On homogenization of the mixed boundary value problem for the heat equation in a domain whose part contains channels arranged in a periodic way
The asymptotic behaviour of solutions \(u_\varepsilon\) of an initial boundary value problem for the heat equation in a partially perforated domain \(\Omega_\varepsilon\) is studied. The domain \(\Omega_\varepsilon\) is of the form \(\Omega_\varepsilon= \Omega^-\cup \Omega_\varepsilon^+\cup\gamma\), where \(\Omega^-\), \(\Omega^+_\varepsilon\) are disjunct regular domains, having a contact hypersurface \(\gamma\) on the boundary; \(\Omega^+_\varepsilon\) is obtained by removing, from a domain \(\Omega\subset \mathbb{R}^n\), a collection of periodically distributed and identical obstacles of size \(\varepsilon\). The Neumann condition on boundaries of obstacles and the Dirichlet condition on the other part of \(\partial\Omega_\varepsilon\) is posed. The homogenized problem is the Dirichlet problem in the domain \(\Omega\cup\Omega^-\cup\gamma\), with the transmission conditions on \(\gamma\). The authors found out the estimates for the difference between \(u_\varepsilon\) and the solution of the homogenized problem.
Neumann condition on boundaries of obstacles
scientific article; zbMATH DE number 1526522