Advertisement

manuscripta mathematica

, Volume 156, Issue 3–4, pp 371–381 | Cite as

Weighted Cheeger sets are domains of isoperimetry

  • Giorgio Saracco
Article

Abstract

We consider a generalization of the Cheeger problem in a bounded, open set \(\Omega \) by replacing the perimeter functional with a Finsler-type surface energy and the volume with suitable powers of a weighted volume. We show that any connected minimizer A of this weighted Cheeger problem such that \(\mathcal {H}^{n-1}(A^{(1)} \cap \partial A)=0\) satisfies a relative isoperimetric inequality. If \(\Omega \) itself is a connected minimizer such that \(\mathcal {H}^{n-1}(\Omega ^{(1)} \cap \partial \Omega )=0\), then it allows the classical Sobolev and BV embeddings and the classical BV trace theorem. The same result holds for any connected minimizer whenever the weights grant the regularity of perimeter-minimizer sets and \(\Omega \) is such that \(|\partial \Omega |=0\) and \(\mathcal {H}^{n-1}(\Omega ^{(1)} \cap \partial \Omega )=0\).

Mathematics Subject Classification

Primary: 46E35 Secondary: 49Q10 28A75 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Ambrosio, L., Fusco, N., Pallara, D.: Functions of Bounded Variation and Free Discontinuity Problems. Oxford Mathematical Monographs. Clarendon Press, Oxford (2000)zbMATHGoogle Scholar
  2. 2.
    Brasco, L., Lindgren, E., Parini, E.: The fractional Cheeger problem. Interfaces Free Bound. 16(3), 419–458 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Carlier, G., Comte, M.: On a weighted total variation minimization problem. J. Funct. Anal. 250(1), 214–226 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Caselles, V., Facciolo, G., Meinhardt, E.: Anisotropic Cheeger sets and applications. SIAM J. Imaging Sci. 2(4), 1211–1254 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Caselles, V., Miranda Jr., M., Novaga, M.: Total variation and Cheeger sets in Gauss space. J. Funct. Anal. 259, 1491–1516 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Cheeger, J.: A lower bound for the smallest eigenvalue of the Laplacian. In: Problems in Analysis (Papers dedicated to Salomon Bochner, 1969), pp. 195–199. Princeton University Press, Princeton (1970)Google Scholar
  7. 7.
    Cinti, E., Pratelli, A.: The \(\epsilon -\epsilon ^\beta \) property, the boundedness of isoperimetric sets in \({\mathbb{R}}^n\) with density, and some applications. J. Reine Angew. Math. 728, 65–103 (2017). doi: 10.1515/crelle-2014-0120 MathSciNetzbMATHGoogle Scholar
  8. 8.
    David, G., Semmes, S.: Quasiminimal surfaces of codimension \(1\) and John domains. Pac. J. Math. 183(2), 213–277 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Federer, H.: Geometric Measure Theory. Die Grundlehren der mathematischen Wissenschaften, vol. 153. Springer, New York (1969)Google Scholar
  10. 10.
    Hassani, R., Ionescu, I.R., Lachand-Robert, T.: Optimization techniques in landslides modelling. Ann. Univ. Craiova Ser. Math. Inform. 32, 158–169 (2005)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Ionescu, I.R., Lachand-Robert, T.: Generalized Cheeger sets related to landslides. Calc. Var. Partial Differ. Equ. 23(2), 227–249 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Kawohl, B., Fridman, V.: Isoperimetric estimates for the first eigenvalue of the \(p\)-Laplace operator and the Cheeger constant. Comment. Math. Univ. Carol. 44(4), 659–667 (2003)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Krejčiřík, D., Kříz, J.: On the spectrum of curved quantum waveguides. Publ. Res. Inst. Math. Sci. 41(3), 757–791 (2015)zbMATHGoogle Scholar
  14. 14.
    Krejčiřík, D., Pratelli, A.: The Cheeger constant of curved strips. Pac. J. Math. 254(2), 309–333 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Leonardi, G.P.: An overview on the Cheeger problem. In: Pratelli, A., Leugering, G. (eds.) New Trends in Shape Optimization. International Series of Numerical Mathematics, vol. 166, pp. 117–139. Birkhäuser, Cham (2015). doi: 10.1007/978-3-319-17563-8_6
  16. 16.
    Leonardi, G.P., Pratelli, A.: On the Cheeger sets in strips and non-convex domains. Calc. Var. Partial Differ. Equ. 55(1), 1–28 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Leonardi, G.P., Saracco, G.: The prescribed mean curvature equation in weakly regular domains. Submitted (2016). http://cvgmt.sns.it/paper/3089/
  18. 18.
    Leonardi, G.P., Saracco, G.: Two examples of minimal Cheeger sets in the plane. Submitted (2017). http://cvgmt.sns.it/paper/3565/
  19. 19.
    Maggi, F.: Sets of Finite Perimeter and Geometric Variational Problems. Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (2012)CrossRefGoogle Scholar
  20. 20.
    Maz’ya, V.: Sobolev Spaces with Applications to Elliptic Partial Differential Equations, vol. 342, 2nd edn. Springer, Berlin (2011)zbMATHGoogle Scholar
  21. 21.
    Morgan, F.: Regularity of isoperimetric hypersurfaces in riemannian manifolds. Trans. Am. Math. Soc. 355(12), 5041–5052 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Parini, E.: An introduction to the Cheeger problem. Surv. Math. Appl. 6, 9–21 (2011)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Pratelli, A., Saracco, G.: On the generalized Cheeger problem and an application to 2d strips. Rev. Mat. Iberoam. 33(1), 219–237 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Reshetnyak, Y.G.: Weak convergence of completely additive vector functions on a set. Sib. Math. J. 9, 1039–1045 (1968)CrossRefzbMATHGoogle Scholar
  25. 25.
    Rudin, L.I., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Phys. D. Nonlinear Phenom. 60(1), 256–268 (1992)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.Department MathematikUniversität Erlangen-NürnbergErlangenGermany

Personalised recommendations