A bilinear estimate for biharmonic functions in Lipschitz domains

We show that a bilinear estimate for biharmonic functions in a Lipschitz domain Ω is equivalent to the solvability of the Dirichlet problem for the biharmonic equation in Ω. As a result, we prove that for any given bounded Lipschitz domain Ω in ℝ^d and 1 < q < ∞, the solvability of the L^q Dirichlet problem for Δ²u = 0 in Ω with boundary data in WA^1,q(∂Ω) is equivalent to that of the L^p regularity problem for Δ²u = 0 in Ω with boundary data in WA^2,p(∂Ω), where 1/p + 1/q = 1. This duality relation, together with known results on the Dirichlet problem, allows us to solve the L^p regularity problem for d ≥ 4 and p in certain ranges.