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A singular local minimizer for the volume- constrained minimal surface problem in a nonconvex domain


Authors: Peter Sternberg and Kevin Zumbrun
Journal: Proc. Amer. Math. Soc. 146 (2018), 5141-5146
MSC (2010): Primary 49K40
DOI: https://doi.org/10.1090/proc/14257
Published electronically: September 4, 2018
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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.


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Additional Information

Peter Sternberg
Affiliation: Department of Mathematics, Indiana University, Bloomington, Indiana 47405
Email: sternber@indiana.edu

Kevin Zumbrun
Affiliation: Department of Mathematics, Indiana University, Bloomington, Indiana 47405
Email: kzumbrun@indiana.edu

DOI: https://doi.org/10.1090/proc/14257
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
Article copyright: © Copyright 2018 American Mathematical Society

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