Weak-type Estimates and Potential Estimates for Elliptic Equations with Drift Terms

Abstract

We consider divergence form elliptic equations with a strongly singular drift term −div(A∇u)+b⋅∇u=μ in a domain \({\Omega } \subset \mathbb {R}^{n}\) (n≥3). We give a weak-type \(L^{1} - L^{n / (n - 2), \infty }\) estimate for a solution to the Dirichlet problem with homogeneous boundary condition. Moreover, we give a two-sided pointwise potential estimate for a weak solution and its applications.