A Relationship between the Dirichlet and Regularity Problems for Elliptic Equations

Abstract. 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 R. We study the relationship between the solvability of the L Dirichlet problem (D)p with boundary data in L(∂Ω) and that of the L regularity problem (R)q with boundary data in W (∂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 < ∞.