An alternative proof of the $L^p$-regularity problem for Dahlberg-Kenig-Pipher operators on $\mathbb R^n_+$

Abstract

In this article, we present a simpler and alternative proof of the solvability of the regularity problem - that is, the Dirichlet problem with boundary data in $\dot W^{1,p}$ - for uniformly elliptic operators on $\mathbb{R}^n_+$ under a (possibly large) Carleson measure condition. In addition, we slightly expand the class of operators for which the regularity problem is solvable, and establish an analogous result for weighted uniformly elliptic operators on $\mathbb{R}^n \setminus \mathbb{R}^d$, where $d < n - 1$.