We prove a local splitting theorem for three-manifolds with mean convex boundary and scalar curvature bounded from below that contain certain locally area-minimizing free boundary surfaces. Our methods are based on those of Micallef and Moraru. We use this local result to establish a global rigidity theorem for area-minimizing free boundary disks. In the negative scalar curvature case, this global result implies a rigidity theorem for solutions of the Plateau problem with length-minimizing boundary.

2013-09-02math.APmath.DG

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}.

2022-02-16math.APmath.DG

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…

2020-01-17math.APmath.DG

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

We consider the evolution of sets by nonlocal mean curvature and we discuss the preservation along the flow of two geometric properties, which are the mean convexity and the outward minimality. The main tools in our analysis are the level set formulation and the minimizing movement scheme for the nonlocal flow. When the initial set is outward minimizing, we also show the convergence of the (time integrated) nonlocal perimeters of the discrete evolutions to the nonlocal perimeter of the limit flow.

2020-11-25math.AP

缩放不变非局部Willmore能量的凸极小曲线

Convex minimizing curves of the scaling-invariant nonlocal Willmore energy

本文研究缩放不变非局部Willmore能量，证明了在凸$C^1$-曲线类中极小元的存在性。

2024-02-07math.AP

Convexity criteria and uniqueness of absolutely minimizing functions

We show that absolutely minimizing functions relative to a convex Hamiltonian $H:\mathbb{R}^n \to \mathbb{R}$ are uniquely determined by their boundary values under minimal assumptions on $H.$ Along the way, we extend the known equivalences between comparison with cones, convexity criteria, and absolutely minimizing properties, to this generality. These results perfect a long development in the uniqueness/existence theory of the archetypal problem of the calculus of variations in $L^\infty.$

2010-03-16math.APmath.OC

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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.

2017-11-21math.APmath.DG

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.

2005-10-16math.AP

The area minimizing problem in conformal cones, II

In this paper we continue to study the connection among the area minimizing problem, certain area functional and the Dirichlet problem of minimal surface equations in a class of conformal cones with a similar motivation from \cite{GZ20}. These cones are certain generalizations of hyperbolic spaces. We describe the structure of area minimizing $n$-nteger multiplicity currents in bounded $C^2$ conformal cones with prescribed $C^1$ graphical boundary via a minimizing problem of these area functionals. As an application we solve the corresponding Dirichlet problem of minimal surface equations unde…

2020-09-19math.APmath.DG

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.

2020-06-06math.AP

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。

2024-05-30math.AP

约束映射下Alt-Phillips能量最小化映射的部分正则性

Partial regularity for minimizing constraint maps for the Alt-Phillips energy

KTH Royal Institute of Technology（瑞典皇家理工学院）

本文建立了一种Alt-Phillips型功能最小化约束映射的ε正则性定理，证明在能量足够小时，最小化映射具有正则性，从而证明这些映射的光滑性，并进一步推导出最优正则性。

2026-03-31math.AP

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.

2017-06-10math.AP

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

University of Wisconsin Madison（威斯康星大学麦迪逊分校）

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。

2026-08-13math.AP

Energy Minimizing Configurations for Single-Director Cosserat Shells

We consider a class of single-director Cosserat shell models accounting for both curvature and finite mid-plane strains. We assume a polyconvexity condition for the stored-energy function that reduces to a physically correct membrane model in the absence of bending. With appropriate growth conditions, we establish the existence of energy minimizers. The local orientation of a minimizing configuration is maintained via the blowup of the stored energy as a version of the local volume ratio approaches zero. Finally, we specialize our results to three constrained versions of the theory commonly em…

2022-12-08math.AP

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。

2024-08-06math.AP

最小化自由能

Minimizing the free energy

本文证明了径向各向同性Almgren问题的尖锐定量稳定性，在极小假设下发展了估计尖锐模数的理论并在任意维度得到尖锐$ε^2$，同时解决了临界质量的计算问题。

2023-04-04math-phmath.APmath.CA