A Bilinear Estimate for Biharmonic Functions in Lipschitz Domains

Abstract

We show that a bilinear estimate for biharmonic functions in a Lipschitz domain $\Omega$is equivalent to the solvability of the Dirichlet problem for the biharmonic equationin $\Omega$. As a result, we prove that for any given bounded Lipschitz domain $\Omega$ in $\rn{d}$ and $1<q<\infty$, the solvability of the $L^q$ Dirichlet problem for ${\Delta }^{2}u=0$ in $\Omega$ with boundary data in ${\text{WA}}^{1,q}(\partial \Omega )$ is equivalent to that of the $L^p$ regularity problem for ${\Delta }^{2}u=0$ in $\Omega$ with boundary data in ${\text{WA}}^{2,p}(\partial \Omega )$, 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 problemfor $d\ge 4$ and $p$ in certain ranges.

Corrected the proofs of Thm. 5.1 and Thm 6.1. Added Thm 2.3 on approximation scheme for Lipschitz domain. Modified Lemma 2.5 (previously Lemma 2.4) to reflect changes in proofs of Thms. 5.1 & 6.1. 24 pages