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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.

##### MSC:
 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)
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