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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 ofthe Dahlberg-Kenig-Pipher condition of approximation by constant coefficientmatrices, with a small enough Carleson norm, and show that the elliptic measurewith pole at infinity associated to $L$ is $A_\infty$-absolutely continuouswith respect to the surface measure on $\Gamma$, with a small $A_\infty$constant. In other words, we show that for relatively flat uniformlyrectifiable sets and for operators with slowly oscillating coefficients theelliptic measure satisfies the $A_\infty$ condition with a small constant andthe logarithm of the Poisson kernel has small oscillations.

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