Compact manifolds with fixed boundary and large Steklov eigenvalues
HTML articles powered by AMS MathViewer
- by Bruno Colbois, Ahmad El Soufi and Alexandre Girouard PDF
- Proc. Amer. Math. Soc. 147 (2019), 3813-3827 Request permission
Abstract:
Let $(M,g)$ be a compact Riemannian manifold with boundary. Let $b>0$ be the number of connected components of its boundary. For manifolds of dimension $\geq 3$, we prove that for $j=b+1$ it is possible to obtain an arbitrarily large Steklov eigenvalue $\sigma _j(M,e^\delta g)$ using a conformal perturbation $\delta \in C^\infty (M)$ which is supported in a thin neighbourhood of the boundary, with $\delta =0$ on the boundary. For $j\leq b$, it is also possible to obtain arbitrarily large eigenvalues, but the conformal factor must spread throughout the interior of $M$. In fact, when working in a fixed conformal class and for $\delta =0$ on the boundary, it is known that the volume of $(M,e^\delta g)$ has to tend to infinity in order for some $\sigma _j$ to become arbitrarily large. This is in stark contrast with the situation for the eigenvalues of the Laplace operator on a closed manifold, where a conformal factor that is large enough for the volume to become unbounded results in the spectrum collapsing to 0. We also prove that it is possible to obtain large Steklov eigenvalues while keeping different boundary components arbitrarily close to each other, by constructing a convenient Riemannian submersion.References
- M. S. Agranovich, On a mixed Poincaré-Steklov type spectral problem in a Lipschitz domain, Russ. J. Math. Phys. 13 (2006), no. 3, 239–244. MR 2262827, DOI 10.1134/S1061920806030010
- Rodrigo Bañuelos, Tadeusz Kulczycki, Iosif Polterovich, and Bartłomiej Siudeja, Eigenvalue inequalities for mixed Steklov problems, Operator theory and its applications, Amer. Math. Soc. Transl. Ser. 2, vol. 231, Amer. Math. Soc., Providence, RI, 2010, pp. 19–34. MR 2758960, DOI 10.1090/trans2/231/04
- Bruno Colbois, Ahmad El Soufi, and Alexandre Girouard, Isoperimetric control of the Steklov spectrum, J. Funct. Anal. 261 (2011), no. 5, 1384–1399. MR 2807105, DOI 10.1016/j.jfa.2011.05.006
- B. Colbois, A. Girouard, and A. Hassannezhad, The Steklov and Laplacian spectra of Riemannian manifolds with boundary, Preprint arXiv:1810.00711.
- B. Colbois, A. Girouard, and B. Raveendran, The Steklov spectrum and coarse discretizations of manifolds with boundary, Pure Appl. Math. Q., to appear.
- Jozef Dodziuk, Eigenvalues of the Laplacian on forms, Proc. Amer. Math. Soc. 85 (1982), no. 3, 437–443. MR 656119, DOI 10.1090/S0002-9939-1982-0656119-2
- José F. Escobar, The geometry of the first non-zero Stekloff eigenvalue, J. Funct. Anal. 150 (1997), no. 2, 544–556. MR 1479552, DOI 10.1006/jfan.1997.3116
- Alexandre Girouard, Leonid Parnovski, Iosif Polterovich, and David A. Sher, The Steklov spectrum of surfaces: asymptotics and invariants, Math. Proc. Cambridge Philos. Soc. 157 (2014), no. 3, 379–389. MR 3286514, DOI 10.1017/S030500411400036X
- Alexandre Girouard and Iosif Polterovich, Spectral geometry of the Steklov problem (survey article), J. Spectr. Theory 7 (2017), no. 2, 321–359. MR 3662010, DOI 10.4171/JST/164
- Asma Hassannezhad, Conformal upper bounds for the eigenvalues of the Laplacian and Steklov problem, J. Funct. Anal. 261 (2011), no. 12, 3419–3436. MR 2838029, DOI 10.1016/j.jfa.2011.08.003
- Pierre Jammes, Une inégalité de Cheeger pour le spectre de Steklov, Ann. Inst. Fourier (Grenoble) 65 (2015), no. 3, 1381–1385 (French, with English and French summaries). MR 3449183, DOI 10.5802/aif.2960
- Gerasim Kokarev, Variational aspects of Laplace eigenvalues on Riemannian surfaces, Adv. Math. 258 (2014), 191–239. MR 3190427, DOI 10.1016/j.aim.2014.03.006
- Luigi Provenzano and Joachim Stubbe, Weyl-type bounds for Steklov eigenvalues, J. Spectr. Theory 9 (2019), no. 1, 349–377. MR 3900789, DOI 10.4171/JST/250
- Qiaoling Wang and Changyu Xia, Sharp bounds for the first non-zero Stekloff eigenvalues, J. Funct. Anal. 257 (2009), no. 8, 2635–2644. MR 2555007, DOI 10.1016/j.jfa.2009.06.008
- Changwei Xiong, Comparison of Steklov eigenvalues on a domain and Laplacian eigenvalues on its boundary in Riemannian manifolds, J. Funct. Anal. 275 (2018), no. 12, 3245–3258. MR 3864501, DOI 10.1016/j.jfa.2018.09.012
Additional Information
- Bruno Colbois
- Affiliation: Institut de Mathématiques, Université de Neuchâtel, Rue Emile-Argand 11, CH-2000 Neuchâtel, Switzerland
- MR Author ID: 50460
- Email: bruno.colbois@unine.ch
- Ahmad El Soufi
- Affiliation: Laboratoire de Mathématiques et Physique Théorique, UMR-CNRS 6083, Université François Rabelais de Tours, Parc de Grandmont, 37200 Tours, France
- Alexandre Girouard
- Affiliation: Département de mathématiques et de statistique, Pavillon Alexandre-Vachon, Université Laval, Québec, Québec, G1V 0A6, Canada
- MR Author ID: 832728
- Email: alexandre.girouard@mat.ulaval.ca
- Received by editor(s): February 2, 2017
- Received by editor(s) in revised form: October 14, 2018
- Published electronically: May 17, 2019
- Additional Notes: During the first week of 2017, the first and third authors were supposed to travel to Tours and work with Ahmad El Soufi to complete this paper. We learned just a few days before our visit of his untimely death. Ahmad was a colleague and a friend. He will be dearly missed.
- Communicated by: Michael Wolf
- © Copyright 2019 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 147 (2019), 3813-3827
- MSC (2010): Primary 35P15, 53C21, 58J50; Secondary 53C23, 53C20
- DOI: https://doi.org/10.1090/proc/14426
- MathSciNet review: 3993774