Topics in the Theory of Elliptic Boundary Value Problems

椭圆边值问题理论专题

基本信息

  • 批准号:
    1201104
  • 负责人:
  • 金额:
    $ 9.9万
  • 依托单位:
  • 依托单位国家:
    美国
  • 项目类别:
    Standard Grant
  • 财政年份:
    2012
  • 资助国家:
    美国
  • 起止时间:
    2012-06-01 至 2014-09-30
  • 项目状态:
    已结题

项目摘要

This mathematics research project focuses on problems in harmonic analysis and partial differential equations, with an emphasis on the study of elliptic boundary value problems for second and higher order operators and for systems in non-smooth domains. A non-smooth domain refers to a domain with either isolated singularities or a Lipschitz domain. While elliptic boundary value problems for the Laplacian have been well understood for over twenty years, the theory for second order elliptic systems and higher order elliptic equations is incomplete. Another generalization of the classical Dirichlet and Neumann boundary value problems that is not yet fully understood is the mixed boundary value problem. This project addresses open problems that are categorized by the following research themes: boundary value problems for higher order elliptic operators; the spectral radius conjecture on Besov spaces; well-posedness of the mixed problem in Lipschitz domains. This mathematics research project is motivated by problems that naturally arise in mathematical physics and engineering. In this regard, the non-smooth setting in which the problems are posed is fundamental since most realistic physical models involve irregular domains. For example, boundary value problems for higher order elliptic operators have applications to engineering in the context of modeling shallow shells, beam bending, and clamped plates. Mixed boundary value problems model the behavior of several physical quantities such as the temperature in a metallurgical melting process, the thermo-elastic potential of an elastic solid punched or stamped by a heated object, or the seepage through a porous material. This mathematics research project will contribute to the aforementioned disciplines as well as produce new mathematical techniques. While pursuing the research directions outlined above, Ott will initiate activities aimed at increasing the participation of women and other under-represented groups in mathematics. These activities will include outreach activities with local middle and high schools, and research and networking opportunities for undergraduate and graduate students.
这个数学研究项目的重点是调和分析和偏微分方程的问题,重点是研究二阶和高阶算子的椭圆边值问题以及非光滑域中的系统。非光滑域是指具有孤立奇点或Lipschitz域的域。虽然拉普拉斯算子的椭圆边值问题已经有二十多年的历史了,但二阶椭圆方程组和高阶椭圆方程的理论还不完善。经典的狄利克雷和诺依曼边值问题的另一个推广是混合边值问题。该项目解决了按以下研究主题分类的开放问题:高阶椭圆算子的边值问题; Besov空间上的谱半径猜想; Lipschitz域中混合问题的适定性。 这个数学研究项目的动机是数学物理和工程中自然出现的问题。在这方面,提出的问题是根本的,因为大多数现实的物理模型涉及不规则域的非光滑设置。例如,高阶椭圆算子的边值问题在工程中有应用,如模拟扁壳、梁弯曲和固支板。混合边值问题模拟了几个物理量的行为,例如冶金熔化过程中的温度,被加热物体冲压或冲压的弹性固体的热弹性势,或通过多孔材料的渗流。这个数学研究项目将有助于上述学科以及产生新的数学技术。 在追求上述研究方向的同时,奥特将发起旨在增加妇女和其他代表性不足的群体参与数学的活动。这些活动将包括与当地初中和高中的外联活动,以及为本科生和研究生提供研究和建立联系的机会。

项目成果

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Katharine Ott其他文献

Katharine Ott的其他文献

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{{ truncateString('Katharine Ott', 18)}}的其他基金

Topics in the Theory of Elliptic Boundary Value Problems
椭圆边值问题理论专题
  • 批准号:
    1458138
  • 财政年份:
    2014
  • 资助金额:
    $ 9.9万
  • 项目类别:
    Standard Grant
PostDoctoral Research Fellowship
博士后研究奖学金
  • 批准号:
    0802938
  • 财政年份:
    2008
  • 资助金额:
    $ 9.9万
  • 项目类别:
    Fellowship Award

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