中国科学院数学与系统科学研究院期刊网

15 November 2021, Volume 37 Issue 11
    

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  • Xiao Li HAN, Meng Qiu SHAO
    Acta Mathematica Sinica. 2021, 37(11): 1645-1678. https://doi.org/10.1007/s10114-021-9523-5
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    This paper is mainly concerned with the following nonlinear p-Laplacian equation
    pu(x) + (λa(x) + 1)|u|p-2(x)u(x)=f(x, u(x)), in V
    on a locally finite graph G=(V, E) with more general nonlinear term, where Δp is the discrete p-Laplacian on graphs, p ≥ 2. Under some suitable conditions on f and a(x), we can prove that the equation admits a positive solution by the Mountain Pass theorem and a ground state solution uλ via the method of Nehari manifold, for any λ > 1. In addition, as λ → +∞, we prove that the solution uλ converge to a solution of the following Dirichlet problem

    where Ω={xV:a(x)=0} is the potential well and ∂Ω denotes the the boundary of Ω.

  • Wei Hua HE, Jun LUO, Chao YANG, Wei YUAN, Hui Chun ZHANG
    Acta Mathematica Sinica. 2021, 37(11): 1679-1691. https://doi.org/10.1007/s10114-021-9546-y
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    Lin-Lu-Yau introduced a notion of Ricci curvature for graphs and obtained a complete classification for all Ricci-flat graphs with girth at least five. In this paper, we characterize all Ricci-flat graphs of girth four with vertex-disjoint 4-cycles.

  • Yu Han JIN, Xian Feng WANG, Yong WEI
    Acta Mathematica Sinica. 2021, 37(11): 1692-1708. https://doi.org/10.1007/s10114-021-0015-4
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    We consider the inverse curvature flows of smooth, closed and strictly convex rotation hypersurfaces in space forms Mκn+1 with speed function given by F-α, where α ∈ (0, 1] for κ=0, -1, α=1 for κ=1 and F is a smooth, symmetric, strictly increasing and 1-homogeneous function of the principal curvatures of the evolving hypersurfaces. We show that the curvature pinching ratio of the evolving hypersurface is controlled by its initial value, and prove the long time existence and convergence of the flows. No second derivatives conditions are required on F.

  • Jia Xiang WANG, Bin ZHOU
    Acta Mathematica Sinica. 2021, 37(11): 1709-1720. https://doi.org/10.1007/s10114-021-0062-x
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    In this paper, we study the regularity of the complex Hessian equation when the right hand has pole singularity. We show the H¨ older continuity of the solution to the Dirichlet problem. In particular, for the complex Monge-Amp` ere equation, we improve a result of[7].

  • Hong CHEN, Jian Quan GE, Kai JIA, Zhi Qin LU
    Acta Mathematica Sinica. 2021, 37(11): 1721-1742. https://doi.org/10.1007/s10114-021-0027-0
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    In order to study the Yamabe changing-sign problem, Bahri and Xu proposed a conjecture which is a universal inequality for p points in Rm. They have verified the conjecture for p ≤ 3. In this paper, we first simplify this conjecture by giving two sufficient and necessary conditions inductively. Then we prove the conjecture for the basic case m=1 with arbitrary p. In addition, for the cases when p=4, 5 and m ≥ 2, we manage to reduce them to the basic case m=1 and thus prove them as well.

  • Victor GUILLEMIN, Zuo Qin WANG
    Acta Mathematica Sinica. 2021, 37(11): 1743-1750. https://doi.org/10.1007/s10114-021-0148-5
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    In this paper we will show how the Bohr-Sommerfeld levels of a quantum completely integrable system can be computed modulo O(?) by an inductive procedure starting at stage zero with the Bohr-Sommerfeld levels of the corresponding classical completely integrable system.

  • Wen Shuai JIANG
    Acta Mathematica Sinica. 2021, 37(11): 1751-1767. https://doi.org/10.1007/s10114-021-0149-4
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    In this paper, we consider Ricci flow on four dimensional closed manifold with bounded scalar curvature, noncollasping volume and bounded diameter. Under such conditions, we can show that the manifold has finitely many diffeomorphism types, which generalizes Cheeger-Naber's result to bounded scalar curvature along Ricci flow. In particular, this implies the manifold has uniform L2 Riemann curvature bound. As an application, we point out that four dimensional Ricci flow would not have uniform scalar curvature upper bound if the initial metric only satisfying lower Ricci curvature bound, lower volume bound and upper diameter bound.

  • Xiang LI, Jun SUN
    Acta Mathematica Sinica. 2021, 37(11): 1768-1782. https://doi.org/10.1007/s10114-021-0162-7
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    In this paper, we prove gradient estimates for the positive solutions of Lu=0 and Lu=∂u/∂t on conformal solitons, where L(·)=Δ(·) +

  • Shu Jing PAN
    Acta Mathematica Sinica. 2021, 37(11): 1783-1793. https://doi.org/10.1007/s10114-021-0395-5
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    In this paper, the curve shortening flow in a general Riemannian manifold is studied, Altschuler's results about the flow for space curves are generalized. For any n-dimensional (n ≥ 2) Riemannian manifold (M, g) with some natural assumptions, we prove the planar phenomenon when the curve shortening flow blows up.

  • Yong Sheng ZHANG
    Acta Mathematica Sinica. 2021, 37(11): 1794-1802. https://doi.org/10.1007/s10114-020-0061-3
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    This short note is concerned with a measure version criterion for hypersurfaces to be minimal. Certain natural flows and associated reflections for many minimal hypercones, including minimal isoparametric hypercones and area-minimizing hypercones, are studied.