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On the theory of nonlocal boundary value problems. (English. Russian original) Zbl 0586.30036

Sov. Math., Dokl. 30, 8-10 (1984); translation from Dokl. Akad. Nauk SSSR 277, 17-19 (1984).
Two nonlocal boundary value problems are considered. (1) Find a function \(u\in C^{0,\alpha}(\bar D)\) harmonic in the unit disk D such that \(u(t)-u(\delta t)=f(t)\), \(t=e^{i\phi}\), \(0\leq \phi \leq 2\pi\), where \(\delta\in (0,1)\) and \(f: \delta\) \(D\to {\mathbb{R}}\) are given. (2) Find a similar function u biharmonic in D with an additional condition \(du/dv=f_ 1(t)\) on \(\partial D\); v being the outer normal. For (1) the condition is \(\int f(t)d\phi =0\) and for (2) \[ \int f(t)d\phi =(1- \delta^ 2)/2\int f_ 1(t)d\phi. \] The solutions are given explicitely.
Reviewer: O.Martio

MSC:

30E25 Boundary value problems in the complex plane
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