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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

The behavior of the spectral gap under growing drift


Authors: B. Franke, C.-R. Hwang, H.-M. Pai and S.-J. Sheu
Journal: Trans. Amer. Math. Soc. 362 (2010), 1325-1350
MSC (2000): Primary 35P15, 60H30
DOI: https://doi.org/10.1090/S0002-9947-09-04939-3
Published electronically: October 1, 2009
MathSciNet review: 2563731
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Abstract: We analyze the behavior of the spectral gap of the Laplace- Beltrami operator on a compact Riemannian manifold when a divergence-free drift vector field is added. We increase the drift by multiplication with a large constant $ c $ and ask the question how the spectral gap behaves as $ c $ goes to infinity. It turns out that the spectral gap stays bounded if and only if the drift-vector field has eigenfunctions in $ H^1 $. In that case the spectral gaps converge and we determine the limit.


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Additional Information

B. Franke
Affiliation: Fakultät für Mathematik, Ruhr-Universität Bochum, Universitätsstrasse 150, 44780 Bochum, Germany
Email: Brice.Franke@ruhr-uni-bochum.de

C.-R. Hwang
Affiliation: Institute of Mathematics, Academia Sinica, Nankang, Taipei 11529, Taiwan
Email: crhwang@math.sinica.edu.tw

H.-M. Pai
Affiliation: Department of Statistics, National Taipei University, No. 151, University Rd., San Shia, Taipei 237, Taiwan
Email: hpai@mail.ntpu.edu.tw

S.-J. Sheu
Affiliation: Institute of Mathematics, Academia Sinica, Nankang, Taipei 11529, Taiwan
Email: sheusj@math.sinica.edu.tw

DOI: https://doi.org/10.1090/S0002-9947-09-04939-3
Keywords: Laplace-Beltrami operator, divergence free drift, spectral gap
Received by editor(s): October 25, 2007
Published electronically: October 1, 2009
Additional Notes: The first author was supported by the DFG, Förderungsnummer: FR2481/1-1.
The second author was supported by the NSC Grant of Republic of China NSC95-2115-M-001-012.
The second, third, and fourth authors were partially supported by the Mathematics Division, NCTS (Taipei Office).
The fourth author was supported by the NSC Grant of Republic of China NSC96-2119-M-001-002.
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.