A class of Riemann boundary value problems with parametric unknown functions (Q2751174)

The author considers the following problem: Find a pair of functions \((\varphi(z), \psi(t))\), satisfying the boundary conditions: NEWLINE\[NEWLINEG_{j1} (t)\Phi^+(t) =G_{j2}(t) \Phi^-(t)+g_j (t)\Psi(t)+ h_j(t), \quad j=1,2,\;t\in L,NEWLINE\]NEWLINE where \(\Phi(z)\) is a piecewise holomorphic function with jump curve \(L\), \(\psi (t)\) is the unknown Hölder continuous function on \(L\) and \(G_{j1}(t)\), \(g_j (t)\) \(h_j(t)\) are Hölder continuous functions given on \(L\). The above problem is called normal if \(G_1(t)=G_{1j} (t)-G_{21} (t)g_1(t)\neq 0\), \(G_2(t)= G_{12} (t)g_1(t)- G_{22}(t)g_2(t)\neq 0\) and nonnormal otherwise. Using the results from the well known Riemann boundary value problem, the author gives description for solving this problem in multiply connected domains in the normal and the non-normal cases as well, when \(1/G_1(t)\) or \(1/G_2(t)\) admits zero points on \(L\).NEWLINENEWLINEFor the entire collection see [Zbl 0966.00026].