Revista Matemática Complutense

, Volume 24, Issue 2, pp 277–322 | Cite as

Spectral optimization problems



In this survey paper we present a class of shape optimization problems where the cost function involves the solution of a PDE of elliptic type in the unknown domain. In particular, we consider cost functions which depend on the spectrum of an elliptic operator and we focus on the existence of an optimal domain. The known results are presented as well as a list of still open problems. Related fields as optimal partition problems, evolution flows, Cheeger-type problems, are also considered.


Optimization problems for eigenvalues Shape optimization Capacity Integral functionals Sobolev spaces Optimality conditions Calculus of variations Relaxation 

Mathematics Subject Classification (2000)

49J45 49R05 35P15 47A75 35J25 


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  1. 1.
    Adams, R.A.: Sobolev Spaces. Academic Press, New York (1975) MATHGoogle Scholar
  2. 2.
    Adams, D.R., Hedberg, L.I.: Function Spaces and Potential Theory. Springer, Berlin (1996) Google Scholar
  3. 3.
    Allaire, G.: Shape Optimization by the Homogenization Method. Applied Mathematical Sciences, vol. 146. Springer, New York (2002) MATHGoogle Scholar
  4. 4.
    Alter, F., Caselles, V.: Uniqueness of the Cheeger set of a convex body. Nonlinear Anal. 70, 32–44 (2009) MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Ambrosio, L., Braides, A.: Functionals defined on partitions of sets of finite perimeter. I: integral representation and Γ-convergence, II: semicontinuity, relaxation and homogenization. J. Math. Pures Appl. 69, 285–305 (1990) and 307–333 MathSciNetMATHGoogle Scholar
  6. 6.
    Ambrosio, L., Buttazzo, G.: An optimal design problem with perimeter penalization. Calc. Var. 1, 55–69 (1993) MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Ambrosio, L., Fusco, N., Pallara, D.: Functions of Bounded Variation and Free Discontinuity Problems. Oxford Mathematical Monographs. Clarendon, Oxford (2000) MATHGoogle Scholar
  8. 8.
    Ambrosio, L., Gigli, N., Savaré, G.: Gradient Flows in Metric Spaces and in the Space of Probability Measures. Lectures in Mathematics ETH Zürich. Birkhäuser, Basel (2005) MATHGoogle Scholar
  9. 9.
    Arrieta, J.M.: Neumann eigenvalue problems on exterior perturbations of the domain. J. Differ. Equ. 118, 54–103 (1995) MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Ashbaugh, M.S.: Open problems on eigenvalues of the Laplacian. In: Analytic and Geometric Inequalities and Applications. Math. Appl., vol. 478, pp. 13–28. Kluwer Academic, Dordrecht (1999) Google Scholar
  11. 11.
    Ashbaugh, M.S., Benguria, R.D.: A sharp bound for the ratio of the first two eigenvalues of Dirichlet Laplacians and extensions. Ann. Math. 135(3), 601–628 (1992) MathSciNetCrossRefGoogle Scholar
  12. 12.
    Athanasopoulos, I., Caffarelli, L.A., Kenig, C., Salsa, S.: An area-Dirichlet integral minimization problem. Commun. Pure Appl. Math. 54, 479–499 (2001) MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Attouch, H.: Variational Convergence for Functions and Operators. Pitman, Boston (1984) MATHGoogle Scholar
  14. 14.
    Attouch, H., Buttazzo, G., Michaille, G.: Variational Analysis in Sobolev and BV Spaces: Applications to PDEs and Optimization. MPS-SIAM Book Series on Optimization, vol. 6. SIAM, Philadelphia (2006) MATHGoogle Scholar
