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Applied Mathematics & Optimization

, Volume 69, Issue 2, pp 199–231 | Cite as

Existence and Regularity of Minimizers for Some Spectral Functionals with Perimeter Constraint

  • Guido De Philippis
  • Bozhidar Velichkov
Article

Abstract

In this paper we prove that the shape optimization problem
$$\min \bigl\{\lambda_k(\varOmega):\ \varOmega\subset \mathbb{R}^d,\ \varOmega\ \hbox{open},\ P(\varOmega)=1,\ |\varOmega|<+\infty \bigr\}, $$
has a solution for any \(k\in \mathbb{N}\) and dimension d. Moreover, every solution is a bounded connected open set with boundary which is C 1,α outside a closed set of Hausdorff dimension d−8. Our results are more general and apply to spectral functionals of the form \(f(\lambda_{k_{1}}(\varOmega),\dots,\lambda_{k_{p}}(\varOmega))\), for increasing functions f satisfying some suitable bi-Lipschitz type condition.

Keywords

Shape optimization Eigenvalues Free boundary Concentration-compactness 

References

  1. 1.
    Almgren, F.: Existence and regularity almost everywhere of solutions to elliptic variational problems with constraints. Mem. Am. Math. Soc. 4(165) (1976) Google Scholar
  2. 2.
    Alt, H.W., Caffarelli, L.A.: Existence and regularity for a minimum problem with free boundary. J. Reine Angew. Math. 325, 105–144 (1981) zbMATHMathSciNetGoogle Scholar
  3. 3.
    Ambrosio, L., Caselles, V., Masnou, S., Morel, J.M.: Connected components of sets of finite perimeter and applications to image processing. J. Eur. Math. Soc. 3(1), 39–92 (2001) CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Briançon, T., Hayouni, M., Pierre, M.: Lipschitz continuity of state functions in some optimal shaping. Calc. Var. Partial Differ. Equ. 23(1), 13–32 (2005) CrossRefzbMATHGoogle Scholar
  5. 5.
    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(4), 1149–1163 (2009) CrossRefzbMATHGoogle Scholar
  6. 6.
    Bucur, D.: Uniform concentration-compactness for Sobolev spaces on variable domains. J. Differ. Equ. 162, 427–450 (2000) CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Bucur, D.: Minimization of the kth eigenvalue of the Dirichlet Laplacian. Arch. Ration. Mech. Anal. 206(3), 1073–1083 (2012) CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Bucur, D., Buttazzo, G.: Variational Methods in Shape Optimization Problems. Progress in Nonlinear Differential Equations, vol. 65. Birkhäuser, Basel (2005) zbMATHGoogle Scholar
  9. 9.
    Bucur, D., Buttazzo, G., Henrot, A.: Minimization of λ 2(Ω) with a perimeter constraint. Indiana Univ. Math. J. 58(6), 2709–2728 (2009) CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Bucur, D., Buttazzo, G., Velichkov, B.: Spectral optimization problems with internal constraint. Ann. Inst. H. Poincaré 30(3), 477–495 (2013) CrossRefMathSciNetGoogle Scholar
  11. 11.
    Bucur, D., Freitas, P.: Asymptotic behaviour of optimal spectral planar domains with fixed perimeter (to appear) Google Scholar
  12. 12.
    Bucur, D., Mazzoleni, D., Pratelli, A., Velichkov, B.: Lipschitz regularity of eigenfunctions at optimal shapes (in preparation) Google Scholar
  13. 13.
    Bucur, D., Velichkov, B.: Multiphase shape optimization problems Preprint available at: http://cvgmt.sns.it/paper/2114/
  14. 14.
    Buttazzo, G.: Spectral optimization problems. Rev. Mat. Complut. 24(2), 277–322 (2011) CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Buttazzo, G., Dal Maso, G.: Shape optimization for Dirichlet problems: relaxed formulation and optimality conditions. Appl. Math. Optim. 23, 17–49 (1991) CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Buttazzo, G., Dal Maso, G.: An existence result for a class of shape optimization problems. Arch. Ration. Mech. Anal. 122, 183–195 (1993) CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Caffarelli, L., Cabré, X.: Fully Nonlinear Elliptic Equations. Am. Math. Soc. Colloquium Publications, vol. 43 (1995) zbMATHGoogle Scholar
