A Relationship Between the Dirichlet and Regularity Problems for Elliptic Equations
Zhongwei Shen · Mathematical Research Letters · 2007
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 n .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 < ∞.