Keisuke Takasao
We consider the obstacle problem of the weak solution for the mean curvature flow, in the sense of Brakke's mean curvature flow. We prove the global existence of the weak solution with obstacles which have $C^{1,1}$ boundaries, in two and three space dimensions. To obtain the weak solution, we use the Allen-Cahn equation with forcing term.
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On an obstacle problem for the Brakke flow with a generalized right-angle boundary condition
本文研究带障碍及广义直角边界条件的 Brakke 平均曲率流，利用相场方法分析带强迫项与齐次 Neumann 条件的 Allen-Cahn 方程奇异极限，并构造相应下、上解，证明弱解的时间整体存在性。
Brakke inequality and the existence of Brakke-flow for volume preserving mean curvature flow
本文提出保体积平均曲率流的新Brakke不等式概念，用相场方法证明积分可变体全局Brakke流的存在性，扩展了相关前期研究结果。
We prove convergence of solutions to the parabolic Allen-Cahn equation to Brakke's motion by mean curvature in space forms, generalizing previous results from [15] in Euclidean space. We show that a sequence of measures, associated to energy density of solutions of the parabolic Allen-Cahn equation, converges in the limit to a family of rectifiable Radon measures, which evolves by mean curvature flow in the sense of Brakke. A key role is played by a local almost monotonicity formula (a weak counterpart of Huisken's monotonicity formula) proved in [22], to get various density bounds for the lim…
We study the convergence of the system of the Allen-Cahn equations to the weak solution for the multi-phase mean curvature flow in the sense of Brakke. The Landau-Lifshitz equation in this paper can be regarded as a system of Allen-Cahn equations with the Lagrange multiplier, which is a phase field model of the multi-phase mean curvature flow. Under an assumption that the limit of the energies of the solutions for the equations matches with the total variation for the singular limit of the solutions, we show that the family of the varifolds derived from the energies is a Brakke flow.
In this paper we consider the Allen-Cahn equation with constraint. In 1994, Chen and Elliott studied the asymptotic behavior of the solution of the Allen-Cahn equation with constraint. They proved that the zero level set of the solution converges to the classical solution of the mean curvature flow under the suitable conditions on initial data. In 1993, Ilmanen proved the existence of the mean curvature flow via the Allen-Cahn equation without constraint in the sense of Brakke. We proved the same conclusion for the Allen-Cahn equation with constraint.
Convergence of a vector-valued Allen-Cahn system to Brakke's multiphase mean curvature flow
该研究首次证明耦合Allen-Cahn方程系统全局无条件收敛到Brakke意义下的多相平均曲率流，解决了该领域长期存在的重大开放问题。
New advances on the existence and regularity of Brakke's mean curvature flows
本文综述Brakke平均曲率流存在性与正则性的最新进展，涵盖局部正则性定理拓展、多相流全局存在性及分支奇异性动力学不稳定性证明，是研讨会报告的扩展版本。
On the existence of canonical multi-phase Brakke flows
本文构造从任意闭可求长初始数据出发、同时满足 Brakke 流与 BV 解定义的全局多相平均曲率流，并通过体积变化恒等式解决其非唯一性问题。
Brakke flow is defined with a variational inequality, which means it may have discontinuous mass over time, i.e. have mass drop. It has long been conjectured that the Brakke flow associated to a nonfattening level set flow has no mass drop and achieves equality in the Brakke inequality. Under natural assumptions, we show that a flow has no mass drop if and only if it satisfies the multiplicity one conjecture $\mathcal{H}^n$-a.e. One application is that there is no mass drop for level set flows with mean convex neighborhoods of singularities, and a generic flow has no mass drop until there is a…
Parabolic rectifiability of the Brakke flow
证明Brakke流的规范时空测度支撑集是抛物(k+2)-可整流集，从而在几乎所有点处存在唯一静态平面切流，并发展了时空-格拉斯曼测度方法，证明Brakke流收敛等价于该测度的收敛。
End-time regularity theorem for Brakke flows
本文证明了接近平面的 Brakke 流在终端时间的光滑图像表示与导数估计，并将 White 的局部正则性定理推广至一般及带强制项的 Brakke-like flows。
In this paper we prove an analogue of the Brakke's $\varepsilon$-regularity theorem for the parabolic Allen-Cahn equation. In particular, we show uniform $C^{2,α}$ regularity for the transition layers converging to smooth mean curvature flows as $\varepsilon\rightarrow0$. The proof utilises Allen-Cahn versions of the monotonicity formula, parabolic Lipschitz approximation and blowups. A corresponding gap theorem for entire eternal solutions of the parabolic Allen-Cahn is also obtained. As an application of the regularity theorem, we give an affirmative answer to a question of Ilmanen that ther…
A possible evolution of a compact hypersurface in R^n by mean curvature past singularities is defined via the level set flow. In the case that the initial hypersurface has positive mean curvature, we show that the Brakke flow associated to the level set flow is actually a Brakke flow with equality. We obtain as a consequence that no mass drop can occur along such a flow. As a further application of the techniques used above we give a new variational formulation for mean curvature flow of mean convex hypersurfaces.
