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Mathematische Annalen

, Volume 336, Issue 3, pp 697–725 | Cite as

Necessary and Sufficient Conditions for the Solvability of the L p Dirichlet Problem on Lipschitz Domains

  • Zhongwei ShenEmail author
Article

Abstract

We study the homogeneous elliptic systems of order \(2\ell\) with real constant coefficients on Lipschitz domains in\(\mathbb{R}^n\), \(n\ge 4\). 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 \(2-\epsilon< p<\frac{2(n-1)}{n-3} +\epsilon\). 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 \(\ell\ge 3\).

Mathematics Subject Classification (2000)

35J40 35J55 

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Copyright information

© Springer-Verlag 2006

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of KentuckyLexingtonUSA

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