  We study the relationship between the solvability of the $L^p$ Dirichlet
problem $(D)_p$ and that of the $L^q$ regularity problem $(R)_q$ for second
order elliptic equations with bounded measurable coefficients. It is known that
the solvability of $(R)_p$ implies that of $(D)_{p^\prime}$. In this note we
show that if $(D)_{p^\prime}$ is solvable, then either $(R)_p$ is solvable or
$(R)_q$ is not solvable for any $1<q<\infty$.
