On Estimates of Biharmonic Functions on Lipschitz and Convex Domains

Abstract

Using Maz&#39;ya type integral identities with power weights, we obtain new boundary estimates for biharmonic functions on Lipschitz and convex domains in $R^n$. For $n\ge 8$, combined with a result in \cite{S2}, these estimates lead to the solvability of the $L^p$ Dirichlet problem for the biharmonic equation on Lipschitz domains for a new range of $p$. In the case of convex domains, the estimates allow us to show that the $L^p$ Dirichlet problem is uniquely solvable for any $2-\e<p<\infty$ and $n\ge 4$.