The L^p Dirichlet problem for elliptic systems on Lipschitz domains
By Zhongwei Shen. 2004
55 citations recorded by the source corpus.
We develop a new approach to the $L^p$ Dirichlet problem via $L^2$ estimates and reverse Holder inequalities. We apply this approach to second order elliptic systems and the polyharmonic equation on a bounded Lipschitz domain $\Omega$ in $R^n$. For $n\ge 4$ and $2-\epsilon<p<2(n-1)/(n-3)+\epsilon$, we establish the solvability of the Dirichlet problem with boundary value data in $L^p(\partial\Omega)$. In the case of the polyharmonic equation $\Delta^{\ell}u=0$ with $\ell\ge 2$, the range of $p$ is sharp if $4\le n \le 2\ell +1$.
Subfields: Applied Mathematics, Computational Theory and Mathematics
View Paper: https://www.semanticscholar.org/paper/8e8d3b28b2dc4c55841e132eefe123df6d5589bf
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In Applied Mathematics
- G. Verchota
- Zhongwei Shen
- Carlos E. Kenig
- S. Armstrong
- Jean-Christophe Mourrat
- Tuomo Kuusi
- Jinping Zhuge
In Computational Theory and Mathematics
- G. Verchota
- Zhongwei Shen
- Carlos E. Kenig
- S. Armstrong
- Jean-Christophe Mourrat
- Tuomo Kuusi
- Jinping Zhuge
Citations in the corpus
- The biharmonic Neumann problem in Lipschitz domains (2005, result): “2 ( n - l ) c < p < 2 + c n + l given the corresponding known results on the Dirichlet and regularity problems [31] and [32], and most remarkably [37].”
- Necessary and Sufficient Conditions for the Solvability of the Lp Dirichlet Problem on Lipschitz Domains (2005, background): “As a consequence, we obtain the following extension of the main results in [36].”