A Bilinear Estimate for Biharmonic Functions in Lipschitz Domains

TL;DR

This paper establishes a duality between bilinear estimates and solvability of Dirichlet and regularity problems for biharmonic functions in Lipschitz domains, enabling solutions for certain $L^p$ problems in higher dimensions.

Contribution

It proves the equivalence between bilinear estimates and the solvability of Dirichlet and regularity problems for biharmonic functions in Lipschitz domains, extending results to higher dimensions.

Findings

Bilinear estimate is equivalent to Dirichlet problem solvability.

Duality relates $L^q$ Dirichlet and $L^p$ regularity problems.

Results enable solving $L^p$ regularity problems in dimensions $d extgreater 3$.

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.