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Yangqin Fang

Publications and source records attributed to Yangqin Fang.

9 recordsLinked to original sources
arXiv2022

Quasiminimal sets and Plateau's problem with Čech homological conditions on $C^2$ submanifold

Let $Ω\subseteq \mathbb{R}^n$ be an $m$-dimensional closed submanifold of class $C^2$, $d$ be a positive integer between 1 and $m$. We will study the geometric and topological proprieties of quasiminimal sets in $Ω$, and show that a minimizing sequence of $d$-sets converges to a minimal set in the sense of weak topology. Following from that, we can solve the Plateau's problem of dimension $d$ on $Ω$ with Čech homological conditions.

math.CA↗
arXiv2018

Existence of solutions to a general geometric elliptic variational problem

We consider the problem of minimising an inhomogeneous anisotropic elliptic functional in a class of closed $m$ dimensional subsets of $\mathbf{R}^n$ which is stable under taking smooth deformations homotopic to the identity and under local Hausdorff limits. We prove that the minimiser exists inside the class and is an $(\mathscr{H}^m,m)$~rectifiable set in the sense of Federer. The class of competitors encodes a notion of spanning a boundary. We admit unrectifiable and non-compact competitors and boundaries, and we make no restrictions on the dimension $m$ and the co-dimension $n-m$ other than $1 \le m < n$. An important tool for the proof is a novel smooth deformation theorem. The skeleton of the proof and the main ideas follow Almgren's 1968 paper. In the end we show that classes of sets spanning some closed set $B$ in homological and cohomological sense satisfy our axioms.

math.AP↗
arXiv2015

Hölder regularity at the boundary of two-dimensional sliding almost minimal sets

In [15], Jean Taylor has proved a regularity theorem away from boundary for Almgren almost minimal sets of dimension two in $\mathbb{R}^{3}$. It is quite important for understanding the soap films and the solutions of Plateau's problem away from boundary. In this paper, we will give a regularity result on the boundary for two dimensional sliding almost minimal sets in $\mathbb{R}^{3}$. It will be of use for understanding their boundary behavior.

math.CA↗
arXiv2014

Existence of Minimizers for the Reifenberg Plateau problem

That is, given a compact set $B \subset \mathbb{R}^n$ (the boundary) and a subgroup $L$ of the Čech homology group $\check{H}_{d-1}(B;G)$ of dimension $d$ over some commutative group $G$, we find a compact set $E \supset B$ such that the image of $L$ by the natural map $\check{H}_{d-1}(B;G)\to\check{H}_{d-1}(S;G)$ induced by the inclusion $B \to E$, is reduced to $\{ 0 \}$, and such that the Hausdorff measure $\mathcal{H}^{d}(E \setminus B)$ is minimal under these constraints. Thus we have no restriction on the group $G$ or the dimensions $0 < d < n$.

math.CA↗