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Zhongwei Tang

Publications and source records attributed to Zhongwei Tang.

18 recordsLinked to original sources
arXiv2026

A Sharp Sobolev Inequality on Compact CR Manifolds

For any \(n\geq 1\), let \((M,\theta)\) be a compact strictly pseudoconvex hypersurface type CR manifold with dimension \(2n+1\), \(Q=2n+2\) be its homogeneous dimension. We prove the sharp Folland--Stein Sobolev inequality on \(M\): if \(Q^*=2Q/(Q-2)\), then there exists a constant \(A>0\), depending only on \((M,\theta)\), such that \[ \|u\|_{L^{Q^*}(M)}^2 \le K(n)^2\|\nabla_b u\|_{L^2(M)}^2 + A\|u\|_{L^2(M)}^2, \qquad u\in S^{1,2}(M). \] Here \(K(n)\) is the sharp constant in the Folland--Stein Sobolev inequality on the Heisenberg group \(\mathbb H^n\).

math.AP↗
arXiv2026

Compactness of Solutions to Sub-Elliptic Equations with Potential on the Heisenberg Group

In this paper, we investigate the compactness of nonnegative solutions to a critical sub-elliptic equation with a nonnegative potential on the Heisenberg group. We establish compactness in all dimensions. Moreover, we show that if a sequence of solutions blows up, both the potential and its sub-Laplacian must vanish at the blow-up point. Our analysis overcomes the inherent geometric and analytical challenges posed by the Heisenberg group, including the degeneracy of the sub-Laplacian, its non-commutative structure, and the anisotropic dilation symmetry.

math.AP↗
arXiv2024

New type of bubbling solutions to a critical fractional Schrödinger equation with double potentials

In this paper, we study the following critical fractional Schrödinger equation: \begin{equation} (-Δ)^s u+V(|y'|,y'')u=K(|y'|,y'')u^{\frac{n+2s}{n-2s}},\quad u>0,\quad y =(y',y'') \in \mathbb{R}^3\times\mathbb{R}^{n-3}, \qquad(0.1)\end{equation} where $n\geq 3$, $s\in(0,1)$, $V(|y'|,y'')$ and $K(|y'|,y'')$ are two bounded nonnegative potential functions. Under the conditions that $K(r,y'')$ has a stable critical point $(r_0,y_0'')$ with $r_0>0$, $K(r_0,y_0'')>0$ and $V(r_0,y_0'')>0$, we prove that equation (0.1) has a new type of infinitely many solutions that concentrate at points lying on the top and the bottom of a cylinder. In particular, the bubble solutions can concentrate at a pair of symmetric points with respect to the origin. Our proofs make use of a modified finite-dimensional reduction method and local Pohozaev identities.

math.AP↗
arXiv2024

On the CR Nirenberg problem: density and multiplicity of solutions

We prove some results on the density and multiplicity of positive solutions to the prescribed Webster scalar curvature problem on the $(2n+1)$-dimensional standard unit CR sphere $(\mathbb{S} ^{2n+1},θ_0)$. Specifically, we construct arbitrarily many multi-bump solutions via the variational gluing method. In particular, we show the Webster scalar curvature functions of contact forms conformal to $θ_0$ are $C^{0}$-dense among bounded functions which are positive somewhere. Existence results of infinitely many positive solutions to the related equation $-Δ_{\mathbb{H}} u=R(ξ) u^{(n+2) /n}$ on the Heisenberg group $\Hn $ with $R(ξ)$ being asymptotically periodic with respect to left translation are also obtained. Our proofs make use of a refined analysis of bubbling behavior, gradient flow, Pohozaev identity, as well as blow up arguments.

math.AP↗
arXiv2023

On the density and multiplicity of solutions to the fractional Nirenberg problem

This paper is devoted to establishing some results on the density and multiplicity of solutions to the fractional Nirenberg problem which is equivalent to studying the conformally invariant equation $P_σ(v)=K v^{\frac{n+2σ}{n-2σ}}$ on the standard unit sphere $(\mathbb{S}^n,g_0)$ with $σ\in (0,1)$ and $n\geq 2$, where $P_σ$ is the intertwining operator of order $2σ$ and $K$ is the prescribed curvature function. More specifically, by using the variational gluing method, refined analysis of bubbling behavior, extension formula, as well as the blow up analysis arguments, we obtain the existence of infinitely many multi-bump solutions. In particular, we show the smooth curvature functions of metrics conformal to $g_0$ are dense in the $C^{0}$ topology. Moreover, the related fractional Laplacian equations $(-Δ)^σ u=K(x) u^{\frac{n+2σ}{n-2σ}}$ in $\mathbb{R}^n$, with $K(x)$ being asymptotically periodic in one of the variables, are also studied and infinitely many solutions are obtained under natural flatness assumptions.

