Brakke's formulation of velocity and the second order regularity property

TL;DR

This paper investigates the regularity of evolving surfaces governed by Brakke's velocity law, proving that under certain conditions, the surface can be represented as a smooth graph satisfying the PDE pointwise, with applications to short-time existence.

Contribution

It establishes conditions under which a surface evolving by Brakke's law is a $C^{1,eta}$ graph satisfying the PDE pointwise, advancing understanding of regularity in geometric flows.

Findings

Surface graphs satisfy the PDE pointwise when the distributional time derivative is a Radon measure.

Proves $C^{1,eta}$ regularity of the evolving surface under specified conditions.

Provides a short-time existence theorem for the surface evolution problem.

Abstract

Suppose that a family of $k$-dimensional surfaces in $\mathbb R^n$ evolves by the motion law of $v=h+u^\perp$ in the sense of Brakke's formulation of velocity, where $v$ is the normal velocity vector, $h$ is the generalized mean curvature vector and $u^\perp$ is the normal projection of a given vector field $u$ in a dimensionally sharp integrability class. When the flow is locally close to a time-independent $k$-dimensional plane in a weak sense of measure in space-time, it is represented as a graph of a $C^{1,\alpha}$ function over the plane. On the other hand, it is not known if the graph satisfies the PDE of $v=h+u^\perp$ pointwise in general. For this problem, when $k=n-1$ and under the additional assumption that the distributional time derivative of the graph is a signed Radon measure, it is proved that the graph satisfies the PDE pointwise. An application to a short-time existence theorem for a surface evolution problem is given.