Skip to main content
SpringerLink
Log in

The isoperimetric problem in the Heisenberg group \(\pmb {\mathbb {H}^n}\) with density

  • Published:
Analysis and Mathematical Physics Aims and scope Submit manuscript

Abstract

We study the isoperimetric problem for the axially symmetric sets in the Heisenberg group \(\mathbb {H}^n\) with density \(|z|^p\). At first, we prove the existence of weighted isoperimetric sets. Then, we characterize weighted isoperimetric sets uniquely as bubble sets. Finally, we deduce an interesting result that, up to a constant multiplicator, \(|z|^p\) is the only horizontal radial density for which bubble sets can be weighted isoperimetric sets.

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. Alvino, A., Brock, F., Chiacchio, F., Mercaldo, A., Posteraro, M.R.: Some isoperimetric inequalities on \(\mathbb{R}^N\) with respect to weights \(|x|^\alpha \). J. Math. Anal. Appl. 451(1), 280–318 (2017)

    Article  MATH  MathSciNet  Google Scholar 

  2. Alvino, A., Brock, F., Chiacchio, F., Mercaldo, A., Posteraro, M.R.: On weighted isoperimetric inequaliities with nonradial densities. Appl. Anal. 98(10), 1935–1945 (2019)

    Article  MATH  MathSciNet  Google Scholar 

  3. Alvino, A., Brock, F., Chiacchio, F., Mercaldo, A., Posteraro, M.R.: The isoperimetric problem for a class of non-radial weights and applications. J. Differ. Equ. 267(12), 6831–6871 (2019)

    Article  MATH  MathSciNet  Google Scholar 

  4. Bagle, V.: Propriétées de Concavité du profil Isopérmétrique et applications. Thèse de Doctorat. Institut Fourier, Vniv. Josph-Forier, Grenoble1 (2003)

  5. Balogh, Z.M.: Size of characteristic sets and functions with prescibed gradients. J. Reine Angew. Math. 564, 63–83 (2003)

    MATH  MathSciNet  Google Scholar 

  6. Betta, M.F., Brock, F., Mercaldo, A., Posteraro, M.R.: Weighted isoperimetric inequalities on \(\mathbb{R}^n\) and applications to rearrangements. Math. Nachr. 281(4), 466–498 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  7. Brock, F., Chiacchio, F.: Some weighted isoperimetric problems on \(\mathbb{R}^{N}_+\) with stable half balls have no solutions. J. Fourier Anal. Appl. 26(1), Paper No. 15, 19 (2020)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  9. Cañete, A., Miranda Jr., M., Vittone, D.: Some isoperimetric problems in planes with density. J. Geom. Anal. 20(2), 243–290 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  10. Capogna, L., Danielli, D., Pauls, S.D., Tyson, J.: An introduction to the Heisenberg group and the sub-riemannian isoperimetric problem. Progr. Math. 259, Birkhäuser, Basel, MR 2312336 (2007)

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

    Article  MATH  MathSciNet  Google Scholar 

  12. Carroll, C., Jacob, A., Quinn, C., Walters, R.: The isoperimetric problem on planes with density. Bull. Aust. Math. Soc. 78(2), 177–197 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  13. Cianchi, A., Fusco, N., Maggi, F., Pratelli, A.: On the isoperimetric deficit in Gauss space. Am. J. Math. 133(1), 131–186 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  14. Danielli, D., Garofalo, N., Nhieu, D.-M.: A partial solution of the isoperimetric problem for the Heisenberg group. Forum Math. 20(1), 99–143 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  15. Dahlberg, J., Dubbs, A., Newkirk, E., Tran, H.: Isoperimetric regions in the plane with density \(r^p\). New York J. Math. 16, 31–51 (2010)

    MATH  MathSciNet  Google Scholar 

  16. Díaz, A., Harman, N., Howe, S., Thompson, D.: Isoperimetric problems in sectors with density. Adv. Geom. 12(4), 589–619 (2012)

    MATH  MathSciNet  Google Scholar 

  17. Engelstein, M., Marcuccio, A., Maurmann, Q., Pritchard, T.: Isoperimetric problems on the sphere and on surfaces with density. N. Y. J. Math. 15, 97–123 (2009)

    MATH  MathSciNet  Google Scholar 

  18. Franchi, B., Serapioni, R., Serra Cassano, F.: Meyers-Serrin type theorems and relaxation of variational integrals depending on vector fields. Houst. J. Math. 22(4), 859–890 (1996)

