The Dirichlet problem for elliptic operators having a BMO anti-symmetric part

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 $\mathrm{div}(A\nabla u)=0$ 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.

61 pages. A new theorem (Theorem 1.2) was added. Some typos and omissions were corrected

References in corpus (2)

• Carleson measure estimates and the Dirichlet problem for degenerate elliptic equations
• $L^p$ theory for the square roots and square functions of elliptic operators having a BMO anti-symmetric part

Cited by in corpus (2)

• $L^p$ theory for the square roots and square functions of elliptic operators having a BMO anti-symmetric part
• The Green function with pole at infinity applied to the study of the elliptic measure