Conductivity interface problems. Part I: Small perturbations of an interface

Abstract

Abstract. We derive high-order terms in the asymptotic expansions of boundary perturbations of steady-state voltage potentials resulting from small perturbations of the shape of a conductivity inclusion with C 2 -boundary. Our derivation is rigorous and based on layer potential techniques. The asymptotic expansion in this paper is valid for C 1 -perturbations and inclusions with extreme conductivities. It extends those already derived for small volume conductivity inclusions and leads us to very effective algorithms for determining lower-order Fourier coefficients of the shape perturbation of the inclusion based on boundary measurements. We perform some numerical experiments using the algorithm to test its effectiveness.

Optimization algorithm for reconstructing interface changes of a conductivity inclusion from modal measurements

“…In fact, on one hand, in a series of recent papers [8,6,4,5], we have derived high-order asymptotic expansions of the eigenvalue perturbations due to the presence of small inclusions and used them for locating the inclusions and identifying some of their geometric features. On the other hand, in [7], we have derived high-order terms in the asymptotic expansions of the boundary perturbations of steady-state voltage potentials resulting from small perturbations of the shape of a conductivity inclusion. Based on these derivations, we have designed an effective algorithm to determine some geometric features of the shape perturbation of the inclusion based on boundary measurements.…”

Asymptotic formula for the solution of the Stokes problem with a small perturbation of the domain in two and three dimensions

“…We denote by ν( x) the outward unit normal to ∂Ω δ at x. Then, it is proved in [1] that ν( x) can be expanded uniformly as…”

“…Likewise, denote by dσ δ ( x) the length element of ∂Ω δ at x, which has an uniform expansion (see in [1])…”

“…The asymptotic expansion is derived due to the theory of layer potentials and Fredholm's alternative, and the properties of small perturbation of an interface. In connection with this, we refer to recent works in the context of interface problems [1], [2], [7], the continuity of the solution due to a small perturbation of an interface [7] and layer-potential theory for Stokes problem [3], [4], [5], [6], [8], [9]. In the work of H. Ammari, H. Kang, M. Lim, and H. Zribi [1] , they derived the asymptotic expansion of the boundary perturbations of steady-state voltage potentials, and now in our work, we derive the asymptotic expansion for the Stokes problem with Dirichlet boundary condition.…”

Stability analysis in the inverse Robin transmission problem

“…Such problem arises in many physical situations such as electrical impedance tomography (for instance, [1][2][3][4][5]). In [6], a partial differential equation with Robin-type transmission conditions, which models the situation where the corrosion takes place between two layers of a non-homogenous medium, is considered.…”