Brakke's inequality for the thresholding scheme

Brakke不等式与保体积平均曲率流的Brakke流存在性

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。

Mass Drop and Multiplicity in Mean Curvature Flow

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…

De Giorgi's inequality for the thresholding scheme with arbitrary mobilities and surface tensions

We provide a new convergence proof of the celebrated Merriman-Bence-Osher scheme for multiphase mean curvature flow. Our proof applies to the new variant incorporating a general class of surface tensions and mobilities, including typical choices for modeling grain growth. The basis of the proof are the minimizing movements interpretation of Esedoğlu and Otto and De Giorgi's general theory of gradient flows. Under a typical energy convergence assumption we show that the limit satisfies a sharp energy-dissipation relation.

半调和映射热流的阈值化格式

Thresholding Scheme for the Half-Harmonic Map Heat Flow

本文提出一种基于泊松核卷积与逐点归一化的阈值化格式求解半调和映射热流，证明其收敛到满足能量耗散的弱解，并建立了弱-强唯一性。

测地直径的先验界。第三部分. Sobolev-Poincaré 不等式及其在多种几何变分问题中的应用

A priori bounds for geodesic diameter. Part III. A Sobolev-Poincaré inequality and applications to a variety of geometric variational problems

本文基于新型 Sobolev-Poincaré 不等式，建立了用曲面及边界平均曲率表示任意维广义紧致带边曲面测地直径的有限上界，并适用于多种含奇点的 varifold 与 Plateau 问题解。

加权、多相、保体积平均曲率流作为流形上MBO格式的极限

Weighted, Multiphase, Volume-Preserving Mean Curvature Flow as Limit of the MBO Scheme on Manifolds

本文研究了数据科学背景下加权流形上的多相体积约束MBO阈值化方案，证明了其在光滑闭流形上向加权保体积平均曲率流的收敛性，并提出了新的De Giorgi解概念。