We investigate the relation between the level set approach and the varifold approach for the mean curvature flow with Neumann boundary conditions. With an appropriate initial data, we prove that the almost all level sets of the unique viscosity level set solution satisfy Brakke's inequality and a generalized Neumann boundary condition.
$L^2$ normal velocity implies strong solution for graphical Brakke flows
本文证明在 Brakke 意义下具有 $L^2$ 法向速度的图变差族具有弱导数且与通常法向速度一致，并结合抛物正则性理论证明带强制项的图 Brakke 流是受迫平均曲率流方程的强解和经典解。
The space-time-Grassmann measure of the Brakke flow
针对开集上的Brakke流，构造了典范的时空-Grassmann测度，并基于该测度给出了Brakke流的新定义，证明了与经典定义的等价性。
Existence of BV flow via elliptic regularization
本文证明通过椭圆正则化得到的平均曲率流不仅是Brakke流，也是Stuvard和Tonegawa提出的广义BV流，并给出其体积变化与广义平均曲率的关系。
In this article we study eternal solutions to the Allen-Cahn equation in the 3-sphere, in view of the connection between the gradient flow of the associated energy functional, and the mean curvature flow. We construct eternal integral Brakke flows that connect Clifford tori to equatorial spheres, and study a family of such flows, in particular their symmetry properties. Our approach is based on the realization of Brakke's motion by mean curvature as a singular limit of Allen-Cahn gradient flows, as studied by Ilmanen, and Tonegawa, and it uses the classification of ancient gradient flows in sp…
In [LW], we construct examples of two-dimensional Hamiltonian stationary self-shrinkers and self-expanders for Lagrangian mean curvature flows, which are asymptotic to the union of two Schoen-Wolfson cones. These self-shrinkers and self-expanders can be glued together to yield solutions of the Brakke flow - a weak formulation of the mean curvature flow. Moreover, there is no mass loss along the Brakke flow. In this paper, we generalize these results to higher dimension. We construct new higher dimensional Hamiltonian stationary cones of different topology as generalizations of the Schoen-Wolfs…
We are concerned with solutions to the parabolic Allen-Cahn equation in Riemannian manifolds. For a general class of initial condition we show non positivity of the limiting energy discrepancy. This in turn allows to prove almost monotonicity formula (a weak counterpart of Huisken's monotonicity formula) which gives a local uniform control of the energy densities at small scales. Such results will be used in [40] to extend previous important results from [31] in Euclidean space, showing convergence of solutions to the parabolic Allen-Cahn equations to Brakke's motion by mean curvature in space…
A note on generalized BV mean curvature flow with a critical forcing term
本文注记证明了Liu–Tonegawa的带临界强制项的弱平均曲率流是广义BV流，确定了Brakke流满足面积变化公式的最小条件，推导了相关流的灭绝时间下界并概述了紧性定理的证明。
The epsilon-regularity theorem for Brakke flows near triple junctions
本文建立了 k 维 Brakke flows 在静态三分叉点附近的 ε-正则性定理，证明了在满足温和结构假设下三分叉点的正则结构在弱平均曲率流中得以保持，并证实两类基本流无条件满足该假设。
We give a proof that Brakke's mean curvature flow under the unit density assumption is smooth almost everywhere in space-time. More generally, if the velocity is equal in a weak sense to its mean curvature plus some given α-Hölder continuous vector field, then we show C^{2,α} regularity almost everywhere.
