Generalizations of certain Beurling's theorem with applications for inverse boundary value problem
In the present paper a nonlinear boundary value problem for conformal mappings is studied. Let \(\Phi = \Phi (w)\) be positive and continuous, where \(\Phi\) may tend to 0 or \(\infty\) in the points \(w = 0\) or \(w = \infty\). The problem is to determine a univalent conformal mapping \(w = w(z)\) of the unit disk with Riemannian normalization \(w(0) = 0\), \(w'(0) > 0\), which satisfies (1) \(|w' (e^{it}) |= \Phi (w(e^{it}))\) for all \(t \in [0,2 \pi]\). In the first part of the paper the author proves some existence- and uniqueness results by studying extremal problems in suitable classes of functions. In the second part of the paper the above problem will be studied for more general \(\Phi = \Phi (w)\). This generalization can be expressed by the presence of isolated singularities of \(\ln \Phi (w)\). Then two effects appear: 1. In opposite to continuous functions \(\ln \Phi (w)\) the solution does not exist everywhere. 2. The solution is in general not univalent. Example: \(\Phi (w) = |(w + 1) \ln (w + 1) |\), \(w = w (\tau) = e^{Cz} - 1\) are satisfying (1) for \(C > 0\), \(w\) is univalent for \(C < \pi\). Some existence- and non-existence results will be proved.
scientific article; zbMATH DE number 794589
scientific article