On an area problem in conformal mapping
An area problem in a conformal mapping is considered. The author presents this problem in a simpler version in the form: let K denote the class of quadrilaterals Q with a fixed module \(m(Q)=m\). That is, each quadrilateral \(Q\in K\) is a Jordan region G with four distinguished boundary points \(z_ 1,z_ 2,z_ 3,z_ 4\) and G can be mapped conformally onto a rectangle with vertices 0, a, \(a+ib\), ib corresponding to the \(z_ 1,z_ 2,z_ 3,z_ 4\), and \(a/b=m\). With a suitable normalization and restrictions of the geometry of Q, we ask: For which \(Q\in K\) is the area F of G minimal ? The results obtained for the above problem can be applied to the following area problem: ''Let \(G_ z\) be a ring domain whose interior (or exterior) boundary component is given. We are looking for an exterior (or interior) boundary component \(\Gamma_ z\) such that (i) the ring domain \(G_ z\) has a given module M, i.e. can be mapped conformally onto \(\{\) \(w: 1<| w| <M\}\), (ii) among all such ring domains, the area of \(G_ z\) is minimal''. Results of numerical experiments are given.
scientific article; zbMATH DE number 3989706