zbMATH — the first resource for mathematics

An asymptotic formula for the voltage potential in a perturbed \(\varepsilon\)-periodic composite medium containing misplaced inclusions of size \(\varepsilon\). (English) Zbl 1105.35011
The authors consider a composite medium made of an array of inclusions \( \varepsilon \)-periodically disposed at the center of each \(\varepsilon \)-cell, except for some of them which are slightly misplaced. The goal of the paper is to describe the error between the potential \(u_{\varepsilon d}\) of this perturbed problem and that \(u_{\varepsilon }\) obtained for the problem with everywhere perfectly disposed inclusions. The authors indeed consider the elliptic problem \(-\text{ div}(a_{\varepsilon }(x)\nabla u_{\varepsilon })=f\), posed in a smooth domain \(\Omega \) of \(\mathbb{R}^{n}\) with nonhomogeneous Neumann boundary conditions on \(\partial \Omega : a_{\varepsilon }\partial u_{\varepsilon }/\partial \nu =g\) on \(\partial \Omega \). They add the compatibility condition between \(f\) and \(g\) and the further condition \(\int_{\Omega }u_{\varepsilon }\,dx=0\).
The main result proves that the difference between the two potentials in \(\Omega\) (plus a term involving this difference on \(\partial \Omega\)) is of order \( \varepsilon ^{3}\) with a first order involving the Green function associated to the homogenized problem and a polarization tensor built with the solution of local problems. The proof heavily relies on the properties of these Green functions.

35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
35A08 Fundamental solutions to PDEs
35J25 Boundary value problems for second-order elliptic equations
74Q05 Homogenization in equilibrium problems of solid mechanics
82C05 Classical dynamic and nonequilibrium statistical mechanics (general)
Full Text: DOI Link