Gellerstedt problem with nonclassical matching conditions for the solution gradient on the type change line with data on internal characteristics

Abstract

We study the solvability of the Gellerstedt problem for the Lavrent’ev–Bitsadze equation. An initial function is posed in the ellipticity domain of the equation on the boundary of the unit half-circle with center the origin. Zero conditions are posed on characteristics in the hyperbolicity domain of the equation. “Frankl-type conditions” are posed on the type change line of the equation. We show that the problem is either conditionally solvable or uniquely solvable. We obtain a closed-form solvability condition in the case of conditional solvability. We derive integral representations of the solution in all cases.