## Abstract

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 BMO anti-symmetric part. In particular, the coefficients are not necessarily bounded. We prove that the Dirichlet problem for elliptic equation div (A∇ u) = 0 in the upper half-space (x,t)∈R+n+1 is uniquely solvable when n≥ 2 and the boundary data is in L^{p}(R^{n}, dx) for some p∈ (1 , ∞). This result is equivalent to saying that the elliptic measure associated to L belongs to the A_{∞} class with respect to the Lebesgue measure dx, a quantitative version of absolute continuity.

Original language | English (US) |
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Pages (from-to) | 103-168 |

Number of pages | 66 |

Journal | Mathematische Annalen |

Volume | 382 |

Issue number | 1-2 |

DOIs | |

State | Published - Jun 17 2021 |

### Bibliographical note

Funding Information:S. Hofmann acknowledges support of the National Science Foundation (Grant number DMS-1664047, DMS-2000048). S. Mayboroda is supported in part by the NSF RAISE-TAQS Grant DMS-1839077 and the Simons foundation Grant 563916, SM.

Publisher Copyright:

© 2021, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.