We propose a new weak solution concept for (two-phase) mean curvature flow which enjoys both (unconditional) existence and (weak-strong) uniqueness properties. These solutions are evolving varifolds, just as in Brakke's formulation, but are coupled to the phase volumes by a simple transport equation. First, we show that, in the exact same setup as in Ilmanen's proof [J. Differential Geom. 38, 417-461, (1993)], any limit point of solutions to the Allen-Cahn equation is a varifold solution in our sense. Second, we prove that any calibrated flow in the sense of Fischer et al. [arXiv:2003.05478] -…
In this article we use the mean curvature flow with surgery to derive regularity estimates for the level set flow going past Brakke regularity in certain special conditions allowing for 2-convex regions of high density. We also show a stability result for the plane under the level set flow.
Generic Mean Curvature Flow With Rough Weights
该研究证明了带粗糙权重空间中平均曲率流的一般存在性，其弱解兼具单位密度Brakke流与分布解的性质，所考虑的权重因过于粗糙无法在每个超曲面上定义明确定义的迹，结论为最优。
This paper presents a new partial differential equation to build Brakke's motion, which is a weak notion of mean curvature flow. We call the equation a modified Allen-Cahn equation abbreviated to MAC. After that we introduce its benefit: An improved estimate on the discrepant energy that means the equipartition of the first energy. This equation is obtained by rendering the parameter in Allen-Cahn equation time-dependent and adding a term like Z-transform. Furthermore, we mention the existence of a Brakke's motion through our MAC.
Brakke's formulation of velocity and the second order regularity property
研究Brakke速度公式下曲面演化的正则性：当流弱接近静态平面时可表为$C^{1,\boldsymbol{\beta}}$图，在$k=n-1$且分布时间导数为符号Radon测度时，证明图逐点满足PDE，并给出短时存在定理应用。
Time global existence of generalized BV flow via the Allen--Cahn equation
本文证明Allen-Cahn方程极限得到的平均曲率流同时是Brakke流和Stuvard-Tonegawa广义BV流，从而确立其时间全局存在性。
Geometric flows of branched transportation networks
本文针对分支输运问题，通过修改Brakke变分逼近格式构造输运网络的几何流，证明极限关于平坦范数Hölder连续且成本递减，并分析1-varifold逼近流的可求长性与运动定律。
Non-uniqueness in Mean Curvature Flow: Non-canonical solutions via the parabolic Allen--Cahn
本文通过抛物型 Allen--Cahn 方程的极限构造了严格位于增厚区域内部的非最外层弱平均曲率流，首次给出闭的非典范积分 Brakke 运动例子，并证明了从分形集出发的此类运动存在。
Diffusion-aggregation equations and volume-preserving mean curvature flows
本文针对椭圆-抛物型PKS模型，在二维和三维中首次无条件证明其解收敛到保体积平均曲率流，通过Brakke型公式和带非局部项的Allen-Cahn方程重释，避免了相分离无能量损失的假设。
We introduce a level-set formulation for the mean curvature flow with obstacles and show existence and uniqueness of a viscosity solution. These results generalize a well known viscosity approach for the mean curvature flow without obstacle by Evans and Spruck and Chen, Giga and Goto in 1991. In addition, we show that this evolution is consistent with the variational scheme introduced by Almeida, Chambolle and Novaga (2012) and we study the long time behavior of our viscosity solutions.
Equivalence of weak solution concepts for mean curvature flow
本文证明了均曲线流中粘性解与变分解概念的等价性，并展示了变分解的普遍唯一性。
We give a new proof of Brakke's partial regularity theorem up to C^{1,ς} for weak varifold solutions of mean curvature flow by utilizing parabolic monotonicity formula, parabolic Lipschitz approximation and blow-up technique. The new proof extends to a general flow whose velocity is the sum of the mean curvature and any given background flow field in a dimensionally sharp integrability class. It is a natural parabolic generalization of Allard's regularity theorem in the sense that the special time-independent case reduces to Allard's theorem.
We consider the sharp interface limit of the Allen-Cahn equation with Dirichlet or dynamic boundary conditions and give a varifold characterization of its limit which is formally a mean curvature flow with Dirichlet or dynamic boundary conditions. In order to show the existence of the limit, we apply the phase field method under the vanishing on the boundary and the boundedness of the discrepancy measure. For this purpose, we extend the usual Brakke flow under these boundary conditions by the first variations for varifolds on the boundary.
