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Singular Integral Operators for Higher-Order Systems in Non-Smooth Domains

非光滑域高阶系统的奇异积分算子

基本信息

批准号：
1900938
负责人：
Irina Mitrea
金额：
$ 18万
依托单位：
Temple University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2019
资助国家：
美国
起止时间：
2019-08-01 至 2022-07-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1900938&HistoricalAwards=false
关键词：
Singular Integral Operators Higher Order

项目摘要

The vast majority of physical phenomena are described in the language of partial differential equations. When coupled with measurements on the interface separating the region where these phenomena take place from the rest of the space, one arrives at what is commonly called a boundary value problem for the phenomenon in question. One of the most powerful techniques employed in the treatment of such boundary value problems is the method of layer potentials. This has had tremendous success both at theoretical and numerical levels for second order operators such as phenomena involving gravitational forces, heat at equilibrium, phenomena involving elastic deformations, acoustic and electromagnetic scattering, and wave propagation. By way of contrast, considerable less is presently understood for phenomena involving higher-order systems such as beam bending, plate vibrations and elastic plate deflections. The overall goal of this project is to address this issue by developing a systematic treatment of higher-order elliptic boundary value problems using singular integral operators, in a very general class of non-smooth domains. The emphasis on non-smooth structures is crucial for the viability of the theory for practical applications as physical domains exhibit asperities and irregularities of a very intricate nature. In turn, these significantly affect the properties of solutions of the partial differential equation problems. The successful completion of this project is expected to have significant impact in theoretical and applied mathematics, numerical methods, mathematical physics, and engineering, by establishing a theoretical framework which is just as effective as the traditional technologies dealing with simpler models.The project is for research in the areas of Harmonic Analysis, Geometric Measure Theory and Partial Differential Equations, and its overall aim is to develop a systematic treatment of higher-order elliptic boundary value problems using singular integral operators, in a very general class of non-smooth domains, which is in the nature of best possible from the geometric measure theoretic point of view. The main objectives of the current project are to identify: (1) the fullest family of multi-layer potential operators associated with a given homogeneous constant coefficient higher-order elliptic system; (2) geometric measure theoretic settings for which boundary multi-layer potential operators of double and single type are bounded on appropriate spaces of Whitney arrays (function spaces on the boundary suitably adapted to the higher-order setting); and (3) algebraic and geometric measure theoretic settings for which boundary multi-layer potential operators are invertible on appropriate spaces of Whitney arrays. The tools are rooted in Harmonic Analysis, Geometric Measure Theory, Partial Differential Equations, and Functional Analysis. A key step is to develop a new generation of Calderon-Zygmund theory for multi-layers acting on Whitney arrays, starting with the case when the domain is merely of locally finite perimeter and then progressively strengthening the hypotheses by ultimately assuming that the domain is uniformly rectifiable. There are inherent difficulties in carrying out this program, such as lack of uniqueness of Green?s formula, algebraic difficulties, the necessity of developing a suitable function space theory (including trace, extension, and interpolation theory) for function spaces of Whitney arrays in uniformly rectifiable domains, the failure in the non-Lipschitz context of a number of key ingredients in the theory of singular integrals on Lipschitz domains such as Rellich-type estimates.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

绝大多数的物理现象都是用偏微分方程来描述的。当再加上将这些现象发生的区域与空间其余部分分开的界面上的测量时，就会得出通常所说的所讨论现象的边界值问题。在处理这类边值问题时，最有力的技术之一是层势法。这已经取得了巨大的成功，在理论和数值水平的二阶运营商，如现象涉及引力，热平衡，现象涉及弹性变形，声和电磁散射，和波的传播。相比之下，对于涉及高阶系统的现象，如梁弯曲、板振动和弹性板偏转，目前了解得相当少。这个项目的总体目标是解决这个问题，通过开发一个系统的处理高阶椭圆边值问题，使用奇异积分算子，在一个非常一般的类非光滑域。强调非光滑结构是至关重要的理论的可行性，为实际应用的物理域表现出粗糙和不规则性的一个非常复杂的性质。反过来，这些显着影响的性质的解决方案的偏微分方程问题。该项目的成功完成将对理论和应用数学、数值方法、数学物理和工程产生重大影响，通过建立一个理论框架，该框架与处理简单模型的传统技术一样有效。该项目的研究领域是调和分析、几何测度理论和偏微分方程，其总体目标是在一个非常一般的非光滑域类中，利用奇异积分算子系统地处理高阶椭圆边值问题，从几何测度理论的角度来看，这是最好的。本课题的主要目标是：（1）确定与给定的齐次常系数高阶椭圆型方程组相联系的多层位势算子的最完整族;（2）双层和单层边界多层位势算子在适当的Whitney阵列空间上有界的几何测度论条件（函数空间的边界上适当地适应高阶设置）;和（3）代数和几何测量理论的设置，其中边界多层潜在的运营商是可逆的适当空间的惠特尼阵列。这些工具植根于调和分析，几何测度理论，偏微分方程和泛函分析。一个关键的步骤是开发新一代的卡尔德龙-Zygmund理论的多层作用于惠特尼阵列，开始的情况下，当域仅仅是局部有限周长，然后逐步加强假设，最终假设域是一致的rectifiable。开展这一方案存在固有的困难，如缺乏绿色的独特性？的公式，代数困难，发展一个合适的函数空间理论的必要性（包括迹，扩张和插值理论）的函数空间的惠特尼阵列在一致求长域，失败的非Lipschitz背景下的一些关键成分的理论奇异积分的Lipschitz域，如Rellich-该奖项反映了NSF的法定使命，并被认为是值得通过使用基金会的知识价值和更广泛的影响审查标准进行评估的支持。