  18. 18.
    Caffarelli, L., Cordoba, A.: An elementary regularity theory of minimal surfaces. Differ. Integral Equ. 6, 1–13 (1993) zbMATHMathSciNetGoogle Scholar
  19. 19.
    De Philippis, G., Paolini, E.: A short proof of the minimality of Simons cone. Rend. Semin. Mat. Univ. Padova 12, 233–241 (2009) CrossRefGoogle Scholar
  20. 20.
    Dal Maso, G., Garroni, A.: New results on the asymptotic behaviour of Dirichlet problems in perforated domains. Math. Models Methods Appl. Sci. 3, 373–407 (1994) CrossRefMathSciNetGoogle Scholar
  21. 21.
    Evans, L.C., Gariepy, R.F.: Measure Theory and Fine Properties of Functions. Studies in Advanced Mathematics. CRC Press, Boca Raton (1992) zbMATHGoogle Scholar
  22. 22.
    Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Classics in Mathematics. Springer, Berlin (2001). Reprint of the 1998 edition zbMATHGoogle Scholar
  23. 23.
    Giusti, E.: Minimal Surfaces and Functions of Bounded Variation. Monographs in Mathematics, vol. 80. Birkhäuser, Boston (1984) CrossRefzbMATHGoogle Scholar
  24. 24.
    Henrot, A.: Minimization problems for eigenvalues of the Laplacian. J. Evol. Equ. 3(3), 443–461 (2003) CrossRefzbMATHMathSciNetGoogle Scholar
  25. 25.
    Henrot, A.: Extremum Problems for Eigenvalues of Elliptic Operators. Frontiers in Mathematics. Birkhäuser, Basel (2006) zbMATHGoogle Scholar
  26. 26.
    Henrot, A., Pierre, M.: Variation et Optimisation de Formes. Une Analyse Géométrique, Mathématiques & Applications, vol. 48. Springer, Berlin (2005) zbMATHGoogle Scholar
  27. 27.
    Jiang, H., Larsen, C., Silvestre, L.: Full regularity of a free boundary problem with two phases. Calc. Var. Partial Differ. Equ. 42, 301–321 (2011) CrossRefzbMATHMathSciNetGoogle Scholar
  28. 28.
    Lieb, E.H.: On the lowest eigenvalue of the Laplacian for the intersection of two domains. Invent. Math. 74(3), 441–448 (1983) CrossRefzbMATHMathSciNetGoogle Scholar
  29. 29.
    Lions, P.L.: The Concentration-Compactness Principle in the Calculus of Variations. The Locally Compact Case, Part 1. Annales de l’I. H. P., Section C, tome 1 (1984) Google Scholar
  30. 30.
    Maggi, F.: Sets of Finite Perimeter and Geometric Variational Problems: an Introduction to Geometric Measure Theory. Cambridge Studies in Advanced Mathematics, vol. 135. Cambridge University Press, Cambridge (2012) CrossRefGoogle Scholar
  31. 31.
    Mazzoleni, D., Pratelli, A.: Existence of minimizers for spectral problems. J. Math. Pures Appl. 100(3), 433–453 (2013) CrossRefzbMATHMathSciNetGoogle Scholar
  32. 32.
    Mazzone, F.: A single phase variational problem involving the area of level surfaces. Commun. Partial Differ. Equ. 28, 991–1004 (2003) CrossRefzbMATHMathSciNetGoogle Scholar
  33. 33.
    Simon, L.: Lectures on geometric measure theory. In: Proceedings of the Centre for Mathematical Analysis, Australian National University, vol. 3. Australian National University, Centre for Mathematical Analysis, Canberra (1983) Google Scholar
  34. 34.
    Talenti, G.: Elliptic equations and rearrangements. Ann. Sc. Norm. Super. Pisa 3(4), 697–718 (1976) zbMATHMathSciNetGoogle Scholar
  35. 35.
    Tamanini, I.: Boundaries of Caccioppoli sets with Hölder-continuous normal vector. J. Reine Angew. Math. 334, 27–39 (1982) zbMATHMathSciNetGoogle Scholar
  36. 36.
    Tamanini, I.: Regularity results for almost minimal hyperurfaces in \(\mathbb{R}^{n}\). Quaderni del Dipartimento di Matematica dell’ Università di Lecce (1984) Google Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Hausdorff Center for MathematicsBonnGermany
  2. 2.Scuola Normale Superiore di PisaPisaItaly

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