Problem \(R\) for the Lavrent'ev--Bitsadze system in weighted Hardy classes (Q1873551)

The author considers boundary-value problems for the Lavrent'ev-Bitsadze equation \[ (\text{sgn} y)u_{xx} +u_{yy} = 0 \] which, being considered in a domain \(D\) such that \(D\cap \{y>0\}\neq\emptyset\) and \(D\cap \{y<0\}\neq\emptyset\), is a model example of a mixed type equation. The analysis of the problem is based on reducing it to a singular integral equation, the solution of which is thought in the weighted Hardy spaces. Under a general form of the boundary conditions the author obtains formulas for the Fredholm index of the problem and formulates necessary and sufficient conditions of solvability. It is shown that in the cases of standard boundary conditions, the analysis is simplified and the results are improved.