On a Boundary Value Problem for a Third Order Elliptic-hyperbolic Equation with Superposition Operators of the First and Second Orders in a Rectangular Domain

This article proposes a method for solving a Dirichlet typeproblem for a third-order equation of elliptic-hyperbolic typewith a superposition operators of first and second orders in arectangular domain. It is shown that the correctness of theformulated problem significantly depends on the ratio of the sidesof the rectangle from the hyperbolic part of the mixed domain. Anexample is given in which the well posed problem with homogeneousconditions has a nontrivial solution. A solution to the problem isconstructed in the form of a sum of a series of eigenfunctions ofthe corresponding one-dimensional spectral problem. A criterionfor the uniqueness of a solution is established. When justifyingthe uniform convergence of a series, the problem of smalldenominators arises. In this connection, estimates of smalldenominators about the distance from zero with the correspondingasymptotic have been established. These estimates made it possibleto prove the convergence of the series in the class of regularsolutions to this equation. Estimates on the stability of thesolution from given boundary functions are proved.