On the Dirichlet problemfor the Beltrami equation

Abstract

We show that homeomorphic Wloc1,1 solutions of the Beltrami equations \(\overline \partial f = \mu \partial f\) satisfy certain modular inequalities. On this basis, we develop the theory of the boundary behavior of such solutions and prove a series of criteria for the existence of regular, pseudoregular and multi-valued solutions for the Dirichlet problem to the Beltrami equation in Jordan domains and finitely connected domains, respectively. These results have important applications to various problems of mathematical physics.