The regularity problem for second order elliptic operators with complex-valued bounded measurable coefficients

Abstract

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 $$L^{p'}$$ , subject to the square function and non-tangential maximal function estimates, if and only if the corresponding Regularity problem is solvable in $$L^p$$ . 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 $$L^p$$ .