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On the spectrum of
deformations of compact double-sided flat hypersurfaces
Denis Borisov and Pedro Freitas |
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Vol. 6 (2013), No. 5, 1051–1088
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Abstract |
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We study the asymptotic behavior of the eigenvalues of the Laplace–Beltrami operator on a compact hypersurface in as it is flattened into a singular double-sided flat hypersurface. We show that the limit spectral problem corresponds to the Dirichlet and Neumann problems on one side of this flat (Euclidean) limit, and derive an explicit three-term asymptotic expansion for the eigenvalues where the remaining two terms are of orders and . |
Keywords
Laplace–Beltrami operator, eigenvalue, flat manifolds
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Mathematical Subject Classification 2000
Primary: 35P15
Secondary: 35J05
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Milestones
Received: 2 May 2012
Revised: 4 October 2012
Accepted: 14 February 2013
Published: 3 November 2013
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Authors |
| Denis Borisov | |
| Department of Physics and
Mathematics Bashkir State Pedagogical University Ufa 3a Oktyabrskoy Revolutsii st. 450000 Russian Federation |
Institute of Mathematics USC
RAS Ufa 112 Chernyshevsky str. 450000 Russian Federation |
| Pedro Freitas | |
| Department of Mathematics, Faculty
of Human Kinetics, and Group of Mathematical Physics University of Lisbon Complexo Interdisciplinar Av. Prof. Gama Pinto 2 P-1649-003 Lisboa Portugal |
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