$K$ mean-convex and $K$-outward minimizing sets

A note on the smoothness of energy-minimizing incompressible deformations

In this note we prove that any $W^{1,2}$ mapping $u$ in the plane that minimizes an appropriate quasiconvex energy functional subject to the Jacobian constraint ${\rm det} \na u=1$ a.e., are necessarily Lipschitz. Furthermore we show that the minimizers corresponding to uniformly convex energy are affine and give an example of non-affine minimizers subject to affine boundary data corresponding to a convex energy. We also discuss the regularity issues in dimension greater than or equal to 3.

关于最小化非局部能量的小液滴形状

On the Shape of Small Liquid Drops Minimizing Nonlocal Energies

本文研究势能作用下分数周长极小化问题中小液滴的平衡形状，并通过定量估计证明小体积极小元具有凸性且一致逼近球体。

Mean Convex Smoothing of Mean Convex Cones

We show that any minimizing hypercone can be perturbed into one side to a properly embedded smooth minimizing hypersurface in the Euclidean space, and every viscosity mean convex cone admits a properly embedded smooth mean convex self-expander asymptotic to it near infinity. These two together confirm a conjecture of Lawson \cite[Problem 5.7]{Brothers86_OpenProblems}.

The area minimizing problem in conformal cones

In this paper we study the area minimizing problem in some kinds of conformal cones. This concept is a generalization of the cones in Eulcidean spaces and the cylinders in product manifolds. We define a non-closed-minimal (NCM) condition for bounded domains. Under this assumption and other necessary conditions we establish the existence of bounded minimal graphs in mean convex conformal cones. Moreover those minimal graphs are the solutions to corresponding area minizing problems. We can solve the area minimizing problem in non-mean convex translating conformal cones if these cones are contain…

A Ginzburg-Landau type energy with weight and with convex potential near zero

In this paper, we study the asymptotic behaviour of minimizing solutions of a Ginzburg-Landau type functional with a positive weight and with convex potential near $0$ and we estimate the energy in this case. We also generalize a lower bound for the energy of unit vector field given initially by Brezis-Merle-Rivière.

非线性弹性映射：预定曲面上膜的能量最小化构型

Nonlinearly Elastic Maps: Energy Minimizing Configurations of Membranes on Prescribed Surfaces

研究受限于预定曲面的非线性弹性膜模型，提出在应变对中凸的能量密度函数，证明了约束于给定曲面时能量最小化构型的存在性，并验证了最小化变形的同胚性及平衡方程的弱欧拉形式。

Minimizing movements for mean curvature flow of partitions

We prove the existence of a weak global in time mean curvature flow of a bounded partition of space using the method of minimizing movements. The result is extended to the case when suitable driving forces are present. We also prove some consistency results for a minimizing movement solution with smooth and viscosity solutions when the evolution starts from a partition made by a union of bounded sets at a positive distance. In addition, the motion starting from the union of convex sets at a positive distance agrees with the classical mean curvature flow and is stable with respect to the Hausdo…

A duality based approach to the minimizing total variation flow in the space $H^{-s}$

We consider a gradient flow of the total variation in a negative Sobolev space $H^{-s}$ $(0\leq s \leq 1)$ under the periodic boundary condition. If $s=0$, the flow is nothing but the classical total variation flow. If $s=1$, this is the fourth order total variation flow. We consider a convex variational problem which gives an implicit-time discrete scheme for the flow. By a duality based method, we give a simple numerical scheme to calculate this minimizing problem numerically and discuss convergence of a forward-backward splitting scheme. Several numerical experiments are given.

无穷维下重排不变凸哈密顿-雅可比方程的周期均匀化最优收敛率

Optimal Convergence Rate for Periodic Homogenization of Rearrangement-Invariant Convex Hamilton--Jacobi Equations in Infinite Dimensions

该研究证明了无穷维重排不变凸哈密顿-雅可比方程周期均匀化的最优收敛率为$O(\varepsilon)$，推广了有限维结果，并通过新方法克服了无穷维空间的拓扑限制，且验证了该收敛率的尖锐性。

任意余维数和重数的二维面积最小化流的边界正则集的密度

Density of the boundary regular set of 2d area minimizing currents with arbitrary codimension and multiplicity

本文在任意余维数和任意边界重数且无凸障碍假设的条件下，证明了二维面积最小化流的正则集在边界中是开稠集，并将部分结果推广到几乎面积最小化流。

On The Phase Connectedness Of The Volume-Constrained Area Minimizing Partitioning Problem

We study the stability of partitions in convex domains involving simultaneous coexistence of three phases, viz. triple junctions. We present a careful derivation of the formula for the second variation of area, written in a suitable form with particular attention to boundary and spine terms, and prove, in contrast to the two phase case, the existence of stable partitions involving a disconnected phase.