Gellerstedt Problem with a Nonlocal Oddness Boundary Condition for the Lavrent’ev–Bitsadze Equation

Abstract

We study the Gellerstedt problem for the Lavrent’ev–Bitsadze equation with the oddness boundary condition on the boundary of the ellipticity domain. All eigenvalues and eigenfunctions are obtained in closed form. It is proved that the system of eigenfunctions is complete in the elliptic part of the domain and incomplete in the entire domain. The unique solvability of the problem is also proved; the solution is written in the form of a series if the spectral parameter is not equal to an eigenvalue. For the spectral parameter coinciding with an eigenvalue, solvability conditions are obtained under which the family of solutions is found in the form of a series. A condition for the solvability of the problem depending on the eigenvalues is obtained. The constructed analytical solutions can be used efficiently in numerical modeling of transonic gas dynamics problems.