On the non-uniqueness of the solution of the inner Neumann–Gellerstedt problem for the Lavrent'ev–Bitsadze equation

It is known that the Gellerstedt problem for the Lavrent'ev–Bitsadze equation under classical conditions of gluing a solution on the line of degeneracy of the equation has only trivial solutions. In particular, in the monograph A. V. Bitsadze “Some classes of partial differential equations” such kind of problem was investigated by the method of reduction to singular integral equations. In the works of T. E. Moiseyev, it was shown for the first time that the homogeneous problem of Gellerstedt with data on external characteristics has a nontrivial solution under the Frankl-type gluing condition for the solution on the line of degeneracy of the equation. In this paper we consider the homogeneous Neumann–Gellerstedt problem with data on internal characteristics. It is proved that this problem has a nontrivial solution under the Frankl-type gluing conditions for the solution on the line of degeneracy of the equation.