The oscillating boundary phenomenon in the homogenization of a climatization problem
Let \(\Omega\subset\mathbb{R}^2\) be a bounded domain, \(\varepsilon>0\) a small parameter and \(\Omega_\varepsilon\) the domain obtained by \(\varepsilon\)-periodic perturbation \(\partial\Omega_\varepsilon\) (with the amplitude \(\varepsilon)\) of the boundary \(\partial\Omega\). The author considers a homogenization of the stationary heat conduction problem in \(\Omega_\varepsilon \), assuming that on \(\partial\Omega_\varepsilon\) the heat exchange is governed by the Newton's law, that the temperature never falls below some given level and that the heat flux is one-directional. The problem is formulated as a variational inequality; the existence and uniqueness of a solution is established. The homogenized problem is found out and the convergence of the homogenization process is proved. A similar problem was considered in the paper of \textit{W. Jäger}, \textit{O. A. Oleinik} and \textit{T. A. Shaposhnikova} [Differ. Equ. 36, 833-846 (2000; Zbl 0993.35013)].
scientific article; zbMATH DE number 1915932