  15. 15.
    Belhachmi, Z., Bucur, D., Buttazzo, G., Sac-Epée, J.M.: Shape optimization problems for eigenvalues of elliptic operators. Z. Angew. Math. Mech. 86(3), 171–184 (2006) MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Bendsøe, M., Sigmund, O.: Topology Optimization. Theory, Methods and Applications. Springer, Berlin (2003) Google Scholar
  17. 17.
    Bossel, M.H.: Membranes élastiquement liées: extension du théorème de Rayleigh-Faber-Krahn et de l’inégalité de Cheeger. C. R. Math. 302-I, 47–50 (1986) MathSciNetGoogle Scholar
  18. 18.
    Bossel, M.H.: Membranes élastiquement liées inhomogènes ou sur une surface: une nouvelle extension du théorème isopérimétrique de Rayleigh-Faber-Krahn. Z. Angew. Math. Phys. 39, 733–742 (1988) MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Bourdin, B., Bucur, D., Oudet, E.: Optimal partition for eigenvalues. SIAM J. Sci. Comput. 31(6), 4100–4114 (2009) MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Braides, A.: Γ-Convergence for Beginners. Oxford Lecture Series in Mathematics and its Applications, vol. 22. Oxford University Press, Oxford (2002) MATHCrossRefGoogle Scholar
  21. 21.
    Briançon, T.: Regularity of optimal shapes for the Dirichlet’s energy with volume constraint. ESAIM Control Optim. Calc. Var. 10, 99–122 (2004) MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Briançon, T., Hayouni, M., Pierre, M.: Lipschitz continuity of state functions in some optimal shaping. Calc. Var. Partial Differ. Equ. 23, 13–32 (2005) MATHCrossRefGoogle Scholar
  23. 23.
    Briançon, T., Lamboley, J.: Regularity of the optimal shape for the first eigenvalue of the Laplacian with volume and inclusion constraints. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 26, 1149–1163 (2009) MATHCrossRefGoogle Scholar
  24. 24.
    Brock, F.: Continuous Steiner-symmetrization. Math. Nachr. 172, 25–48 (1995) MathSciNetMATHCrossRefGoogle Scholar
  25. 25.
    Brock, F.: Continuous symmetrization and symmetry of solutions of elliptic problems. Proc. Indian Acad. Sci. Math. Sci. 110, 157–204 (2000) MathSciNetMATHCrossRefGoogle Scholar
  26. 26.
    Bucur, D., Buttazzo, G.: Variational Methods in Shape Optimization Problems. Progress in Nonlinear Differential Equations, vol. 65. Birkhäuser, Basel (2005) MATHGoogle Scholar
  27. 27.
    Bucur, D., Buttazzo, G.: On the characterization of the compact embedding of Sobolev spaces. Calc. Var. (2011, to appear). Available at and at
  28. 28.
    Bucur, D., Buttazzo, G., Figueiredo, I.: On the attainable eigenvalues of the Laplace operator. SIAM J. Math. Anal. 30, 527–536 (1999) MathSciNetMATHCrossRefGoogle Scholar
  29. 29.
    Bucur, D., Buttazzo, G., Henrot, A.: Existence results for some optimal partition problems. Adv. Math. Sci. Appl. 8, 571–579 (1998) MathSciNetMATHGoogle Scholar
  30. 30.
    Bucur, D., Buttazzo, G., Henrot, A.: Minimization of λ 2(Ω) with a perimeter constraint. Indiana Univ. Math. J. 58(6), 2709–2728 (2009) MathSciNetMATHCrossRefGoogle Scholar
  31. 31.
    Bucur, D., Buttazzo, G., Lux, A.: Quasistatic evolution in debonding problems via capacitary methods. Arch. Ration. Mech. Anal. 190(2), 281–306 (2008) MathSciNetMATHCrossRefGoogle Scholar
  32. 32.
    Bucur, D., Daners, D.: An alternative approach to the Faber-Krahn inequality for Robin problems. Calc. Var. Partial Differ. Equ. 37, 75–86 (2010) MathSciNetMATHCrossRefGoogle Scholar
