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Grafakos, Loukas
(Ed.)
We establish 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 BMO anti- symmetric part. In particular, the coefficients are not necessarily bounded. We prove that for some $$p\in (1,\infty)$$, the Dirichlet problem for the elliptic equation $$Lu= \dv A\nabla u=0$$ in the upper half-space $$\mathbb{R}^{n+1},\, n\geq 2,$$ is uniquely solvable when the boundary data is in $$L^p(\mathbb{R}^n,dx)$$, provided that the coefficients are independent of the vertical variable. 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.
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