On the asymptotic behaviour of elliptic problems with periodic data (Q704258)

Let \(V_n\) be a subspace of \(H^1(\Omega_n)\) such that, \(V_n\) is closed in \(H^1(\Omega)\), \(H^1_0(\Omega_n)\subset V_n\subset H^1(\Omega_n)\), with \(\Omega_n= (-nT, nT)^k\). By \(u_n\) is denoted the weak solution to \[ u_n\in V_n,\;\int_{\Omega_n} a_{ij}(x) \partial_{x_i} u_n\partial_{x_j}v+ a(x) u_nv\, dx= \int_{\Omega_n} f_0v+ f_i\partial_{x_i} v\,dx,\;\forall v\in V_n\tag{1} \] and by \(u_\infty\) the solution to \[ \begin{multlined} u_\infty\in H^1_{\text{per}}(Q) \int_Q [a_{ij}(x)\partial_{x_i} u_\infty\partial_{x_j}v+ a(x) u_\infty v]\,dx= \int_Q f_0v+ f_i\partial_{x_i} v\,dx,\;\forall v\in H^1_{\text{per}}(Q),\end{multlined}\tag{2} \] where \(Q= (0,T)^k\) and \[ H^1_{\text{per}}(Q)= \{v\in H^1(Q)\mid v\text{ is }T\text{-periodic in all directions}\}. \] Under suitable assumptions on the data (1) and (2) the authors prove existence and uniqueness of (1) and (2). They show also converges \(u_n\) towards \(u_\infty\) as well.