Dorodnyi, M. A.

Homogenization of one-dimensional hyperbolic equations with corrector.
Approximation of solutions in the Sobolev class $H^1 (\mathbb{R})$.

35B27 Homogenization; partial differential equations in media with periodic
structure

periodic differential operators, hyperbolic equations, spectral bands, homogenization, effective operator, corrector, operator error estimates

    We consider an elliptic differential operator $A_\varepsilon = - \frac{d}{dx} g(x/\varepsilon) \frac{d}{dx}$, $\varepsilon > 0$, with a periodic coefficient acting in $L_2(\mathbb{R})$. The behavior of the solution of the Cauchy problem for the hyperbolic equation $\partial_\tau^2 u_\varepsilon (x,\tau) = - (A_\varepsilon u_\varepsilon) (x,\tau)$ is studied, as $\varepsilon \to 0$. We obtain an approximation of the solution $u_\varepsilon (\cdot,\tau)$ in the $H^1 (\mathbb{R})$-norm with error $O(\varepsilon)$. In this approximation a corrector is taken into account.

Russian, 26 pp.

mdorodni@yandex.ru