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

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 ($\mathrm{KP}$). 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 $\mathrm{KP}$ and the $A_{\infty }$ condition break down in the limit. (b) There are sequences of matrices for which $\mathrm{KP}$ breaks down but $A_{\infty }$ is preserved in the limit.

This note is not intended for publication