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Hodge-Laplace eigenvalues of convex bodies


Author: Alessandro Savo
Journal: Trans. Amer. Math. Soc. 363 (2011), 1789-1804
MSC (2010): Primary 58J50
DOI: https://doi.org/10.1090/S0002-9947-2010-04844-5
Published electronically: November 17, 2010
MathSciNet review: 2746665
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Abstract: We give upper and lower bounds of the first eigenvalue of the Hodge Laplacian acting on smooth $ p$-forms on a convex Euclidean domain for the absolute and relative boundary conditions. In particular, for the absolute conditions we show that it behaves like the squared inverse of the $ p$-th longest principal axis of the ellipsoid of maximal volume included in the domain (the John ellipsoid). Using John's theorem, we then give a spectral geometric interpretation of the bounds and relate the eigenvalues with the largest volume of a $ p$-dimensional section of the domain.


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Additional Information

Alessandro Savo
Affiliation: Dipartimento di Scienze di Base e Applicate per l’Ingegneria - Sezione di Matematica, Sapienza Università di Roma, Via Antonio Scarpa 14, 00161 Roma, Italy
Email: savo@dmmm.uniroma1.it

DOI: https://doi.org/10.1090/S0002-9947-2010-04844-5
Keywords: Laplacian on forms, eigenvalues, convex bodies, John ellipsoid
Received by editor(s): June 10, 2008
Published electronically: November 17, 2010
Additional Notes: This work was partially supported by the COFIN program of MIUR and by GNSAGA (Italy)
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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