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

Abstract

In the present paper, we consider elliptic operators $L=-\text{div}(A\nabla )$ 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.

58 pages