Operator estimates for the averaging of the Riemann-Hilbert problem for the Beltrami equation with a locally periodic coefficient

Abstract

Local characteristics of mathematical models of strongly inhomogeneous media are usually described by functions of the form $a(\varepsilon^{-1} x)$, $b(x,\varepsilon^{-1} x)$, $c(\varepsilon^{-1} x,\delta^{-1} x)$, $d(\varepsilon^{-1} x,\delta^{-1} x,\gamma^{-1} x)$, etc., where $\varepsilon$, $\delta$, $\gamma,\ldots>0$ --- small parameters, while functions $a$, $b$, $c$, $d$, $\ldots$ have an ordered structure (for example, they are periodic in variables $y=\varepsilon^{-1} x$, $z=\delta^{-1} x$, etc.). Consequently, the corresponding mathematical models are differential equations with rapidly oscillating coefficients. This work is devoted to estimates of the averaging error. We study the generalized Beltrami equation with a locally periodic coefficient $\mu(x,\varepsilon^{-1} x)$.