On the Dirichlet problem for degenerate Beltrami equations

Abstract

We show that every homeomorphic $W_{\mathrm{loc}}^{1,1}$ solution $f$ to a Beltrami equation $\bar{\partial }f=\mu \partial f$ in a domain $D\subset \mathbb{C}$ is the so--called lower $Q-$homeomorphism with $Q(z)=K_{\mu }^T(z,z_0)$ where $K_{\mu }^T(z,z_0)$ is the tangent dilatation of $f$ with respect to an arbitrary point $z_0\in \bar{D}$ and develop the theory of the boundary behavior of such solutions. Then, on this basis, we show that, for wide classes of degenerate Beltrami equations $\bar{\partial }f=\mu \partial f$, there exist regular solutions of the Dirichlet problem in arbitrary Jordan domains in $\mathbb{C}$ and pseudoregular and multi-valued solutions in arbitrary finitely connected domains in $\mathbb{C}$ bounded by mutually disjoint Jordan curves.

35 pages. arXiv admin note: substantial text overlap with arXiv:1201.5570