arXiv subjects

Changlin Xiang

Publications and source records attributed to Changlin Xiang.

4 recordsLinked to original sources

arXiv2026

Quantitative stratification and global regularity for 1/2-harmonic mappings

In this paper, we extend the celebrated global regularity theory of Naber-Valtorta [Ann. Math. 2017] to 1/2-harmonic mappings into manifolds. Inspired by their work, we first adapt Lin's defect measure theory [Ann. Math. 1999] to such maps building on the partial regularity established by Millot-Pegon-Schikorra [Arch. Ration. Mech. Anal. 2021]. Then apply it to show that the set of singular points of such maps can be quantitatively stratified via a new notion of boundary symmetry with the aid of {the celebrated harmonic extension method by Caffarelli-Silverstre}. As in that of Naber-Valtorta, developing the necessary quantitative regularity estimates, and then combining it with the Reifenberg type theorems and a delicate covering argument allow us to get sharp growth estimates on the volume of tubular neighborhood around singular points and establish the rectifiability of each singular stratum.

math.AP↗

arXiv2016

Liouville Type Theorem For A Nonlinear Neumann Problem

Consider the following nonlinear Neumann problem \[ \begin{cases} \text{div}\left(y^{a}\nabla u(x,y)\right)=0, & \text{for }(x,y)\in\mathbb{R}_{+}^{n+1}\\ \lim_{y\rightarrow0+}y^{a}\frac{\partial u}{\partial y}=-f(u), & \text{on }\partial\mathbb{R}_{+}^{n+1},\\ u\ge0 & \text{in }\mathbb{R}_{+}^{n+1}, \end{cases} \] $a\in(-1,1)$. A Liouville type theorem and its applications are given under suitable conditions on $f$. Our tool is the famous moving plane method.

math.AP↗

arXiv2015

Infinitely many solutions for p-Laplacian equation involving double critical terms and boundary geometry

Let $1 p^{2}+p,a(0)>0$ and $Ω$ satisfies some geometry conditions if $0\in\partialΩ$, say, all the principle curvatures of $\partialΩ$ at $0$ are negative, then the above problem has infinitely many solutions.

math.AP↗

arXiv2014

Quantitative Properties on the Steady States to A Schrödinger-Poisson-Slater System

A relatively complete picture on the steady states of the following Schr$\ddot{o}$dinger-Poisson-Slater (SPS) system \[ \begin{cases} -ΔQ+Q=VQ-C_{S}Q^{2}, & Q>0\text{ in }\mathbb{R}^{3}\\ Q(x)\to0, & \mbox{as }x\to\infty,\\ -ΔV=Q^{2}, & \text{in }\mathbb{R}^{3}\\ V(x)\to0 & \mbox{as }x\to\infty. \end{cases} \] is given in this paper: existence, uniqueness, regularity and asymptotic behavior at infinity, where $C_{S}>0$ is a constant. To the author's knowledge, this is the first uniqueness result on SPS system.

math.AP↗