Uniform asymptotic expansion of the voltage potential in the presence of thin inhomogeneities with arbitrary conductivity
Denote by \(\Omega\) a bounded domain in \(\mathbb R^2\), with smooth boundary and assume that it is formed by a conductive material, and it has a thin inhomogeneity of width depending on \(\epsilon\), \({\omega}_{\epsilon}\), with conductivity \(a_{\epsilon}\). The conductivity in the entire domain will be denoted by \({\gamma}_{\epsilon}\). One defines the electric potential \(u_{\epsilon}\) as the solution of the Dirichlet problem for the second order elliptic equation \[ -div(\gamma_\epsilon\nabla u_\epsilon)=f\quad\text{in}\quad\Omega,\quad u_{\epsilon} = \varphi\quad\text{on}\quad {\partial}{\Omega}. \] The goal of the paper is to investigate the uniform asymptotic behavior of the potential \(u_{\epsilon}\) as the thin inhomogeneity converges to zero. The authors intend to find an approximate solution uniformly close to \(u_{\epsilon}\) which is independent of the behavior of \(a_{\epsilon}\). After presenting preliminaries, the statement of the problem and notations, they introduce the main tool that will be used for studying the convergence of minimizers in variational problems. In the next section the authors construct a \(0\)-th order approximation to \(u_{\epsilon}\). Further, there an extensive study of this approximation is presented. Investigated are uniform energy and regularity estimates, asymptotic exactness and the limit behavior. Using the same strategy, the derivation of the uniform first-order approximation to \(u_{\epsilon}\) is discussed. An appendix is devoted to the proof of uniform regularity estimates for the 0-th order approximation.
scientific article; zbMATH DE number 6707257
scientific article