The Conormal Derivative Problem for Elliptic Equations with BMO Coefficients on Reifenberg flat domains

Abstract

We study the inhomogeneous conormal derivative problem for the divergence form elliptic equation, assuming that the principal coefficients belong to the BMO space with small BMO semi-norms and that the boundary is $\delta$-Reifenberg flat. These conditions for the $W^{1, p}$-theory not only weaken the requirements on the coefficients but also lead to a more general geometric condition on the domain. In fact, the Reifenberg flatness is the minimal regularity condition for the $W^{1, p}$-theory.