On Quasilinear Beltrami-Type Equations with Degeneration

We consider the solvability problem for the equation $f_{\overline{z}}=\nu(z,f(z)) f_z$, where the function $\nu(z,w)$ of two variables can be close to unity. Such equations are called quasilinear Beltrami-type equations with ellipticity degeneration. We prove that, under some rather general conditions on $\nu(z,w)$, the above equation has a regular homeomorphic solution in the Sobolev class $W_{\operatorname{loc}}^{1,1}$. Moreover, such solutions $f$ satisfy the inclusion $f^{\,-1}\in W_{\operatorname{loc}}^{1,2}$.