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A generalized Poincaré inequality for a class of constant coefficient differential operators

Author: Derek Gustafson
Journal: Proc. Amer. Math. Soc. 139 (2011), 2721-2728
MSC (2010): Primary 35A99; Secondary 35B45, 58J10
Published electronically: March 23, 2011
MathSciNet review: 2801612
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Abstract | References | Similar Articles | Additional Information

Abstract: We study first order differential operators $ \mathcal{P} = \mathcal{P}(D)$ with constant coefficients. The main question is under what conditions the following full gradient $ L^p$ estimate holds:

$\displaystyle \Vert D(f-f_0)\Vert _{L^p} \leq C \Vert\mathcal{P} f\Vert _{L^p}, \textrm{for some } f_0 \in \ker \mathcal{P}.$

We show that the constant rank condition is sufficient. The concept of the Moore-Penrose generalized inverse of a matrix comes into play.

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Additional Information

Derek Gustafson
Affiliation: Department of Mathematics, Syracuse University, Syracuse, New York 13210

Keywords: Elliptic complexes, Poincaré inequality, constant rank
Received by editor(s): October 11, 2009
Received by editor(s) in revised form: February 15, 2010, and May 13, 2010
Published electronically: March 23, 2011
Communicated by: Matthew J. Gursky
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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