Tricomi-type boundary value problem in an unbounded domain

Vagapov V. Z.

The existence and uniqueness of a solution to a Tricomi-type boundary value problem for a single mixed-type equation in an unbounded domain in a special class is proved. In the region of ellipticity of the equation, the auxiliary Holmgren problem is solved by the Green's function method, and in the region of hyperbolicity of the equation, a problem of Darboux type. Moreover, to obtain a solution to the Darboux-type problem, first in the characteristic triangle by the method of the Riemann-Hadamard function, first proposed by Professor S.P. Pulkin, the Darboux problem was solved, and then in the constructed solution one of the characteristics was directed to infinity. By sewing together the solutions of these auxiliary problems with respect to the function and the normal derivative on the line of change of the equation type, the Tricomi-type problem is reduced to an equivalent singular integral equation, which is uniquely reduced by the Carleman-Vekua method to a Fredholm integral equation of the second kind, the unconditional solvability of which follows from the corresponding uniqueness theorem for the solution . The uniqueness of the solution of the problem under study is proved using the local extremum principle. Note that the formulation of this problem is based on the well-known works of M.S. Keldysh, I.L. Karolya.

Keywords: tricomi-type problem; unlimited area; Holmgren's problem; Darboux problem; singular integral equation; Carleman-Vekua method.