Remarks on Hilbert generalized problem (Q5929625)

Let \(\Gamma\) be a simple closed curve contained in the ring \(D=\{z: r<| z| <R\}\) and divided this ring on the domains \(D^+\) and \(D^-\) such that \(\partial D^+=\Gamma_1\cup \Gamma\), where \(\Gamma_1=\{z: | z| =r\}\) and \(D^-=D\backslash D^+\). The authors consider the generalized Hilbert problem for two functions \(\varphi (z)\) and \(\psi (z)\), analytic in \(D^+\) and \(D^-\) respectively, under the following boundary conditions: \[ a(z)\varphi (z)=\psi (z)+f(z),\;z\in \Gamma; \] \[ \varphi (\gamma_1z) =\varphi(\gamma_1\overline{z}),\;z\in \Gamma_1;\;\;\psi (\gamma_2z) =\psi(\gamma_2\overline{z}),\;z\in \Gamma_2. \] Here \(\gamma_j, j=1,2\) are constants such that \(| \gamma_1| =| \gamma_2| =1\) and \(a(z)\neq 0, z\in \Gamma\). They study the relations between classical and generalized Hilbert problem and give an effective method of resolution of the problem above. The application to the Poincaré problem for the Laplace equation is also given.