#### Vol. 6, No. 5, 2013

 Download this article For screen For printing
 Recent Issues
 The Journal About the Journal Editorial Board Editors’ Interests Subscriptions Submission Guidelines Submission Form Policies for Authors Ethics Statement ISSN: 1948-206X (e-only) ISSN: 2157-5045 (print) Author Index To Appear Other MSP Journals
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 ${ℝ}^{n+1}$ 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 ${\epsilon }^{2}log\epsilon$ and ${\epsilon }^{2}$.

##### Keywords
Laplace–Beltrami operator, eigenvalue, flat manifolds
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