Dirichlet and Neumann problems for elliptic equations with singular drifts on Lipschitz domains

Abstract

We consider the Dirichlet and Neumann problems for second-order linear elliptic equations: \[ − △ u + div ⁡ ( u b ) = f and − △ v − b ⋅ ∇ v = g -\triangle u +\operatorname {div}(u\mathbf {b}) =f \quad \text { and }\quad -\triangle v -\mathbf {b} \cdot \nabla v =g \] in a bounded Lipschitz domain Ω \Omega in R n \mathbb {R}^n ( n ≥ 3 ) (n\geq 3) , where b : Ω → R n \mathbf {b}:\Omega \rightarrow \mathbb {R}^n is a given vector field. Under the assumption that b ∈ L n ( Ω ) n \mathbf {b} \in L^{n}(\Omega )^n , we first establish existence and uniqueness of solutions in L α p ( Ω ) L_{\alpha }^{p}(\Omega ) for the Dirichlet and Neumann problems. Here L α p ( Ω ) L_{\alpha }^{p}(\Omega ) denotes the Sobolev space (or Bessel potential space) with the pair ( α , p ) (\alpha ,p) satisfying certain conditions. These results extend the classical works of Jerison-Kenig [J. Funct. Anal. 130 (1995), pp. 161–219] and Fabes-Mendez-Mitrea [J. Funct. Anal. 159 (1998), pp. 323–368] for the Poisson equation. We also prove existence and uniqueness of solutions of the Dirichlet problem with boundary data in L 2 ( ∂ Ω ) L^{2}(\partial \Omega ) . Our results for the Dirichlet problems hold even for the case n = 2 n=2 .