Abstract
In a recent paper (Carlier et al. in ESAIM Control Optim Calc Var 18(3):611–620, 2012), an interpolation flow between an evolution by convexity and the geometric flow of motion by principal negative curvature was informally proposed. It is also expected that the geometric flow will eventually convexify the sub-level sets of the initial function \(u_0\), yielding the quasiconvex envelope of \(u_0\). In this note, we establish existence and uniqueness of the interpolation flow under appropriate conditions and provide a rigorous proof for its limit behaviour. In addition, we show by example that, contrary to intuition, the proposed geometric flow does not always convexify the sub-level sets of \(u_0\).
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We are grateful to the anonymous referee for reading the manuscript carefully and offering constructive suggestions which have helped us improve the paper.
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The author is funded by the National Natural Science Foundation of China No. 11201016 (Grant No. 11126035).
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Ruan, Y.L. A convergence result related to the geometric flow of motion by principal negative curvature. Arch. Math. 114, 585–594 (2020). https://doi.org/10.1007/s00013-020-01446-3
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DOI: https://doi.org/10.1007/s00013-020-01446-3