On the theory of convergence and compactness for Beltrami equations (Q765388)

The present paper is devoted to the study of the Beltrami equation \(f_{\overline{z}}=\mu(z)f_z,\) where \(\mu(z):D\rightarrow {\mathbb C}\) is a measurable function with \(|\mu(z)|<1\) almost everywhere, \(D\) is a domain in the complex plane \({\mathbb C}\), and \[ f_{\overline{z}}=(f_x+if_y)/2, \] \[ f_z=(f_x-if_y)/2, z=x+iy. \] Set \[ K_{\mu}(z)=\frac{1+|\mu(z)|}{1-|\mu(z)|}. \] Let \(\Omega\) be an open set in \({\mathbb C}\), let \(M(\Omega)\) be a certain locally finite set function, and let \[ \Phi:\overline{{\mathbb R}^{\,+}}\rightarrow \overline{{\mathbb R}^{\,+}} \] be a continuous strictly convex function; set \(dS(z)=\frac{dm(z)}{\left(1+|z|^2\right)^2}\). One of the main results of the paper states that the class of regular solutions of the Beltrami equation with constraints of the type \[ \int\limits_{\Omega}\Phi(K_{\mu}(z))dS(z)\leq M(\Omega) \] is closed in the space of homeomorphisms with respect to locally uniform convergence. Denote by \(\mathfrak{F}^M_{\Phi, \Delta}\) the class of regular solutions \(f:D\rightarrow \overline{{\mathbb C}}\) of the Beltrami equationsatisfying \(h(\overline{\mathbb C}\setminus f(D))\geq \Delta\) and \[ \int\limits_{\Omega}\Phi(K_{\mu}(z))dS(z)\leq M(\Omega) \] for any open set \(\Omega\) in \(D,\) where \(h(E)\) denotes the spherical diameter of \(E\). Another result states that \(\mathfrak{F}^M_{\Phi, \Delta}\) forms a normal family of mappings provided that \[ \int\limits_{\delta}^{\infty}\frac{d\tau}{\tau\Phi^{\,-1}(\tau)}=\infty \] for some \(\delta>\Phi(0),\) where \(\Phi\) is convex and non-decreasing, and \[ \Phi^{\,-1}(\tau)=\inf\limits_{\Phi(t)\geq \tau} t. \] Finally it is shown that \(\mathfrak{F}^M_{\Phi, \Delta}\) is compact if, in addition, \(\Phi\) is continuous and strongly convex and \(M(\Omega)\) is bounded.