On the degenerate Beltrami equation and hydrodynamic normalization

Abstract

The linear Beltrami equation on the Riemann sphere is studied under the assumption that its measurable complex-valued coefficient $\mu(z)$ has a compact support in $\mathbb{C}$ and $\Vert\mu\Vert_{\infty}=1.$ Sufficient conditions for the existence of regular homeomorphic $W_{\mathrm{loc}}^{1,1}$ solutions to the Beltrami equation with hydrodynamic normalization at infinity are given, in particular, provided that either the dilatation $K_{\mu}$ has the {bounded-mean-oscillation} majorant or the so-called tangent dilatations $K_{\mu}^{T}$ satisfy the integral divergence conditions of the Lehto type. The corresponding applications to the degenerate $A$-harmonic equation associated with the Beltrami equation have also been formulated.