A generalization of Dahlberg's theorem concerning the regularity of harmonic Green potentials期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

按检索

A generalization of Dahlberg's theorem concerning the regularity of harmonic Green potentials

Authors:	Dorina Mitrea

Institution:	Department of Mathematics, University of Missouri-Columbia, Columbia, Missouri 65211

Abstract:	Let $\mathbb{G}_D$ be the solution operator for $\Delta u = f$ in $\Omega$ , Tr on $\partial\Omega$ , where $\Omega$ is a bounded domain in $\mathbb{R}^n$ . B. E. J. Dahlberg proved that for a bounded Lipschitz domain $\Omega, \nabla \mathbb{G}_D$ maps $L^1 (\Omega)$ boundedly into weak- $L^1(\Omega)$ and that there exists such that $\nabla\mathbb{G}_D : L^p (\Omega)\rightarrow L^{p^{}} (\Omega)$ is bounded for $1 < p < n, \frac{1}{p^} = \frac {1}{p} - \frac {1}{n}$ . In this paper, we generalize this result by addressing two aspects. First we are also able to treat the solution operator $\mathbb{G}_N$ corresponding to Neumann boundary conditions and, second, we prove mapping properties for these operators acting on Sobolev (rather than Lebesgue) spaces.

Keywords:	Green potentials Poisson problem Lipschitz domain Sobolev spaces

	点击此处可从《Transactions of the American Mathematical Society》浏览原始摘要信息
	点击此处可从《Transactions of the American Mathematical Society》下载免费的PDF全文