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Proceedings of the Yerevan State University, series Physical and Mathematical Sciences, 2017, Volume 51, Issue 3, Pages 250–254
(Mi uzeru418)
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Mathematics
Dirichlet boundary value problem in the weighted spaces $L^{1}(\rho)$
V. G. Petrosyan Institute of Mathematics, National Academy of Sciences of Armenia, Yerevan
Abstract:
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.
Received: 05.06.2017 Accepted: 12.07.2017
Citation:
V. G. Petrosyan, “Dirichlet boundary value problem in the weighted spaces $L^{1}(\rho)$”, Proceedings of the YSU, Physical and Mathematical Sciences, 51:3 (2017), 250–254
Linking options:
https://www.mathnet.ru/eng/uzeru418 https://www.mathnet.ru/eng/uzeru/v51/i3/p250
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Abstract page: | 84 | Full-text PDF : | 21 | References: | 18 |
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