Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

   
 

 

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: 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.


References [Enhancements On Off] (What's this?)

  • 1. Stephen L. Campbell and C. D. Meyer Jr., Generalized inverses of linear transformations, Surveys and Reference Works in Mathematics, vol. 4, Pitman (Advanced Publishing Program), Boston, Mass.-London, 1979. MR 533666
  • 2. Luigi D’Onofrio and Tadeusz Iwaniec, Interpolation theorem for the 𝑝-harmonic transform, Studia Math. 159 (2003), no. 3, 373–390. Dedicated to Professor Aleksander Pełczyński on the occasion of his 70th birthday (Polish). MR 2052229, 10.4064/sm159-3-3
  • 3. Irene Fonseca and Stefan Müller, 𝒜-quasiconvexity, lower semicontinuity, and Young measures, SIAM J. Math. Anal. 30 (1999), no. 6, 1355–1390 (electronic). MR 1718306, 10.1137/S0036141098339885
  • 4. Flavia Giannetti and Anna Verde, Variational integrals for elliptic complexes, Studia Math. 140 (2000), no. 1, 79–98. MR 1763883
  • 5. G. H. Golub and V. Pereyra, The differentiation of pseudo-inverses and nonlinear least squares problems whose variables separate, SIAM J. Numer. Anal. 10 (1973), 413–432. Collection of articles dedicated to the memory of George E. Forsythe. MR 0336980
  • 6. Derek Gustafson, Elliptic complexes and generalized Poincaré inequalities, Ph.D. thesis, Syracuse University, February 2010.
  • 7. Lars Hörmander, Linear partial differential operators, Third revised printing. Die Grundlehren der mathematischen Wissenschaften, Band 116, Springer-Verlag New York Inc., New York, 1969. MR 0248435
  • 8. Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095
  • 9. Nikolai N. Tarkhanov, Complexes of differential operators, Mathematics and its Applications, vol. 340, Kluwer Academic Publishers Group, Dordrecht, 1995. Translated from the 1990 Russian original by P. M. Gauthier and revised by the author. MR 1368856
  • 10. K. Uhlenbeck, Regularity for a class of non-linear elliptic systems, Acta Math. 138 (1977), no. 3-4, 219–240. MR 0474389

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Additional Information

Derek Gustafson
Affiliation: Department of Mathematics, Syracuse University, Syracuse, New York 13210
Email: degustaf@syr.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-2011-10607-5
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.