Some properties of the solutions of the Dirichlet problem for a second-order elliptic equation

The paper is devoted to the identification of the properties of the solution of the Dirichlet problem for a second-order elliptic equation with boundary function in $L_2$ that characterize its behaviour near the boundary of the domain under consideration. In particular, we study the behaviour of the integrals of the derivatives of the solution with respect to measures concentrated to a considerable extent on sets of various dimensions approaching the boundary. The corresponding description is given in terms of special function spaces that reflect the interior regularity of the solution and some of its integral properties. The results obtained are applied to the study of the Fredholm property for a wide class of non-local problems, in which the boundary values of a solution are related to its values and the values of its derivatives at interior points.