  33. 33.
    Bucur, D., Giacomini, A.: A variational approach to the isoperimetric inequality for the Robin eigenvalue problem. Preprint available at
  34. 34.
    Bucur, D., Henrot, A.: Minimization of the third eigenvalue of the Dirichlet Laplacian. Proc. R. Soc. Lond. Ser. A, Math. Phys. Sci. 456, 985–996 (2000) MathSciNetMATHCrossRefGoogle Scholar
  35. 35.
    Bucur, D., Trebeschi, P.: Shape optimization problem governed by nonlinear state equation. Proc. R. Soc. Edinb. 128 A, 945–963 (1998) MathSciNetGoogle Scholar
  36. 36.
    Bucur, D., Zolésio, J.P.: N-dimensional shape optimization under capacitary constraints. J. Differ. Equ. 123(2), 504–522 (1995) MATHCrossRefGoogle Scholar
  37. 37.
    Buttazzo, G.: Semicontinuity, Relaxation and Integral Representation in the Calculus of Variations. Pitman Res. Notes Math. Ser., vol. 207. Longman, Harlow (1989) MATHGoogle Scholar
  38. 38.
    Buttazzo, G., Dal Maso, G.: Shape optimization for Dirichlet problems: relaxed solutions and optimality conditions. Bull. Am. Math. Soc. 23, 531–535 (1990) MathSciNetMATHCrossRefGoogle Scholar
  39. 39.
    Buttazzo, G., Dal Maso, G.: Shape optimization for Dirichlet problems: relaxed formulation and optimality conditions. Appl. Math. Optim. 23, 17–49 (1991) MathSciNetMATHCrossRefGoogle Scholar
  40. 40.
    Buttazzo, G., Dal Maso, G.: An existence result for a class of shape optimization problems. Arch. Ration. Mech. Anal. 122, 183–195 (1993) MathSciNetMATHCrossRefGoogle Scholar
  41. 41.
    Buttazzo, G., Giaquinta, M., Hildebrandt, S.: One-Dimensional Calculus of Variations: An Introduction. Oxford University Press, Oxford (1998). Russian translation: Tamara Rozhkovskaya, Novosibirsk (2002) Google Scholar
  42. 42.
    Buttazzo, G., Timofte, C.: On the relaxation of some optimal partition problems. Adv. Math. Sci. Appl. 12(2), 509–520 (2002) MathSciNetMATHGoogle Scholar
  43. 43.
    Buttazzo, G., Wagner, A.: On some rescaled shape optimization problems. Adv. Calc. Var. 3(2), 213–232 (2010) MathSciNetMATHCrossRefGoogle Scholar
  44. 44.
    Caffarelli, L.A., Lin, F.H.: An optimal partition problem for eigenvalues. J. Sci. Comput. 31, 5–18 (2007) MathSciNetCrossRefGoogle Scholar
  45. 45.
    Caffarelli, L.A., Lin, F.H.: Analysis on the junction of domain walls. Discrete Contin. Dyn. Syst. 28(3), 915–929 (2010) MathSciNetMATHCrossRefGoogle Scholar
  46. 46.
    Caselles, V., Chambolle, A., Novaga, M.: Uniqueness of the Cheeger set of a convex body. Pac. J. Math. 232, 77–90 (2007) MathSciNetMATHCrossRefGoogle Scholar
  47. 47.
    Chipot, M., Dal Maso, G.: Relaxed shape optimization: the case of nonnegative data for the Dirichlet problem. Adv. Math. Sci. Appl. 1, 47–81 (1992) MathSciNetMATHGoogle Scholar
  48. 48.
    Cioranescu, D., Murat, F.: Un terme étrange venu d’ailleurs. In: Nonlinear Partial Differential Equations and Their Applications. Coll. de France Semin. Vol. II, Res. Notes Math., vol. 60, pp. 98–138. Pitman, Boston (1982) Google Scholar
  49. 49.
    Conti, M., Terracini, S., Verzini, G.: An optimal partition problem related to nonlinear eigenvalues. J. Funct. Anal. 198, 160–196 (2003) MathSciNetMATHCrossRefGoogle Scholar
  50. 50.
    Cox, S.J., Uhlig, P.X.: Where best to hold a drum fast. SIAM J. Optim. 9(4), 948–964 (1999) MathSciNetMATHCrossRefGoogle Scholar
  51. 51.
