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All issues Volume 23 / No 2 (April-June 2017) ESAIM: COCV, 23 2 (2017) 685-720 References
Issue
ESAIM: COCV
Volume 23, Number 2, April-June 2017
Page(s) 685 - 720
DOI https://doi.org/10.1051/cocv/2016008
Published online 26 January 2017
  1. P.R.S. Antunes and P. Freitas, Numerical optimization of low eigenvalues of the Dirichlet and Neumann Laplacians. J. Optim. Theory Appl. 154 (2012) 235–257. [CrossRef] [MathSciNet] [Google Scholar]
  2. M.G. Armentano and R.G. Durán, Asymptotic lower bounds for eigenvalues by nonconforming finite element methods. Electron. Trans. Numer. Anal. 17 (2004) 93–101. [MathSciNet] [Google Scholar]
  3. M.S. Ashbaugh and R.D. Benguria, Isoperimetric inequalities for eigenvalues of the Laplacian. Proc. Symp. Pure Math. 76 (2007) 105–139. [Google Scholar]
  4. P.H. Bérard and G. Besson, Lectures on Spectral Geometry. Conselho Nacional de Desenvolvimento Cientifico e Tecnologico, Instituto de Matematica Pura e Aplicada (1985). [Google Scholar]
  5. M. Berger, Sur les premiéres valeurs propres des variétés Riemanniennes. Comp. Math. 26 (1973) 129–149. [Google Scholar]
  6. D. Boffi, Finite element approximation of eigenvalue problems. Acta Numer. 19 (2010) 1–120. [CrossRef] [MathSciNet] [Google Scholar]
  7. P. Buser, Geometry and spectra of compact Riemann surfaces. Springer (2010). [Google Scholar]
  8. B. Colbois and J. Dodziuk, Riemannian metrics with large λ1, Proc. Am. Math. Soc. 122 (1994) 905–906. [Google Scholar]
  9. B. Colbois and A. El Soufi, Extremal eigenvalues of the Laplacian in a conformal class of metrics: the ‘conformal spectrum’. Ann. Global Anal. Geom. 24 (2003) 337–349. [CrossRef] [MathSciNet] [Google Scholar]
  10. B. Colbois and A. El Soufi, Extremal eigenvalues of the Laplacian on Euclidean domains and closed surfaces. Math. Z. 278 (2014) 529–546. [CrossRef] [MathSciNet] [Google Scholar]
  11. B. Colbois, A. El Soufi and A. Girouard, Isoperimetric control of the spectrum of a compact hypersurface. J. Reine Angew. Math. 2013 (2013) 49–65. [CrossRef] [Google Scholar]
  12. B. Colbois, E.B. Dryden and A. El Soufi, Bounding the eigenvalues of the Laplace–Beltrami operator on compact submanifolds. Bull. London Math. Soc. 42 (2010) 96–108. [CrossRef] [MathSciNet] [Google Scholar]
  13. I. Chavel, Eigenvalues in Riemannian geometry. Academic Press (1984). [Google Scholar]
  14. S. Cox and J. McLaughlin, Extremal eigenvalue problems for composite membranes. Part I and II. Appl. Math. Optim. 22 (1990) 153–167 and 169–187. [CrossRef] [MathSciNet] [Google Scholar]
  15. T.A. Driscoll, N. Hale and L.N. Trefethen, Chebfun guide. Pafnuty Publications, Oxford (2014). [Google Scholar]
  16. A. El Soufi and S. Ilias, Laplacian eigenvalue functionals and metric deformations on compact manifolds. J. Geom. Phys. 58 (2008) 89–104. [CrossRef] [MathSciNet] [Google Scholar]
  17. S. Friedland, Extremal eigenvalue problems defined on conformal classes of compact Riemannian manifolds. Comment. Math. Helvetici 54 (1979) 494–507. [CrossRef] [MathSciNet] [Google Scholar]
  18. O. Giraud and K. Thas, Hearing shapes of drums-mathematical and physical aspects of isospectrality. Rev. Mod. Phys. 82 (2010) 2213–2255. [CrossRef] [Google Scholar]
  19. A. Girouard, N. Nadirashvili and I. Polterovich, Maximization of the second positive Neumann eigenvalue for planar domains. J. Differ. Geom. 83 (2009) 637–661. [Google Scholar]
