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Kiichi Tashiro

Publications and source records attributed to Kiichi Tashiro.

6 recordsLinked to original sources
arXiv2026

A note on generalized BV mean curvature flow with a critical forcing term

In this note, we show that a weak mean curvature flow with critical forcing term obtained by Liu--Tonegawa (2024) satisfies the BV-type area change formula, that is, their flow is not only a Brakke flow but also a generalized BV flow, which is proposed by Stuvard--Tonegawa (2024). To establish this result, we identify minimal conditions under which a Brakke flow satisfies the area-change formula. As an application of our main theorem, we derive a lower bound for the extinction time of generalized BV flows with a critical forcing term. We also outline the proof of a compactness theorem for generalized BV flows.

math.AP↗
arXiv2025

Parabolic free boundary phase transition and mean curvature flow

It is known that there is a strong relation between the parabolic Allen--Cahn equation and the mean curvature flow, in the sense that the parabolic Allen--Cahn equation can be considered as a "diffused" mean curvature flow. In this work, we derive a forced mean curvature flow \[ v=-H-\partial_\nu\log |\nabla u|+f(u)/|\nabla u|, \] satisfied by level surfaces of any solution to the nonlinear parabolic equation \[ \partial_tu=\Delta u-f(u). \] Moreover, we introduce the notion of the inner gradient flow, and unify parabolic free boundary problems in the gradient flow framework. Finally, we consider the parabolic free boundary Allen--Cahn equation \[ \left\{\begin{alignedat}{2} \partial_tu&=\Delta u\quad&&\text{in}\quad\{|u|<1\} |\nabla u|&=1/\epsilon\quad&&\text{on}\quad\partial\{|u|<1\}, \end{alignedat} \right. \] and confirm that under reasonable assumptions, the $C^{\alpha}$ norm of the forcing term $\partial_\nu\log|\nabla u|$ converges to zero at an algebraic rate as $\epsilon\to 0$, uniformly in time. This implies that the parabolic free boundary Allen--Cahn equation converges to the mean curvature flow, uniformly (in $\epsilon$ and in time) in the $C^{2,\alpha}$ sense.

math.AP↗
arXiv2025

Varifold convergence of free boundary Allen--Cahn equation

The free boundary Allen--Cahn equation $\Delta u=0$ in $\{|u|<1\}$, $|\nabla u|=1/\varepsilon$ on $\partial\{|u|<1\}$ has recently attracted considerable attention because it retains the essential features of the classical Allen--Cahn equation while being significantly more tractable. In this work, we establish the free boundary analogue of the seminal Hutchinson--Tonegawa theory, developing the varifold convergence framework for solutions of the free boundary Allen--Cahn equation to minimal surfaces. In addition, we provide the $\Gamma$-convergence of the free boundary Allen--Cahn energy to the area functional, and the conservation of local minimization property. This foundation is expected to be used in further applications of the free boundary Allen--Cahn equation in the study of minimal surfaces, such as providing an alternative proof of celebrated Yau's conjecture, possibly with simpler and more complete arguments.

math.AP↗
arXiv2024

Existence of weak mean curvature flow with prescribed contact angle via elliptic regularization

In the present paper, we study the existence of Brakke-type weak mean curvature flow satisfying a prescribed contact angle condition for a general angle $ \theta \in ( 0 , \pi ) $ via Ilmanen's regularization. The main ingredients of the result are the extension of Ilmanen's regularization to the capillarity and the derivation of the first variation estimates for the interior and wetted boundary varifolds separately.

math.DG↗
arXiv2023

Existence of BV flow via elliptic regularization

We investigate a mean curvature flow obtained via elliptic regularization, and prove that it is not only a Brakke flow, but additionally a generalized BV flow proposed by Stuvard and Tonegawa. In particular, we show that the change in volume of the evolving phase can be expressed in terms of the generalized mean curvature of the Brakke flow.

math.DG↗