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

TL;DR

This paper demonstrates that in nonconvex domains, volume-constrained minimal surfaces can have singular local minimizers, contrasting with convex cases where minimizers are regular, highlighting the importance of domain convexity.

Contribution

The paper provides a counterexample showing singular local minimizers in nonconvex domains, emphasizing the role of convexity in regularity of volume-constrained minimal surfaces.

Findings

Singular local minimizer exists in a nonconvex domain.

Convexity condition is crucial for regularity of minimizers.

Counterexample uses the Simons cone near the unit ball.

Abstract

It has recently been established byWang and Xia [WX] 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.