## The behavior of the spectral gap under growing drift

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- by B. Franke, C.-R. Hwang, H.-M. Pai and S.-J. Sheu PDF
- Trans. Amer. Math. Soc.
**362**(2010), 1325-1350 Request permission

## 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.## References

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## Additional Information

**B. Franke**- Affiliation: Fakultät für Mathematik, Ruhr-Universität Bochum, Universitätsstrasse 150, 44780 Bochum, Germany
- MR Author ID: 728183
- 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
- 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. - © Copyright 2009
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 2563731