Cheeger N-clusters

Abstract

In this paper we introduce a Cheeger-type constant defined as a minimization of a suitable functional among all the N-clusters contained in an open bounded set \(\Omega \). Here with N-Cluster we mean a family of N sets of finite perimeter, disjoint up to a set of null Lebesgue measure. We call any N-cluster attaining such a minimum a Cheeger N-cluster. Our purpose is to provide a non trivial lower bound on the optimal partition problem for the first Dirichlet eigenvalue of the Laplacian. Here we discuss the regularity of Cheeger N-clusters in a general ambient space dimension and we give a precise description of their structure in the planar case. The last part is devoted to the relation between the functional introduced here (namely the N-Cheeger constant), the partition problem for the first Dirichlet eigenvalue of the Laplacian and the Caffarelli and Lin’s conjecture.

This is a preview of subscription content, log in to check access.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7

References

  1. 1.

    Aleksandrov, A.D.: Uniqueness theorems for surfaces in the large. I. Am. Math. Soc. Transl. 2(21), 341–354 (1962)

    MathSciNet  MATH  Google Scholar 

  2. 2.

    Bucur, D., Buttazzo, G.: Variational Methods in Shape Optimization Problems. Progress in Nonlinear Differential Equations 65, Birkhäuser Verlag, Basel (2005), viii+216 p. ISBN 978-0-8176-4359-1

  3. 3.

    Bucur, D., Buttazzo, G., Henrot, A.: Existence results for some optimal partition problems. Adv. Math. Sci. Appl. 8, 571–579 (1998)

    MathSciNet  MATH  Google Scholar 

  4. 4.

    Bourdin, B., Bucur, D., Oudet, É.: Optimal partitions for eigenvalues. SIAM J. Sci. Comput. 31(6), 4100–4114 (2009)

    MathSciNet  Article  MATH  Google Scholar 

  5. 5.

    Bombieri, E., De Giorgi, E., Giusti, E.: Minimal cones and the Bernstein problem. Invent. Math. 7(3), 243–268 (1969)

    MathSciNet  Article  MATH  Google Scholar 

  6. 6.

    Buttazzo, G., Maso, G.D.: An existence result for a class of shape optimization problems. Arch. Ration. Mech. Anal. 122(2), 183–195 (1993)

    MathSciNet  Article  MATH  Google Scholar 

  7. 7.

    Barozzi, E., Massari, U.: Regularity of minimal boundaries with obstacles. Rend. Sem. Mat. Univ. Padova 66, 129–135 (1982)

    MathSciNet  MATH  Google Scholar 

  8. 8.

    Bucur, D.: Minimization of the k-th eigenvalue of the Dirichlet Laplacian. Arch. Ration. Mech. Anal. 206(3), 1073–1083 (2012)

    MathSciNet  Article  MATH  Google Scholar 

  9. 9.

    Buttazzo, G.: Spectral optimization problems. Rev. Mat. Complut. 24(2), 277–322 (2011)

    MathSciNet  Article  MATH  Google Scholar 

  10. 10.

    Bucur, D., Velichkov, B.: Multiphase Shape Optimization Problems. SIAM J. Control Optim. 52(6), 3556–3591 (2014)

  11. 11.

    Cheeger, J.: A lower bound for the smallest eigenvalue of the Laplacian. Probl. Anal. 625, 195–199 (1970)

    MathSciNet  MATH  Google Scholar 

  12. 12.

    Caffarelli, L., Lin, F.: An optimal partition problem for eigenvalues. J. Sci. Comput. 31(1), 5–18 (2007)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Cicalese, M., Leonardi, G.P., Maggi, F.: Improved Convergence Theorems for Bubble Clusters. I. The planar case. p 50 arXiv:1409.6652 (2015)

  14. 14.

    De Philippis, G., Paolini, E.: A short proof of the minimality of Simons cone. Rend. Sem. Mat. Univ. Padova 121, 233–241 (2009)

    MathSciNet  Article  MATH  Google Scholar 

  15. 15.

    Evans, L., Gariepy, R.: Measure Theory and Fine Properties of Functions, vol. 5. CRC press, Boca Raton (1991)

    Google Scholar 

  16. 16.

    Figalli, A., Maggi, F., Pratelli, A.: A note on Cheeger sets. In: Proceedings of the American Mathematical Society, vol. 137, pp. 2057–2062 (2009)

  17. 17.

    Henrot, A.: Extremum Problems for Eigenvalues of Elliptic Operators: Antoine Henrot. Springer, New York (2006)

    Google Scholar 

  18. 18.

    Kawohl, B., Lachand-Robert, T.: Characterization of cheeger sets for convex subsets of the plane. Pac. J. Math. 225(1), 103–118 (2006)

    MathSciNet  Article  MATH  Google Scholar 

  19. 19.

    Kawohl, B., Novaga, M.: The p-laplace eigenvalue problem as \(p \rightarrow 1\) and Cheeger sets in a Finsler metric. J. Convex Anal. 15(3), 623 (2008)

    MathSciNet  MATH  Google Scholar 

  20. 20.

    Leonardi, G.P.: An overview on the Cheeger problem In: New trends in shape optimization, pp. 117–139, Springer (2015)

  21. 21.

    Leonardi, G.P., Pratelli, A.: On the Cheeger sets in strips and non-convex domains. Calc. Var. Part. Differ. Equ. 55(1), 1–28 (2016)

  22. 22.

    Maggi, F.: Sets of Finite Perimeter and Geometric Variational Problems, Volume 135 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (2012). (An introduction to geometric measures theory)

    Google Scholar 

  23. 23.

    Naber, A., Valtorta, D.: The Singular Structure and Regularity of Stationary and Minimizing Varifolds. arXiv:1505.03428 (2015)

  24. 24.

    Parini, E.: The second eigenvalue of the p-Laplacian as p goes to 1. Int. J. Differ. Equ. 2010(2010), 23. doi:10.1155/2010/984671 (2009)

  25. 25.

    Parini, E.: An introduction to the Cheeger problem. Surv. Math. Appl 6, 9–22 (2011)

    MathSciNet  Google Scholar 

Download references

Acknowledgements

The author is grateful to professor Giovanni Alberti for his useful comments and for the useful discussions about this subject. Thanks to Gian Paolo Leonardi and Aldo Pratelli for having carefully read this work as a part of the Ph.D thesis of the author. The author is also grateful to Enea Parini for the useful comments about Proposition 4.4. The work of the author was partially supported by the project 2010A2TFX2-Calcolo delle Variazioni, funded by the Italian Ministry of Research and University. This work has been partially edited while the author was already a post-doc fellowship at the Carnegie Mellon University-Center for Nonlinear Analysis to which he is grateful.

Author information

Affiliations

Authors

Corresponding author

Correspondence to M. Caroccia.

Additional information

Communicated by L. Caffarelli.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Caroccia, M. Cheeger N-clusters. Calc. Var. 56, 30 (2017). https://doi.org/10.1007/s00526-017-1109-9

Download citation

Mathematics Subject Classification

  • 49Q15