A~generalized Carleman boundary value problem

G. S. Litvinchuk ; A. P. Nechaev

Sbornik. Mathematics, Tome 11 (1970) no. 1, pp. 25-45

Résumé

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.