Solvability of the Dirichlet problem for a general second-order elliptic equation

The paper is concerned with studying the solvability of the Dirichlet problem for the second-order elliptic equation -div(A(x)del u) + ((b) over bar (x), del u) - div((c) over barc(x) u) + d(x) u = f(x) - div F(x), x is an element of Q, u vertical bar(partial derivative Q) = u(0), in a bounded domain Q subset of R-n, n >= 2, with C-1-smooth boundary and boundary condition u(0) is an element of L-2(partial derivative Q). Conditions for the existence of an (n -1)-dimensionally continuous solution are obtained, the resulting solvability condition is shown to be similar in form to the solvability condition in the conventional generalized setting (in W-2(1)(Q)). In particular, the problem is shown to have an (n - 1)-dimensionally continuous solution for all u(0) is an element of L-2(partial derivative Q) and all f and F from the appropriate function spaces, provided that the homogeneous problem (with zero boundary conditions and zero right-hand side) has no nonzero solutions in W-2(1) (Q).