arXiv subjects

Kotaro Motegi

Publications and source records attributed to Kotaro Motegi.

2 recordsLinked to original sources

arXiv2026

Non-uniqueness of Brakke flows starting from minimal surfaces with singularities

We prove the existence of a genuinely time-dependent Brakke flow starting from $\Gamma_0 \subset \mathbb{R}^{n+1}$ whose associated multiplicity-one varifold is stationary, provided that, at some singular point, the scale-invariant $L^2$ distance of $\Gamma_0$ from an $n$-dimensional plane has sufficiently small limsup as the scale tends to zero. This yields the dynamical instability of $\Gamma_0$, a notion recently introduced by Stuvard and Tonegawa, and hence the non-uniqueness of Brakke flows starting from $\Gamma_0$. A notable feature of our result is that it holds without assuming the uniqueness of tangent cones at the singular point.

math.AP↗

arXiv2025

$L^2$ normal velocity implies strong solution for graphical Brakke flows

We prove that if a one-parameter family of varifolds has an $L^2$ normal velocity $v$ in the sense of Brakke, and if the family is represented as the graph of a continuous function $f$ with continuous spatial derivative $\nabla f$, then $f$ has weak derivatives $\partial_t f, \nabla^2 f \in L^2$, and $v$ coincides with the usual normal velocity of the graph. Moreover, by combining this result with parabolic regularity theory, we show that graphical Brakke flows with forcing term in $L^{p,q}$ and $C^{0,\alpha}$ are strong and classical solutions to the forced mean curvature flow equation, respectively.

math.AP↗