Eventual regularity of the volume-preserving mean curvature flow in three and two dimensions
本文研究了体积保持均曲率流在三维和二维中的最终正则性，证明了流在有限时间后变得光滑，并以经典意义满足方程，且在所有C^k范数下以指数速度收敛到极限球。
We prove Taylor scaling for dislocation lines characterized by line-tension and moving by curvature under the action of an applied shear stress in a plane containing a random array of obstacles. Specifically, we show--in the sense of optimal scaling--that the critical applied shear stress for yielding, or percolation-like unbounded motion of the dislocation, scales in proportion to the square root of the obstacle density. For sufficiently small obstacle densities, Taylor scaling dominates the linear-scaling that results from purely energetic considerations and, therefore, characterizes the dom…
Finding the convex hull of a set using the flow by minimal curvature with an obstacle. A game theoretical approach
本文研究带障碍的最小主曲率几何流，通过水平集障碍问题的博弈论近似，证明正超水平集随时间收敛到障碍的凸包。
Allen--Cahn and mean curvature flows from Clifford hypersurfaces to equators
该研究借助OpenAI对Solomon--Yau猜想的解决，将与Pedro Gaspar的合作成果推广至n≥2的所有维度，在S^{n+1}上构造了连接Clifford超曲面近似值与赤道球面的永恒Allen--Cahn流及无穷多永恒积分Brakke流。
Generic level sets in mean curvature flow are BV solutions
本文证明平均曲率流粘性解的一般水平集是有限周长集框架下的分布 BV 解，并达到最优能量耗散率，从而推广 Evans–Spruck 关于粘性解与 Brakke 流联系的结果。
Mean curvature bounds for the obstacle problem
该研究针对带光滑右端项的经典障碍问题，证明其解的最优$(n-1)$-半凸性估计，推导得所有维度自由边界平均曲率有通用上界，且三维及以上完整曲率界不成立，$n=2$常右端项情形下结果新颖。
Suppose that $Γ_0\subset\mathbb R^{n+1}$ is a closed countably $n$-rectifiable set whose complement $\mathbb R^{n+1}\setminus Γ_0$ consists of more than one connected component. Assume that the $n$-dimensional Hausdorff measure of $Γ_0$ is finite or grows at most exponentially near infinity. Under these assumptions, we prove a global-in-time existence of mean curvature flow in the sense of Brakke starting from $Γ_0$. There exists a finite family of open sets which move continuously with respect to the Lebesgue measure, and whose boundaries coincide with the space-time support of the mean curva…
A game-theoretic approach to the asymptotic behavior of solutions to an obstacle problem for the mean curvature flow equation
本文扩展 Kohn–Serfaty 确定性零和博弈逼近，证明平均曲率流在障碍作用下渐近收敛至障碍凸包边界，并分析含驱动力项的情形。
A family of hypersurfaces evolves by mean curvature flow if the velocity at each point is given by the mean curvature vector. Mean curvature flow is the most natural evolution equation in extrinsic geometry, and has been extensively studied ever since the pioneering work of Brakke and Huisken. In the last 15 years, White developed a far-reaching regularity and structure theory for mean convex mean curvature flow, and Huisken-Sinestrari constructed a flow with surgery for two-convex hypersurfaces. In this course, I first give a general introduction to the mean curvature flow of hypersurfaces an…
Existence of weak solutions to volume-preserving mean curvature flow with obstacles
通过相场方法证明了所有维度下带障碍的体积保持平均曲率流的全局弱解存在性，基于Allen-Cahn方程与乘子的收敛，并处理了障碍引起的空间依赖强迫项。
Convergence of thresholding energies for anisotropic mean curvature flow on inhomogeneous obstacle
本文扩展了Esedoglu和Otto的阈值能量分析至空间依赖各向异性情形，针对障碍问题构造单调近似能量并证明其Γ-收敛性。
Mean curvature flow of clusters of n-dimensional surfaces in R^{n+k} that meet in triples at equal angles along smooth edges and higher order junctions on lower dimensional faces is a natural extension of classical mean curvature flow. We call such a flow a mean curvature flow with triple edges. We show that if a smooth mean curvature flow with triple edges is weakly close to a static union of three n-dimensional unit density half-planes, then it is smoothly close. Extending the regularity result to a class of integral Brakke flows, we show that this implies smooth short-time existence of the …
Existence of curvature flow with forcing in a critical Sobolev space
研究给定闭1-可求积集\(\Gamma_0\)及临界Sobolev空间向量场\(u\)时，从\(\Gamma_0\)出发的曲线流的存在性，其法向速度由曲率与\(u\)之和给出，运动定律在Brakke意义下成立且流在奇点处存在。
Dynamical instability of minimal surfaces at flat singular points
该研究针对具有平坦奇点的一重平稳Varifold，在适当衰减率假设下证明存在非常值Brakke流，揭示Brakke流的非唯一性，为用平均曲率流稳定性排除平坦奇点提供依据。
We show short time existence and uniqueness of $\C^{1,1}$ solutions to the mean curvature flow with obstacles, when the obstacles are of class $\C^{1,1}$. If the initial interface is a periodic graph we show long time existence of the evolution and convergence to a minimal constrained hypersurface.
