On solvability of Dirichlet problem for second order elliptic equation

The paper gives some solvability conditions of the Dirichlet problem for the second order elliptic equation $$ - div(A(x)nabla u) + (bar b(x),nabla u) - div(bar c(x)u) + d(x)u = f(x) - divF(x),x in Q,u|_{partial Q} = u_0 in L_2 (partial Q) $$ in bounded domain Q ? R _n (n ≥ 2) with smooth boundary ?Q ∈ C ₁. In particular, it is proved that if the homogeneous problem has only the trivial solution, then for any u ₀ ∈L ₂(?Q) and f, F from the corresponding functional spaces the solution of the non-homogeneous problem exists, from Gushchin’s space $ C_{n - 1} (bar Q) $ and the following inequality is true: $$ begin{gathered} left| u right|_{C_{n - 1} (bar Q)}^2 + mathop smallint limits_Q rleft| {nabla u} right|^2 dx leqslant hfill leqslant Cleft( {left| {u_0 } right|_{L_2 (partial Q)}^2 + mathop smallint limits_Q r^3 (1 + |ln r|)^{3/2} f^2 dx + mathop smallint limits_Q r(1 + |ln r|)^{3/2} |F|^2 dx} right) hfill end{gathered} $$ where r(x) is the distance from a point x ∈ Q to the boundary ?Q and the constant C does not depend on u ₀, f and F.