  20. R. Glowinski and D.C. Sorensen, Computing the eigenvalues of the Laplace-Beltrami operator on the surface of a torus: A numerical approach. Partial Differential Equations. In vol. 16 of Comput. Methods Appl. Sci. Springer (2008) 225–232. [Google Scholar]
  21. A. Henrot, Extremum Problems for Eigenvalues of Elliptic Operators. Verlag, Birkhäuser (2006). [Google Scholar]
  22. J. Hersch, Quatre propriétés isopérimétriques de membranes sphériques homogenes. (French). C. R. Acad. Sci. Paris Sér. AB 270 (1970) A1645–A1648. [Google Scholar]
  23. Y. Imayoshi and M. Taniguchi, An Introduction to Teichmüller Spaces. Springer-Verlag (1992). [Google Scholar]
  24. D. Jakobson, M. Levitin, N. Nadirashvili, N. Nigam and I. Polterovich, How large can the first eigenvalue be on a surface of genus two? Int. Math. Res. Not. 2005 (2005) 3967–3985. [CrossRef] [Google Scholar]
  25. D. Jakobson, N. Nadirashvili and I. Polterovich, Extremal metric for the first eigenvalue on a Klein bottle. Canadian J. Math. 58 (2006) 381–400. [CrossRef] [MathSciNet] [Google Scholar]
  26. M.A. Karpukhin, Nonmaximality of known extremal metrics on torus and Klein bottle. Sb. Math. 204 (2013) 1–17. [CrossRef] [MathSciNet] [Google Scholar]
  27. M.A. Karpukhin, Spectral properties of bipolar surfaces to Otsuki tori. J. Spectr. Theory 4 (2014) 87–111. [CrossRef] [MathSciNet] [Google Scholar]
  28. G. Kokarev, Variational aspects of Laplace eigenvalues on Riemannian surfaces. Adv. Math. 258 (2014) 191–239. [CrossRef] [MathSciNet] [Google Scholar]
  29. N. Korevaar, Upper bounds for eigenvalues of conformal metrics. J. Differ. Geom. 37 (1993) 73–93. [Google Scholar]
  30. P. Kroger, On the spectral gap for compact manifolds. J. Differ. Geom. 36 (1992), 315–330. [Google Scholar]
  31. R. Lai, Z. Wen, W. Yin, X. Gu and L.M. Lui, Folding-free global conformal mapping for genus-0 surfaces by harmonic energy minimization, J. Sci. Comput. 18 (2014) 705–725. [CrossRef] [Google Scholar]
  32. H. Lapointe, Spectral properties of bipolar minimal surfaces in S4. Differ. Geom. Appl. 26 (2008) 9–22. [CrossRef] [Google Scholar]
  33. S. Larsson and V. Thomée, Partial Differential Equations with Numerical Methods. New York (2005). [Google Scholar]
  34. R.S. Laugesen and B.A. Siudeja, Sums of Laplace eigenvalues: Rotations and tight frames in higher dimensions. J. Math. Phys. 52 (2011) 093703. [CrossRef] [MathSciNet] [Google Scholar]
  35. R.B. Lehoucq and D.C. Sorensen, Deflation techniques for an implicitly restarted Arnoldi iteration. SIAM J. Matrix Anal. Appl. 17 (1996) 789–821. [CrossRef] [MathSciNet] [Google Scholar]
  36. A.S. Lewis and M.L. Overton, Nonsmooth optimization via quasi-Newton methods. Math. Program. 141 (2013) 135–163. [CrossRef] [MathSciNet] [Google Scholar]
  37. J. Ling and Z. Lu, Bounds of eigenvalues on Riemannian manifolds. ALM 10 (2010) 241–264. [Google Scholar]
  38. J. Milnor, Eigenvalues of the Laplace operator on certain manifolds. Proc. Nat. Acad. Sci. USA 51 (1964) 542. [CrossRef] [Google Scholar]
  39. N. Nadirashvili, Berger’s isoperimetric problem and minimal immersions of surfaces. Geom. Funct. Anal. 6 (1996) 877–897. [CrossRef] [MathSciNet] [Google Scholar]
  40. N. Nadirashvili, Isoperimetric inequality for the second eigenvalue of a sphere. J. Differ. Geom. 61 (2002) 335–340. [Google Scholar]
  41. N. Nadirashvili and Y. Sire, Conformal spectrum and harmonic maps. Preprint arXiv:1007.3104 (2014). [Google Scholar]
