DIRICHLET BOUNDARY VALUE PROBLEM IN THE WEIGHTED SPACES $L^{1}(ρ)$ - Petrosyan, V.G. (Proceedings of the YSU A: Physical and Mathematical Sciences, 2017)
The Dirichlet boundary value problem in the weighted spaces $L^{1}(ρ)$ on the unit circle $T=\{z: |z|=1\}$ is investigated, where $ρ(t)={|t-t_{k}|}^{α_{k}}$,~~$k=1,…,m$, \lb $t_{k}\in T$ and $α_{k}$ are arbitrary real numbers. The problem is to determine a function $Φ(z)$ analytic in unit disc such that: $ \lim_{r→ 1-0}\|ReΦ(rt)-f(t)\|_{L^{1}(ρ_{r})}=0, $ where $f\in L^{1}(ρ)$. In the paper necessary and sufficient conditions for solvability of the problem are given and the general solution is written in the explicit form.
Keywords: Dirichlet problem, weighted spaces, Cauchy type integral.