Elliptic equations with BMO coefficients in Lipschitz domains.

In this paper, we study inhomogeneous Dirichlet problems for elliptic equations in divergence form. Optimal regularity requirements on the coefficients and domains for the $W^{1,p} (1<p<\infty)$ estimates are obtained. The principal coefficients are supposed to be in the John-Nirenberg space with small BMO semi-norms. The domain is supposed to have Lipschitz boundary with small Lipschitz constant. These conditions for the $W^{1,p}$ theory do not just weaken the requirements on the coefficients; they also lead to a more general geometric condition on the domain. [ABSTRACT FROM AUTHOR]