Abstract:
In the two centuries that have passed since K. Gauss posed the Dirichlet problem for the Laplace equation (1828), many famous mathematicians have devoted their research to this problem and its various generalizations, and they have obtained a large number of interesting and important results that have already become classical. Nevertheless, the purpose of this report is to convince the listener that not everything is known in this "main" problem of mathematical physics and that this area of research should be given special attention.
It is well known that not every classical solution is generalized. The main content of this report is an extension of the concept of a solution to the Dirichlet problem: an increase in the set of boundary functions and a change in the definition of accepting a boundary condition, which include classical and generalized solutions. To describe the place of the discussed results among the known ones, it will be necessary to cite a number of well-known statements, discuss the main formulations of the Dirichlet problem, their advantages and disadvantages. The focus will be on the conditions of the problem data: the right side of the equation, the boundary function, the coefficients of the equation and the domain in which the problem is considered.