Mathematische Annalen

, Volume 336, Issue 3, pp 697–725

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

Article

DOI: 10.1007/s00208-006-0022-x

Cite this article as:
Shen, Z. Math. Ann. (2006) 336: 697. doi:10.1007/s00208-006-0022-x

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 Lp. We also obtain a simple sufficient condition. As a consequence, we establish the solvability of the Lp 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)

35J4035J55

Copyright information

© Springer-Verlag 2006

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of KentuckyLexingtonUSA