Noncoercive convection–diffusion elliptic problems with Neumann boundary conditions

Abstract

We study the existence and uniqueness of solutions of the convective-diffusive elliptic equation −div(D∇u) + div(V u) = f posed in a bounded domain Ω ⊂ RN , with pure Neumann boundary conditions D∇u * n = (V * n) u on ∂Ω. Under the assumption that V ∈ Lp (Ω)N with p = N if N ≥ 3 (resp. p > 2 if N = 2), we prove that the problem has a solution u ∈ H 1 (Ω) if Ω f dx = 0, and also that the kernel is generated by a function u ∈ H 1 (Ω), unique up to a multiplicative constant, which satisfies u > 0 a.e. on Ω. We also prove that the equation −div(D∇u) + div(V u) + ν u = f has a unique solution for all ν > 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 ∇v) − V * ∇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.