Homogenization Estimates in the Riemann–Hilbert Problem for the General Beltrami Equation on the Plane

Abstract

We study homogenization for the Beltrami equation \( {A}_{\varepsilon }{u}_{\varepsilon}\equiv {\partial}_{\overline{z}}{u}_{\varepsilon }+{\mu}^{\varepsilon }{\partial}_z{u}_{\varepsilon }+{\nu}^{\varepsilon}\overline{\partial_z{u}_{\varepsilon }}=f \) with measurable ε-periodic coefficients μ^ε and ν^ε, where ε is a small parameter. The coefficients of the equation satisfy the uniform ellipticity condition. The equation is considered in a bounded domain Ω of the complex plane with the Riemann–Hilbert condition on the boundary ∂Ω. For the resolvent \( {A}_{\varepsilon}^{-1} \) of this boundary value problem we obtain an approximation in the operator norm of the Sobolev space W^1,2(Ω) with approximation error of order O(\( \sqrt{\varepsilon } \)).