Nonlinear elliptic equations

非线性椭圆方程

基本信息

  • 批准号:
    1362168
  • 负责人:
  • 金额:
    $ 27.64万
  • 依托单位:
  • 依托单位国家:
    美国
  • 项目类别:
    Continuing Grant
  • 财政年份:
    2014
  • 资助国家:
    美国
  • 起止时间:
    2014-07-01 至 2018-06-30
  • 项目状态:
    已结题

项目摘要

The research activity into this proposed project will deepen our understanding of two intimately connected mathematical fields, partial differential equations and differential geometry, which may be viewed as extensions of advanced calculus. Simultaneously, the project will also have impact on the areas on which the equations studied in the project rest: some equations provide the mathematical foundation for mirror symmetry in the string theory of modern physics, which is a unified way to describe our physical universe; another equation is an effective model in material science; solutions to the so called Isaacs equations lead to the optimal strategy for certain random processes, for example, in engineering and finance; Also Hessian equations are related to nonlinear elasticity theory in mechanics, which studies the mechanisms whereby a material that is stretched returns to its original size and shape.The objectives for special Lagrangian equations are to derive Schauder and Calderon-Zygmund estimates for the equations with critical and supercritical phases, to answer the question whether any homogeneous order two solution in dimension five or higher is trivial or not, and to study low regularity of continuous viscosity solutions to the equations with subcritical phases. The purposes for self similar solutions to mean curvature flows are to classify Lagrangian translating solutions and study uniqueness of embedded sphere shrinker in 3-d Euclidean space. The aim for symmetric Hessian equations is to investigate Hessian estimates for quadratic Hessian equations in dimension four and higher and also scalar curvature equations, to obtain Schauder and Calderon-Zygmund estimates for 3-d quadratic Hessian equations, and to study the Liouville problem for k-symmetric Hessian equations. The attempt for fully nonlinear elliptic equations such as Isaacs equations in 3-d is to study the regularity for general fully nonlinear elliptic equations in 3-d, in particular for equations in the form of linear combinations of k-symmetric Hessians and finitely piecewise linear Isaacs equations. The plan for complex Monge-Ampere equations is to show the triviality of any global solution to complex Monge-Ampere equations including self-shrinking equations for the Kahler Ricci flow with certain necessary restrictions.
该项目的研究活动将加深我们对两个密切相关的数学领域的理解,即偏微分方程和微分几何,它们可以被视为高等微积分的扩展。同时,该项目也将对项目中所研究的方程所依赖的领域产生影响:一些方程为现代物理学弦理论中的镜像对称提供了数学基础,这是描述我们的物理宇宙的统一方式;另一个方程是材料科学中的有效模型;所谓的Isaacs方程的解导致某些随机过程的最优策略,例如,在工程和金融中;海森方程也与力学中的非线性弹性理论有关,它研究的机制,使材料被拉伸返回到其原始大小和形状。特殊的拉格朗日方程的目标是推导Schauder和Calderon-Zygmund估计,回答了高维齐次二阶解是否平凡的问题,并研究了次临界方程连续粘性解的低正则性.平均曲率流自相似解的目的是对拉格朗日平移解进行分类,并研究三维欧氏空间中嵌入球收缩器的唯一性。本文的主要目的是研究四维及高维二次Hessian方程和标量曲率方程的Hessian估计,得到三维二次Hessian方程的Schauder和Calderon-Zygmund估计,以及k-对称Hessian方程的Liouville问题.对于完全非线性椭圆型方程如三维Isaacs方程的研究,主要是研究一般的三维完全非线性椭圆型方程,特别是k-对称Hessian方程和k-分段线性Isaacs方程的线性组合形式的方程的正则性。复杂的Monge-Ampere方程的计划是显示复杂的Monge-Ampere方程的任何全局解的平凡性,包括具有某些必要限制的Kahler Ricci流的自收缩方程。

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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Yu Yuan其他文献