    Dacorogna, B.: Direct Methods in the Calculus of Variations. Appl. Math. Sciences, vol. 78. Springer, Berlin (1989) MATHGoogle Scholar
  52. 52.
    Dal Maso, G.: Γ-convergence and μ-capacities. Ann. Sc. Norm. Super. Pisa, Cl. Sci. 14, 423–464 (1988) MathSciNetGoogle Scholar
  53. 53.
    Dal Maso, G.: An Introduction to Γ-Convergence. Birkhäuser, Boston (1993) Google Scholar
  54. 54.
    Dal Maso, G., Mosco, U.: Wiener’s criterion and Γ-convergence. Appl. Math. Optim. 15, 15–63 (1987) MathSciNetMATHCrossRefGoogle Scholar
  55. 55.
    Dal Maso, G., Murat, F.: Asymptotic behavior and correctors for Dirichlet problems in perforated domains with homogeneous monotone operators. Ann. Sc. Norm. Super. Pisa, Cl. Sci. 24, 239–290 (1997) MathSciNetMATHGoogle Scholar
  56. 56.
    Daners, D.: A Faber-Krahn inequality for Robin problems in any space dimension. Math. Ann. 335(4), 767–785 (2006) MathSciNetMATHCrossRefGoogle Scholar
  57. 57.
    De Giorgi, E.: New problems on minimizing movements. In: Boundary Value Problems for Partial Differential Equations. Res. Notes Appl. Math., vol. 29, pp. 81–98. Masson, Paris (1993) Google Scholar
  58. 58.
    Doob, J.L.: Classical Potential Theory and Its Probabilistic Counterpart. Springer, Berlin (1984) MATHGoogle Scholar
  59. 59.
    Ekeland, I., Temam, R.: Convex Analysis and Variational Problems. Studies in Mathematics and Its Applications, vol. 1. North-Holland, Amsterdam (1976) MATHGoogle Scholar
  60. 60.
    Faber, G.: Beweis, dass unter allen homogenen Membranen von gleicher Fläche und gleicher Spannung die kreisförmige den tiefsten Grundton gibt. Sitz. Ber. Bayer. Akad. Wiss. 169–172 (1923) Google Scholar
  61. 61.
    Fuglede, B.: Finely Harmonic Functions. Lecture Notes in Math., vol. 289. Springer, Berlin (1972) MATHGoogle Scholar
  62. 62.
    Garabedian, P.R., Schiffer, M.: Variational problems in the theory of elliptic partial differential equations. J. Ration. Mech. Anal. 2, 137–171 (1953) MathSciNetMATHGoogle Scholar
  63. 63.
    Giusti, E.: Minimal Surfaces and Functions of Bounded Variation. Birkhäuser, Boston (1984) MATHGoogle Scholar
  64. 64.
    Giusti, E.: Metodi Diretti nel Calcolo delle Variazioni. Unione Matematica Italiana, Bologna (1994) MATHGoogle Scholar
  65. 65.
    Hayouni, M.: Sur la minimisation de la première valeur propre du laplacien. C. R., Math. 330(7), 551–556 (2000) MathSciNetMATHGoogle Scholar
  66. 66.
    Heionen, J., Kilpelainen, T., Martio, O.: Nonlinear Potential Theory of Degenerate Elliptic Equations. Clarendon, Oxford (1993) Google Scholar
  67. 67.
    Henrot, A.: Minimization problems for eigenvalues of the Laplacian. J. Evol. Equ. 3(3), 443–461 (2003) MathSciNetMATHCrossRefGoogle Scholar
  68. 68.
    Henrot, A.: Extremum Problems for Eigenvalues of Elliptic Operators. Frontiers in Mathematics. Birkhäuser, Basel (2006) MATHGoogle Scholar
  69. 69.
    Henrot, A., Oudet, E.: Le stade ne minimise pas λ 2 parmi les ouverts convexes du plan. C. R. Math. 332, 417–422 (2001) MathSciNetMATHGoogle Scholar
  70. 70.
    Henrot, A., Oudet, E.: Minimizing the second eigenvalue of the Laplace operator with Dirichlet boundary conditions. Arch. Ration. Mech. Anal. 169, 73–87 (2003) MathSciNetMATHCrossRefGoogle Scholar
  71. 71.
    Henrot, A., Pierre, M.: Variation et Optimisation de Formes. Une Analyse Géométrique. Mathématiques & Applications, vol. 48. Springer, Berlin (2005) MATHGoogle Scholar
