Advertisement

Cheeger N-clusters

  • M. Caroccia
Article

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.

Mathematics Subject Classification

49Q15 

Notes

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.

References

  1. 1.
    Aleksandrov, A.D.: Uniqueness theorems for surfaces in the large. I. Am. Math. Soc. Transl. 2(21), 341–354 (1962)MathSciNetMATHGoogle 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 978-0-8176-4359-1Google Scholar
  3. 3.
    Bucur, D., Buttazzo, G., Henrot, A.: Existence results for some optimal partition problems. Adv. Math. Sci. Appl. 8, 571–579 (1998)MathSciNetMATHGoogle Scholar
  4. 4.
    Bourdin, B., Bucur, D., Oudet, É.: Optimal partitions for eigenvalues. SIAM J. Sci. Comput. 31(6), 4100–4114 (2009)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Bombieri, E., De Giorgi, E., Giusti, E.: Minimal cones and the Bernstein problem. Invent. Math. 7(3), 243–268 (1969)MathSciNetCrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Barozzi, E., Massari, U.: Regularity of minimal boundaries with obstacles. Rend. Sem. Mat. Univ. Padova 66, 129–135 (1982)MathSciNetMATHGoogle Scholar
  8. 8.
    Bucur, D.: Minimization of the k-th eigenvalue of the Dirichlet Laplacian. Arch. Ration. Mech. Anal. 206(3), 1073–1083 (2012)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Buttazzo, G.: Spectral optimization problems. Rev. Mat. Complut. 24(2), 277–322 (2011)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Bucur, D., Velichkov, B.: Multiphase Shape Optimization Problems. SIAM J. Control Optim. 52(6), 3556–3591 (2014)Google Scholar
  11. 11.
    Cheeger, J.: A lower bound for the smallest eigenvalue of the Laplacian. Probl. Anal. 625, 195–199 (1970)MathSciNetMATHGoogle Scholar
  12. 12.
    Caffarelli, L., Lin, F.: An optimal partition problem for eigenvalues. J. Sci. Comput. 31(1), 5–18 (2007)MathSciNetCrossRefGoogle 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)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Evans, L., Gariepy, R.: Measure Theory and Fine Properties of Functions, vol. 5. CRC press, Boca Raton (1991)MATHGoogle 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)Google Scholar
  17. 17.
    Henrot, A.: Extremum Problems for Eigenvalues of Elliptic Operators: Antoine Henrot. Springer, New York (2006)MATHGoogle 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)MathSciNetCrossRefMATHGoogle 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)MathSciNetMATHGoogle Scholar
  20. 20.
    Leonardi, G.P.: An overview on the Cheeger problem In: New trends in shape optimization, pp. 117–139, Springer (2015)Google Scholar
  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)Google Scholar
  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)CrossRefGoogle 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)MathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Dipartmento di MatematicaUniversità di PisaPisaItaly

Personalised recommendations