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

绝大多数物理现象都是用偏微分方程式来描述的。当与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，人们就到达了所讨论现象的通常所说的边值问题。在处理这类边值问题时，最有力的技术之一是层势法。这在理论和数值层面上对二阶算符都取得了巨大的成功，例如涉及引力的现象、平衡热、涉及弹性变形的现象、声和电磁散射以及波的传播。相比之下，目前对高阶系统，如梁的弯曲、板的振动和弹性板的挠度等现象的了解要少得多。这个项目的总体目标是通过在一类非常一般的非光滑区域中使用奇异积分算子来系统地处理高阶椭圆边值问题来解决这个问题。对非光滑结构的强调对于该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粗糙和不规则。反过来，这些又显著地影响偏微分方程解的性质。该项目的成功完成有望在理论和应用数学、数值方法、数学物理和工程方面产生重大影响，因为它建立了一个与处理简单模型的传统技术一样有效的理论框架。该项目是为了研究调和分析、几何测量理论和偏微分方程领域，其总体目标是发展一种系统的处理高阶椭圆边值问题的奇异积分算子，在非常一般的非光滑区域中，从几何测量理论的角度来看，这是最好的。本项目的主要目标是确定：(1)与给定齐次常系数高阶椭圆型系统相关的最全多层势算子族；(2)边界多层势算子在Whitney阵列的适当空间(边界上的函数空间适合于高阶设置)上有界的几何测度论设置；以及(3)边界多层势算子在Whitney阵列的适当空间上可逆的代数和几何测度论设置。这些工具植根于调和分析、几何测量理论、偏微分方程式和泛函分析。一个关键的步骤是发展新一代Calderon-Zygmund理论，用于作用于Whitney阵列的多层结构，从区域只是局部有限周长的情况开始，然后通过最终假设区域是一致可校正的来逐步加强假设。执行这一计划存在固有的困难，例如格林？S公式缺乏唯一性，代数困难，为一致可校正区域中的惠特尼阵列的函数空间发展一个合适的函数空间理论(包括迹、扩张和内插理论)的必要性，在非Lipschitz背景下，Lipschitz域上的奇异积分理论的一些关键成分的失败，如Rellich-型估计。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智力优势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.