Noncoercive convection-diffusion elliptic problems with Neumann boundary conditions

We study the existence and uniqueness of solutions of the convective-diffusive elliptic equation -div(D del u) + div(V u) = f posed in a bounded domain Omega subset of R(N), with pure Neumann boundary conditions D del u . n = (V . n) u on partial derivative Omega. Under the assumption that V is an element of L(p)(Omega)(N) with p = N if N >= 3 (resp. p > 2 if N = 2), we prove that the problem has a solution u is an element of H(1)(Omega) if f(Omega) f dx = 0, and also that the kernel is generated by a function (u) over cap is an element of H(1)(Omega), unique up to a multiplicative constant, which satisfies (u) over cap > 0 a.e. on Omega. We also prove that the equation -div(D del u) + div(Vu) +v u = f has a unique solution for all v > 0 and the map f -> u is an isomorphism of the respective spaces. The study is made in parallel with the dual problem, with equation -div(D(T)del v) - V .del v = g. The dependence on the data is also examined, and we give applications to solutions of nonlinear elliptic PDE with measure data and to parabolic problems.