On the Dirichlet problem for the Beltrami equations in finitely connected domains (Q1937950)

The Beltrami equation has the form \(f_{\bar z}=\mu(z) f_z\), where \(\mu\) is a measurable complex-valued function such that \(|\mu|<1\) almost everywhere in some domain \(D\) in the complex plane. A map \(f: D\to\mathbb C\) is called a regular solution of the Beltrami equation if it is continuous, discrete and open, is locally in the Sobolev class \(W^{1,1}\), satisfies the equation a.e., and has Jacobian determinant that is nonzero a.e. The authors give several sufficient conditions for the existence of regular solutions with prescribed boundary values in a Jordan domain \(D\subset \mathbb C\). To state one of them, let \(\|K_\mu\|_1(a,r)\) denote the average of the dilatation \(K_\mu=(1+|\mu|)/(1-|\mu|)\) over the circle of radius \(r\) centered at \(a\). Suppose that \(K_\mu\) is integrable in \(D\) and the improper integral \(\int_0 \frac{dr}{\|K_\mu(a,r)\|_1(r)}\) diverges for all \(a\in \overline{D}\). Theorem 2 of the present article asserts that under these conditions, for any continuous function \(\varphi : \partial D\to\mathbb R\), the Beltrami equation has a regular solution \(f\) such that \(\lim_{z\to \zeta} \mathrm{Re}\, f(z)=\varphi(z)\) for all \(\zeta\in \partial D\).