Necessary and Sufficient Conditions for the Solvability of the $L^p$ Dirichlet Problem On Lipschitz Domains
| dc.creator | Shen, Zhongwei | |
| dc.date | 2005-09-26 | |
| dc.date.accessioned | 2026-07-07T06:19:11Z | |
| dc.date.available | 2026-07-07T06:19:11Z | |
| dc.description | We study the homogeneous elliptic systems of order $2\ell$ with real constant coefficients on Lipschitz domains in $R^n$, $n\ge 4$. For any fixed $p>2$, we show that a reverse Hölder condition with exponent $p$ is necessary and sufficient for the solvability of the Dirichlet problem with boundary data in $L^p$. We also obtain a simple sufficient condition. As a consequence, we establish the solvability of the $L^p$ Dirichlet problem for $n\ge 4$ and $2-\e< p<\frac{2(n-1)}{n-3} +\e$. The range of $p$ is known to be sharp if $\ell\ge 2$ and $4\le n\le 2\ell +1$. For the polyharmonic equation, the sharp range of $p$ is also found in the case $n=6$, 7 if $\ell=2$, and $n=2\ell+2$ if $\ell\ge 3$. | |
| dc.identifier | https://arxiv.org/abs/math/0509606 | |
| dc.identifier | http://arxiv.org/abs/math/0509606 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/94990 | |
| dc.subject | Analysis of PDEs | |
| dc.subject | 35J40, 35J55 | |
| dc.title | Necessary and Sufficient Conditions for the Solvability of the $L^p$ Dirichlet Problem On Lipschitz Domains | |
| dc.type | text |