Numerical minimization of eigenmodes of a membrane with respect to the domain

 Access to full text

 Full (PDF)

1. [1] G. Allaire, Shape optimization by the homogenization method. Springer-Verlag, New York (2001). Zbl0990.35001MR1859696
2. [2] G. Allaire, F. Jouve and A.M. Toader, A level-set method for shape optimization. C. R. Acad. Sci. Paris 334 (2002) 1125-1130. Zbl1115.49306MR1911658
3. [3] M. Bendsoe, Optimization of structural Topology, Shape and Material. Springer (1995). Zbl0822.73001MR1350791
4. [4] M. Bendsoe and C. Mota Soares, Topology optimization of structures. Kluwer Academic Press, Dordrechts (1993). MR1250185
5. [5] G. Buttazzo and G. Dal Maso, An Existence Result for a Class of Shape Optimization Problems. Arch. Ration. Mech. Anal. 122 (1993) 183-195. Zbl0811.49028MR1217590
6. [6] M.G. Crandall and P.L. Lions, Viscosity Solutions of Hamilton-Jacobi Equations. Trans. Amer. Math. Soc. 277 (1983) 1-43. Zbl0599.35024MR690039
7. [7] 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. Zbl49.0342.03JFM49.0342.03
8. [8] S. Finzi Vita, Constrained shape optimization for Dirichlets problems: discretization via relaxation. Adv. Math. Sci. Appl. 9 (1999) 581-596. Zbl0962.49023MR1725674
9. [9] H. Hamda, F. Jouve, E. Lutton, M. Schoenauer and M. Sebag, Représentations non structurées en optimisation topologique de formes par algorithmes évolutionnaires. Actes du 32${}^{\mathrm{e}}$ Congrès d’Analyse Numérique, Canum. ESAIM Proc. 8 (2000). 
10. [10] A. Henrot, Minimization problems for eigenvalues of the Laplacian. J. Evol. Eq. 3 (2003) 443-461. Zbl1049.49029MR2019029
11. [11] A. Henrot and E. Oudet, Le stade ne minimise pas ${\lambda }_2$ parmi les ouverts convexes du plan. C. R. Acad. Sci. Paris Sér. I Math. 332 (2001) 417-422. Zbl1011.35100MR1826627
12. [12] A. Henrot and E. Oudet, Minimizing the second eigenvalue of the Laplace operator with Dirichlet boundary conditions. Arch. Ration. Mech. Anal. 169 (2003) 73-87. Zbl1055.35080MR1996269
13. [13] A. Henrot and M. Pierre, Optimisation de forme (in preparation). 
14. [14] E. Krahn, Über eine von Rayleigh formulierte Minimaleigenshaft des Kreises. Math. Ann. 94 (1925) 97-100. Zbl51.0356.05MR1512244JFM51.0356.05
15. [15] E. Krahn, Über Minimaleigenshaften der Kugel in drei und mehr Dimensionen. Acta Comm. Univ. Dorpat. A9 (1926) 1-44. Zbl52.0510.03JFM52.0510.03
16. [16] S. Osher and F. Santosa, Level set methods for optimization problems involving geometry and constraints: frequencies of a two-density inhomogeneous drum. J. Comput. Phys. 171 (2001) 272-288. Zbl1056.74061MR1843648
17. [17] S. Osher and J.A. Sethian, Front propagation with curvature-dependant speed: Algorithms based on Hamilton-Jacobi formulations J. Comput. Phys. 79 (1988) 12-49. Zbl0659.65132MR965860
18. [18] E. Oudet, $Quelques$ résultats en optimisation de forme et stabilisation. Prépublication de l’Institut de recherche mathématique avancée, Strasbourg (2002). 
19. [19] M. Pierre and J.M. Roche, Numerical simulation of tridimensional electromagnetic shaping of liquid metals. Numer. Math. 65 (1993) 203-217. Zbl0792.65096MR1222619
20. [20] G. Pólya and G. Szegö, Isoperimetric Inequalities in Mathematical Physics. Ann. Math. Stud. 27 (1952). Zbl0044.38301MR43486
21. [21] J.A. Sethian, Level Set Methods and Fast Marching Methods. Cambridge University Press (1999). Zbl0973.76003MR1700751
22. [22] J. Sokolowski and J.P. Zolesio, Introduction to shape optimization: shape sensitivity analysis. Springer, Berlin, Springer Ser. Comput. Math. 10 (1992). Zbl0761.73003MR1215733
23. [23] B.A. Troesch, Elliptical membranes with smallest second eigenvalue. Math. Comp. 27 (1973) 767-772. Zbl0271.35018MR421277
24. [24] S.A. Wolf and J.B. Keller, Range of the first two eigenvalues of the laplacian. Proc. Roy. Soc. Lond. A 447 (1994) 397-412. Zbl0816.35097MR1312811

1. Pedro Ricardo Simão Antunes, Pedro Freitas, James Bernard Kennedy, Asymptotic behaviour and numerical approximation of optimal eigenvalues of the Robin laplacian
2. Davide Buoso, Pier Domenico Lamberti, Eigenvalues of polyharmonic operators on variable domains
3. Édouard Oudet, Marc Oliver Rieger, Local minimizers of functionals with multiple volume constraints