Multiple Solutions for Resonant Elliptic Equations via Local Linking Theory and Morse Theory

AbstractWe consider two classes of elliptic resonant problems. First, by local linking theory, we study the double-double resonant case and obtain three solutions. Second, we introduce some new conditions and compute the critical groups both at zero and at infinity precisely. Combining Morse theory, we get three solutions for the completely resonant case

Explicit3D: Graph Network with Spatial Inference for Single Image 3D Object Detection

Indoor 3D object detection is an essential task in single image scene understanding, impacting spatial cognition fundamentally in visual reasoning. Existing works on 3D object detection from a single image either pursue this goal through independent predictions of each object or implicitly reason over all possible objects, failing to harness relational geometric information between objects. To address this problem, we propose a dynamic sparse graph pipeline named Explicit3D based on object geometry and semantics features. Taking the efficiency into consideration, we further define a relatedness score and design a novel dynamic pruning algorithm followed by a cluster sampling method for sparse scene graph generation and updating. Furthermore, our Explicit3D introduces homogeneous matrices and defines new relative loss and corner loss to model the spatial difference between target pairs explicitly. Instead of using ground-truth labels as direct supervision, our relative and corner loss are derived from the homogeneous transformation, which renders the model to learn the geometric consistency between objects. The experimental results on the SUN RGB-D dataset demonstrate that our Explicit3D achieves better performance balance than the-state-of-the-art

Sign-changing solution for logarithmic elliptic equations with critical exponent

In this paper, we consider the logarithmic elliptic equations with critical exponent \begin{equation} \begin{cases} -\Delta u=\lambda u+ |u|^{2^*-2}u+\theta u\log u^2, \\ u \in H_0^1(\Omega), \quad \Omega \subset \R^N. \end{cases} \end{equation} Here, the parameters $N\geq 6$, $\lambda\in \R$, $\theta>0$ and $2^*=\frac{2N}{N-2}$ is the Sobolev critical exponent. We prove the existence of sign-changing solution with exactly two nodal domain for an arbitrary smooth bounded domain $\Omega\subset \mathbb{R}^{N}$. When $\Omega=B_R(0)$ is a ball, we also construct infinitely many radial sign-changing solutions with alternating signs and prescribed nodal characteristic

On the stability of critical points of the Hardy-Littlewood-Sobolev inequality

This paper is concerned with the quantitative stability of critical points of the Hardy-Littlewood-Sobolev inequality. Namely, we give quantitative estimates for the Choquard equation: $$-\Delta u=(I_{\mu}\ast|u|^{2_\mu^*}) u^{2_\mu^*-1}\ \ \text{in}\ \ \R^N,$$ where $u>0,\ N\geq 3,\ \mu\in(0,N)$, $I_{\mu}$ is the Riesz potential and $2_\mu^* \coloneqq \frac{2N-\mu}{N-2}$ is the upper Hardy-Littlewood-Sobolev critical exponent. The Struwe's decomposition (see M. Struwe: Math Z.,1984) showed that the equation $\Delta u + u^{\frac{N+2}{N-2 }}=0$ has phenomenon of ``stable up to bubbling'', that is, if $u\geq0$ and $\|\Delta u+u^{\frac{N+2}{N-2}}\|_{(\mathcal{D}^{1,2})^{-1}}$ approaches zero, then $d(u)$ goes to zero, where $d(u)$ denotes the $\mathcal{D}^{1,2}(\R^N)$-distance between $u$ and the set of all sums of Talenti bubbles. Ciraolo, F{}igalli and Maggi (Int. Math. Res. Not.,2017) obtained the f{}irst quantitative version of Struwe's decomposition with single bubble in all dimensions $N\geq 3$, i.e, $\displaystyle d(u)\leq C\|\Delta u+u^{\frac{N+2}{N-2}}\|_{L^{\frac{2N}{N+2}}}.$ For multiple bubbles, F{}igalli and Glaudo (Arch. Rational Mech. Anal., 2020) obtained quantitative estimates depending on the dimension, namely $$d(u)\leq C\|\Delta u+u^{\frac{N+2}{N-2}}\|_{(\mathcal{D}^{1,2})^{-1}}, \hbox{ where } 3\leq N\leq 5,$$ which is invalid as $N\geq 6.$ \vskip0.1in \quad In this paper, we prove the quantitative estimate of the Hardy-Littlewood-Sobolev inequality, we get $$d(u)\leq C\|\Delta u +(I_{\mu}\ast|u|^{2_\mu^*})|u|^{2_\mu^*-2}u\|_{(\mathcal{D}^{1,2})^{-1}}, \hbox{ when } N=3 \hbox{ and } 5/2< \mu<3.$

Seeking Salient Facial Regions for Cross-Database Micro-Expression Recognition

Cross-Database Micro-Expression Recognition (CDMER) aims to develop the Micro-Expression Recognition (MER) methods with strong domain adaptability, i.e., the ability to recognize the Micro-Expressions (MEs) of different subjects captured by different imaging devices in different scenes. The development of CDMER is faced with two key problems: 1) the severe feature distribution gap between the source and target databases; 2) the feature representation bottleneck of ME such local and subtle facial expressions. To solve these problems, this paper proposes a novel Transfer Group Sparse Regression method, namely TGSR, which aims to 1) optimize the measurement and better alleviate the difference between the source and target databases, and 2) highlight the valid facial regions to enhance extracted features, by the operation of selecting the group features from the raw face feature, where each region is associated with a group of raw face feature, i.e., the salient facial region selection. Compared with previous transfer group sparse methods, our proposed TGSR has the ability to select the salient facial regions, which is effective in alleviating the aforementioned problems for better performance and reducing the computational cost at the same time. We use two public ME databases, i.e., CASME II and SMIC, to evaluate our proposed TGSR method. Experimental results show that our proposed TGSR learns the discriminative and explicable regions, and outperforms most state-of-the-art subspace-learning-based domain-adaptive methods for CDMER