math.AP↗
arXiv2023

Multiple solutions for a fractional Choquard problem with slightly subcritical exponents on bounded domains

This paper is devoted to study a fractional Choquard problem with slightly subcritical exponents on bounded domains. When the exponent of the convolution type nonlinearity tends to the fractional critical one in the sense of Hardy-Littlewood-Sobolev inequality, we obtain the existence of multiple positive solutions via Lusternik-Schnirelmann category and nonlocal global compactness. Moreover, we prove that the topology of the domain furnishes a lower bound for the number of positive solutions.

math.AP↗
arXiv2023

Interior estimates of derivatives and a Liouville type theorem for Parabolic $k$-Hessian equations

In this paper, we establish the gradient and Pogorelov estimates for $k$-convex-monotone solutions to parabolic $k$-Hessian equations of the form $-u_tσ_k(λ(D^2u))=ψ(x,t,u)$. We also apply such estimates to obtain a Liouville type result, which states that any $k$-convex-monotone and $C^{4,2}$ solution $u$ to $-u_tσ_k(λ(D^2u))=1$ in $\mathbb{R}^n\times(-\infty,0]$ must be a linear function of $t$ plus a quadratic polynomial of $x$, under some growth assumptions on $u$.

math.AP↗
arXiv2022

Uniqueness of positive solutions to the higher order Brezis-Nirenberg problem

In this paper, we study the higher order Brezis-Nirenberg problem under the Navier boundary condition \be\label{eq} \begin{cases} (-Δ)^m u=\varepsilon u+u^{p} & \text { in }\, Ω, \\ u>0 & \text { in }\, Ω, \\ u=-Δu=\cdots=(-Δ)^{m-1} u=0 & \text { on }\, \partial Ω, \end{cases} \ee where $Ω$ is a strictly convex smooth bounded domain in $\mathbb{R}^n$ with $n \geq 4m$, $m \in \mathbb{N}_{+}$, $\varepsilon\in (0,λ_{1})$, $λ_{1}$ is the first Navier eigenvalue for $(-Δ)^{m}$ in $Ω$, and $p=\frac{n+2m}{n-2m}$. We prove that the solutions of \eqref{eq} are unique if either $\varepsilon$ close to $λ_1$ or $\varepsilon$ close to 0 and $Ω$ satisfies some symmetry assumptions. The proof is mainly based on our previous works about the blow up analysis and compactness result for solutions to higher order critical elliptic equations and the asymptotic behavior of solutions to \eqref{eq}.

math.AP↗
arXiv2022

Unified results of compactness and existence for prescribing fractional $Q$-curvatures problem

In this paper we study the problem of prescribing fractional $Q$-curvature of order $2σ$ for a conformal metric on the standard sphere $\Sn$ with $σ\in (0,n/2)$ and $n\geq2$. Compactness and existence results are obtained in terms of the flatness order $β$ of the prescribed curvature function $K$. Making use of integral representations and perturbation result, we develop a unified approach to obtain these results when $β\in [n-2σ,n)$ for all $σ\in (0,n/2)$. This work generalizes the corresponding results of Jin-Li-Xiong [Math. Ann. 369: 109--151, 2017] for $β\in (n-2σ,n)$.

math.AP↗
arXiv2022

A weighted Sobolev-Poincaré type trace inequality on Riemannian manifolds

Given $(M, g)$ a smooth compact $(n+1)$-dimensional Riemannian manifold with boundary $\partial M$. Let $ρ$ be a defining function of $M$ and $σ\in(0,1)$. In this paper we study a weighted Sobolev-Poincaré type trace inequality corresponding to the embedding of $W^{1,2}(ρ^{1-2 σ}, M) \hookrightarrow L^{p}(\partial M)$, where $p=\frac{2 n}{n-2 σ}$. More precisely, under some assumptions on the manifold, we prove that there exists a constant $B>0$ such that, for all $u \in W^{1,2}(ρ^{1-2σ}, M)$, $$ \Big(\int_{\partial M}|u|^{p} \,\ud s_{g}\Big)^{2/p} \leq μ^{-1} \int_{M} ρ^{1-2 σ}|\nabla_{g} u|^{2} \,\ud v_{g}+B \Big|\int_{\partial M} |u|^{p-2}u \,\ud s_{g}\Big|^{2/(p-1)}. $$ This inequality is sharp in the sense that $μ^{-1}$ cannot be replaced by any smaller constant. Moreover, unlike the classical Sobolev inequality, $μ^{-1}$ does not depend on $n$ and $σ$ only, but depends on the manifold.

math.AP↗
arXiv2022

Compactness and existence results of the prescribing fractional $Q$-curvatures problem on $\mathbb{S}^n$

