The present paper establishes the first result on the absolute continuity of elliptic measure with respect to the Lebesgue measure for a divergence form elliptic operator with non-smooth coefficients that have a \({{\,\mathrm{BMO}\,}}\) anti-symmetric part. In particular, the coefficients are not necessarily bounded. We prove that the Dirichlet problem for elliptic equation \(\mathrm{div}(A\nabla u)=0\) in the upper half-space \((x,t)\in {\mathbb {R}}^{n+1}_+\) is uniquely solvable when \(n\ge 2\) and the boundary data is in \(L^p({\mathbb {R}}^n,dx)\) for some \(p\in (1,\infty )\). This result is equivalent to saying that the elliptic measure associated to L belongs to the \(A_\infty \) class with respect to the Lebesgue measure dx, a quantitative version of absolute continuity.

本文建立了关于具有\({{\,\mathrm{BMO}\,}}\ )反对称部分。特别地，系数不一定是有界的。我们证明了椭圆方程的狄利克雷问题\(\mathrm{div}(A\nabla u)=0\)在上半空间\((x,t)\in {\mathbb {R}}^{ n+1}_+\)是唯一可解的当\(n\ge 2\)并且边界数据在\(L^p({\mathbb {R}}^n,dx)\)对于某些\( p\in (1,\infty )\)。这个结果相当于说与L相关的椭圆测度属于\(A_\infty \)类关于 Lebesgue 测度dx，绝对连续性的定量版本。