Small $A_\infty$ results for Dahlberg-Kenig-Pipher operators in sets with uniformly rectifiable boundaries

TL;DR

This paper demonstrates that elliptic operators with coefficients close to constant on flat, uniformly rectifiable boundaries have elliptic measures that are absolutely continuous with small $A_ abla$ constants, extending small-scale regularity results.

Contribution

It establishes small $A_ abla$ results for elliptic operators with coefficients approximating constants on sets with uniformly rectifiable boundaries.

Findings

Elliptic measure is $A_ abla$-absolutely continuous with small constants.

Logarithm of the Poisson kernel exhibits small oscillations.

Results apply to operators with slowly oscillating coefficients on flat sets.

Abstract

In the present paper, we consider elliptic operators $L=-\textrm{div}(A\nabla)$ in a domain bounded by a chord-arc surface $\Gamma$ with small enough constant, and whose coefficients $A$ satisfy a weak form of the Dahlberg-Kenig-Pipher condition of approximation by constant coefficient matrices, with a small enough Carleson norm, and show that the elliptic measure with pole at infinity associated to $L$ is $A_\infty$-absolutely continuous with respect to the surface measure on $\Gamma$, with a small $A_\infty$ constant. In other words, we show that for relatively flat uniformly rectifiable sets and for operators with slowly oscillating coefficients the elliptic measure satisfies the $A_\infty$ condition with a small constant and the logarithm of the Poisson kernel has small oscillations.