On a fundamental variational lemma for extremal quasiconformal mapping (Q1092200)

This paper deals with a classical extremal problem concerning quasiconformal self-mappings of the unit disk D. Let \(\sigma\) be a closed set on the boundary \(\partial D\) of D containing at least four points, and let E be a measurable set in D such that \(D\setminus E\) has positive area. Let furthermore \(h: \partial D\to \partial D\) be a quasisymmetric mapping and \(b: E\to {\mathbb{R}}^ a \)measurable non-negative function with \( \sup_{w\in E}b(w)<1\). The author considers the class \(Q=Q(h,\sigma,E,b)\) of all quasiconformal mappings \(F: D\to D\) such that \(F|_{\sigma}=h|_{\sigma}\) and \(| \kappa_ F(w)| \leq b(w)\) a.e. in E. Here \(\kappa_ F\) is the complex dilatation of F. The problem considered here is to find a mapping in the class Q having the smallest maximal dilatation. A necessary and sufficient criteria for extremality is given in the case \(b(w)=0\) for all \(w\in E\). The proof is based on an interesting variational lemma which solves also the general case, where the function b does not vanish. This has been shown by \textit{K. Sakan} [J. Math. Kyoto Univ. 26, 31-37 (1986; Zbl 0604.30023)].