Discontinuous Riemann boundary problem in weighted spaces

V. G. Petrosyan

Proceedings of the Yerevan State University. Physical and mathematical sciences, Tome 51 (2017) no. 1, pp. 38-41

Résumé

The Riemann boundary problem in weighted spaces $L^{1}(\rho)$ on $T=\{t, |t|=1\}$, where $\rho(t)={|t-t_{0}|}^{\alpha}$,   $ t_{0}\in T$ and $\alpha>-1$, is investigated. The problem is to find analytic functions $\Phi^{+}(z)$ and $\Phi^{-}(z),\,\,\Phi^{-}(\infty)=0$ defined on the interior and exterior domains of $T$ respectively, such that:  $ \lim\limits_{r\rightarrow 1-0}\|\Phi^{+}(rt)-a(t)\Phi^{-}(r^{-1}t)-f(t)\|_{L^{1}(\rho)}=0, $ where $f\in L^{1}(\rho),\,\,a(t)\in H_{0}(T;t_{1},t_{2},\dots,t_{m})$. The article gives necessary and sufficient conditions for solvability of the problem and with explicit form of the solutions.

Détail

Keywords: Riemann boundary problem, weighted spaces, Cauchy type integral, Hölder classes.