Problem with Shift on Internal Characteristics and an Analogue of the Frankl Condition for the Gellerstedt Equation with a Singular Coefficient

In this paper, for a degenerate equation of elliptic-hyperbolic type with a singular coefficient in a certain mixed domain, the correctness of the JN problem (Zhegalov, Nakhushev) with displacement conditions on the internal characteristics and an additional condition of the Frankl condition type on a segment of the degeneracy line is investigated. In proving the uniqueness of the solution to the problem, the extreme principle of Bitsadze is used. Proof of the existence of a solution to the problem by virtue of integral representations of solutions to the Dirichlet problem and the modified Cauchy problem, is reduced to solving a non-standard singular integral Tricomi equation, an algorithm for regularizing this equation to a Fredholm integral equation of the second kind is developed, the unique solvability of which follows from the uniqueness of the solution to the problem.

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