Temperature-driven wear behavior of Si3N4-based ceramic reinforced by in situ formed TiC0.3N0.7 particles
原位形成的 TiC0.3N0.7 颗粒增强 Si3N4 基陶瓷的温度驱动磨损行为
  • DOI:
    10.1111/jace.16283
  • 发表时间:
    2019
  • 期刊:
  • 影响因子:
    3.9
  • 作者:
    Liu Jiongjie;Yang Jun;Zhu Shengyu;Cheng Jun;Yu Yuan;Qiao Zhuhui;Liu Weimin
  • 通讯作者:
    Liu Weimin
Regularity for the Monge–Ampère equation by doubling
Monge-Ampère 方程的加倍正则性
  • DOI:
    10.1007/s00209-024-03508-6
  • 发表时间:
    2024
  • 期刊:
  • 影响因子:
    0.8
  • 作者:
    Ravi Shankar;Yu Yuan
  • 通讯作者:
    Yu Yuan
Associations of the PTEN -9C>G polymorphism with insulin sensitivity and central obesity in Chinese.
PTEN -9C>G 多态性与中国人胰岛素敏感性和中心性肥胖的关系。
  • DOI:
    10.1016/j.gene.2013.06.026
  • 发表时间:
    2013
  • 期刊:
  • 影响因子:
    3.5
  • 作者:
    Qiu Yang;Hongyi Cao;Shugui Xie;Yuzhen Tong;Qibo Zhu;Fang Zhang;Q. Lü;Yan Yang;Daigang Li;Mei Chen;Chang;W. Jin;Yu Yuan;N. Tong
  • 通讯作者:
    N. Tong
The correlation between intestinal mucosal lesions and hepatic dysfunction in patients without chronic liver disease
非慢性肝病患者肠黏膜病变与肝功能障碍的相关性
  • DOI:
  • 发表时间:
    2020
  • 期刊:
  • 影响因子:
    1.6
  • 作者:
    Li;Mei;Jie Cai;Yu Yuan;Li;Hui;Lan Li;Kayiu Wan;Xingxiang He
  • 通讯作者:
    Xingxiang He
Extraction of 3D quantitative maps using EDS-STEM tomography and HAADF-EDS bimodal tomography.
使用 EDS-STEM 断层扫描和 HAADF-EDS 双峰断层扫描提取 3D 定量图
  • DOI:
    10.1016/j.ultramic.2020.113166
  • 发表时间:
    2020
  • 期刊:
  • 影响因子:
    2.2
  • 作者:
    Yu Yuan;Katherine MacArthur;Sean M Collins;Nicolas Brodusch;Frederic Voisard;Rafal E Dunin-Borkowski;Raynald Gauvin
  • 通讯作者:
    Raynald Gauvin

Yu Yuan的其他文献

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{{ truncateString('Yu Yuan', 18)}}的其他基金

Fully Nonlinear Elliptic Equations
完全非线性椭圆方程
  • 批准号:
    2054973
  • 财政年份:
    2021
  • 资助金额:
    $ 27.64万
  • 项目类别:
    Standard Grant
Fully Nonlinear Elliptic and Parabolic Equations
完全非线性椭圆和抛物线方程
  • 批准号:
    1800495
  • 财政年份:
    2018
  • 资助金额:
    $ 27.64万
  • 项目类别:
    Continuing Grant
Conference on Geometric Analysis
几何分析会议
  • 批准号:
    1707760
  • 财政年份:
    2017
  • 资助金额:
    $ 27.64万
  • 项目类别:
    Standard Grant
Fully nonlinear elliptic and parabolic equations
完全非线性椭圆和抛物线方程
  • 批准号:
    1100966
  • 财政年份:
    2011
  • 资助金额:
    $ 27.64万
  • 项目类别:
    Continuing Grant
Fully nonlinear elliptic equations
全非线性椭圆方程
  • 批准号:
    0758256
  • 财政年份:
    2008
  • 资助金额:
    $ 27.64万
  • 项目类别:
    Standard Grant
Fully Nonlinear Equations
完全非线性方程
  • 批准号:
    0500808
  • 财政年份:
    2005
  • 资助金额:
    $ 27.64万
  • 项目类别:
    Standard Grant
Regularity for Fully Nonlinear Equations
完全非线性方程的正则性
  • 批准号:
    0200784
  • 财政年份:
    2002
  • 资助金额:
    $ 27.64万
  • 项目类别:
    Standard Grant
A Priori Estimates for Linear and Nonlinear Partial Differential Equations
线性和非线性偏微分方程的先验估计
  • 批准号:
    0296153
  • 财政年份:
    2001
  • 资助金额:
    $ 27.64万
  • 项目类别:
    Continuing Grant
A Priori Estimates for Linear and Nonlinear Partial Differential Equations
线性和非线性偏微分方程的先验估计
  • 批准号:
    9970367
  • 财政年份:
    1999
  • 资助金额:
    $ 27.64万
  • 项目类别:
    Continuing Grant

