The existence theorem of a generalized solution for the Dirichlet problem

Spaces of functions with derivatives of, called Sobolev spaces, occupy an important place in modern analysis, for example, in the theory of partial differential equations. Since the 30s of the last century, these functional classes have been intensively studied, and by now many of the problems associated with them have been solved. Sobolev spaces are a convenient and natural mathematical apparatus in the theory of Uch. They are widely used, for example, in the theory of boundary value problems for elliptic equations, in particular for the Poisson equation.