A relationship between the dirichlet and regularity problems for elliptic equations

Let ℒ = divA∇ be a real, symmetric second order elliptic operator with bounded measurable coefficients. Consider the elliptic equation ℒu = 0 in a bounded Lipschitz domain Ω of ℝⁿ. We study the relationship between the solvability of the L^p Dirichlet problem (D)_p with boundary data in L^p(∂Ω) and that of the L^q regularity problem (R)_q with boundary data in W^1,q(∂Ω), where 1 < p, q < ∞. It is known that the solvability of (R)_p implies that of (D)_p′. In this note we show that if (D)_p′ is solvable, then either (R)_p is solvable or (R)_q is not solvable for any 1 < q < ∞.