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 ${\emph{WA}}^{1,q}(\partial\Omega)$ is equivalent to that of the $L^p$ regularity problem for $\Delta^2 u=0$ in $\Omega$ with boundary data in ${\emph{WA}}^{2,p}(\partial\Omega)$, 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.