The asymptotic expansion of the solution of a boundary-value problem with rapidly alternating type of boundary conditions
Considered is the Poisson equation \(\Delta u^\varepsilon(x)+ f(x)= 0\) in \(\Omega\subset\mathbb{R}^n\) between hyperplanes \(\Gamma_1\) and \(\Gamma_2\). Boundary conditions are \(u^\varepsilon(x)= 0\) on \(\Gamma^\varepsilon_D\cup \Gamma_2\) and \({\partial u^\varepsilon(x)\over \partial\nu}= 0\) on \(\Gamma^\varepsilon_N= \Gamma_1\setminus \Gamma^\varepsilon_D\) where \(\nu\) is normal to \(\partial\Omega\) and \(u^\varepsilon\), \(f\), \(\Gamma^\varepsilon_D\) are of some periodicities. \(\varepsilon>0\) effects that the boundary conditions are rapidly alternating for small \(\varepsilon\). Here it is constructed an asymptotic expansion for the solution \(u^\varepsilon\) in powers of \(\varepsilon\) with terms of some boundary layers and solutions of certain homogenized problems in \(\Omega\). For the remainder term of this asymptotic expansion for the solution \(u^\varepsilon\) it is given an estimation.
scientific article; zbMATH DE number 1259436
scientific article