On the Robin eigenvalues of the Laplacian in the exterior of a convex polygon
https://doi.org/10.17586/222080542015614656
Abstract
Let R2 be the exterior of a convex polygon whose side lengths are `1; : : : ; `M. For a real constant , let H denote the Laplacian in , u 7! u, with the Robin boundary conditions @u=@ = u at @ , where is the outer unit normal. We show that, for any fixed m 2 N, the mth eigenvalue E m ( ) of H behaves as E m ( ) = 2 + D m + O( 1=2) as ! +1, where D m stands for the mth eigenvalue of the operator D1 DM and Dn denotes the onedimensional Laplacian f 7! f00 on (0; `n) with the Dirichlet boundary conditions.
Supporting agencies: The work was partially supported by ANR NOSEVOL (ANR 2011 BS01019 01) and GDR Dynamique quantique (GDR CNRS 2279 DYNQUA).
About the Author
K. Pankrashkin
Laboratoire de math´ematiques, Universit´e ParisSud
France
France
Bˆatiment 425, 91405 Orsay Cedex
References
1. F. Cakoni, N. Chaulet, H. Haddar. On the asymptotics of a Robin eigenvalue problem. Comptes Rendus Math., 351 (13–14), P. 517–521 (2013).
2. D. Daners, J.B. Kennedy. On the asymptotic behaviour of the eigenvalues of a Robin problem. Differential Integr. Equ., 23 (7–8), P. 659–669 (2010).
3. P. Exner, A. Minakov. Curvatureinduced bound states in Robin waveguides and their asymptotical properties. J. Math. Phys., 55 (12), P. 122101 (2014).
4. P. Exner, A. Minakov, L. Parnovski. Asymptotic eigenvalue estimates for a Robin problem with a large parameter. Portugal. Math., 71 (2), P. 141–156 (2014).
5. P. Exner, O. Post. Convergence of spectra of graphlike thin manifolds. J. Geom. Phys., 54 (1), P. 77–115 (2005).
6. T. Giorgi, R. Smits. Eigenvalue estimates and critical temperature in zero fields for enhanced surface superconductivity. Z. Angew. Math. Phys., 58 (2), P. 224–245 (2007).
7. P. Grisvard. Singularities in boundary value problems. Recherches en Math´ematiques Appliques, No. 22 (1992).
8. B. Helffer, A. Kachmar. Eigenvalues for the Robin Laplacian in domains with variable curvature.arXiv:1411.2700 (2014).
9. B. Helffer, K. Pankrashkin. Tunneling between corners for Robin Laplacians. J. London Math. Soc., 91 (1), P. 225248 (2015).
10. P. Freitas, D. Krejˇciˇr´ık. The first Robin eigenvalue with negative boundary parameter.arXiv:1403.6666, (2014).
11. A.A. Lacey, J.R. Ockendon, J. Sabina. Multidimensional reaction diffusion equations with nonlinear boundary conditions. SIAM J. Appl. Math., 58 (5), P. 1622–1647 (1998).
12. M. Levitin, L. Parnovski. On the principal eigenvalue of a Robin problem with a large parameter. Math. Nachr., 281 (2), P. 272–281 (2008).
13. Y. Lou, M. Zhu. A singularly perturbed linear eigenvalue problem in C1 domains. Pacific J. Math., 214 (2), P. 323–334 (2004).
14. K. Pankrashkin. On the asymptotics of the principal eigenvalue for a Robin problem with a large parameter in planar domains. Nanosystems: Phys. Chem. Math., 4 (4), P. 474–483 (2013).
15. K. Pankrashkin, N. Popoff. Mean curvature bounds and eigenvalues of Robin Laplacians. arXiv:1407.3087, (2014).
16. O. Post. Branched quantum wave guides with Dirichlet boundary conditions: the decoupling case. J. Phys. A, 38 (22), P. 4917–4932 (2005).
Review
For citations:
Pankrashkin K. On the Robin eigenvalues of the Laplacian in the exterior of a convex polygon. Nanosystems: Physics, Chemistry, Mathematics. 2015;6(1):46-56. https://doi.org/10.17586/222080542015614656
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