, Volume 336, Issue 3, pp 697-725

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

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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 $\ell\ge 3$ .

Research supported in part by the NSF.