G-convergence, Dirichlet to Neumann maps and invisibility

Abstract

We establish optimal conditions under which the Gconvergence of linear elliptic operators implies the convergence of the corresponding Dirichlet to Neumann maps. As an application we show that the approximate cloaking isotropic materials from [19] are independent of the source.

The inverse conductivity problem via the calculus of functions of bounded variation

“…We mention here that the necessity of the implication of the parameter and the zone Ω is due to the desire to have the ability to take strong limits of the Calderón's operator Λ , when follows specific limit processes. 19 The fourth term of the functional, offering the necessary regularization, is a term properly referring to the selection over conductivity profiles obeying to restriction rules concerning their total masses and some reference compatibility. This last term of the functional involves two regularization parameters , and obtains the following form:…”

“…On the basis of the convergences (19) and (20), the limit processes declared by Propositions 5 and 6, and the lower semicontinuity of the L 2 (Ω) norm in the weak topology, we obtain…”

“…At this point, we exploit the main result of Faraco et al, 19 according to which when a very weak assumption is made for the conductivity profiles n , then the strong convergence ||Λ n − Λ ||  − → n→∞ 0 is valid. Actually, the mentioned weak assumption requires a specific convergence of the profiles n to in a very thin region around the boundary Ω.…”

“…This is expected, since extreme oscillations of a sequence of conductivities create an instability of the Calderón problem. This kind of asymptotics is well described via the implication of H or G convergence analysis, [18][19][20] where the homogenization theory assigns the suitable convergence regime. 21,22 As already mentioned, the investigation method depends strongly on the assumptions made for the regularity of the sought conductivity profile.…”

A multiscale theory for image registration and nonlinear inverse problems

“…Here we just remark that for symmetric conductivity tensors H-convergence reduces to the more usual G-convergence and that M(a, b) and M sym (a, b) are compact with respect to Hconvergence. We also recall that G-or H-convergence has already been shown to be a useful tool in the context of the inverse conductivity problem, see for instance [22,2,16,34,35].…”