Geometric properties of solutions of exterior inverse boundary value problem
The list of references shows that one of the main interests of mathematicians of Kazan university working on the field of function theory is the main exterior inverse boundary value problem for analytic functions. This is the following problem: one has to determine a \((n+1)\)- connected exterior domain \(D_ z\) of Smirnov's class in the closed complex plane and an analytic function \(w=w(z)\) in \(D_ z\) satisfying \((n+1)\) boundary conditions \[ w(z)| _{L_ k}= w_ k (s/l_ k), \] where \(L_ k\), \(k=0,\dots, n\), are closed rectifiable Jordan curves forming the boundary of \(D_ z\). The lengths of curves \(L_ k\) are denoted by \(l_ k\). The given functions \(w_ k= w_ k(\tau)\), \(\tau\in [0,1]\), are absolutely continuous and satisfy \(0< | w_ k (\tau) |\leq M<\infty\). Moreover, the functions \(w_ k= w_ k (\tau)\) fulfil conditions providing existence of schlicht image \(D_ w\) of \(D_ z\) under conformal mapping \(w= w(z)\). The point \(w_ 0\in D_ w\) which is not fixed is the image of infinity under \(w= w(z)\). The points \(w_ 0\) are determined by so-called Gakhov's equation. The author proves: The condition \(\gamma (w_ 0) =1\) is necessary that the corresponding solution of the inverse problem is schlicht. Here \(\gamma (w_ 0)\) denotes a suitable winding number. From this result one concludes that if the starting problem has more than one solution, then even one of these solutions is not schlicht. Examples show that \(\gamma (w_ 0) =1\) is not a sufficient condition.
scientific article; zbMATH DE number 536618