Ellipticheskie zadachi i integral'nye uravneniya v prostranstvakh razlichnoy gladkosti po peremennym

Abstract

We consider a model elliptic pseudodifferential equation and the simplest boundary value problems in a quadrant in a Sobolev–Slobodetsky space of different orders of smoothness in different variables. In the case of a special representation of the symbol, we describe a general solution of the equation and consider the simplest boundary value problem with the Dirichlet and Neumann conditions on the sides of the quadrant. This boundary value problem is reduced to a system of integral equations, which, under additional assumptions about the structure of the symbol, can also be reduced to a system of first-order difference equations