SOME ASYMPTOTIC PROPERTIES OF THE SOLUTIONS OF LAPLACE EQUATIONS IN A UNIT DISK

The authors consider the optimization problem related to the integral representation of the deviation of positive linear operators on the classes of (psi, beta) -differentiable functions in the integral metric. The Poisson integral, which is the solution of the Laplace equation in polar coordinates with the corresponding initial conditions given on the boundary of the unit disk, is taken as a positive linear operator. The Poisson integral is an operator with a delta-like kernel; therefore, it is the best apparatus for solving many problems of applied mathematics, namely: methods of optimization and variational calculus, mathematical control theory, theory of dynamical systems and game problems of dynamics, applied nonlinear analysis and moving objects search. The classes of (psi, beta) -differentiable functions on which the asymptotic properties of the solutions to Laplace equations in a unit disk are analyzed are generalizations of the Sobolev, Weyl-Nagy, etc. classes well-known in optimization problems. The problem solved in the article will make it possible to generate high-quality mathematical models of many natural and social processes.