On homogenization problem for elliptic equation with rapidly oscillating coefficients in partially perforated domain with the Neumann conditions on the boundary of cavities
For the semi-perforated composite the problem \[ \begin{gathered} \frac{\partial}{\partial x_k}\bigg(a_{ks}\bigg(\frac{x}{\varepsilon}\bigg) \,\frac{\partial u_{\varepsilon}}{\partial x_s}\bigg) = f(x),\quad\,\, x\in\Omega_\varepsilon;\\ a_{ks}\bigg(\frac{x}{\varepsilon}\bigg) \,\frac{\partial u_{\varepsilon}}{\partial x_s}\, \nu_k = 0,\quad x\in S_\varepsilon; \quad\,\, u_\varepsilon =0,\quad x\in\Gamma_\varepsilon \end{gathered}\tag{1} \] is considered, where \(\,\nu = (\nu_!,\dots,\nu_n)\,\) is a unit vector of the outer normal to \(S_\varepsilon\), \(\,f(x)\in C^\infty(\overline{\Omega^+})\,\) and \(\,f(x)\in C^\infty(\overline{\Omega^-})\), \(\,a_{ks}(y)\) are the functions 1-periodic in \(y\) in \(\,R^n_y\cap \{y\: y_1>0\}\,\) and in \(\,R^n_y\cap \{y\: y_1<0\}\,\) which satisfy the uniform ellipticity condition. The author substantiates the averaging of the Laplace operator for the problem (1).
scientific article; zbMATH DE number 1974897