Vol. 6, No. 5, 2013

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On the spectrum of deformations of compact double-sided flat hypersurfaces

Denis Borisov and Pedro Freitas

Vol. 6 (2013), No. 5, 1051–1088
Abstract

We study the asymptotic behavior of the eigenvalues of the Laplace–Beltrami operator on a compact hypersurface in n+1 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 ε2 logε and ε2.

Keywords
Laplace–Beltrami operator, eigenvalue, flat manifolds
Mathematical Subject Classification 2000
Primary: 35P15
Secondary: 35J05
Milestones
Received: 2 May 2012
Revised: 4 October 2012
Accepted: 14 February 2013
Published: 3 November 2013
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