Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Analysis and Geometry in Metric Spaces

Ed. by Ritoré, Manuel

Covered by SCOPUS, Web of Science - Science Citation Index Expanded, MathSciNet, and Zentralblatt Math (zbMATH)


IMPACT FACTOR 2018: 0.536

CiteScore 2018: 0.83

SCImago Journal Rank (SJR) 2018: 1.041
Source Normalized Impact per Paper (SNIP) 2018: 0.801

Mathematical Citation Quotient (MCQ) 2018: 0.83

Open Access
Online
ISSN
2299-3274
See all formats and pricing
More options …

Double Bubbles on the Real Line with Log-Convex Density

Eliot Bongiovanni / Leonardo Di Giosia / Alejandro Diaz / Jahangir Habib / Arjun Kakkar / Lea Kenigsberg / Dylanger Pittman / Nat Sothanaphan / Weitao Zhu
Published Online: 2018-06-15 | DOI: https://doi.org/10.1515/agms-2018-0004

Abstract

The classic double bubble theorem says that the least-perimeter way to enclose and separate two prescribed volumes in ℝN is the standard double bubble. We seek the optimal double bubble in ℝN with density, which we assume to be strictly log-convex. For N = 1 we show that the solution is sometimes two contiguous intervals and sometimes three contiguous intervals. In higher dimensions we think that the solution is sometimes a standard double bubble and sometimes concentric spheres (e.g. for one volume small and the other large).

Keywords : double bubble; density; isoperimetric

MSC 2010: 49Q10

References

  • [1] V. Bayle (2004). Propriétés de concavité du profil isopérimétrique et applications, PhD thesis, Institut Joseph Fourier, Grenoble.Google Scholar

  • [2] E. Bongiovanni, Alejandro Diaz, Arjun Kakkar, Nat Sothanaphan. Isoperimetry in surfaces of revolutionwith density, preprint. https://arxiv.org/abs/1709.06040.Google Scholar

  • [3] K. Brakke. The surface evolver, http://facstaff.susqu.edu/brakke/evolver/evolver.html.Google Scholar

  • [4] S.G. Bobkov, Christian Houdré (1997). Some connections between isoperimetric and Sobolev-type inequalities, Mem. Amer. Math. Soc. 129, no. 616, viii+111.Google Scholar

  • [5] G.R. Chambers (2015). J. Eur. Math. Soc., to appear.Google Scholar

  • [6] F. Morgan (2016). Geometric Measure Theory: A Beginner’s Guide, Academic press.Google Scholar

  • [7] E. Milman, Joe Neeman (2018). The Gaussian double-bubble conjecture, preprint. https://arxiv.org/abs/1801.09296.Google Scholar

  • [8] F. Morgan, Aldo Pratelli (2013). Existence of isoperimetric regions in Rn with density, Ann. Glob. Anal. Geom. 43, 331-365.Web of ScienceGoogle Scholar

  • [9] I. McGillivray (2017). An isoperimetric inequality in the planewith a log-convex density, preprint. https://arxiv.org/abs/1612.07052.Google Scholar

  • [10] C. Rosales, Antonio Cañete, Vincent Bayle, Frank Morgan (2008). On the isoperimetric problem in Euclidean space with density, Calc. Var. 31, 27-46.Web of ScienceGoogle Scholar

  • [11] N. Sothanaphan (2018). 1D Triple Bubble Problem with Log-Convex Density. https://arxiv.org/abs/1805.08377.Google Scholar

About the article

Received: 2017-09-20

Revised: 2018-03-30

Accepted: 2018-05-02

Published Online: 2018-06-15


Citation Information: Analysis and Geometry in Metric Spaces, Volume 6, Issue 1, Pages 64–88, ISSN (Online) 2299-3274, DOI: https://doi.org/10.1515/agms-2018-0004.

Export Citation

© 2018 Nat Sothanaphan, published by De Gruyter. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License. BY-NC-ND 4.0

Comments (0)

Please log in or register to comment.
Log in