Construction by similarity method of the fundamental solution of the Dirichlet problem for Keldysh type equation in the half-space
For elliptic in the half-space and parabolic degenerating on the boundary equation of Keldysh type we construct by similarity method the self-similar solution, which is the approximation to the identity in the class of integrable functions. This solution is the fundamental solution of the Dirichlet problem, i.e. the solution of the Dirichlet problem with the Dirac delta-function in the boundary condition. Solution of the Dirichlet problem with an arbitrary function in the boundary condition can be written as the convolution of the function with the fundamental solution of the Dirichlet problem, if a convolution exists. For a bounded and piecewise continuous boundary function convolution exists and is written in the form of an integral, which gives the classical solution of the Dirichlet problem, and is a generalization of the Poisson integral for the Laplace equation. If the boundary function is a generalized function, the convolution is a generalized solution of the Dirichlet problem.