Elliptic equations in divergence form with drifts in 𝐿²

Abstract

We consider the Dirichlet problem for second-order linear elliptic equations in divergence form − d i v ( A ∇ u ) + b ⋅ ∇ u + λ u = f + d i v F in Ω and u = 0 on ∂ Ω , \begin{equation*} -div(A\nabla u)+\mathbf {b} \cdot \nabla u+\lambda u=f+div\mathbf {F}\quad \text {in } \Omega \quad \text {and}\quad u=0\quad \text {on } \partial \Omega , \end{equation*} in bounded Lipschitz domain Ω \Omega in R 2 \mathbb {R}^2 , where A : R 2 → R 2 2 A:\mathbb {R}^2\rightarrow \mathbb {R}^{2^2} , b : Ω → R 2 \mathbf {b} : \Omega \rightarrow \mathbb {R}^2 , and λ ≥ 0 \lambda \geq 0 are given. If 2 > p > ∞ 2>p>\infty and A A has a small mean oscillation in small balls, Ω \Omega has small Lipschitz constant, and d i v A , b ∈ L 2 ( Ω ; R 2 ) divA,\,\mathbf {b} \in L^{2}(\Omega ;\mathbb {R}^2) , then we prove existence and uniqueness of weak solutions in W 0 1 , p ( Ω ) W^{1,p}_0(\Omega ) of the problem. Similar result also holds for the dual problem.

Scalar elliptic equations with a singular drift

“…There are many papers devoted to the investigation of the problem (1.1) in the case of divergence free drift, [1], [9], [11], [12], [13], [26], [27], [28], [34], [36], [41], [42] and references there. Papers devoted to the non-divergence free drifts are not so numerous (see [3], [4], [6], [15], [16], [17], [18], [20], [21], [22], [23], [24], [28] for the references). Our present contribution can be viewed as a 3D analogue of the results obtained earlier in [6] in the 2D case.…”