On homogenization of the first initial-boundary value problem for periodic hyperbolic systems

Abstract

Let $\mathcal{O}\subset {\mathbb{R}}^d$ a bounded domain of class $C^{1,1}$. In $L_2(\mathcal{O};{\mathbb{C}}^n)$, we consider a self-adjoint matrix strongly elliptic second order differential operator $B_{D,\epsilon }$, $0<\epsilon ⩽1$, with the Dirichlet boundary condition. The coefficients of the operator $B_{D,\epsilon }$ are periodic and depend on $\mathbf{x}/\epsilon$. We are interested in the behavior of the operators $\cos (t{B}_{D,\epsilon }^{1/2})$ and $B_{D,\epsilon }^{-1/2}\sin (t{B}_{D,\epsilon }^{1/2})$, $t\in \mathbb{R}$, in the small period limit. For these operators, approximations in the norm of operators acting from some subspace $\mathcal{H}$ of the Sobolev space $H^4(\mathcal{O};{\mathbb{C}}^n)$ to $L_2(\mathcal{O};{\mathbb{C}}^n)$ are found. Moreover, for $B_{D,\epsilon }^{-1/2}\sin (t{B}_{D,\epsilon }^{1/2})$, the approximation with the corrector in the norm of operators acting from $\mathcal{H}\subset H^4(\mathcal{O};{\mathbb{C}}^n)$ to $H^1(\mathcal{O};{\mathbb{C}}^n)$ is obtained. The results are applied to homogenization for the solution of the first initial-boundary value problem for the hyperbolic equation ${\partial }_t^2{\mathbf{u}}_{\epsilon }=-B_{D,\epsilon }{\mathbf{u}}_{\epsilon }$.

32 pages. arXiv admin note: text overlap with arXiv:1801.05035