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

TL;DR

This paper establishes a duality between Dirichlet and Regularity problems for elliptic operators with complex coefficients, showing their equivalence under certain estimates and providing layer potential representations.

Contribution

It proves the equivalence of solvability for Dirichlet and Regularity problems in $L^p$ spaces for elliptic operators with complex coefficients, including real non-symmetric cases.

Findings

Duality between Dirichlet and Regularity problems established

Layer potential representations for solutions proved

Existence of $p>1$ for well-posedness in real coefficient case

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 $L^p$.