Existence of weak mean curvature flow with prescribed contact angle via elliptic regularization
本文通过Ilmanen正则化，将其扩展到毛细现象领域并推导对应一阶变分估计，证明了满足θ∈(0,π)指定接触角的Brakke型弱平均曲率流的存在性。
Second order estimates for transition layers and a curvature estimate for the parabolic Allen-Cahn
本文证明了抛物型Allen-Cahn方程向平均曲率流改进的收敛性质（即相变水平集在$C^2$中收敛则必在$C^{2,θ}$中收敛），并以此推导出该方程的曲率估计，可视为平均曲率流Brakke和White正则性定理的扩散版本。
An Intersection Principle for Mean Curvature Flow
本研究推广平均曲率流的回避原理，证明多类平均曲率流交集/自相交维数的非递增性，明确单调性适用范围并给出有限奇点水平集流的胖化判据。
We consider the problem of evolving hypersurfaces by mean curvature flow in the presence of obstacles, that is domains which the flow is not allowed to enter. In this paper, we treat the case of complete graphs and explain how the approach of M. Saez and the second author yields a global weak solution to the original problem for general initial data and onesided obstacles.
In this paper, we study an obstacle problem associated with the mean curvature flow with constant driving force. Our first main result concerns interior and boundary regularity of the solution. We then study in details the large time behavior of the solution and obtain the convergence result. In particular, we give full characterization of the limiting profiles in the radially symmetric setting.
Gauss curvature flow with shrinking obstacle
该研究针对带障碍约束的高斯曲率幂次流，在全维度正幂次下证得解的最优曲率界与长时间存在性，并在一致厚度假设下得到自由边界的$C^1$正则性，丰富了带约束曲率流理论。
In this paper, we prove the mean-convex neighborhood conjecture for neck singularities of the mean curvature flow in $\mathbb{R}^{n+1}$ for all $n\geq 3$: we show that if a mean curvature flow $\{M_t\}$ in $\mathbb{R}^{n+1}$ has an $S^{n-1}\times \mathbb{R}$ singularity at $(x_0,t_0)$, then there exists an $\varepsilon=\varepsilon(x_0,t_0)>0$ such that $M_t\cap B(x_0,\varepsilon)$ is mean-convex for all $t\in(t_0-\varepsilon^2,t_0+\varepsilon^2)$. As in the case $n=2$, which was resolved by the first three authors in arXiv:1810.08467, the existence of such a mean-convex neighborhood follows fr…
Generic mean curvature flow with obstacles
本文研究带障碍的平均曲率流，通过添加奇异扰动惩罚约束违反并取极限，证明通用水平集是障碍问题的分布解。
Let $N$ be a smooth $(n+l)$-dimensional Riemannian manifold. We show that if $V$ is an area-stationary union of three or more $C^{1,μ}$ $n$-dimensional submanifolds-with-boundary $M_k \subset N$ with a common boundary $Γ$, then $Γ$ is smooth and each $M_k$ is smooth up to $Γ$ (real-analytic in the case $N$ is real-analytic). This extends a previous result of the author for codimension $l = 1$. We additionally show that if $\{V_t\}_{t \in (-1,1)}$ is a Brakke flow such that each time-slice $V_t$ is a union of three or more $n$-dimensional submanifolds-with-boundary $M_{k,t} \subset N$ with a co…
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