On boundary values of solutions of the dirichlet problem for second order elliptic equation

Abstract

The paper suggests some conditions on the lower order terms, which provide that the solution of the Dirichlet problem for the general elliptic equation of the second order $$ \begin{gathered} - \sum\limits_{i,j = 1}^n {\left( {a_{i j} \left( x \right)u_{x_i } } \right)_{x_j } + } \sum\limits_{i = 1}^n {b_i \left( x \right)u_{x_i } - } \sum\limits_{i = 1}^n {\left( {c_i \left( x \right)u} \right)_{x_i } + d\left( x \right)u = f\left( x \right) - divF\left( x \right), x \in Q,} \hfill \\ \left. u \right|_{\partial Q} = u_0 \in L_2 \left( {\partial Q} \right) \hfill \\ \end{gathered} $$ , possesses the property of (n − 1)-dimensional continuity, i.e. is from A. K. Gushchin’s space Cn−1(\( \bar Q \)), and the boundary function u0 is the L2-limit of the solution traces on some surfaces from a rather wide class, which are not necessarily “parallel” to the boundary.