  42. B. Osting, Optimization of spectral functions of Dirichlet–Laplacian eigenvalues. J. Comp. Phys. 229 (2010) 8578–8590. [CrossRef] [Google Scholar]
  43. B. Osting and C.Y. Kao, Minimal convex combinations of sequential Laplace-Dirichlet eigenvalues. SIAM J. Sci. Comput. 35 (2013) B731–B750. [CrossRef] [Google Scholar]
  44. B. Osting and C.Y. Kao, Minimal convex combinations of three sequential Laplace–Dirichlet eigenvalues. Appl. Math. Optim. 69 (2014) 123–139. [CrossRef] [MathSciNet] [Google Scholar]
  45. E. Oudet, Numerical minimization of eigenmodes of a membrane with respect to the domain. ESAIM: COCV 10 (2004) 315–335. [CrossRef] [EDP Sciences] [Google Scholar]
  46. F. Pacard and P. Sicbaldi, Extremal domains for the first eigenvalue of the Laplace-Beltrami operator. Ann. Inst. Fourier 59 (2009) 515–542. [CrossRef] [MathSciNet] [Google Scholar]
  47. A.V. Penskoi, Extremal spectral properties of Lawson tau-surfaces and the Lamé equation. Moscow Math. J. 12 (2012) 173–192. [MathSciNet] [Google Scholar]
  48. A.V. Penskoi, Extremal metrics for eigenvalues of the Laplace-Beltrami operator on surfaces. Russian Math. Surveys 68 (2013) 1073. [CrossRef] [MathSciNet] [Google Scholar]
  49. A.V. Penskoi, Extremal spectral properties of Otsuki tori. Math. Nachr. 286 (2013) 379–391. [CrossRef] [MathSciNet] [Google Scholar]
  50. A.V. Penskoi, Generalized Lawson tori and Klein bottles. J. Geom. Anal. 25 (2015) 2645–2666. [CrossRef] [MathSciNet] [Google Scholar]
  51. R. Petrides, Existence and regularity of maximal metrics for the first Laplace eigenvalue on surfaces. Preprint arXiv:1310.4697 (2013). [Google Scholar]
  52. R. Petrides, Maximization of the second conformal eigenvalue of spheres. Proc. Am. Math. Soc. 142 (2014) 2385–2394. [CrossRef] [Google Scholar]
  53. A. Qiu, D. Bitouk and M.I. Miller, Smooth functional and structural maps on the neocortex via orthonormal bases of the Laplace–Beltrami operator. IEEE Trans. Med. Imaging 25 (2006) 1296–1306. [CrossRef] [PubMed] [Google Scholar]
  54. J.W.S. Rayleigh, The Theory of Sound. Vol. 1. Dover Publications New York (1877). [Google Scholar]
  55. M. Reuter, F.-E. Wolter and N. Peinecke, Laplace–Beltrami spectra as ‘Shape-DNA’ of surfaces and solids. Comput. Aid. Design 38 (2006) 342–366. [CrossRef] [Google Scholar]
  56. R. Schoen and S.-T. Yau, Lectures on differential geometry. International Press (1994). [Google Scholar]
  57. Y. Shi, R. Lai, R. Gill, D. Pelletier, D. Mohr, N. Sicotte and A.W. Toga, Conformal metric optimization on surface (cmos) for deformation and mapping in Laplace-Beltrami embedding space, Medical Image Computing and Computer-Assisted Intervention–MICCAI 2011. Springer (2011) 327–334. [Google Scholar]
  58. D. C. Sorensen, Implicit application of polynomial filters in a k-step Arnoldi method. SIAM J. Matrix Anal. Appl. 13 (1992) 357–385. [CrossRef] [MathSciNet] [Google Scholar]
  59. L. N. Trefethen, Spectral methods in MATLAB. Vol. 10. SIAM (2000). [Google Scholar]
  60. H. Urakawa, On the least positive eigenvalue of the Laplacian for compact group manifolds. J. Math. Soc. Jpn 31 (1979) 209–226. [CrossRef] [Google Scholar]
  61. P.C. Yang and S.-T. Yau, Eigenvalues of the Laplacian of compact Riemann surfaces and minimal submanifolds. Ann. Sc. Norm. Super. Pisa-Cl. Sci. 7 (1980) 55–63. [Google Scholar]

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