The Regularity Problem for Second Order Elliptic Operators with Complex-valued Bounded Measurable Coefficients

The present paper establishes a certain duality between the Dirichlet and Regularity problems for elliptic operators with t-independent complex bounded measurable coefficients (t being the transversal direction to the boundary). To be precise, we show that the Dirichlet boundary value problem is solvable in Lp ′ , subject to the square function and non-tangential maximal function estimates, if and only if the corresponding Regularity problem is solvable in Lp. Moreover, the solutions admit layer potential representations. In particular, we prove that for any elliptic operator with t-independent real (possibly non-symmetric) coefficients there exists a p > 1 such that the Regularity problem is well-posed in Lp.