Correctors for some asymptotic problems (Q630220)

Let \(\omega\) be a bounded open subset of the Euclidean space \({\mathbb{R}}^{p}\) with \(p\geq 1\). Let \(x\equiv (X_{1},X_{2})\equiv (x_{1},x_{1}',\dots,x_{p}')\) denote the points of \({\mathcal{O}}\equiv (-1,1)\times\omega\). Let \(f\in L^{2}({\mathcal{O}})\). Let \(u_{\epsilon}\in H^{1}_{0}({\mathcal{O}})\) denote the weak solution of the problem \(-\epsilon^{2}\partial^{2}_{X_{1}}u_{\epsilon}-\Delta_{X_{2}} u_{\epsilon}=f\) in \({\mathcal{O}}\) for each \(\epsilon>0\). In a previous paper [Commun. Pure Appl. Anal. 8, No. 1, 179--193 (2009; Zbl 1152.35309)], the authors have shown that \(u_{\epsilon}\) converges in \( L^{2}({\mathcal{O}})\) to the solution \(u_{0}\) of a limiting problem. Such a convergence however is not expected to hold in \(H^{1}_{0}({\mathcal{O}})\). In this paper the authors investigate the existence of a correction term \(w_{\epsilon}\) such that \(u_{\epsilon}-w_{\epsilon}\) converges to \(u_{0}\) in \(H^{1}_{0}({\mathcal{O}})\). Then the authors consider applications of their results to problems in domains of the form \((0,l)^{m}\times\omega\), where \(m\) is an integer and the parameter \(l\) tends to infinity.