Failure to slide: a brief note on the interplay between the Kenig-Pipher condition and the absolute continuity of elliptic measures

TL;DR

This paper investigates the relationship between the Kenig-Pipher condition and absolute continuity of elliptic measures, demonstrating scenarios where these properties fail or are preserved through specific matrix sequences.

Contribution

It provides new examples showing the breakdown and preservation of absolute continuity and the Kenig-Pipher condition in elliptic measure approximations.

Findings

Sequences where both KP and A_infinity fail

Sequences where KP fails but A_infinity holds

Insights into the limitations of KP for absolute continuity

Abstract

In this note, we explore some consequences of the Modica-Mortola construction of a singular elliptic measure, as regards the link between the quantitative absolute continuity ($A_{\infty}$) of their approximations and the suitability of a well-known tool, the so-called Kenig-Pipher condition ($\operatorname{KP}$). The Kenig-Pipher condition is used to ascertain absolute continuity in the presence of some mild regularity of the coefficient matrix. We perform some modifications of the Modica-Mortola example to show the following two statements: (a) There are sequences of matrices for which both $\operatorname{KP}$ and the $A_{\infty}$ condition break down in the limit. (b) There are sequences of matrices for which $\operatorname{KP}$ breaks down but $A_{\infty}$ is preserved in the limit.