Partial Differential Equations in Geometry

几何中的偏微分方程

基本信息

  • 批准号:
    1105321
  • 负责人:
  • 金额:
    $ 15.33万
  • 依托单位:
  • 依托单位国家:
    美国
  • 项目类别:
    Standard Grant
  • 财政年份:
    2011
  • 资助国家:
    美国
  • 起止时间:
    2011-09-15 至 2015-08-31
  • 项目状态:
    已结题

项目摘要

Under this award, the principal investigator plans to continue his work in geometric partial differential equations. In particular he will investigate the role of level sets of solutions of differential equations. Special cases of level sets are given by the nodal sets, the singular sets and the branch sets. An important part of the study is the investigation of the asymptotic behavior of solutions near these sets or the asymptotic behavior of these sets themselves. The differential equations may be elliptic, hyperbolic or of mixed type. Some problems have close connections with other fields in mathematics, including several complex variables and algebraic geometry. Another class of problems that the PI will continue to work on involves the effect of level sets of known functions in the equations on the existence and properties of solutions. A particular problem is the isometric embedding of 2-dim Riemannian manifolds in 3-space when the zero set of Gauss curvature is well behaved. This problem is in the form of degenerate Monge-Ampere equations. The problems involving nodal sets or the singular sets originate from materials science and control theory. Singular sets, as the name suggests, are those sets where singularities occur. Precise definitions vary depending on the problems where they arise. It often is impossible to eliminate the singular sets, the so-called "bad sets". One of the central tasks is to identify the conditions under which the singular sets can be controlled and the conditions under which the singular sets are small. The proposed problems concerning singular sets in the project are in their simplest forms. They are related to the Erickson's model for liquid crystals and the Ginzburg-Landau equation in the superconductivity. The PI believes that the discussion of these mathematical problems will help scientists work with singular sets in various applications.
根据这一奖项,首席研究员计划继续他在几何偏微分方程式方面的工作。特别是,他将研究微分方程解的水平集的作用。水平集的特例由节点集、奇异集和分支集给出。研究的一个重要部分是研究解在这些集合附近的渐近行为或这些集合本身的渐近行为。微分方程可以是椭圆型、双曲型或混合型。有些问题与数学中的其他领域有着密切的联系,包括多个复变量和代数几何。PI将继续研究的另一类问题涉及方程中已知函数的水平集对解的存在和性质的影响。一个特殊的问题是当高斯曲率的零点集表现良好时,二维黎曼流形在三维空间中的等距嵌入问题。这个问题是退化的Monge-Ampere方程的形式。涉及节点集或奇异集的问题起源于材料科学和控制理论。奇异集合,顾名思义,就是奇点出现的集合。准确的定义取决于它们出现的问题。要消除奇异集,即所谓的“坏集”,往往是不可能的。中心任务之一是确定奇异集可以控制的条件和奇异集较小的条件。项目中提出的关于奇异集的问题都是最简单的形式。它们与液晶的Erickson模型和超导中的Ginzburg-Landau方程有关。PI相信,对这些数学问题的讨论将有助于科学家在各种应用中处理奇异集。

项目成果

期刊论文数量(0)
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科研奖励数量(0)
会议论文数量(0)
专利数量(0)

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Qing Han其他文献

Understanding the Impact of HIV on MPOX Transmission in an MSM Population: A Mathematical Modeling Study
了解 HIV 对 MSM 人群中 MPOX 传播的影响:数学模型研究
  • DOI:
  • 发表时间:
    2024
  • 期刊:
  • 影响因子:
    0
  • 作者:
    Andrew Omame;Qing Han;S. Iyaniwura;Adeniyi Ebenezer;N. Bragazzi;Xiaoying Wang;Jude Dzevela Kong;W. A. Woldegerima
  • 通讯作者:
    W. A. Woldegerima
Optimal regularity of minimal graphs in the hyperbolic space
双曲空间中最小图的最优正则性
Design and Synthesis of 60 degrees Dendritic Donor Ligands and Their Coordination-Driven Self-Assembly into Supramolecular Rhomboidal Metallodendrimers
60度树枝状供体配体的设计与合成及其配位驱动自组装成超分子菱形金属树枝状聚合物
  • DOI:
  • 发表时间:
    2011
  • 期刊:
  • 影响因子:
    3.6
  • 作者:
    Qing Han;Quan-Jie Li;Jiuming He;Bingjie Hu;Hongwei Tan;Zeper Abliz;Cui-Hong Wang;Yihua Yu;Hai-Bo Yang
  • 通讯作者:
    Hai-Bo Yang
Interior estimates for the n-dimensional Abreu?s equation
n 维 Abreuï¤s 方程的内部估计
  • DOI:
  • 发表时间:
    2014
  • 期刊:
  • 影响因子:
    1.7
  • 作者:
    Bohui Chen;Qing Han;An-Min Li;Li Sheng
  • 通讯作者:
    Li Sheng
Metal-Organic Frameworks with Organogold(III) Complexes for Photocatalytic Amine Oxidation with Enhanced Efficiency and Selectivity
具有有机金 (III) 配合物的金属有机框架可提高光催化胺氧化效率和选择性
  • DOI:
    10.1002/chem.201803161
  • 发表时间:
    2018
  • 期刊:
  • 影响因子:
    0
  • 作者:
    Qing Han;Yue Lin Wang;Min Sun;Chun Yi Sun;Shan Shan Zhu;Xin Long Wang;Zhong Min Su
  • 通讯作者:
    Zhong Min Su

