On homogenization problems for the Laplace operator in partially perforated domains with Neumann's condition on the boundary of cavities
Consider the problem: \[ \Delta u_\varepsilon= f\quad \text{in} \quad \Omega_\varepsilon,\quad {\partial u_\varepsilon\over \partial v}= 0\quad \text{on} \quad S_\varepsilon,\quad u_\varepsilon= 0\quad \text{on} \quad \Gamma_\varepsilon. \] Here, \(\Omega\) is a bounded domain in \(\mathbb{R}^n\) with a smooth boundary \(\partial\Omega\), \(\Omega^+\) [resp. \(\Omega^-\), \(\gamma\)] is the intersection of \(\Omega\) with \(\{x; x_1> 0\}\) (resp. \(\{x_1< 0\}\), \(\{x_1= 0\}\)), \(\Omega_\varepsilon= \Omega^+_\varepsilon\cup \gamma\cup \Omega^-\), where \(\Omega^+_\varepsilon\) is a perforated domain contained in \(\Omega^+\), \(S_\varepsilon= \partial \Omega_\varepsilon\cap \Omega\), \(\Gamma_\varepsilon= \partial\Omega_\varepsilon\backslash S_\varepsilon\). The authors study the behavior, as \(\varepsilon\to 0\), of solutions and eigenvalues of the problem.
scientific article; zbMATH DE number 852037