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

TL;DR

This paper provides a simpler proof for the solvability of the $L^p$ regularity problem for Dahlberg-Kenig-Pipher operators on half-spaces, expanding the class of operators and extending results to weighted cases.

Contribution

It introduces an alternative, simpler proof for the $L^p$ regularity problem and broadens the class of operators for which solvability is established, including weighted cases.

Findings

Simplified proof of the $L^p$ regularity problem.

Extended solvability to a larger class of operators.

Established results for weighted elliptic operators on $ r^n ackslash r^d$.

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 $d < n - 1$.