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Chuanxi Wu

Publications and source records attributed to Chuanxi Wu.

11 recordsLinked to original sources
arXiv2026

On eigenfunctions and nodal sets of the Witten-Laplacian

In this paper, we successfully establish a Courant-type nodal domain theorem for both the Dirichlet eigenvalue problem and the closed eigenvalue problem of the Witten-Laplacian. Moreover, we also characterize the properties of the nodal lines of the eigenfunctions of the Witten-Laplacian on smooth Riemannian $2$-manifolds. Besides, for a Riemann surface with genus $g$, an upper bound for the multiplicity of closed eigenvalues of the Witten-Laplacian can be provided.

math.DG↗
arXiv2021

The $L_p$ Gauss image problem

In this paper we study the $L_p$ Gauss image problem, which is a generalization of the $L_p$ Aleksandrov problem and the Gauss image problem in convex geometry. We obtain the existence result for the $L_p$ Gauss image problem in two cases (i) $p>0$ or (ii) $p<0$ with the given even measures.

math.AP↗
arXiv2021

A stability result for translating space-like graphs in Lorentz manifolds

In this paper, we investigate space-like graphs defined over a domain $Ω\subset M^{n}$ in the Lorentz manifold $M^{n}\times\mathbb{R}$ with the metric $-ds^{2}+σ$, where $M^{n}$ is a complete Riemannian $n$-manifold with the metric $σ$, $Ω$ has piecewise smooth boundary, and $\mathbb{R}$ denotes the Euclidean $1$-space. We can prove an interesting stability result for translating space-like graphs in $M^{n}\times\mathbb{R}$ under a conformal transformation.

math.DG↗
arXiv2020

Eigenvalue inequalities for the buckling problem of the drifting Laplacian of arbitrary order

In this paper, we investigate the buckling problem of the drifting Laplacian of arbitrary order on a bounded connected domain in complete smooth metric measure spaces (SMMSs) supporting a special function, and successfully get a general inequality for its eigenvalues. By applying this general inequality, if the complete SMMSs considered satisfy some curvature constraints, we can obtain a universal inequalities for eigenvalues of this buckling problem.

math.DG↗
arXiv2019

Eigenvalue comparisons in Steklov eigenvalue problem and some other eigenvalue estimates

In this paper, two interesting eigenvalue comparison theorems for the first non-zero Steklov eigenvalue of the Laplacian have been established for manifolds with radial sectional curvature bounded from above. Besides, sharper bounds for the first non-zero eigenvalue of the Wentzell eigenvalue problem of the weighted Laplacian, which can be seen as a natural generalization of the classical Steklov eigenvalue problem, have been obtained.

math.DG↗
arXiv2018

The distributional hyper-Jacobian determinants in fractional Sobolev spaces

In this paper we give a positive answer to a question raised by Baer-Jerison in connection with hyper-Jacobian determinants and associated minors in fractional Sobolev spaces. Inspired by recent works of Brezis-Nguyen and Baer-Jerison on the Jacobian and Hessian determinants, we show that the distributional $m$th-Jacobian minors of degree $r$ are weak continuous in fractional Sobolev spaces $W^{m-\frac{m}{r},r}$, and the result is optimal, satisfying the necessary conditions, in the frame work of fractional Sobolev spaces. In particular, the conditions can be removed in case $m=1,2$, i.e., the $m$th-Jacobian minors of degree $r$ are well defined in $W^{s,p}$ if and only if $W^{s,p} \subseteq W^{m-\frac{m}{r},m}$ in case $m=1,2$.

math.AP↗
arXiv2018

Cartesian currents in fractional Sobolev spaces and applications to functions of bounded higher variation

In this paper we establish weak continuity results for the distribution Jacobian minors in fractional sobolev spaces, which can be seen as a extension of recent work of Brezis and Nguyen on the distributional Jacobian determinant. Then we apply results to introduce the currents associated with graphs of maps in fractional Sobolev spaces and study some relevant properties such as structure properties, weak continuity and so on. As another application, we extend the definition of functions of bounded higher variation, which defined by Jerrard and Soner in $W^{1,N-1}\cap L^{\infty}(Ω,\mathbb{R}^N)$ ( $\mbox{dim}Ω\geq N$), to $W^{1-\frac{1}{N},N}(Ω,\mathbb{R}^N)$ and give some results including weak coarea formula, strong coarea formula and chain rule.

math.AP↗
arXiv2015

The Gagliardo-Nirenberg inequality on metric measure spaces

In this paper, we prove that if a metric measure space satisfies the volume doubling condition and the Gagliardo-Nirenberg inequality with the same exponent $n$ $(n\geq 2)$, then it has exactly the $n$-dimensional volume growth. Besides, two interesting applications have also been given. The one is that we show that if a complete $n$-dimensional Finsler manifold of nonnegative $n$-Ricci curvature satisfies the Gagliardo-Nirenberg inequality with the sharp constant, then its flag curvature is identically zero. The other one is that we give an alternative proof to Mao's main result in [23] for smooth metric measure spaces with nonnegative weighted Ricci curvature.

math.DG↗
arXiv2013

Volume-Preserving flow by powers of the mth mean curvature in the hyperbolic space

This paper concerns closed hypersurfaces of dimension $n(\geq 2)$ in the hyperbolic space ${\mathbb{H}}_κ^{n+1}$ of constant sectional curvature $κ$ evolving in direction of its normal vector, where the speed is given by a power $β(\geq 1/m)$ of the $m$th mean curvature plus a volume preserving term, including the case of powers of the mean curvature and of the $\mbox{Gauß}$ curvature. The main result is that if the initial hypersurface satisfies that the ratio of the biggest and smallest principal curvature is close enough to 1 everywhere, depending only on $n$, $m$, $β$ and $κ$, then under the flow this is maintained, there exists a unique, smooth solution of the flow for all times, and the evolving hypersurfaces exponentially converge to a geodesic sphere of ${\mathbb{H}}_κ^{n+1}$, enclosing the same volume as the initial hypersurface.

math.DG↗
arXiv2009

Self-Similar Solutions to Curvature Flow of Convex Hypersurfaces

We classify the self-similar solutions to a class of Weingarten curvature flow of connected compact convex hypersurfaces, isometrically immersed into space forms with non-positive curvature, and obtain a new characterization of a sphere in a Euclidean space $\R^{n+1}$.

math.DG↗