On the convergence of a class of degenerate parabolic equations (Q1277401)

The authors study the convergence of the Cauchy-Dirichlet problems for a sequence of parabolic operators in the divergence form \(P_h =\frac{\partial}{\partial t} - \text{div} (a_h(x,t) D)\), as \(h\to\infty\), where \(a_h(x,t) =[a_{h,ij}(x,t)]_{i,j =1,\dots n}\) are matrices of measurable functions defined on a bounded open cylinder \(\Omega\times (0,T)\) of \(\mathbb{R}^{n+1}\) with \(a_h(\cdot,t)\)'s satisfy the degenerate ellipticity condition \[ \lambda_h(x)| \xi| ^2\leq\sum_{i,j = 1}^na_{h,ij}(\cdot,t)\xi_i\xi_j\leq L\lambda_h (x)| \xi| ^2\quad \text{a.e. } x\in\Omega. \] The authors prove that there exist a subsequence \((a_{h_j})_j\), a matrix \(a_{\infty}(x,t)\) and a weight \(\lambda_{\infty}\) verifying some reasonable conditions (which complete the degenerate ellipticity condition) for which the sequence of the unique solutions \(u_j\) of the Cauchy-Dirichlet problems converges in \(L^2((0,T);L^2(\Omega))\) to the unique solution \(u_{\infty}\) of the corresponding problem. Moreover the weak convergence in \(L^2((0,T); (L^1(\Omega))^n)\) of the momenta \(a_h Du_h\) to the momentum \(a_{\infty} Du_{\infty}\) and the convergence in \(D'(\Omega\times (0,T))\) of the energies \((a_{h_j} Du_j, Du_j)\) to the energy hold.