A change of variable for Dahlberg-Kenig-Pipher operators

TL;DR

This paper introduces a method to handle Dahlberg-Kenig-Pipher (DKP) operators in boundary value problems, using bi-Lipschitz changes of variables to simplify analysis and extend existing results.

Contribution

It identifies a subclass of weak DKP operators that generate all such operators under bi-Lipschitz transformations fixing the boundary, providing an alternative proof and extending previous results.

Findings

Subclass of weak DKP operators generates all under bi-Lipschitz changes

Method simplifies proofs of properties stable under bi-Lipschitz transformations

Extends the class of known results for DKP operators

Abstract

In the present article, we purpose a method to deal with Dahlberg-Kenig-Pipher (DPK) operators in boundary value problems on the upper half plane. We give a nice subclass of the weak DKP operators that generates the full class of weak DKP operators under bi-Lipschitz changes of variable on $\mathbb R^n_+$ that fixe the boundary $\mathbb R^{n-1}$. Therefore, if one wants to prove a property on DKP operators which is stable by bi-Lipschitz transformations, one can directly assume that the operator belongs to the subclass. Our method gives an alternative proof to some past results and self-improves others beyond the existing literature.