Asymptotics for the level set equation near a maximum

TL;DR

This paper analyzes the asymptotic behavior of the level set equation for mean curvature flow near a maximum point on convex domains, revealing non-smooth solutions and constructing solutions with prescribed local behavior.

Contribution

It provides detailed asymptotic analysis near maxima, constructs non-smooth solutions, and prescribes solution behaviors, advancing understanding of mean curvature flow near critical points.

Findings

Solutions are not necessarily $C^3$ near maxima.

Constructed non-smooth $C^3$ solutions.

Developed solutions with prescribed asymptotic behavior.

Abstract

We give asymptotics for the level set equation for mean curvature flow on a convex domain near the point where it attains a maximum. It is known that solutions are not necessarily $C^3,$ and we recover this result and construct non-smooth solutions which are $C^3.$ We also construct solutions having prescribed behavior near the maximum. We do this by analyzing the asymptotics for rescaled mean curvature flow converging to a stationary sphere.