A refined convergence result in homogenization of second order parabolic systems
In the paper under review, the authors investigate the convergence rate in periodic homogenization of second order parabolic systems. Let $\Omega$ be a bounded domain in $\mathbb{R}^d$ and $T>0$. The authors consider the homogenization of the parabolic system \[ \begin{cases} \partial_t u_\epsilon+\mathcal{L}_\epsilon u_\epsilon =f & \text{in } \Omega \times (0,T),\\ u_\epsilon=0 & \text{on } \partial \Omega \times (0,T),\\ u_\epsilon=h & \text{on } \Omega \times \{t=0\}, \end{cases} \] where \[ \mathcal{L}_\epsilon=-\partial_\alpha (A_{ij}^{\alpha\beta}(x/\epsilon,t/\epsilon^2)\partial_\beta), \qquad 1\leq i,j\leq n,1\leq \alpha,\beta\leq d,\] where the coefficient matrix $A(y,s)=(A_{ij}^{\alpha\beta}(y,s))$ is real, bounded, measurable, $1$-periodic in $(y,s)$ and \[\mu |\xi|^2\leq A^{\alpha \beta}(y,s)\xi_\alpha \xi_\beta,\qquad |A(y,s)|\leq 1/\mu \quad \text{for a.e. } (y,s)\in \mathbb{R}^{d+1}. \] The unique solution $u^\epsilon$ of the problem above is well known to converge to the unique solution $u_0$ of the following parabolic equation with constant coefficients \[ \begin{cases} \partial_t u_0+\mathcal{L}_0 u_0 =f & \text{in } \Omega \times (0,T),\\ u_0=0 & \text{on } \partial \Omega \times (0,T),\\ u_0=h & \text{on } \Omega \times \{t=0\}, \end{cases} \] where $\mathcal{L}_0$ is the homogenized operator. The authors show the optimal $O(\epsilon)$ convergence rate of $u_\epsilon$ to $u_0$ in $L^2(0,T;L^{q_0}(\Omega))$, with $q_0=2d/(d-1)$, $d\geq 2$.
scientific article; zbMATH DE number 7039978
scientific article