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

绝大多数的物理现象都是用偏微分方程来描述的。当与将这些现象发生的区域与空间的其余部分分开的界面上的测量相结合时，人们就得到了通常所说的有关现象的边值问题。处理这类边值问题最有力的技术之一是层势法。这在理论和数值层面上都取得了巨大的成功，用于二阶算符，如涉及重力、平衡热、涉及弹性变形、声波和电磁散射以及波传播的现象。相比之下，目前对涉及高阶系统的现象，如梁弯曲、板振动和弹性板挠度的理解相当少。本项目的总体目标是通过在非常一般的非光滑域中使用奇异积分算子对高阶椭圆边值问题进行系统处理来解决这个问题。强调非光滑结构对于理论在实际应用中的可行性至关重要，因为物理领域表现出非常复杂的性质的凹凸不平和不规则性。反过来，这些显著地影响了偏微分方程问题解的性质。该项目的成功完成有望在理论和应用数学、数值方法、数学物理和工程领域产生重大影响，建立一个与处理简单模型的传统技术一样有效的理论框架。本课题的研究方向为调和分析、几何测度理论和偏微分方程，其总体目标是在一类非常一般的非光滑域上，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。当前项目的主要目标是确定：(1)与给定齐次常系数高阶椭圆系统相关的多层势算子的最完整族；(2)双、单型边界多层势算子在Whitney阵列的适当空间（边界上适合高阶设置的函数空间）上有界的几何测度理论设置；(3)惠特尼阵列适当空间上边界多层势算子可逆的代数和几何测度理论设置。这些工具植根于谐波分析、几何测量理论、偏微分方程和泛函分析。关键的一步是发展新一代的卡尔德隆-齐格蒙德理论，用于作用于惠特尼阵列的多层，从区域仅仅是局部有限周长的情况开始，然后逐步加强假设，最终假设区域是均匀可校正的。这个项目在实施过程中存在着固有的困难，比如缺乏绿色的独特性。在非Lipschitz环境下，Lipschitz域上的奇异积分理论（如rellich型估计）中的一些关键成分在非Lipschitz环境下的失效。该奖项反映了美国国家科学基金会的法定使命，并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，被认为值得支持。

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.