Certain types of Bitsadze-Samarskij problems for analytic functions (Q1916567)

Let \(D\) be a bounded domain in \(C\) which boundary consists of two smooth arcs \(\Gamma_1\) and \(\Gamma_2\) with common endpoints \(\tau_1\), \(\tau_2\). Let \(\Gamma_0\) be the image of the arc \(\Gamma_1\) under the diffeomorphism \(\alpha\) such that \(\alpha(\tau_1)= \tau_1\), \(\alpha(\tau_2)= \tau_2\) and \(\Gamma_0\) should be contained in \(D\). The problem is to study the following boundary value problem of Bitzadze-Samarski type: To determine an analytic function \(\Phi= \Phi(z)\) in \(D\) satisfying the boundary conditions \hskip17mm \(\text{Re}(G_1(t) \Phi^+(t)- G_0(t) \Phi(\alpha(t)))= \phi_1(t),\;t\in \Gamma_1\), \hskip17mm \(\text{Re}(G_2(t) \Phi^+(t))= \phi_2(t),\;t\in \Gamma_2\), where \(G_j= G_j(t)\neq 0\) for \(t\in \Gamma_j\), \(j= 1,2\). Then this problem is called of normal type. In this case one defines two auxiliary entire functions \(Y_j= Y_j(\zeta)\) which can be used to characterize the properties of our starting problem. The author studies: 1. the Fredholm property, 2) the index of the problem, 3) the solvability by the aid of the adjoint problem, 4) the zeros of \(Y_j\) to calculate the index of the problem.