Skip to main content
SpringerLink
Log in

Some Isoperimetric Problems in Planes with Density

  • Published:
Journal of Geometric Analysis Aims and scope Submit manuscript

Abstract

We study the isoperimetric problem in Euclidean space endowed with a density. We first consider piecewise constant densities and examine particular cases related to the characteristic functions of half-planes, strips and balls. We also consider continuous modification of Gauss density in ℝ2. Finally, we give a list of related open questions.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Ambrosio, L., Fusco, N., Pallara, D.: Functions of Bounded Variation and Free Discontinuity Problems. Oxford University Press, New York (2000)

    MATH  Google Scholar 

  2. Bobkov, S.: Extremal properties of half-spaces for log-concave distributions. Ann. Probab. 24(1), 35–48 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  3. Bobkov, S.: An isoperimetric property on the discrete cube, and an elementary proof of the isoperimetric inequality in Gauss space. Ann. Probab. 25(1), 206–214 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  4. Borell, C.: The Brunn-Minkowski inequality in Gauss space. Invent. Math. 30(2), 207–216 (1975)

    Article  MATH  MathSciNet  Google Scholar 

  5. Borell, C.: The Ornstein-Uhlenbeck velocity process in backward time and isoperimetry. Chalmers University of Technology (1986)

  6. Brock, F., Chiacchio, F., Mercaldo, A.: A class of degenerate elliptic equations and a Dido’s problem with respect to a measure. J. Math. Anal. Appl. 348, 356–365 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  7. Burago, Y.D., Zalgaller, V.A.: Geometric Inequalities. Springer Series in Soviet Mathematics, vol. 285. Springer, Berlin (1988). Translated from the Russian by A.B. Sosinskiĭ. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]

    MATH  Google Scholar 

  8. Carlen, E.A., Kerce, C.: On the cases of equality in Bobkov’s inequality and Gaussian rearrangement. Calc. Var. 13, 1–18 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  9. Carroll, C., Jacob, A., Quinn, C., Walters, R.: The isoperimetric problem on planes with density. Bull. Aust. Math. Soc. 78, 177–197 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  10. Chlebík, M., Cianchi, A., Fusco, N.: The perimeter inequality under Steiner symmetrization: Cases of equality. Ann. of Math. 162, 545–555 (2005)

    Article  Google Scholar 

  11. De Giorgi, E.: Su una teoria generale della misura (r−1)-dimensionale in uno spazio ad r dimensioni. Ann. Mat. Pura Appl. 36(4), 191–213 (1954)

    MATH  MathSciNet  Google Scholar 

  12. De Giorgi, E.: Nuovi teoremi relativi alle misure (r−1)-dimensionali in uno spazio ad r dimensioni. Ric. Mat. 4, 95–113 (1955)

    MATH  Google Scholar 

  13. De Giorgi, E.: Sulla proprietà isoperimetrica dell’ipersfera, nella classe degli insiemi aventi frontiera orientata di misura finita. Atti Accad. Naz. Lincei. Mem. Cl. Sci. Fis. Mat. Nat. Sez. I 5(8), 33–44 (1958)

    MathSciNet  Google Scholar 

  14. De Giorgi, E.: Selected Papers. Springer, Berlin (2006)

    Google Scholar 

  15. Federer, H.: Geometric Measure Theory. Die Grundlehren der mathematischen Wissenschaften, vol. 153. Springer, New York (1969)

    MATH  Google Scholar 

  16. Foisy, J., Alfaro, M., Brock, J., Hodges, N., Zimba, J.: The standard double soap bubble in R 2 uniquely minimizes perimeter. Pac. J. Math. 159(1), 47–59 (1993)

    MathSciNet  Google Scholar 

  17. Fusco, N., Maggi, F., Pratelli, A.: The sharp quantitative isoperimetric inequality. Ann. of Math. 168(3), 941–980 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  18. Giusti, E.: Minimal Surfaces and Functions of Bounded Variation. Monographs in Mathematics, vol. 80, Birkhäuser, Basel (1984)

    MATH  Google Scholar 

  19. Harvey, R., Lawson, H.B. Jr.: Calibrated geometries. Acta Math. 148, 47–157 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  20. Morgan, F.: Regularity of isoperimetric hypersurfaces in Riemannian manifolds. Trans. Am. Math. Soc. 355(12), 5041–5052 (2003) (electronic)

    Article  MATH  Google Scholar 

  21. Morgan, F.: Manifolds with density. Notices Am. Math. Soc. 52(8), 853–858 (2005)

    MATH  Google Scholar 

  22. Morgan, F.: Geometric Measure Theory: A Beginner’s Guide, 4th edn. Academic Press, San Diego (2008)

    Google Scholar 

  23. Ros, A.: The isoperimetric problem. In: Global Theory of Minimal Surfaces. Clay Math. Proc., vol. 2, pp. 175–209. Am. Math. Soc., Providence (2005)

    Google Scholar 

  24. Rosales, C., Cañete, A., Bayle, V., Morgan, F.: On the isoperimetric problem in Euclidean space with density. Calc. Var. Partial Differ. Equ. 31(1), 27–46 (2008)

    Article  MATH  Google Scholar 

  25. Sudakov, V.N., Tsirel’son, B.S.: Extremal properties of half-spaces for spherically invariant measures. J. Sov. Math. 9–18 (1978)

  26. Talenti, G.: The standard isoperimetric theorem. In: Gruber, P.M., Wills, J.M. (eds.) Handbook of Convex Geometry, vol. A, pp. 73–123. North-Holland, Amsterdam (1993)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Departamento de Matemática Aplicada I, Universidad de Sevilla, Campus de Reina Mercedes, 24071, Sevilla, Spain

    Antonio Cañete

  2. Dipartimento di Matematica, Università degli Studi di Ferrara, via Machiavelli 35, 44100, Ferrara, Italy

    Michele Miranda Jr

  3. Dipartimento di Matematica Pura ed Applicata, Università degli Studi di Padova, via Trieste 63, 35100, Padova (PD), Italy

    Davide Vittone

Authors
  1. Antonio Cañete
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Michele Miranda Jr
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Davide Vittone
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Antonio Cañete.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Cañete, A., Miranda, M. & Vittone, D. Some Isoperimetric Problems in Planes with Density. J Geom Anal 20, 243–290 (2010). https://doi.org/10.1007/s12220-009-9109-4

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12220-009-9109-4

Keywords

Mathematics Subject Classification (2000)

Search

Navigation