zbMATH — the first resource for mathematics

The asymptotic behaviour of the third boundary-value problem solutions in domains with fine-grained boundaries. (English) Zbl 0892.35018
Cioranescu, Doina (ed.) et al., Homogenization and applications to material sciences. Proceedings of the international conference, Nice, France, June 6–10, 1995. Tokyo: Gakkotosho. GAKUTO Int. Ser., Math. Sci. Appl. 9, 203-213 (1995).
Summary: We consider the initial boundary-value problem \[ {\partial u^\varepsilon \over \partial t}- \Delta u^\varepsilon= f^\varepsilon (x,t),\;x\in \Omega^\varepsilon,\;t>0;\quad {\partial u^\varepsilon \over\partial n}+ \sigma^\varepsilon (x,u^\varepsilon) =0,\;x\in \partial F^\varepsilon ,\;t>0; \] \[ u^\varepsilon(x,t)=0,\;x\in\partial \Omega^\varepsilon,\;t>0;\;u^\varepsilon (x,0)= \varphi (x),\;x\in\Omega^\varepsilon; \] where \(\Omega^\varepsilon = \Omega \setminus \cup^{N (\varepsilon)}_{j=1}\); \(\{F^\varepsilon_j\), \(j=1,2, \dots, N (\varepsilon)\}\) is a system of balls of radius \(\varepsilon^\alpha\), \(2<\alpha\leq 3\). The centers of the balls form a periodical lattice with period \(\varepsilon\). A homogenized model of diffusion is obtained, and a corrector is constructed.
For the entire collection see [Zbl 0873.00028].

35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
35K20 Initial-boundary value problems for second-order parabolic equations