A bilinear estimate for biharmonic functions in Lipschitz domains

Abstract

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 \({\mathbb{R}^{d}}\) and 1 < q < ∞, the solvability of the L q Dirichlet problem for Δ 2 u = 0 in Ω with boundary data in WA 1,q(∂Ω) is equivalent to that of the L p regularity problem for Δ 2 u = 0 in Ω with boundary data in WA 2,p(∂Ω), where \({\frac{1}{p} + \frac{1}{q}=1}\). 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.