  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 $L^p$.
