On a pore-scale stationary diffusion equation: scaling effects and correctors for the homogenization limit (Q2033579)

The authors consider homogenization of the following semilinear elliptic equation in a periodically perforated domain \(\Omega_{\varepsilon}\) with a Fourier-type condition on the internal boundary \(\Gamma_{\varepsilon}\): \[-\nabla\cdot(A(x/\varepsilon)\nabla u_{\varepsilon})+\varepsilon^{\alpha}\mathcal{R}(u_{\varepsilon})=f(x) \mbox{ on }\Omega_{\varepsilon},\] \[-A(x/\varepsilon)\nabla u_{\varepsilon}\cdot n = \varepsilon^{\beta}\mathcal{S}(u_\epsilon) \mbox{ on }\Gamma_{\varepsilon},\] \[ u_{\varepsilon}=0\mbox{ on }\Gamma^{\text{ext}},\] where \(\mathcal{R}\) models a volume reaction and \(\mathcal{S}\) models a surface reaction and \(\Gamma^{\text{ext}}\) refers to the external boundary. They prove well-posedness of the problem by a linearization scheme, under suitable hypothesis on the nonlinear terms \(\mathcal{R}\) and \(\mathcal{S}\). Further, they construct asymptotic expansions in appropriate powers of \(\varepsilon\) for the solutions \(u_\varepsilon\) and prove higher order corrector estimates in three different regimes: (i) \(\alpha<0\) or \(\beta<0\), (ii) \(\alpha>0\) \& \(\beta>1\), and (iii) \(\alpha>0\) \& \(\beta\in[0,1)\). The construction of asymptotic expansions for \(u_{\varepsilon}\) requires the reaction terms \(\mathcal{R}\) and \(\mathcal{S}\) to have a matching asymptotic expansion. The theoretical results are supported by a numerical study.