Geometric De Giorgi Theory

We prove De Giorgi-Nash-Moser Theory using a geometric approach.Comment: 14 pages, 1 figur

‘The century of humiliation’ and the politics of memory in China

Lihe Wang looks at how selective national memories play a part in the construction of Chinese identity in the context of increasingly nationalistic politics in the country

Partial Regularity of Navier-Stokes Equations

We prove, with a more geometric approach, that the solutions to the Navier-Stokes equations are regular up to a set of Hausdorff dimension 1. The main tool for the proof is a new compactness lemma and the monotonicity property of harmonic functions.Comment: 16 page

Boundary first order derivative estimates for fully nonlinear elliptic equations

AbstractIn this paper, we prove the estimates of modulus of continuity up to the boundary for the first order derivative of viscosity solutions for fully nonlinear uniformly elliptic equations under Dini boundary data with the domain in the same class. As a corollary we derive C1,α boundary regularity

Elliptic equations with measurable coefficients in Reifenberg domains

AbstractWe prove W1,p estimates for elliptic equations in divergence form under the assumption that for each point and for each sufficiently small scale there is a coordinate system so that the coefficients have small oscillation in (n−1) directions. We assume the boundary to be δ-Reifenberg flat and the coefficients having small oscillation in the flat direction of the boundary

Parabolic equations in time dependent Reifenberg domains

AbstractIn this paper we define time dependent parabolic Reifenberg domains and study Lp estimates for weak solutions of uniformly parabolic equations in divergence form on these domains. The basic assumption is that the principal coefficients are of parabolic BMO space with small parabolic BMO seminorms. It is shown that Lp estimates hold for time dependent parabolic δ-Reifenberg domains

C0C^{0}-regularity for solutions of elliptic equations with distributional coefficients

In this paper, the continuity of solutions for elliptic equations in divergence form with distributional coefficients is considered. Inspired by the discussion on necessary and sufficient conditions for the form boundedness of elliptic operators by Maz'ya and Verbitsky (Acta Math., 188, 263-302, 2002 and Comm. Pure Appl. Math., 59, 1286-1329, 2006), we propose two kinds of sufficient conditions, which are some Dini decay conditions and some integrable conditions named Kato class or $K^{1}$ class, to show that the weak solution of the Schr\"{o}dinger type elliptic equation with distributional coefficients is continuous and give an almost optimal priori estimate. These estimates can clearly show that how the coefficients and nonhomogeneous terms influence the regularity of solutions. The $\ln$-Lipschitz regularity and H\"{o}lder regularity are also obtained as corollaries which cover the classical De Giorgi's H\"{o}lder estimates

FusionRCNN: LiDAR-Camera Fusion for Two-stage 3D Object Detection

3D object detection with multi-sensors is essential for an accurate and reliable perception system of autonomous driving and robotics. Existing 3D detectors significantly improve the accuracy by adopting a two-stage paradigm which merely relies on LiDAR point clouds for 3D proposal refinement. Though impressive, the sparsity of point clouds, especially for the points far away, making it difficult for the LiDAR-only refinement module to accurately recognize and locate objects.To address this problem, we propose a novel multi-modality two-stage approach named FusionRCNN, which effectively and efficiently fuses point clouds and camera images in the Regions of Interest(RoI). FusionRCNN adaptively integrates both sparse geometry information from LiDAR and dense texture information from camera in a unified attention mechanism. Specifically, it first utilizes RoIPooling to obtain an image set with a unified size and gets the point set by sampling raw points within proposals in the RoI extraction step; then leverages an intra-modality self-attention to enhance the domain-specific features, following by a well-designed cross-attention to fuse the information from two modalities.FusionRCNN is fundamentally plug-and-play and supports different one-stage methods with almost no architectural changes. Extensive experiments on KITTI and Waymo benchmarks demonstrate that our method significantly boosts the performances of popular detectors.Remarkably, FusionRCNN significantly improves the strong SECOND baseline by 6.14% mAP on Waymo, and outperforms competing two-stage approaches. Code will be released soon at https://github.com/xxlbigbrother/Fusion-RCNN.Comment: 7 pages, 3 figure