Mathematical Research Letters

Volume 13 (2006)

Number 1

The $L^p$ Dirichlet Problem for Elliptic Systems on Lipschitz Domains

Pages: 143 – 159

DOI: http://dx.doi.org/10.4310/MRL.2006.v13.n1.a11

Author

Zhongwei Shen (University of Kentucky)

Abstract

We develop a new approach to the $L^p$ Dirichlet problem via $L^2$ estimates and reverse Hölder inequalities. We apply this approach to second order elliptic systems and the polyharmonic equation on a bounded Lipschitz domain $\OO$ in $\br^n$. For $n\ge 4$ and $2-\e<p<\frac{2(n-1)}{n-3} +\e$, we establish the solvability of the Dirichlet problem with boundary data in $L^p(\partial\OO)$. In the case of the polyharmonic equation $\Delta^\ell u=0$ with $\ell\ge 2$, the range of $p$ is sharp if $4\le n\le 2\ell +1$.

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