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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
Posted:
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  -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  -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  -dimensional section of the domain.
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- 2.
- I. Chavel, Eigenvalues in Riemannian Geometry (Appendix by J. Dodziuk), Academic Press Inc., 1984. MR 768584 (86g:58140)
- 3.
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- 4.
- S. Gallot and D. Meyer, D'un résultat hilbertien à un principe de comparaison entre spectres. Applications, Ann. Scient. Ec. Norm. Sup., 4a sèrie, 21 (1988) 561-591. MR 982334 (90k:58236)
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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:
http://dx.doi.org/10.1090/S0002-9947-2010-04844-5
PII:
S 0002-9947(2010)04844-5
Keywords:
Laplacian on forms,
eigenvalues,
convex bodies,
John ellipsoid
Received by editor(s):
June 10, 2008
Posted:
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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