This paper is devoted to establishing the compactness and existence results of the solutions to the prescribing fractional $Q$-curvatures problem of order $2σ$ on $n$-dimensional standard sphere when $ n-2σ=2$, $σ=1+m/2,$ $m\in \mathbb{N}_{+}.$ The compactness results are novel and optimal. In addition, we prove a degree-counting formula of all solutions to achieve the existence. From our results, we can know where blow up occur. Furthermore, the sequence of solutions that blow up precisely at any finite distinct location can be constructed. It is worth noting that our results include the case of multiple harmonic.

math.AP↗
arXiv2022

On a Fractional Nirenberg problem involving the square root of the Laplacian on $\mathbb{S}^{3}$

In this paper, we are devoted to establishing the compactness and existence results of the solutions to the fractional Nirenberg problem for $n=3,$ $σ=1/2,$ when the prescribing $σ$-curvature function satisfies the $(n-2σ)$-flatness condition near its critical points. The compactness results are new and optimal. In addition, we obtain a degree-counting formula of all solutions. From our results, we can know where blow up occur. Moreover, for any finite distinct points, the sequence of solutions that blow up precisely at these points can be constructed. We extend the results of Li in \cite[CPAM, 1996]{LYY} from the local problem to nonlocal cases.

math.AP↗
arXiv2021

Compactness of solutions to higher order elliptic equations

We use blow up analysis for local integral equations to prove compactness of solutions to higher order critical elliptic equations provided the potentials only have non-degenerate zeros. Secondly, corresponding to Schoen's Weyl tensor vanishing conjecture for the Yamabe equation on manifolds, we establish a Laplacian vanishing rate of the potentials at blow up points of solutions.

math.AP↗
arXiv2017

Solutions for biharmonic equations with steep potential wells

In this paper, we are concerned with the existence of least energy solutions for the following biharmonic equations: $$Δ^2 u+(λV(x)-δ)u=|u|^{p-2}u \quad in\quad \mathbb{R}^N$$ where $N\geq 5, 2 0$ is a parameter, $V(x)$ is a nonnegative potential function with nonempty zero sets $\mbox{int} V^{-1}(0)$, $0<δ<μ_0$ and $μ_0$ is the principle eigenvalue of $Δ^2$ in the zero sets $\mbox{int} V^{-1}(0)$ of $V(x)$. Here $\mbox{int} V^{-1}(0)$ denotes the interior part of the set $V^{-1}(0):=\{x\in \mathbb{R}^N: V(x)=0\}$. We prove that the above equation admits a least energy solution which is trapped near the zero sets $\mbox{int} V^{-1}(0)$ for $λ>0$ large.

math.AP↗
arXiv2012

High-precision camera distortion measurements with a "calibration harp"

This paper addresses the high precision measurement of the distortion of a digital camera from photographs. Traditionally, this distortion is measured from photographs of a flat pattern which contains aligned elements. Nevertheless, it is nearly impossible to fabricate a very flat pattern and to validate its flatness. This fact limits the attainable measurable precisions. In contrast, it is much easier to obtain physically very precise straight lines by tightly stretching good quality strings on a frame. Taking literally "plumb-line methods", we built a "calibration harp" instead of the classic flat patterns to obtain a high precision measurement tool, demonstrably reaching 2/100 pixel precisions. The harp is complemented with the algorithms computing automatically from harp photographs two different and complementary lens distortion measurements. The precision of the method is evaluated on images corrected by state-of-the-art distortion correction algorithms, and by popular software. Three applications are shown: first an objective and reliable measurement of the result of any distortion correction. Second, the harp permits to control state-of-the art global camera calibration algorithms: It permits to select the right distortion model, thus avoiding internal compensation errors inherent to these methods. Third, the method replaces manual procedures in other distortion correction methods, makes them fully automatic, and increases their reliability and precision.

cs.CV↗
arXiv2012

Are You Imitating Me? Unsupervised Sparse Modeling for Group Activity Analysis from a Single Video

A framework for unsupervised group activity analysis from a single video is here presented. Our working hypothesis is that human actions lie on a union of low-dimensional subspaces, and thus can be efficiently modeled as sparse linear combinations of atoms from a learned dictionary representing the action's primitives. Contrary to prior art, and with the primary goal of spatio-temporal action grouping, in this work only one single video segment is available for both unsupervised learning and analysis without any prior training information. After extracting simple features at a single spatio-temporal scale, we learn a dictionary for each individual in the video during each short time lapse. These dictionaries allow us to compare the individuals' actions by producing an affinity matrix which contains sufficient discriminative information about the actions in the scene leading to grouping with simple and efficient tools. With diverse publicly available real videos, we demonstrate the effectiveness of the proposed framework and its robustness to cluttered backgrounds, changes of human appearance, and action variability.

cs.CV↗