Asymptotics for steady‐state voltage potentials in a bidimensional highly contrasted medium with thin layer

Abstract

To cite this version:Clair Poignard. Asymptotics for steady state voltage potentials in a bidimensional highly contrasted medium with thin layer. Mathematical Methods in the Applied Sciences, Wiley, 2008, 31 (4) CLAIR POIGNARDAbstract. We study the behavior of steady state voltage potentials in two kinds of bidimensional media composed of material of complex permittivity equal to 1 (respectively α) surrounded by a thin membrane of thickness h and of complex permittivity α (respectively 1). We provide in both cases a rigorous derivation of the asymptotic expansion of steady state voltage potentials at any order as h tends to zero, when Neumann boundary condition is imposed on the exterior boundary of the thin layer. Our complex parameter α is bounded but may be very small compared to 1, hence our results describe the asymptotics of steady state voltage potentials in all heterogeneous and highly heterogeneous media with thin layer. The asymptotic terms of the potential in the membrane are given explicitly in local coordinates in terms of the boundary data and of the curvature of the domain, while these of the inner potential are the solutions to the so-called dielectric formulation with appropriate boundary conditions. The error estimates are given explicitly in terms of h and α with appropriate Sobolev norm of the boundary data. We show that the two situations described above lead to completely different asymptotic behaviors of the potentials.

Generalized impedance boundary condition at high frequency for a domain with thin layer: the circular case

“…It might be seen as an extension of the hypothesis δω → c 0 > 0 of Lafitte [8]. Moreover if δω 1−γ/2 is bounded, we expect that the analysis performed by Ammari, Vogelius et al (see for example [1], [5], [3], [2] or [10]) could be applied. Actually, in this case δ might be replaced by c × ω −1+γ/2 , for an appropriate constant c > 0, and only one parameter appears in our problem.…”

“…Approximated boundary conditions for domains with thin layer have been extensively studied for Helmholtz equation at given frequency during the last twenty years; see for instance [1], [5], [3] or [2], and [10] for highly contrasted domains. In these previous works a bounded domain O surrounded by a thin layer is considered.…”

About the transmembrane voltage potential of a biological cell in time-harmonic regime.

“…This method, which leads to precise a priori error estimates was extensively presented in previous papers [9][10][11] in a bidimensional domain. It is based on an appropriate change of variables in the thin layer in order to make appear the small parameter (the thickness of the membrane) in the equations satisfied by the electric potential.…”

“…The high frequency case, which corresponds to non-highly heterogeneous domains, was treated in [8][9][10], therefore we refer the reader to these papers for a precise description of the asymptotic method. We focus here on the low and mid-frequency approximations for a threedimensional cell.…”

Spectroscopic conductivity imaging of a cell culture

“…By the uniqueness of the solution to (15) up to a constant, v(x) = c, ∀x ∈ Y. Then, we have ϕ 1 = 0 on ∂C and ϕ 2 = 0 on ∂B because they are equal to the jumps of v (resp.…”