On extremal quasiconformal mappings with varying dilatation bounds (Q1820272)

The authors consider solutions to a general type of extremal problem introduced by \textit{E. Reich} in [Arch. Ration. Mech. Anal. 68, 99-112 (1978; Zbl 0421.30013)] for quasiconformal selfmappings of the closed unit disk with prescribed values on a closed subset \(\sigma\) of the boundary of the unit disk. For the given extremal problem a subset E of D is given with certain regularity properties. Also, a positive real-valued function b(w) defined on E is given with \(\| b\|_{\infty}<1\). One considers the set \(Q=Q(h,\sigma,E,b)\) of all quasiconformal mappings F from D onto D with \(F| \sigma =h| \sigma\) and complex dilatation \(K_ F\) satisfying \[ | K_ F(w)| \leq b(w)\text{ almost everywhere on }E. \] An element F in \(Q(h,\sigma,E,b)\) is extremal if \( \sup_{w\in D-E}| K_ F(w)|\) is minimal among all elements in \(Q(h,\sigma,E,b)\). In certain instances the authors have already shown that there exists a unique extremal in \(Q(h,\sigma,E,b)\) [\textit{R. Fehlmann}, Ann. Acad. Sci. Fenn. Ser. A I Math. 7, 337-347 (1982; Zbl 0487.30008); Comment. Math. Helv. 61, 565-580 (1986); the reviewer, Mich. Math. J. 29, 237-242 (1982; Zbl 0476.30015) and \textit{K. Sakan}, J. Math. Kyoto Univ. 26, 31-37 (1986; Zbl 0604.30023) and loc. cit. above]. In this paper, the authors show how the extremal value in \(Q(h,\sigma,E,b)\) and the extremal dilatation coefficient vary with respect to the function b under certain regularity conditions.