## 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:

 3e+26 Boundary value problems in the complex plane

### Keywords:

nonlocal boundary value problems