Shape optimization problems for eigenvalues of elliptic operators: Plenary lecture presented at the 76th Annual GAMM Conference, Luxembourg, 29 March -1 April 2005

ZAMM Zeitschrift fur Angewandte Mathematik und Mechanik

Volumn 86, Issue 3, 2006, Pages 171-184

  Belhachmi, Zakaria ^a   Bucur, Dorin ^a   Buttazzo, Giuseppe ^b   Sac Epée, Jean Marc ^a  

^a UNIVERSITÉ DE LORRAINE   (France)

^b UNIVERSITY OF PISA   (Italy)

Author keywords

Eigenvalues; Finite elements method; Laplace operator; Shape optimization

Indexed keywords

EID: 33644928050     PISSN: 00442267     EISSN: 15214001     Source Type: Journal    
DOI: 10.1002/zamm.200510259     Document Type: Conference Paper

References (30)

1. M. S. Ashbaugh, Open problems on eigenvalues of the Laplacian. In: Analytic and geometric inequalities and applications, Math. Appl. 478, pp. 13-28 (Kluwer Academic Publishers, Dordrecht, 1999).
2. M. S. Ashbaugh, Isoperimetric and universal inequalities for eigenvalues. Spectral theory and geometry (Edinburgh, 1998), London Math. Soc. Lecture Note Ser., 273, pp. 95-139 (Cambridge University Press, Cambridge, 1999).
3. M.S. Ashbaugh and R.D. Benguria, On Rayleigh's conjecture for the clamped plate and its generalization to three dimensions, Duke Math. J. 78(1), 1-17 (1995).
4. M. S. Ashbaugh and D. Bucur, On the isoperimetric inequality for the buckling of a clamped plate, Z. Angew. Math. Phys. 54(5), 756-770 (2003).
5. Z. Belhachmi, D. Bucur, and J.-M. Sac-Epée, Finite element approximation of the Neumann eigenvalue problem in domains with multiple cracks, Preprint LMAM, 2004-05.
6. D. Bucur, Regularity of optimal convex shapes, J. Convex Anal. 10(2), 501-516 (2003).
7. D. Bucur and G. Buttazzo, Variational Methods in Shape Optimization Problems. Progress in Nonlinear Differential Equations 65 (Birkhäuser Verlag, Basel, 2005).
8. D. Bucur, G. Buttazzo, and I. Figueiredo, On the attainable eigenvalues of the Laplace operator, SIAM J. Math. Anal. 30, 527-536 (1999).
9. D. Bucur, G. Buttazzo, and N. Varchon, On the problem of optimal cutting, SIAM J. Optimization 13(1), 157-167 (2002).
10. D. Bucur and A. Henrot, Minimization of the third eigenvlaue of the Dirichlet Laplacian, Proc. Roy. Soc. London, Ser. A 456, 985-996 (2000).
11. D. Bucur and N. Varchon, A duality approach for the boundary variations of Neumann problems, SIAM J. Math. Anal. 34(2), 460-477 (2002).
12. D. Bucur and J.P. Zolésio, N-Dimensional Shape Optimization under Capacitary Constraints, J. Differ. Equ. 123(2), 504-522 (1995).
13. D. Bucur and J.P. Zolésio, Wiener's criterion and shape continuity for the Dirichlet problem, Boll. Un. Mat. Ital. B11(4), 757-771 (1997).
14. D. Bucur and J.P. Zolésio, Boundary optimization under pseudo curvature constraint, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 23(4), 681-699 (1996).
15. V.I. Burenkov and E.B. Davies, Spectral stability of the Neumann Laplacian, J. Differ. Equ. 186(2), 485-508 (2002).
16. G. Buttazzo and G. Dal Maso, An existence result for a class of shape optimization problems, Arch. Rational Mech. Anal. 122, 183-195 (1993).
17. A. Chambolle and F. Doveri, Continuity of Neumann linear elliptic problems on varying two-dimensional bounded open sets, Commun. Partial Differ. Equations 22(5-6), 811-840 (1997).
18. D. Chenais, On the existence of a solution in a domain identification problem, J. Math. Anal. Appl. 52, 189-219 (1975).
19. N. Dunford and J.T. Schwartz, Linear Operators, Part II, Spectral Theory (Interscience Publishers, New York, London, 1963).
20. M. Hayouni, Sur la minimisation de la première valeur propre du laplacien, C. R. Acad. Sci. Paris Série. I Math. 330(7), 551-556 (2000).
21. A. Henrot, Minimization problems for eigenvalues of the Laplacian. Dedicated to Philippe Bénilan. J. Evol. Equ. 3(3), 443-461 (2003).
22. B. Kawohl, Some nonconvex shape optimization problems, in: Optimal Shape Design, edited by A. Cellina and A. Ornelas, Springer Lecture Notes in Math. 1740, pp. 7-46 (2000).
23. N.S. Nadirashvili, An isoperimetric inequality for the main frequency of a clamped plate (Russian), Dokl. Akad. Nauk 332(4), 436-439 (1993),
24. translation in Phys. Dokl. 38 (1993), no. (10), 419-421 (1994).
25. E. Oudet, Numerical Minimization of eigenmodes of a membrane with respect to the domain, ESAIM COCV 10, 315-330 (2004).
26. G. Szego, Inequalities for certain eigenvalues of a membrane of given area, J. Rational Mech. Anal. 3, 343-356 (1954).
27. H. Weinberger, An isoperimetric inequality for the N-dimensional free membrane problem, J. Rational Mech. Anal. 5, 633-636 (1956).
28. V. Šverák, On optimal shape design, J. Math. Pures Appl. 72, 537-551 (1993).
29. B. Willms, An isoperimetric inequality for the buckling of a clamped plate, Lecture at the Oberwolfach meeting on "Qualitative properties of PDE" (organized by H. Berestycki, B. Kawohl, and G. Talenti), Feb. 1995.
30. S.A. Wolf and J. B. Keller, Range of the first two eigenvalues of the Laplacian, Proc. Roy. Soc. Lond. A 447, 397-412 (1994).