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A generalized Carleman boundary value problem
G. S. Litvinchuk, A. P. Nechaev
Abstract:
In a finite simply connected domain $D^+$ with a Lyapunov boundary $L$ there is considered the following boundary value problem: to find a function $\Phi^+(z)$ analytic in $D^+$ and $H$-continuous in $D^++L$, according to the boundary condition
\begin{equation}
\Phi^+[\alpha(t)]=a(t)\Phi^+(t)+b(t)\overline{\Phi^+(t)}+h(t),
\end{equation}
where $\alpha(t)$ homeomorphically maps $L$ on itself with the preservation $(\alpha=\alpha_+(t))$ or with the change $(\alpha=\alpha_-(t))$ of the direction of the circuit on $L$; $\alpha[\alpha(t)]\equiv t$; $\alpha'(t)\ne0$, $\alpha'(t)\in H(L)$; the functions $a(t),b(t),h(t)\in H(L)$ satisfy the identities
\begin{gather*}
a(t)a[\alpha(t)]+b(t)\overline{b[\alpha(t)]}=1,\\
a(t)b[\alpha(t)]+\overline{a[\alpha(t)]}b(t)=0,\\
a(t)h[\alpha(t)]+b(t)\overline{h[\alpha(t)]}+h(t)=0.
\end{gather*}
The Noether theory of problem (1) is constructed, its index is calculated and theorems of its solvability and stability are proved. An investigation of the problem in the case when $\alpha=\alpha_-(t)$ and $|a(t)|>|b(t)|$ is presented. From it there follows when $b(t)\equiv 0$ the known solvability theory of the Carleman problem.
Bibliography: 10 titles.
Received: 28.10.1968
Citation:
G. S. Litvinchuk, A. P. Nechaev, “A generalized Carleman boundary value problem”, Math. USSR-Sb., 11:1 (1970), 25–45
Linking options:
https://www.mathnet.ru/eng/sm3433https://doi.org/10.1070/SM1970v011n01ABEH002061 https://www.mathnet.ru/eng/sm/v124/i1/p30
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