Mass transport, Geometric inequalities and partial differential systems

质量传递、几何不等式和偏微分系统

基本信息

  • 批准号:
    RGPIN-2015-03951
  • 负责人:
  • 金额:
    $ 3.79万
  • 依托单位:
  • 依托单位国家:
    加拿大
  • 项目类别:
    Discovery Grants Program - Individual
  • 财政年份:
    2016
  • 资助国家:
    加拿大
  • 起止时间:
    2016-01-01 至 2017-12-31
  • 项目状态:
    已结题

项目摘要

Partial Differential Equations and Systems provide major bridges between mathematics and many other disciplines in basic and applied sciences and engineering. This central field of Mathematics is used to model physical phenomena, biological systems, quantum materials and solid state devices, to name only a few areas of applications. My research program is focused on developing novel, preferably general and encompassing methods to deal with the analysis of classes of such systems, including evolution equations. The goal is to either apply these methods to problems that are not amenable to standard techniques, or to exhibit universal features, which could aid in the understanding and the deployment of the often-required hard analysis. The methods normally address questions of existence, multiplicity, regularity and other qualitative properties of solutions of stationary or dynamic differential equations and systems. They include: • Self-dual variational calculus on linear as well as curved spaces, to deal with equations, which are not of Euler-Lagrange type, such as those involving non self-adjoint terms or/and non-linear operators. • Mass transport theory to handle the still elusive variational problems arising in non-linear elasticity theory.  • Monge-Kantorovich theory to decouple certain partial differential systems, such as those of De Giorgi-type arising as a limiting elliptic system describing phase separation of multiple state Bose-Einstein condensates. • Its equivariant form for applications to matching problems in economics, and to Density Functional Theory, a well-established method for tackling the quantum mechanics of many-body systems. • Infinite dimensional critical point theory for borderline problems that lack compactness. These often require a fine analysis of the linear operators involved, such as the second and fourth-order Hardy-Schrodinger operators, the fractional Laplacians, and appropriate perturbations of them. The singularities could lie either in the interior, or on the boundary of the domain under consideration. The latter can be Euclidean or curved. • Novel approaches for the identification of best constants and/or the establishment of the existence of steady states and other extremals for various functional and geometric inequalities, such as those of Caffarelli-Kohn-Nirenberg, Moser-Aubin-Onofri, and Gagliardo-Nirenberg inequalities. • New insight for addressing issues of regularity, stability, and critical dimensions in various non-linear eigenvalue problems (self-adjoint or not), especially those dealing with the biharmonic operator, and the fractional Laplacians.
偏微分方程式和系统在数学和基础科学、应用科学和工程学中的许多其他学科之间架起了重要的桥梁。数学的这个中心领域被用来对物理现象、生物系统、量子材料和固态设备进行建模,仅举几个应用领域。我的研究计划专注于开发新的、最好是通用的和包罗万象的方法来处理这类系统的分析,包括演化方程。目标是要么将这些方法应用于标准技术不适用的问题,要么展示通用功能,这可能有助于理解和部署经常需要的硬分析。这些方法通常解决定常或动态微分方程解的存在性、多解性、正则性和其他定性性质等问题。它们包括: ·线性和弯曲空间上的自对偶变分,用于处理非欧拉-拉格朗日类型的方程,如涉及非自伴项或/和非线性算子的方程。 ·质量传输理论,用于处理非线性弹性理论中仍然难以捉摸的变分问题。 ·Monge-Kantorovich理论将某些偏微分系统解耦,例如描述多态玻色-爱因斯坦凝聚体相分离的极限椭圆系统的de Giorgi型偏微分系统。 ·它的等变形式,用于经济学中的匹配问题和密度泛函理论,密度泛函理论是解决多体系统量子力学的成熟方法。 ·无限维临界点理论,适用于缺乏紧致性的边界问题。这些通常需要对所涉及的线性算子进行精细分析,例如二阶和四阶Hardy-Schrodinger算子、分数拉普拉斯算子以及它们的适当扰动。奇点可以位于内部,也可以位于所考虑的区域的边界上。后者可以是欧几里得的,也可以是曲线的。 ·确定最佳常量和/或确定各种泛函和几何不平等的稳态和其他极值的存在的新方法,如Caffarelli-Kohn-Nirenberg、Moser-Aubin-Onofri和Gagliardo-Nirenberg不等。 ·解决各种非线性特征值问题(是否自伴)中的正则性、稳定性和临界维问题的新见解,特别是处理双调和算子和分数拉普拉斯算子的问题。

