Dirichlet problem for mixed type equation with characteristic degeneration

For a mixed elliptic-hyperbolic type equation with characteristic degeneration, the first boundary value problem in a rectangular region is investigated. The criterion for the uniqueness of the solution of the problem is established. Earlier, in proving the uniqueness of solutions of boundary value problems for equations of mixed type, the extremum principle or the method of integral identities was used. The uniqueness of the solution to this problem is established on the basis of the completeness of the system of eigenfunctions of the corresponding one-dimensional spectral problem. The solution of the problem is constructed as a sum of a series in the system of eigenfunctions. When we proved the convergence of the obtained series, the problem of small denominators of a more complicated structure than in other known works arose. These denominators contain a parameter depending on the lengths of the sides of the rectangle in the hyperbolic part of the domain and the exponent of the degree of degeneration. In this connection, estimates are established about separation from zero with the corresponding asymptotics, in cases where this parameter is a natural, rational and algebraic irrational number of degree two. If this parameter is not an algebraic irrational number of degree two, then the solution of the problem as a sum of a series does not exist. Using the obtained estimates, the uniform convergence of the constructed series in the class of regular solutions is justified under certain sufficient conditions with respect to the boundary functions. The stability of the solution of the problem with respect to the boundary functions in the norms of the space of summable functions and in the space of continuous functions is also proved.

equation of mixed type with characteristic degeneration, Dirichlet problem, criterion of uniqueness, existence, small denominator