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

ABSTRACT

Let $\mathcal{O}\subset {\mathbb{R}}^d$ be a bounded domain of class $C^{3,1}$. 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 $\cos \left(t{B}_{D,\epsilon }^{1/2}\right)$ and $B_{D,\epsilon }^{-1/2}\sin \left(t{B}_{D,\epsilon }^{1/2}\right)$, $t\in \mathbb{R}$, in the small period limit. For these operators, approximations in the norm of operators acting from a certain 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 $B_{D,\epsilon }^{-1/2}\sin \left(t{B}_{D,\epsilon }^{1/2}\right)$, 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 ${\mathrm{\partial }}_t^2{\mathbf{u}}_{\epsilon }=-B_{D,\epsilon }{\mathbf{u}}_{\epsilon }$.

KEYWORDS:

• Periodic differential operators
• hyperbolic systems
• homogenization
• operator error estimates

2010 MATHEMATICS SUBJECT CLASSIFICATIONS:

• 35B27
• 35L53

Acknowledgments

The author is deeply grateful to T. A. Suslina for her attention to this work.

Disclosure statement

No potential conflict of interest was reported by the author.

Additional information

Funding

Research is supported by the Russian Science Foundation [grant number 14-21-00035].