相似海外基金

Nonlinear Elliptic Equations and Systems, and Applications
非线性椭圆方程和系统以及应用
  • 批准号:
    2247410
  • 财政年份:
    2023
  • 资助金额:
    $ 27.64万
  • 项目类别:
    Standard Grant
Singularity and structure of solutions to nonlinear elliptic partial differential equations
非线性椭圆偏微分方程解的奇异性和结构
  • 批准号:
    23K03167
  • 财政年份:
    2023
  • 资助金额:
    $ 27.64万
  • 项目类别:
    Grant-in-Aid for Scientific Research (C)
CAREER: Singular and Global Solutions to Nonlinear Elliptic Equations
职业:非线性椭圆方程的奇异和全局解
  • 批准号:
    2143668
  • 财政年份:
    2022
  • 资助金额:
    $ 27.64万
  • 项目类别:
    Continuing Grant
Singular solutions for nonlinear elliptic and parabolic equations
非线性椭圆方程和抛物方程的奇异解
  • 批准号:
    DP220101816
  • 财政年份:
    2022
  • 资助金额:
    $ 27.64万
  • 项目类别:
    Discovery Projects
Singularity Formations in Nonlinear Elliptic and Parabolic Equations
非线性椭圆方程和抛物线方程中的奇异性形成
  • 批准号:
    RGPIN-2018-03773
  • 财政年份:
    2022
  • 资助金额:
    $ 27.64万
  • 项目类别:
    Discovery Grants Program - Individual
Narrow-Stencil Numerical Methods for Approximating Nonlinear Elliptic Partial Differential Equations
逼近非线性椭圆偏微分方程的窄模板数值方法
  • 批准号:
    2111059
  • 财政年份:
    2021
  • 资助金额:
    $ 27.64万
  • 项目类别:
    Standard Grant
Singularity Formations in Nonlinear Elliptic and Parabolic Equations
非线性椭圆方程和抛物线方程中的奇异性形成
  • 批准号:
    RGPIN-2018-03773
  • 财政年份:
    2021
  • 资助金额:
    $ 27.64万
  • 项目类别:
    Discovery Grants Program - Individual
Fully Nonlinear Elliptic Equations
完全非线性椭圆方程
  • 批准号:
    2054973
  • 财政年份:
    2021
  • 资助金额:
    $ 27.64万
  • 项目类别:
    Standard Grant
Existence and non-existence of blowing-up solutions for nonlinear elliptic equations arising in physics and geometry
物理和几何中非线性椭圆方程的爆炸解的存在性和不存在性
  • 批准号:
    RGPIN-2016-04195
  • 财政年份:
    2021
  • 资助金额:
    $ 27.64万
  • 项目类别:
    Discovery Grants Program - Individual
Singularity Formations in Nonlinear Elliptic and Parabolic Equations
非线性椭圆方程和抛物线方程中的奇异性形成
  • 批准号:
    RGPIN-2018-03773
  • 财政年份:
    2020
  • 资助金额:
    $ 27.64万
  • 项目类别:
    Discovery Grants Program - Individual
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