Poincare–Tricomi problem for the equation of a mixed elliptico-hyperbolic type of second kind

It is known, that if the partial differential equation of the second order belongs to the elliptic type in one part of the domain and to the hyperbolic type in the other part, then such equation is called an equation of mixed type; both parts of the domain are separated by a transition line on which the equation either degenerates into parabolic or is not defined. Equations of mixed elliptic-hyperbolic type are divided into equations of the first and second kind. For equations of the first kind, the line of parabolic degeneracy is the return point of the family of characteristics of the corresponding hyperbolic equation. The equation whose degeneration line is simultaneously the envelope of a family of characteristics, i.e. is itself a characteristic, is an equation of the second kind. Therefore, equations of a mixed type of the second kind in all respects are relatively little studied. For example, the Poincare–Tricomi problem and its various generalizations for equations of the first kind have long been studied. For equations of mixed type of the second kind, depending on the degree of degeneracy, the limiting values of the desired solution and its derivative on the line of change of the type of equation can have singularities. To ensure the necessary smoothness of the desired solution outside the line of characteristic degeneracy, one has to require increased smoothness of the given limit functions. In order to weaken this requirement, in the present paper we introduce a class of generalized solutions. In the hyperbolic part of the mixed domain, we seek a generalized solution; in the elliptic part, a regular solution. This paper is devoted to the study of the Poincare–Tricomi problem for one equation of the mixed elliptic-hyperbolic type of the second kind. The conditions under which the problem has a unique solution are identified.
In the elliptic part of the mixed domain, a classical solution is sought and a similar second functional relationship brought from the ellipticity domain of the equation is derived. Then, after exclusion of one of the two unknown functions from these two functional relationships, the solution of the posed problem is reduced to solving a singular integral equation for the limit value of the sought function on the line separating the types of the equation. Under certain restrictions on the given functions and parameters of the Poincare–Tricomi type problem, this singular integral equation can be reduced to a Fredholm integral equation of the second kind by the Carleman method. The unique solvability of this equation follows from the Fredholm alternative and uniqueness theorem for the posed problem.