项目成果

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Ghoussoub, Nassif其他文献

STRUCTURE OF OPTIMAL MARTINGALE TRANSPORT PLANS IN GENERAL DIMENSIONS
  • DOI:
    10.1214/18-aop1258
  • 发表时间:
    2019-01-01
  • 期刊:
  • 影响因子:
    2.3
  • 作者:
    Ghoussoub, Nassif;Kim, Young-Heon;Lim, Tongseok
  • 通讯作者:
    Lim, Tongseok
Bessel pairs and optimal Hardy and Hardy-Rellich inequalities
  • DOI:
    10.1007/s00208-010-0510-x
  • 发表时间:
    2011-01-01
  • 期刊:
  • 影响因子:
    1.4
  • 作者:
    Ghoussoub, Nassif;Moradifam, Amir
  • 通讯作者:
    Moradifam, Amir
On the best possible remaining term in the Hardy inequality

Ghoussoub, Nassif的其他文献

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{{ truncateString('Ghoussoub, Nassif', 18)}}的其他基金

Mass transfers and Optimal Stochastic Transports
质量传递和最优随机传递
  • 批准号:
    RGPIN-2020-04248
  • 财政年份:
    2022
  • 资助金额:
    $ 3.79万
  • 项目类别:
    Discovery Grants Program - Individual
Mass transfers and Optimal Stochastic Transports
质量传递和最优随机传递
  • 批准号:
    RGPIN-2020-04248
  • 财政年份:
    2021
  • 资助金额:
    $ 3.79万
  • 项目类别:
    Discovery Grants Program - Individual
Mass transfers and Optimal Stochastic Transports
质量传递和最优随机传递
  • 批准号:
    RGPIN-2020-04248
  • 财政年份:
    2020
  • 资助金额:
    $ 3.79万
  • 项目类别:
    Discovery Grants Program - Individual
Mass transport, Geometric inequalities and partial differential systems
质量传递、几何不等式和偏微分系统
  • 批准号:
    RGPIN-2015-03951
  • 财政年份:
    2019
  • 资助金额:
    $ 3.79万
  • 项目类别:
    Discovery Grants Program - Individual
Mass transport, Geometric inequalities and partial differential systems
质量传递、几何不等式和偏微分系统
  • 批准号:
    RGPIN-2015-03951
  • 财政年份:
    2018
  • 资助金额:
    $ 3.79万
  • 项目类别:
    Discovery Grants Program - Individual
Banff International Research Station
班夫国际研究站
  • 批准号:
    245746-2015
  • 财政年份:
    2018
  • 资助金额:
    $ 3.79万
  • 项目类别:
    Thematic Resources Support in Mathematics and Statistics
Mass transport, Geometric inequalities and partial differential systems
质量传递、几何不等式和偏微分系统
  • 批准号:
    RGPIN-2015-03951
  • 财政年份:
    2017
  • 资助金额:
    $ 3.79万
  • 项目类别:
    Discovery Grants Program - Individual
Banff International Research Station
班夫国际研究站
  • 批准号:
    245746-2015
  • 财政年份:
    2017
  • 资助金额:
    $ 3.79万
  • 项目类别:
    Thematic Resources Support in Mathematics and Statistics
Banff International Research Station
班夫国际研究站
  • 批准号:
    245746-2015
  • 财政年份:
    2016
  • 资助金额:
    $ 3.79万
  • 项目类别:
    Thematic Resources Support in Mathematics and Statistics
Banff International Research Station
班夫国际研究站
  • 批准号:
    245746-2010
  • 财政年份:
    2015
  • 资助金额:
    $ 3.79万
  • 项目类别:
    Major Resources Support Program - Infrastructure

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Mass transport, Geometric inequalities and partial differential systems
质量传递、几何不等式和偏微分系统
  • 批准号:
    RGPIN-2015-03951
  • 财政年份:
    2019
  • 资助金额:
    $ 3.79万
  • 项目类别:
    Discovery Grants Program - Individual
Mass transport, Geometric inequalities and partial differential systems
质量传递、几何不等式和偏微分系统
  • 批准号:
    RGPIN-2015-03951
  • 财政年份:
    2018
  • 资助金额:
    $ 3.79万
  • 项目类别:
    Discovery Grants Program - Individual
Mass transport, Geometric inequalities and partial differential systems
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  • 批准号:
    RGPIN-2015-03951
  • 财政年份:
    2017
  • 资助金额:
    $ 3.79万
  • 项目类别:
    Discovery Grants Program - Individual
Mass transport, Geometric inequalities and partial differential systems
质量传递、几何不等式和偏微分系统
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    RGPIN-2015-03951
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Flow, Geometric Motion, Deformation and Mass Transport in Materials Science and Physiological Processes
材料科学和生理过程中的流动、几何运动、变形和质量传递
  • 批准号:
    1261325
  • 财政年份:
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    Discovery Grants Program - Individual
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    Discovery Grants Program - Individual
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