 0. Then, u (1) satisfies the assumptions (i)-(iv) of Lemma 4.1, which implies that u (1) ∈ H 1 0 (R 2 + ). By Assumption 1.1, u (1) vanishes for x 2 > 0, which yields the uniqueness.…”

Inverse acoustic scattering problem in half-space with anisotropic random impedance

“…for all ξ ∈ R 3 0 , where F = F x→ξ is the Fourier transform and ξ ⊥ = (ξ 2 , −ξ 1 ). Our interest lies in the case when the anisotropic local strength is of quadratic form (8) b(x, ξ 0 ) = ξ 0 , A(x)ξ 0 , where the matrix field A : R 2 → R 2×2 is (A5) smooth and symmetric, has uniformly bounded eigenvalues and satisfies supp(A) ⊂ D. The main result regarding the quadratic model is then as follows.…”

“…Let the assumptions in Theorem 2.4 hold. In addition, we assume that the local strength of λ is of the form (8), where A : R 2 → R 2×2 satisfies (A5). Given the backscattering data n 0 (x), x ∈ U, the trace tr(A) can be uniquely determined everywhere.…”