Some remarks on the small electromagnetic inhomogeneities reconstruction problem

Abstract

This work considers the problem of recovering small electromagnetic inhomogeneities in a bounded domain Ω ⊂ R 3 , from a single Cauchy data, at a fixed frequency. This problem has been considered by several authors, in particular in [4]. In this paper, we revisit this work with the objective of providing another identification method and establishing stability results from a single Cauchy data and at a fixed frequency. Our approach is based on the asymptotic expansion of the boundary condition derived in [4] and the extension of the direct algebraic algorithm proposed in [1].

Identification and stability of small-sized dislocations using a direct algorithm

“…In a similar manner and using simple calculations, it can be proved that for i = 1, 2 and j = 1, 2, 3 3 . Now, if we introduce the function w as a solution of the homogenous elasticity equation ( 12)…”

“…In particular, based on the choice of a suitable test function, the reciprocity gap functional and suitable estimation inequalities, we establish a Hölder stability over the centers of the dislocations. This methodology has been initially used in [10] in the problem of recovering monopolar and dipolar sources in Helmholtz equation, then in [3] for the problem of inhomogeneities reconstruction over Helmholtz and Maxwell equations.…”

“…Our objective now is to prove that v ∈ [L 2 (Γ)] 3 and σ(v) 3 . Indeed, to achieve the desired regularities and due to the definition of v in (10), the form of the source term F , defined in (8), and the fundamental tensor (11), it is sufficient to prove that 3 , the other terms are done in the same way. In fact, this amounts to showing that h i * F j ∈ L 2 (Γ) and…”

“…where g is the force density defined in ( 6) and v is the solution of the linear isotropic elasticity equation, defined by the convolution in (10). As g belong to [ 3 . Using the classical theory of elliptic equation (see Lions and Magenes [26]), one can see that problem ( 12) is well posed (up to an additive constant) in the Sobolev space H 1 (Ω) 3 .…”

“…As g belong to [ 3 . Using the classical theory of elliptic equation (see Lions and Magenes [26]), one can see that problem ( 12) is well posed (up to an additive constant) in the Sobolev space H 1 (Ω) 3 .…”