On the boundary behavior of regular solutions of the degenerate Beltrami equations (Q325978)

The behavior of mappings of finite distortion on curves in the plane is studied under pointwise type regularity conditions. A homeomorphism \(f:D \rightarrow \mathbb{C}\) is assumed to be a solution of the Beltrami equation in the class \(W^{1,1}_{\mathrm{loc}}\) such that the dilatation \(\mu\) of \(f\) satisfies \(|\mu| < 1\) a.e. The mapping \(f\), \(f(0) = 0\), is said to be asymptotically homogeneous at \(0\) if for all \(\zeta \in \mathbb{C}\), \[ (*) \qquad\lim_{z \rightarrow 0}\frac{f(z \zeta)}{f(z)} = \zeta, \] see [\textit{B. Bojarski} et al., Infinitesimal geometry of quasiconformal and bi-Lipschitz mappings in the plane. Zürich: European Mathematical Society (EMS) (2013; Zbl 1276.30002)]. If this holds uniformly in \(A \subset D\), with an obvious change in (*), then \(f\) is said to be asymptotically homogeneous on \(A\). Conditions related to asymptotical homogenity in terms of \(\mu\) are given at \(0\) or on a set \(A\). Using these results the author shows that if \(f : \mathbb{C} \rightarrow \mathbb{C}\) is asymptotically homogeneous on \(\partial B = \partial B(0,1)\) and \[ \limsup_{r \rightarrow 0}|B(x,r)|^{-1} \sup_{x \in \partial B} \int_{B(x,r)} K_f \, dm < \infty, \] then the curve \(f(\partial B)\) is assymptotically conformal. Moreover, if \(\Phi\) is a convex function satisfying an integral condition, then functions of the type \(\Phi(K_f)\) are allowed instead of \(K_f\) inside the above integral. Under similar conditions the smoothness of the curve \(f(\partial B)\) is also studied.