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Necessary and sufficient conditions for the solvability of the \(L^p\) Dirichlet problem on Lipschitz domains. (English) Zbl 1194.35131

Summary: We study the homogeneous elliptic systems of order \(2\ell\) with real constant coefficients on Lipschitz domains in \(\mathbb R^n\), \(n \geq 4\). 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 \(n \geq 4\) and \(2- \varepsilon < p <\frac{2(n-1)}{n-3} + \varepsilon\). The range of \(p\) is known to be sharp if \(\ell \geq 2\) and \(4 \leq n \leq 2\ell+1\). 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 \(\ell \geq 3\).

MSC:

35J40 Boundary value problems for higher-order elliptic equations
35J58 Boundary value problems for higher-order elliptic systems
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