Qing Han的其他文献

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{{ truncateString('Qing Han', 18)}}的其他基金

Asymptotic Analysis of Geometric Partial Differential Equations
几何偏微分方程的渐近分析
  • 批准号:
    2305038
  • 财政年份:
    2023
  • 资助金额:
    $ 15.33万
  • 项目类别:
    Standard Grant
Degenerate Partial Differential Equations in Geometry
几何中的简并偏微分方程
  • 批准号:
    1404596
  • 财政年份:
    2014
  • 资助金额:
    $ 15.33万
  • 项目类别:
    Standard Grant
Nonlinear Partial Differential Equations and Applications
非线性偏微分方程及其应用
  • 批准号:
    0654261
  • 财政年份:
    2007
  • 资助金额:
    $ 15.33万
  • 项目类别:
    Standard Grant
Nonlinear Partial Differential Equations
非线性偏微分方程
  • 批准号:
    0354948
  • 财政年份:
    2004
  • 资助金额:
    $ 15.33万
  • 项目类别:
    Standard Grant
Partial Differential Equations and Variational Problems
偏微分方程和变分问题
  • 批准号:
    0100260
  • 财政年份:
    2001
  • 资助金额:
    $ 15.33万
  • 项目类别:
    Standard Grant
Partial Differential Equations and Variational Problems
偏微分方程和变分问题
  • 批准号:
    9801250
  • 财政年份:
    1998
  • 资助金额:
    $ 15.33万
  • 项目类别:
    Standard Grant
Mathematical Sciences: Nonlinear Differential Equations and Variational Problems
数学科学:非线性微分方程和变分问题
  • 批准号:
    9501122
  • 财政年份:
    1995
  • 资助金额:
    $ 15.33万
  • 项目类别:
    Standard Grant

相似海外基金

Conference: Geometric Measure Theory, Harmonic Analysis, and Partial Differential Equations: Recent Advances
会议:几何测度理论、调和分析和偏微分方程:最新进展
  • 批准号:
    2402028
  • 财政年份:
    2024
  • 资助金额:
    $ 15.33万
  • 项目类别:
    Standard Grant
Problems in Regularity Theory of Partial Differential Equations
偏微分方程正则论中的问题
  • 批准号:
    2350129
  • 财政年份:
    2024
  • 资助金额:
    $ 15.33万
  • 项目类别:
    Standard Grant
Conference: Recent advances in nonlinear Partial Differential Equations
会议:非线性偏微分方程的最新进展
  • 批准号:
    2346780
  • 财政年份:
    2024
  • 资助金额:
    $ 15.33万
  • 项目类别:
    Standard Grant
Geometric Techniques for Studying Singular Solutions to Hyperbolic Partial Differential Equations in Physics
研究物理学中双曲偏微分方程奇异解的几何技术
  • 批准号:
    2349575
  • 财政年份:
    2024
  • 资助金额:
    $ 15.33万
  • 项目类别:
    Standard Grant
Regularity Problems in Free Boundaries and Degenerate Elliptic Partial Differential Equations
自由边界和简并椭圆偏微分方程中的正则问题
  • 批准号:
    2349794
  • 财政年份:
    2024
  • 资助金额:
    $ 15.33万
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    Standard Grant
Interfaces, Degenerate Partial Differential Equations, and Convexity
接口、简并偏微分方程和凸性
  • 批准号:
    2348846
  • 财政年份:
    2024
  • 资助金额:
    $ 15.33万
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    Standard Grant
Comparative Study of Finite Element and Neural Network Discretizations for Partial Differential Equations
偏微分方程有限元与神经网络离散化的比较研究
  • 批准号:
    2424305
  • 财政年份:
    2024
  • 资助金额:
    $ 15.33万
  • 项目类别:
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A new numerical analysis for partial differential equations with noise
带有噪声的偏微分方程的新数值分析
  • 批准号:
    DP220100937
  • 财政年份:
    2023
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  • 项目类别:
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Nonlinear Stochastic Partial Differential Equations and Applications
非线性随机偏微分方程及其应用
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    2023
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    $ 15.33万
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Theoretical Guarantees of Machine Learning Methods for High Dimensional Partial Differential Equations: Numerical Analysis and Uncertainty Quantification
高维偏微分方程机器学习方法的理论保证:数值分析和不确定性量化
  • 批准号:
    2343135
  • 财政年份:
    2023
  • 资助金额:
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