Existence of boundary values for solutions of degenerate elliptic equations

The behaviour near the boundary of the solution of a second-order elliptic equation degenerate at some part of the boundary is discussed. The case is considered when the quadratic form corresponding to the principal part of the differential operator vanishes at the (unit) normal vector to the boundary and the setting of the first boundary-value problem (problem D or problem E) depends on the values of the coefficients of the first derivatives (Keldysh-type degeneracy). Conditions on the solution of the equation necessary and sufficient for the existence of its limit on the part of the boundary on which one sets boundary values in the first boundary-value problem are found. A solution satisfying these conditions proves to have limit also at the remaining part of the boundary. In addition, a closely related problem on the unique solubility of the corresponding boundary-value problem with boundary functions in $L_p$ is studied.