A variational approach in weighted Sobolev spaces to the operator −Δ+∂/∂x 1 in exterior domains of ℝ3

Abstract

Consider the Dirichlet problem −vΔu+k∂ 1 u = f withv, k>0 in ℝ3 or in an exterior domain of ℝ3 where the skew-symmetric differential operator −1=∂/∂x1 is a singular perturbation of the Laplacian. Because of the inhomogeneity of the fundamental solution we study existence, uniqueness and regularity in Sobolev spaces with anisotropic weights. In these spaces the operator ∂1 yields an additional positive definite term giving better results than in Sobolev spaces with radial weights. The elliptic equation −vΔu +k∂1 u=f can be taken as a model problem for the Oseen equations, a linearized form of the Navier-Stokes equations.