  72. 72.
    Iversen, M., van den Berg, M.: On the minimization of Dirichlet eigenvalues of the Laplace operator. Preprint, 0905.4812
  73. 73.
    Kohler-Jobin, M.T.: Une méthode de comparaison isopérimétrique de fonctionnelles de domaines de la physique mathématique. I. Première partie: une démonstration de la conjecture isopérimétrique \(P\lambda^{2}\ge\pi j_{0}^{4}/2\) de Pó1ya et Szegö. Z. Angew. Math. Phys. 29, 757–766 (1978) MathSciNetMATHCrossRefGoogle Scholar
  74. 74.
    Kohler-Jobin, M.T.: Une méthode de comparaison isopérimétrique de fonctionnelles de domaines de la physique mathématique. II. Seconde partie: cas inhomogène: une inégalité isopérimétrique entre la fréquence fondamentale d’une membrane et l’énergie d’équilibre d’un problème de Poisson. Z. Angew. Math. Phys. 29, 767–776 (1978) MathSciNetMATHCrossRefGoogle Scholar
  75. 75.
    Krahn, E.: Über eine von Rayleigh formulierte Minimaleigenschaft des Kreises. Math. Ann. 94, 97–100 (1924) MathSciNetCrossRefGoogle Scholar
  76. 76.
    Krahn, E.: Über Minimaleigenschaften der Kugel in drei un mehr Dimensionen. Acta Comment. Univ. Tartu Math. A9, 1–44 (1926) Google Scholar
  77. 77.
    Maz’ja, V.G.: Sobolev Spaces. Springer, Berlin (1985) MATHGoogle Scholar
  78. 78.
    Oudet, E.: Numerical minimization of eigenmodes of a membrane with respect to the domain. ESAIM Control Optim. Calc. Var. 10, 315–330 (2004) MathSciNetMATHCrossRefGoogle Scholar
  79. 79.
    Payne, L.E., Pólya, G., Weinberger, H.F.: On the ratio of consecutive eigenvalues. J. Math. Phys. 35, 289–298 (1956) MATHGoogle Scholar
  80. 80.
    Payne, L.E., Weinberger, H.F.: An optimal Poincaré inequality for convex domains. Arch. Ration. Mech. Anal. 5, 286–292 (1960) MathSciNetMATHCrossRefGoogle Scholar
  81. 81.
    Pironneau, O.: Optimal Shape Design for Elliptic Systems. Springer, Berlin (1984) MATHGoogle Scholar
  82. 82.
    Polya, G.: On the characteristic frequencies of a symmetric membrane. Math. Z. 63, 331–337 (1955) MathSciNetMATHCrossRefGoogle Scholar
  83. 83.
    Sokolowski, J., Zolésio, J.P.: Introduction to Shape Optimization. Shape Sensitivity Analysis. Springer Series in Computational Mathematics, vol. 16. Springer, Berlin (1992) MATHGoogle Scholar
  84. 84.
    Šverák, V.: On optimal shape design. J. Math. Pures Appl. 72, 537–551 (1993) MathSciNetMATHGoogle Scholar
  85. 85.
    Szegö, G.: Inequalities for certain eigenvalues of a membrane of given area. J. Ration. Mech. Anal. 3, 343–356 (1954) MATHGoogle Scholar
  86. 86.
    Troesch, B.A.: Elliptical membranes with smallest second eigenvalue. Math. Comput. 27, 767–772 (1973) MathSciNetMATHCrossRefGoogle Scholar
  87. 87.
    Weinberger, H.F.: An isoperimetric inequality for the N-dimensional free membrane problem. J. Ration. Mech. Anal. 5, 633–636 (1956) MathSciNetMATHGoogle Scholar
  88. 88.
    Wolf, S.A., Keller, J.B.: Range of the first two eigenvalues of the Laplacian. Proc. R. Soc. Lond. Ser. A, Math. Phys. Sci. 447, 397–412 (1994) MathSciNetMATHCrossRefGoogle Scholar
  89. 89.
    Ziemer, W.P.: Weakly Differentiable Functions. Springer, Berlin (1989) MATHCrossRefGoogle Scholar

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© Revista Matemática Complutense 2011

Authors and Affiliations

  1. 1.Dipartimento di MatematicaUniversità di PisaPisaItaly

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