St. Petersburg Mathematical Society

Препринты Санкт-Петербургского математического общества
Preprints of the St. Petersburg Mathematical Society

Аннотация к препринту 2026-01

Author

Dorodnyi, M. A.

Title

Homogenization of one-dimensional hyperbolic equations with corrector.
Approximation of solutions in the Sobolev class

H^1 (\mathbb{R})

AMS Subj-class

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

Keywords

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

Abstract

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.