RII Track-4: Uniqueness and Quantitative Uniqueness of Solutions to Partial Differential Equations

非技术描述数学为描述和研究许多现实生活中的问题提供了必要的工具。变化率是数学中的导数，在这些问题中是基本的。偏微分方程(PDE)是一类涉及未知多变量函数及其偏导数的微分方程。它们描述了各种各样看似截然不同的物理现象，如声、热、静电学、电动力学、流体动力学、弹性或量子力学。正因为如此，偏微分方程组在工程学、物理学、经济学和生物学等许多学科中都扮演着重要的角色。偏微分方程解的唯一性及其定量性质是研究的核心问题之一。该奖学金在路易斯安那州立大学(LSU)和芝加哥大学(Uchi)之间建立了合作关系，使PI能够对Uchi进行长期的研究访问。该项目将促进对偏微分方程组的独特性的理解，激发其研究能力，加强路易斯安那州立大学在偏微分方程组和分析方面的研究计划，并有利于其本科生和研究生教育。在偏微分方程组的研究中，为解提供定量和定性的信息是必不可少的，而偏微分方程组的研究是数学分析的核心。通常，获取此类信息的最有效方法是首先探索方程解的定量和定性属性，然后根据需要开发算法；这是以足够的精度通过计算求解偏微分方程组的挑战的有益替代方案。数量唯一性是近年来发展很快的一个领域，它描述了强唯一延拓性质的数量行为。研究具有奇异权的椭圆型偏微分方程解的数量唯一性。这一结果将加深对这类偏微分方程强的唯一延拓性的理解。研究一个小开集上的信息如何准确地传播到区域中的任何其他开集上是非常重要和有趣的，它量化了弱唯一连续性质。这样的探索将在反问题和控制理论中有许多应用。PI将探索系数函数如何决定唯一性，以及区域的形状如何影响抛物型偏微分方程解的唯一性。该项目将有助于对椭圆型和抛物型偏微分方程式的基本理解。这一奖项反映了NSF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。

Nontechnical DescriptionMathematics provides essential tools for the description and investigation of numerous real-life problems. Rate of change, which is the derivative in mathematics, is fundamental in these problems. Partial differential equations (PDEs) are differential equations that relate unknown multivariable functions and their partial derivatives. They describe a wide variety of seemingly distinct physical phenomena such as sound, heat, electrostatics, electrodynamics, fluid dynamics, elasticity, or quantum mechanics. Because of this, PDEs play a prominent role in many disciplines including engineering, physics, economics, and biology. One of the central problems in the study of PDEs is about uniqueness and its quantitative properties. The fellowship builds a collaboration between Louisiana State University (LSU) and University of Chicago (UChi) by enabling the PI to make extended research visits to UChi. This project will lead to advancements in understanding uniqueness of PDEs, stimulate the PI's research capacity, strengthen the research program in PDEs and analysis at LSU, and benefit its undergraduate and graduate education.Technical DescriptionThe goal of this research is to investigate the quantitative and qualitative properties of uniqueness of solutions to PDEs. Providing quantitative and qualitative information for the solutions is essential in the study of PDEs, which lies in the core of mathematical analysis. Often the most effective way to obtain such information is to first explore the quantitative and qualitative properties of solutions of the equations and then to develop algorithms in accordance; this is a beneficial alternative to the challenge of instead solving PDEs computationally with sufficient accuracy. Quantitative uniqueness, a recent fast-developing area, describes the quantitative behavior of the strong unique continuation property. The proposed research is to study the quantitative uniqueness for elliptic PDEs with singular weights. The outcome will provide a deeper understanding of the strong unique continuation property for this category of PDEs. It is important and interesting to study how exactly the information on a small open set propagates to any other open sets in the domain, which quantifies the weak unique continuation property. Such exploration will lead to many applications in inverse problems and control theory. The PI will explore how the coefficient functions determine the uniqueness and how the shape of the domain influences the uniqueness of solutions for parabolic PDEs. The project will contribute the fundamental understanding of elliptic and parabolic partial differential equations.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.