    MATH  MathSciNet  Google Scholar 

  19. Franceschi, V., Monti, R.: Isoperimetric problem in \(H\)-type groups and Grushin spaces. Rev. Mat. Iberoam. 32(4), 1227–1258 (2016)

    Article  MATH  MathSciNet  Google Scholar 

  20. Franceschi, V., Montefalcone, F., Monti, R.: CMC spheres in the Heisenberg group. Anal. Geom. Metr. Spaces 7(1), 109–129 (2019)

    Article  MATH  MathSciNet  Google Scholar 

  21. He, G., Zhao, P.: The weighted isoperimetric-type and sobolev-type inequalities for hypersurfaces in Carnot groups. Nonlinear Anal. Theor. 135, 35–56 (2016)

    Article  MATH  MathSciNet  Google Scholar 

  22. He, G., Zhao, P.: The isoperimetric problem in Grushin space \(\mathbb{R}^{h+1}\) with density \(|x|^p\). Rend. Sem. Mat. Univ. Padova 139, 241–260 (2018)

    Article  MATH  MathSciNet  Google Scholar 

  23. Leonardi, G.P., Rigot, S.: Isoperimetric sets on Carnot groups. Houston J. Math. 29, 609–637 (2003)

    MATH  MathSciNet  Google Scholar 

  24. Leonardi, G.P., Masnou, S.: On the isoperimetric problem in the Heisenberg group \(\mathbb{H}^n\). Ann. Mat. Pura Appl. 184(4), 533–553 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  25. Maurmann, Q., Morgan, F.: Isoperimetric comparison theorems for manifolds with density. Calc. Var. Partial Differ. Equ. 36(1), 1–5 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  26. Monti, R.: Brunn-Minkowski and isoperimetric inequality in the Heisenberg group. Ann. Acad. Sci. Fenn. Math. 28(1), 99–109 (2003)

    MATH  MathSciNet  Google Scholar 

  27. Monti, R.: Heisenberg isoperimetric problem. The axial case. Adv. Calc. Var. 1(1), 93–121 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  28. Monti, R., Rickly, M.: Convex isoperimetric sets in the Heisenberg group. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 8(2), 391–415 (2009)

    MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  31. Morgan, F.: Manifolds with density and Perelman’s proof of the Poincaré conjecture. Am. Math. Mon. 116(2), 134–142 (2009)

    Article  MATH  Google Scholar 

  32. Morgan, F., Pratelli, A.: Existence of isoperimetric regions in \(\mathbb{R}^n\) with density. Ann. Global Anal. Geom. 43(4), 331–365 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  33. Pansu, P.: An isoperimetric inequality on the Heisenberg group. Conference on differential geometry on homogeneous spaces (Turin, 1983). Rend. Sem. Mat. Univ. Politec. Torino, 159–174 (1983) (Special issue)

  34. Pratelli, A., Saracco, G.: On the isoperimetric problem with double density. Nonlinear Anal. 177 part B, 733–752 (2018)

    Article  MATH  MathSciNet  Google Scholar 

  35. Ritoré, M., Rosales, C.: Area-stationary surfaces in the Heisenberg group \(\mathbb{H}^1\). Adv. Math. 219(2), 633–671 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  36. Ritoré, M.: A proof by calibration of an isoperimetric inequality in the Heisenberg group \(\mathbb{H}^n\). Calc. Var. Partial Differ. Equ. 44(1–2), 47–60 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  37. 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  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematic and Statistics, Anhui Normal University, Wuhu, 241000, People’s Republic of China

    Guoqing He

  2. School of Science, Nanjing University of Science and Technology, Nanjing, 210094, People’s Republic of China

    Peibiao Zhao

Authors
  1. Guoqing He
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Peibiao Zhao
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to Guoqing He or Peibiao Zhao.

Ethics declarations

Conflict of interest

The authors declare that there is no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This work is supported by the National Natural Science Foundation of China (Nos.11871275; 11671193) and the Doctoral Program of Anhui Normal University (CN) (751841).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

He, G., Zhao, P. The isoperimetric problem in the Heisenberg group \(\pmb {\mathbb {H}^n}\) with density. Anal.Math.Phys. 10, 24 (2020). https://doi.org/10.1007/s13324-020-00367-2

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s13324-020-00367-2

Keywords

Mathematics Subject Classification

Search

Navigation