Minimizing under relaxed symmetry constraints: Triple and $N$-junctions

Regularity and relaxed problems of minimizing biharmonic maps into spheres

For $n\ge 5$ and $k\ge 4$, we show that any minimizing biharmonic map from $Ω\subset R^n$ to $S^k$ is smooth off a closed set whose Hausdorff dimension is at most $n-5$. When $n=5$ and $k=4$, for a parameter $λ\in [0,1]$ we introduce a $λ$-relaxed energy $\H_λ$ for the Hessian energy for maps in $W^{2,2}(Ω,S^4)$ so that each minimizer $u_λ$ of $\H_λ$ is also a biharmonic map. We also estabilish the existence and partial regularity of a minimizer of $\H_λ$ for $λ\in [0,1)$.

A note on $n$-axially symmetric harmonic maps from $B^3$ to $S^2$ minimizing the relaxed energy

For any n>1 we give an explicit example of an n-axially symmetric Cartesian current in B^3 x S^2 with non-trivial vertical part and non-constant graph part minimizing the relaxed Dirichlet energy among the n-axially symmetric Cartesian currents with the same boundary. This stands in sharp contrast with a results of Hardt, Lin and Poon for the case n=1.

On the relaxed area of the graph of discontinuous maps from the plane to the plane taking three values with no symmetry assumptions

In this paper we estimate from above the area of the graph of a singular map $u$ taking a disk to three vectors, the vertices of a triangle, and jumping along three $\mathcal{C}^2-$ embedded curves that meet transversely at only one point of the disk. We show that the relaxed area can be estimated from above by the solution of a Plateau-type problem involving three entangled nonparametric area-minimizing surfaces. The idea is to "fill the hole" in the graph of the singular map with a sequence of approximating smooth two-codimensional surfaces of graph-type, by imagining three minimal surfaces,…

三重点问题中 blow-down 极限的唯一性

Uniqueness of the blow-down limit for triple junction problem

本文证明具有三井势的平面 Allen-Cahn 系统整体极小解在无穷远处具有唯一的 $L^1$ blow-down 极限，并可通过三重点映射逼近，证明基于能量上下界的精细分析。

二维艾伦 - 卡恩系统极小化解的分类

Classification of minimizing solutions to a two-dimensional Allen-Cahn system

研究二维艾伦 - 卡恩系统有界整体极小化解，通过特殊代数结构产生的校准恒等式，得到其完整分类，明确无穷远处有三叉结构时解的显式轮廓，并证明\(D_3\)等变类中极小化解与该轮廓一致。

Allen-Cahn 系统三结点解的刚性结果

Rigidity results for a triple junction solution of Allen-Cahn system

本文研究二维三阱势 Allen-Cahn 系统三结点极小化解的渐近刚性，证明其沿三界面呈一维异宿结构、扩散界面保持 $O(1)$ 宽度并趋于平坦，从而将对称情形结果推广至非对称情形。