Convergence theorem for Beltrami equations with integral type restrictions on the dilatation
The present paper is devoted to the study of properties of the well-known Beltrami equation \(f_{\overline z}=\mu(z)\cdot f_z\) having a degeneration. Set \(f_z=(f_x-if_y)/2\) and \(f_{\overline z}=(f_x+if_y)/2\), where \(z=x+iy\). Given a measurable function \(\mu(z):D\rightarrow {\mathbb C}\) such that \(| \mu(z)| <1\) almost everywhere, a maximal dilatation \(K_{\mu}(z)\) can be defined as \(K_{\mu}(z)=\frac{1+| \mu(z)| }{1-| \mu(z)| }\). A solution \(f\) of the Beltrami equation is said to be regular if its Jacobian \(J_f(z)=| f_z| ^2-| f_{\overline{z}}| ^2\) is bigger than \(0\) almost everywhere. The main result of the paper is the following. Suppose that a function of the set \(M_E\) is absolutely continuous and take sequence \(f_n\) of regular solutions of the Beltrami equation that satisfy the inequality \(\int\limits_{E}\Phi(K_{\mu_n}(z))dm(z)\leq M_E\) for some non-decreasing function \(\Phi:[0, +\infty]\rightarrow [0, +\infty]\) which is supposed to be continuous from the right at the point \(Q=\sup\limits_{\Phi(t)<\infty}t\) and satisfies \(\lim\limits_{t\rightarrow\infty}\frac{\Phi(t)}{t}=\infty\). Suppose that \(f_n\rightarrow f\) locally uniformly on compacts in \(D\). Then \(f\) is a regular solution of the same Beltrami equation and satisfies the inequality \(\int\limits_{E}\Phi(K_{\mu}(z))dm(z)\leq M_E\) for a dilatation \(K_{\mu}(z)\) of the limit mapping \(f\).
scientific article; zbMATH DE number 6062272