Some a priori estimates of solutions to the Maxwell equations

Abstract

In this paper, we present a collection of a priori estimates of the electromagnetic field scattered by a general bounded domain. The constitutive relations of the scatterer are in general anisotropic. Surface averages are investigated, and several results on the decay of these averages are presented. The norm of the exterior Calderón operator for a sphere is investigated and depicted as a function of the frequency.

The exterior Calderón operator for non-spherical objects

“…where n l ðzÞ ¼ zh ð1Þ l ðzÞ is the Riccati-Hankel function [17,22]. Notice the result of Lemma 5, i.e., C À1 sn;s 0 n 0 ¼ C sn;s 0 n 0 : Moreover, which is, apart from a different normalization, in agreement with [18], see Figure 3.…”

“…which is identical to the statement in the lemma. Integration by parts gives an alternative form of the matrix A sn;s 0 n 0 , see (18) and use Definition 2.…”

“…This operator is also called the Poincaré-Steklov operator, and its discretization is often called the Schur complement. It has been studied intensively during many years, see e.g., [9,18,20].…”

“…This operator is also called the Poincaré-Steklov operator, and its discretization is often called the Schur complement. It has been studied intensively during many years, see e.g., [9,18,20].It is related to the Dirichlet-to-Neumann map for the scalar Helmholtz equation. The exterior Calderón map is instrumental in the analysis of the solution to the exterior solution of the scattering problem.…”

“…These eigenfunctions and the corresponding eigenvalues are intrinsic to the surface and constitute an excellent tool for further analysis; the literature on this subject of finding these eigenfunctions and eigenvalues is extensive, see, e.g., [4,11,19,27]. Explicit values of the norm of the exterior Calderón operator have only been obtained for the sphere case [18,20] and the planar case [3,9], and we refer to these bibliographical items for the explicit techniques of computing the norm. In this paper, we present a new way to explicitly find the norm for non-spherical obstacles.…”