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


Author: Dorina Mitrea
Journal: Trans. Amer. Math. Soc. 360 (2008), 3771-3793
MSC (2000): Primary 35J05, 46E35; Secondary 42B20, 34B27
DOI: https://doi.org/10.1090/S0002-9947-08-04384-5
Published electronically: February 27, 2008
MathSciNet review: 2386245
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Abstract: Let $ \mathbb{G}_D$ be the solution operator for $ \Delta u = f$ in $ \Omega$, Tr $ u = 0$ 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 $ p_n > 1$ 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.


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Additional Information

Dorina Mitrea
Affiliation: Department of Mathematics, University of Missouri-Columbia, Columbia, Missouri 65211

DOI: https://doi.org/10.1090/S0002-9947-08-04384-5
Keywords: Green potentials, Poisson problem, Lipschitz domain, Sobolev spaces
Received by editor(s): May 22, 2006
Published electronically: February 27, 2008
Additional Notes: The author was supported in part by NSF FRG Grant #0456306
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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