Homogenization of the Higher-Order Hyperbolic Equations with Periodic Coefficients
Tatiana Aleksandrovna Suslina · Lobachevskii Journal of Mathematics · 2021
In $$L_{2}({\mathbb{R}}^{d};{\mathbb{C}}^{n})$$ , we consider a matrix strongly elliptic differential operator $${A}_{\varepsilon}$$ of order $$2p$$ , $$p\geqslant 2$$ . The operator $${A}_{\varepsilon}$$ is given by $${A}_{\varepsilon}=b(\mathbf{D})^{*}g(\mathbf{x}/\varepsilon)b(\mathbf{D})$$ , $$\varepsilon>0$$ , where $$g(\mathbf{x})$$ is a periodic, bounded, and positive definite matrix-valued function, and $$b(\mathbf{D})$$ is a homogeneous differential operator of order $$p$$ . We study the behavior of the operator-valued functions $$\cos(\tau{A}_{\varepsilon}^{1/2})$$ and $${A}_{\varepsilon}^{-1/2}\sin(\tau{A}_{\varepsilon}^{1/2})$$ for small $$\varepsilon$$ and $$\tau\in{\mathbb{R}}$$ . It is shown that these operators converge, as $$\varepsilon\to 0$$ , to the corresponding operator-valued functions of $$A^{0}$$ in the norm of operators acting from the Sobolev space $$H^{s}({\mathbb{R}}^{d};{\mathbb{C}}^{n})$$ (for suitable exponents $$s$$ ) into $$L_{2}({\mathbb{R}}^{d};{\mathbb{C}}^{n})$$ . Here $$A^{0}$$ is the effective operator. Sharp-order error estimates are obtained. The results are applied to homogenization of the Cauchy problem for the hyperbolic gather $$\partial^{2}_{\tau}{\mathbf{u}}_{\varepsilon}=-{A}_{\varepsilon}{\mathbf{u}}_{\varepsilon}+{\mathbf{F}}$$ .