A nonlocal boundary value problem for the Lavrent'ev-Bitsadze equation (Q1930832)

In this short note, the authors consider the following nonlocal boundary value problem of mixed type: \[ u_{xx} +\text{sign}\, y\, u_{yy}=0\qquad \text{in}\quad \Omega_+\cap\Omega_-, \] where \(\Omega_+:=(0,\pi )\times (0,\infty )\) and \(\Omega_- =(0,\pi )\times (-T,0)\), or \(\Omega_+:=(0,\pi )\times (0,T_+)\) and \(\Omega_- =(0,\pi )\times (0,T_-)\) (\(T_-<T_+\)), together with nonlocal boundary condition \(\alpha u(x,0)-u(x,T)=f(x)\), or \(\alpha u(x,0)-u(x,T_-)=f(x)\) (\(x\in [0,\pi ],\; \alpha\in I\!\! R\)), coupled with other classical boundary conditions. For certain values of \(\alpha\), the authors state some solvability results on this initial boundary value problem of mixed type. It is also shown that for some other values of \(\alpha\), the problem is uniquely solvable if and only if a countable set of orthogonality conditions is imposed on the right hand side of the nonlocal condition.