Extremal problems on conformal moduli and estimates for harmonic measures (Q1128157)

The following general problem is considered: ``Let \(D\) be a domain of \(\overline{\mathbb{C}}\) and let \(z\in D\). If \(E\) is a Borel subset in \(\overline D\), the harmonic measure \(\omega_E(z)= \omega(z,E,D)\) will mean the harmonic measure of \(E\cap\partial(D\setminus E)\) with respect to the component of \(D\setminus E\) containing \(z\). Let \(\varepsilon\) be some family of closed sets \(E\) in \(\overline D\). Let \(D\), \(z\in D\), and \(\varepsilon\) be given. The problem is to compute (1) \(\inf\omega_E(z)\) or (2) \(\sup\omega_E(z)\) under the condition \(E\in\varepsilon\) and to find all extremizers of (1) (or (2)) in \(\varepsilon''\). The special versions of the above problem were investigated by many authors. Appeared also a lot of news and open problems connected with the above general problem. The author searched out some of them and obtained a lot of interesting results by using the extremal metric method combined with certain kinds of symmetrizations.