arXiv subjects

Xusheng Du

Publications and source records attributed to Xusheng Du.

22 records · Page 2Linked to original sources

arXiv2022

Sketch-based 3D Shape Modeling from Sparse Point Clouds

3D modeling based on point clouds is an efficient way to reconstruct and create detailed 3D content. However, the geometric procedure may lose accuracy due to high redundancy and the absence of an explicit structure. In this work, we propose a human-in-the-loop sketch-based point cloud reconstruction framework to leverage users cognitive abilities in geometry extraction. We present an interactive drawing interface for 3D model creation from point cloud data with the help of user sketches. We adopt an optimization method in which the user can continuously edit the contours extracted from the obtained 3D model and retrieve the model iteratively. Finally, we verify the proposed user interface for modeling from sparse point clouds. see video here https://www.youtube.com/watch?v=0H19NyXDRJE .

cs.GR↗

arXiv2021

Blow up limits of the fractional Laplacian and their applications to the fractional Nirenberg problem

We show a convergence result of the fractional Laplacian for sequences of nonnegative functions without uniform boundedness near infinity. As an application, we construct a sequence of solutions to the fractional Nirenberg problem that blows up in the region where the prescribed functions are negative. This is a different phenomenon from the classical Nirenberg problem.

math.AP↗

arXiv2020

Large singular solutions for conformal $Q$-curvature equations on $\mathbb{S}^n$

In this paper, we study the existence of positive functions $K \in C^1(\mathbb{S}^n)$ such that the conformal $Q$-curvature equation \begin{equation}\label{001} P_m (v) =K v^{\frac{n+2m}{n-2m}}~~~~~~ {on} ~ \mathbb{S}^n \{equation} has a singular positive solution $v$ whose singular set is a single point, where $m$ is an integer satisfying $1 \leq m < n/2$ and $P_m$ is the intertwining operator of order $2m$. More specifically, we show that when $n\geq 2m+4$, every positive function in $C^1(\mathbb{S}^n)$ can be approximated in the $C^1(\mathbb{S}^n)$ norm by a positive function $K\in C^1(\mathbb{S}^n)$ such that the conformal $Q$-curvature equation has a singular positive solution whose singular set is a single point. Moreover, such a solution can be constructed to be arbitrarily large near its singularity. This is in contrast to the well-known results of Lin \cite{Lin1998} and Wei-Xu \cite{Wei1999} which show that the conformal $Q$-curvature equation, with $K$ identically a positive constant on $\mathbb{S}^n$, $n > 2m$, does not exist a singular positive solution whose singular set is a single point.

math.AP↗

arXiv2020

Local behavior of positive solutions of higher order conformally invariant equations with a singular set

We study some properties of positive solutions to the higher order conformally invariant equation with a singular set $$ (-Δ)^m u = u^{\frac{n+2m}{n-2m}} ~~~~~~ \textmd{in} ~ Ω\backslash Λ, $$ where $Ω\subset \mathbb{R}^n$ is an open domain, $Λ$ is a closed subset of $\mathbb{R}^n$, $1 \leq m < n/2$ and $m$ is an integer. We first establish an asymptotic blow up rate estimate for positive solutions near the singular set $Λ$ when $Λ\subset Ω$ is a compact set with the upper Minkowski dimension $\overline{\textmd{dim}}_M(Λ) < \frac{n-2m}{2}$, or is a smooth $k$-dimensional closed manifold with $k\leq \frac{n-2m}{2}$. We also show the asymptotic symmetry of singular positive solutions suppose $Λ\subset Ω$ is a smooth $k$-dimensional closed manifold with $k\leq \frac{n-2m}{2}$. Finally, a global symmetry result for solutions is obtained when $Ω$ is the whole space and $Λ$ is a $k$-dimensional hyperplane with $k\leq \frac{n-2m}{2}$.

math.AP↗