Homogenization of Solutions of Initial Boundary Value Problems for Parabolic Systems

Let ${\mathcal O}\subset {\mathbb R}^d$ be a bounded $C^{1,1}$ domain. In $L_2({\mathcal O};{\mathbb C}^n)$ we consider strongly elliptic operators $A_{D,\varepsilon}$ and $A_{N,\varepsilon}$ given by the differential expression $b({\mathbf D})^*g({\mathbf x}/\varepsilon)b({\mathbf D})$, $\varepsilon>0$, with Dirichlet and Neumann boundary conditions, respectively. Here $g({\mathbf x})$ is a bounded positive definite matrix-valued function assumed to be periodic with respect to some lattice and $b({\mathbf D})$ is a first-order differential operator. We find approximations of the operators $\exp(-A_{D,\varepsilon} t)$ and $\exp(-A_{N,\varepsilon} t)$ for fixed $t>0$ and small $\varepsilon$ in the $L_2 \to L_2$ and $L_2 \to H^1$ operator norms with error estimates depending on $\varepsilon$ and $t$. The results are applied to homogenize the solutions of initial boundary value problems for parabolic systems.