任意余维的非参数Plateau问题

余维三中的非交换杨-米尔斯-希格斯与普拉托问题

Nonabelian Yang-Mills-Higgs and Plateau's problem in codimension three

本文研究SU(2)-杨-米尔斯-希格斯能量在大质量极限下的渐近行为，证明其Γ-收敛到余维三面积泛函，并在广义磁单极方程场景中推导得能量测度极限与校准流的关系，为相关几何预测提供证据。

Plateau问题的直接解法

A direct approach to Plateau's problem

我们提供了一个紧凑性原则，这是适用于不同配方的高原问题的余维1，这是完全基于理论的Radon措施和小学的比较参数。利用几何测度论中的一些附加技巧，我们可以利用这个原理给出Harrison和Pugh的一个定理的不同证明，并回答Guy David提出的一个问题。

涡旋映射面积的松弛：包含线段的悬链面的非参数 Plateau 问题

Relaxation of the area of the vortex map: a non-parametric Plateau problem for a catenoid containing a segment

本文研究了涡旋映射图面积松弛的奇异部分，证明其对应于包含线段的极小悬链面，并分析了该自由边界面积最小化问题的边界正则性。

Solutions to the Reifenberg Plateau problem with cohomological spanning conditions

We prove existence and regularity of minimizers for Hölder densities over general surfaces of arbitrary dimension and codimension in \(\R^n \), satisfying a cohomological boundary condition, providing a natural dual to Reifenberg's Plateau problem. We generalize and extend methods of Reifenberg, Besicovitch, and Adams, in particular we generalize a particular type of minimizing sequence used by Reifenberg (whose limits have nice properties, including lower bounds on lower density and finite Hausdorff measure,) prove such minimizing sequences exist, and develop cohomological spanning conditions…

具有大奇异集的最小曲面系统的解

Solutions to the minimal surface system with large singular sets

本文构造了最小曲面系统 Dirichlet 问题中 Lipschitz 图极小化序列的 Cartesian current 极限，证明其可具有大的内部垂直和非极小部分，揭示了高余维下参数化与非参数化面积极小化问题的显著差异，且构造达到最小维数 n=3 和余维数 m=2。

Well-posedness properties of geometric variational problems: existence, regularity and uniqueness results

This thesis is devoted to the study of well-posedness properties of some geometric variational problems: existence, regularity and uniqueness of solutions. We study two specific problems arising in the context of geometric calculus of variations and sharing strong analogies: the Plateau's problem and the optimal branched transport problem. The first part of the thesis discusses the existence theory. Both problems are formulated in the language of Federer and Fleming's theory of currents. After an exposition of the main results, we will present the core ideas of the (interior) regularity theory…

Existence results for minimizers of parametric elliptic functionals

We prove a compactness principle for the anisotropic formulation of the Plateau problem in any codimension, in the same spirit of the previous works of the authors \cite{DelGhiMag,DePDeRGhi,DeLDeRGhi16}. In particular, we perform a new strategy for the proof of the rectifiability of the minimal set, based on the new anisotropic counterpart of the Allard rectifiability theorem proved by the authors in \cite{DePDeRGhi2}. As a consequence we provide a new proof of Reifenberg existence theorem.

The One-Sided Isometric Extension Problem

Let $Σ$ be a codimension one submanifold of an $n$-dimensional Riemannian manifold $M$, $n\geqslant 2$. We give a necessary condition for an isometric immersion of $Σ$ into $\mathbb R^q$ equipped with the standard Euclidean metric, $q\geqslant n+1$, to be locally isometrically $C^1$-extendable to $M$. Even if this condition is not met, "one-sided" isometric $C^1$-extensions may exist and turn out to satisfy a $C^0$-dense parametric $h$-principle in the sense of Gromov.

通过 Allen--Cahn 泛函求解 Plateau 问题

Plateau's problem via the Allen--Cahn functional

本文通过研究实线丛截面上的 Allen--Cahn 泛函临界点，证明了其极限收敛于平稳的极小超曲面，从而为光滑版 Plateau 问题提供了新证明。

半线性变分问题自由边界正则性研究：带有拓扑约束

Free boundary regularity for semilinear variational problems with a topological constraint

本文研究带有拓扑约束的半线性自由边界问题，探讨了自由边界在特定奇异结构下的正则性，证明了极小解的存在性及最优Lipschitz正则性，以及远离二维奇异集的自由边界分析正则性。

多圆柱端点流形上的椭圆边值问题

Elliptic boundary problems on manifolds with polycylindrical ends

本文研究了多柱端点流形上的一般Shapiro-Lopatinsky椭圆边值问题.这是通过紧化这样的一个流形的角落，在一般较高的余维流形，然后我们处理的尖微分算子的边值问题。本文介绍了一种适用于伪微分边值问题的Boutet de Monvel演算，并在此演算中构造了椭圆尖算子的参数. Fredholm可解性和椭圆正则性，直到边界和无穷大的边值问题的流形与polycylindrical结束如下。

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,…

关于曲线不连续的平面到平面的分段光滑映射的图的面积

On the area of the graph of a piecewise smooth map from the plane to the plane with a curve discontinuity

在本文中，我们提供了一个估计，从上面的放松面积功能的值定义在一个有界区域的平面上的值和不连续的平面上的正则简单曲线上的两个端点。我们表明，在适当的假设下，放松的区域不超过该地区的经常部分的地图，增加了一个奇异项测量面积的一个磁盘类型的解决方案的高原的问题跨越两个痕迹的地图上的跳跃。当面积最小化曲面有自相交时，该结果同样有效。在我们的论点中的一个关键因素是要显示存在的，我们称之为一个semicartesian参数化这个表面，即一个共形参数化定义在一个合适的参数空间，这是在第一个组件的身份。为了证明我们的结果，使用了参数极小曲面理论的各种工具，以及莫尔斯理论的一些结果。

关于线性增长变分问题中边界数据的可达性

On the attainment of boundary data in variational problems with linear growth

该研究针对线性增长凸变分问题的边界数据可达性展开，证明在边界平均凸性条件下，松弛问题极小元可在迹意义下满足边界数据，方法可处理拟迷向假设下的方程组，还给出了相关应用。