REGULAR SELF-ADJOINT PROBLEMS FOR THE LAPLACE EQUATION

Abstract

The construction of solutions of regular boundary value problems for the Laplace equation has great theoretical and applied significance. Therefore, it is an urgent problem, and numerous studies have been devoted to this problem. Unlike other researches, in the work of T.Sh. Kalmenov, when performing a priori estimates for solving correctly solvable problems, using Riesz's theorem for a Hilbert space with a scalar product with a parameter, the potential of a simple layer was transformed into an integral operator over the domain, depending on the right-hand side. Using this, integral representations of solutions of coercively solvable problems were constructed, including criteria for their boundary value. In this paper, Green's functions of self-adjoint problems are explicitly obtained through the fundamental solution of the Laplace equation.