Mathematics – Analysis of PDEs
Scientific paper
2005-09-26
Mathematics
Analysis of PDEs
Scientific paper
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 H\"older 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 $\ell\ge 3$.
No associations
LandOfFree
Necessary and Sufficient Conditions for the Solvability of the $L^p$ Dirichlet Problem On Lipschitz Domains does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Necessary and Sufficient Conditions for the Solvability of the $L^p$ Dirichlet Problem On Lipschitz Domains, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Necessary and Sufficient Conditions for the Solvability of the $L^p$ Dirichlet Problem On Lipschitz Domains will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-503641