A singular local minimizer for the volume- constrained minimal surface problem in a nonconvex domain
HTML articles powered by AMS MathViewer
- by Peter Sternberg and Kevin Zumbrun PDF
- Proc. Amer. Math. Soc. 146 (2018), 5141-5146 Request permission
Abstract:
It has recently been established by Wang and Xia that local minimizers of perimeter within a ball subject to a volume constraint must be spherical caps or planes through the origin. This verifies a conjecture of the authors and is in contrast to the situation of area-minimizing surfaces with prescribed boundary where singularities can be present in high dimensions. This result lends support to the more general conjecture that volume-constrained minimizers in arbitrary convex sets may enjoy better regularity properties than their boundary-prescribed cousins. Here, we show the importance of the convexity condition by exhibiting a simple example, given by the Simons cone, of a singular volume-constrained locally area-minimizing surface within a nonconvex domain that is arbitrarily close to the unit ball.References
- E. Bombieri, E. De Giorgi, and E. Giusti, Minimal cones and the Bernstein problem, Invent. Math. 7 (1969), 243–268. MR 250205, DOI 10.1007/BF01404309
- Enrico Giusti, Minimal surfaces and functions of bounded variation, Monographs in Mathematics, vol. 80, Birkhäuser Verlag, Basel, 1984. MR 775682, DOI 10.1007/978-1-4684-9486-0
- E. Gonzalez, U. Massari, and I. Tamanini, On the regularity of boundaries of sets minimizing perimeter with a volume constraint, Indiana Univ. Math. J. 32 (1983), no. 1, 25–37. MR 684753, DOI 10.1512/iumj.1983.32.32003
- Michael Grüter, Boundary regularity for solutions of a partitioning problem, Arch. Rational Mech. Anal. 97 (1987), no. 3, 261–270. MR 862549, DOI 10.1007/BF00250810
- Michael Grüter and Jürgen Jost, Allard type regularity results for varifolds with free boundaries, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 13 (1986), no. 1, 129–169. MR 863638
- Leon Simon, Lectures on geometric measure theory, Proceedings of the Centre for Mathematical Analysis, Australian National University, vol. 3, Australian National University, Centre for Mathematical Analysis, Canberra, 1983. MR 756417
- James Simons, Minimal varieties in riemannian manifolds, Ann. of Math. (2) 88 (1968), 62–105. MR 233295, DOI 10.2307/1970556
- James Simons, Minimal cones, Plateau’s problem, and the Bernstein conjecture, Proc. Nat. Acad. Sci. U.S.A. 58 (1967), 410–411. MR 216387, DOI 10.1073/pnas.58.2.410
- Peter Sternberg and Kevin Zumbrun, A Poincaré inequality with applications to volume-constrained area-minimizing surfaces, J. Reine Angew. Math. 503 (1998), 63–85. MR 1650327
- G. Wang and C. Xia, Uniqueness of stable capillary hypersurfaces in a ball, preprint, arXiv:1708.06861.
Additional Information
- Peter Sternberg
- Affiliation: Department of Mathematics, Indiana University, Bloomington, Indiana 47405
- MR Author ID: 167150
- Email: sternber@indiana.edu
- Kevin Zumbrun
- Affiliation: Department of Mathematics, Indiana University, Bloomington, Indiana 47405
- MR Author ID: 330192
- Email: kzumbrun@indiana.edu
- Received by editor(s): November 1, 2017
- Published electronically: September 4, 2018
- Additional Notes: Research of the first author was partially supported under NSF grant no. DMS 1362879.
Research of the second author was partially supported under NSF grant no. DMS-0300487. - Communicated by: Michael Wolf
- © Copyright 2018 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 146 (2018), 5141-5146
- MSC (2010): Primary 49K40
- DOI: https://doi.org/10.1090/proc/14257
- MathSciNet review: 3866853