The boundary values of solutions of an elliptic equation

The paper is devoted to the study of the boundary behaviour of solutions of a second-order elliptic equation. Criteria are established for the existence of a boundary value of a solution of the homogeneous equation under the same conditions on the coefficients of the equation as were used to establish that the Dirichlet problem with a boundary function in $L_p$, $p>1$, has a unique solution. In particular, an analogue of Riesz's well-known theorem (on the boundary values of an analytic function) is proved: if a family of norms in the space $L_p$ of the traces of a solution on surfaces ‘parallel’ to the boundary is bounded, then this family of traces converges in $L_p$. This means that the solution of the equation under consideration is a solution of the Dirichlet problem with a certain boundary value in $L_p$. Estimates of the nontangential maximal function and of an analogue of the Luzin area integral hold for such a solution, which make it possible to claim that the boundary value is taken in a substantially stronger sense.
Bibliography: 57 titles.