Geodesic
Dirichlet boundary value problem in the weighted spaces $L^{1}(\rho)$
Proceedings of the Yerevan State University. Physical and mathematical sciences, Tome 51 (2017) no. 3, pp. 250-254

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The Dirichlet boundary value problem in the weighted spaces $L^{1}(\rho)$ on the unit circle $T=\{z: |z|=1\}$ is investigated, where $\rho(t)={|t-t_{k}|}^{\alpha_{k}}$,  $k=1,\dots,m$, $t_{k}\in T$ and $\alpha_{k}$ are arbitrary real numbers. The problem is to determine a function $\Phi(z)$ analytic in unit disc such that: $\displaystyle\lim_{r\to 1-0}\|Re\Phi(rt)-f(t)\|_{L^{1}(\rho_{r})}=0,$ where $f\in L^{1}(\rho)$. 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. 
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     title = {Dirichlet boundary value problem in the weighted spaces $L^{1}(\rho)$},
     journal = {Proceedings of the Yerevan State University. Physical and mathematical sciences},
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     publisher = {mathdoc},
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     language = {en},
     url = {http://geodesic.mathdoc.fr/item/UZERU_2017_51_3_a6/}
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V. G. Petrosyan. Dirichlet boundary value problem in the weighted spaces $L^{1}(\rho)$. Proceedings of the Yerevan State University. Physical and mathematical sciences, Tome 51 (2017) no. 3, pp. 250-254. http://geodesic.mathdoc.fr/item/UZERU_2017_51_3_a6/