Boundary problem with discontinuous translation for two functions which are analytic in domains of different connectivity (Q752888)

Let \(\Gamma\) (\(\gamma\)) be closed Lyapunov curves which bound a finite double (simply) connected domain respectively and let \(\alpha: \Gamma\to \gamma\) be an orientation preserving piecewise-smooth translation. The author considers the following boundary problem for analytic functions \(\phi\) (\(\psi\)): \[ (1)\quad a(t)\psi (\alpha (t))+b(t)\overline{\psi (\alpha (t))}+c(t)\phi (t)+d(t)\overline{\phi (t)}=h(t),\quad t\in \Gamma, \] where a, b, c, d are discontinuous in some points. The author finds necessary and sufficient conditions for the problem (1) to be Noetherian in the space \(L_ p(\Gamma)\), \(1<p<\infty\). The formula for calculating the index of the problem (1) is also discussed.