Free Access
Volume 19, Number 2, April-June 2013
Page(s) 438 - 459
Published online 16 January 2013
  1. C.J.S. Alves and P.R.S. Antunes, The method of fundamental solutions applied to the calculation of eigenfrequencies and eigenmodes of 2D simply connected shapes. Comput. Mater. Continua 2 (2005) 251–266. [Google Scholar]
  2. P.R.S. Antunes and P. Freitas, Numerical optimization of low eigenvalues of the Dirichlet and Neumann Laplacians. J. Optim. Theory Appl. 154 (2012). DOI: 10.1007/s10957-011-9983-3. [Google Scholar]
  3. M.-H. Bossel, Membranes élastiquement liées : extension du théorème de Rayleigh–Faber–Krahn et de l’inégalité de Cheeger. C. R. Acad. Sci. Paris Sér. I Math. 302 (1986) 47–50. [Google Scholar]
  4. D. Bucur, Minimization of the kth eigenvalue of the Dirichlet Laplacian. Preprint (2012). [Google Scholar]
  5. D. Bucur and D. Daners, An alternative approach to the Faber-Krahn inequality for Robin problems. Calc. Var. Partial Differ. Equ. 37 (2010) 75–86. [CrossRef] [Google Scholar]
  6. D. Bucur and A. Henrot, Minimization of the third eigenvalue of the Dirichlet Laplacian. R. Soc. Lond. Proc. A 456 (2000) 985–996. [CrossRef] [Google Scholar]
  7. B. Colbois and A. El Soufi, Extremal eigenvalues of the Laplacian on Euclidean domains and Riemannian manifolds. Preprint (2012). [Google Scholar]
  8. R. Courant and D. Hilbert, Methods of mathematical physics I. Interscience Publishers, New York (1953). [Google Scholar]
  9. F.E. Curtis and M.L. Overton, A sequential quadratic programming algorithm for nonconvex, nonsmooth constrained optimization. SIAM J. Optim. 22 (2012) 474–500. [CrossRef] [Google Scholar]
  10. E.N. Dancer and D. Daners, Domain perturbation for elliptic equations subject to Robin boundary conditions. J. Differ. Equ. 138 (1997) 86–132. [CrossRef] [Google Scholar]
  11. D. Daners, A Faber-Krahn inequality for Robin problems in any space dimension. Math. Ann. 335 (2006) 767–785. [CrossRef] [MathSciNet] [Google Scholar]
  12. G. Faber, Beweis, dass unter allen homogenen membranen von gleicher Fläche und gleicher spannung die kreisförmige den tiefsten grundton gibt. Sitz. Ber. Bayer. Akad. Wiss. (1923) 169–172. [Google Scholar]
  13. T. Giorgi and R. Smits, Bounds and monotonicity for the generalized Robin problem. Z. Angew. Math. Phys. 59 (2008) 600–618. [CrossRef] [MathSciNet] [Google Scholar]
  14. A. Henrot, Extremum problems for eigenvalues of elliptic operators. Frontiers in Mathematics. Birkhäuser Verlag, Basel (2006). [Google Scholar]
  15. T. Kato, Perturbation theory for linear operators, 2nd edition. Springer-Verlag, Berlin. Grundlehren der Mathematischen Wissenschaften 132 (1976). [Google Scholar]
  16. J.B. Kennedy, An isoperimetric inequality for the second eigenvalue of the Laplacian with Robin boundary conditions. Proc. Amer. Math. Soc. 137 (2009) 627–633. [CrossRef] [MathSciNet] [Google Scholar]
  17. J.B. Kennedy, On the isoperimetric problem for the higher eigenvalues of the Robin and Wentzell Laplacians. Z. Angew. Math. Phys. 61 (2010) 781–792. [CrossRef] [MathSciNet] [Google Scholar]
  18. E. Krahn, Über eine von Rayleigh formulierte minimaleigenschaft des kreises. Math. Ann. 94 (1924) 97–100. [CrossRef] [MathSciNet] [Google Scholar]
  19. E. Krahn, Über Minimaleigenshaften der Kugel in drei und mehr dimensionen. Acta Comm. Univ. Dorpat. A 9 (1926) 1–44. [Google Scholar]
  20. A.A. Lacey, J.R. Ockendon and J. Sabina, Multidimensional reaction-diffusion equations with nonlinear boundary conditions. SIAM J. Appl. Math. 58 (1998) 1622–1647. [CrossRef] [Google Scholar]
  21. D. Mazzoleni and A. Pratelli, Existence of minimizers for spectral problems. Preprint (2012). [Google Scholar]
  22. J. Nocedal and S.J. Wright, Numer. Optim. Springer (1999). [Google Scholar]
  23. E. Oudet, Numerical minimization of eigenmodes of a membrane with respect to the domain. ESAIM : COCV 10 (2004) 315–330. [CrossRef] [EDP Sciences] [Google Scholar]
  24. J.W.S. Rayleigh, The theory of sound, 2nd edition. Macmillan, London (1896) (reprinted : Dover, New York (1945)). [Google Scholar]
  25. W. Rudin, Real and Complex Analysis, 3rd edition. McGraw-Hill, New York (1987). [Google Scholar]
  26. G. Szegö, Inequalities for certain eigenvalues of a membrane of given area. J. Rational Mech. Anal. 3 (1954) 343–356. [MathSciNet] [Google Scholar]
  27. H.F. Weinberger, An isoperimetric inequality for the N-dimensional free membrane problem. J. Rational Mech. Anal. 5 (1956) 633–636. [MathSciNet] [Google Scholar]
  28. S.A. Wolf and J.B. Keller, Range of the first two eigenvalues of the Laplacian. Proc. Roy. Soc. London A 447, (1994) 397–412. [CrossRef] [MathSciNet] [Google Scholar]

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