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The behavior of the spectral gap under growing drift
Author(s):
B.
Franke;
C.-R.
Hwang;
H.-M.
Pai;
S.-J.
Sheu
Journal:
Trans. Amer. Math. Soc.
362
(2010),
1325-1350.
MSC (2000):
Primary 35P15, 60H30
Posted:
October 1, 2009
MathSciNet review:
2563731
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Additional information
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  and ask the question how the spectral gap behaves as  goes to infinity. It turns out that the spectral gap stays bounded if and only if the drift-vector field has eigenfunctions in  . 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:
10.1090/S0002-9947-09-04939-3
PII:
S 0002-9947(09)04939-3
Keywords:
Laplace-Beltrami operator,
divergence free drift,
spectral gap
Received by editor(s):
October 25, 2007
Posted:
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.
Copyright of article:
Copyright
2009,
American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.
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