On Lp-Resolvent Estimates for Second-Order Elliptic Equations in Divergence Form

Abstract

We consider the Dirichlet problems for second-order linear elliptic equations in divergence form. The leading coefficient A has small BMO semi-norm and first-order coefficient b belongs to Lr, where $n \leq r < \infty $ if n ≥ 3 and $2 < r < \infty $ if n = 2. We first establish Lp-resolvent estimates on bounded domains having small Lipschitz constant when $r/(r-1) < p < \infty $ . Under the additional assumption div A ∈ Lr, we also establish Lp-resolvent estimates on bounded domains with C1,1 boundary when 1 < p < r.