Unique Continuation for Geometric Wave Equations, and Applications to Relativity, Holography, and Controllability

在科学、经济学和工程学中，各种各样的现象都是用偏微分方程或偏微分方程进行数学建模的。波动方程是偏微分方程的一个重要子类，存在于许多基本的物理方程中，如麦克斯韦方程（电磁学）、杨-米尔斯方程（粒子物理）、爱因斯坦场方程（引力）和欧拉方程（流体动力学）。特别是，隐藏在爱因斯坦方程中的波结构导致了引力波的预测，比2016年实验探测到引力波早了一个世纪。偏微分方程，特别是波动方程的一个基本问题，是在给定适当数据的情况下找到唯一解。这可以解释为能够在给定系统初始条件的情况下“预测未来”。另一方面，在方程不一定总能解的情况下，有必要问一下，如果解存在，它是否仍然是唯一的；这就是唯一延续的问题。直观上，这个问题问的是初始数据和解之间是否存在一一对应关系。现在有大量的文献围绕着独特延续理论。现代发展始于1939年Carleman的工作，并随着Calderón， Hörmander， Tataru和许多其他人的突破而继续发展。主要的分析技术是一类加权不等式，现在被称为Carleman估计。我最近在这个方向上的贡献围绕线性和非线性波的独特延拓性质；这也是我提出的研究的主要内容。重点是研究“退化”的设置，这是经典理论无法适用的。另一个重要的目标是发展强大的几何技术，适用于各种各样的曲线设置。拟议的研究方案的大部分内容涉及将这种独特的延拓理论的结果和技术应用于其他问题。在过去的几十年里，独特延拓与偏微分方程和物理学的其他方面之间有许多联系。例如，波动方程的唯一延拓结果最近被应用于相对论的对称扩展和刚性结果。此外，Carleman估计，特别是对于波动方程，已经成为在偏微分方程和微分几何背景下研究逆和控制理论问题的宝贵工具。一个正在进行的研究项目是拟议方案的一个主要组成部分，是应用独特的延续技术来研究理论物理学中的全息原理。这在很大程度上是由AdS/CFT对应引起的，它大致假设了反德西特时空中的引力动力学与其边界上的共形场论之间的对应。虽然这个想法在理论物理学中有着巨大的影响力，但在严格的数学公式和结果方面却没有取得多少进展。这个项目的最终目标是为这些物理概念建立数学基础。在经典相对论的背景下，这一努力的第一个主要步骤，是制定并证明一个对应陈述，作为爱因斯坦场方程的唯一延拓问题。我研究的另一个主要方面是应用这些技术，特别是新颖的Carleman估计，来解决几何偏微分方程的其他问题。一个例子是可控性问题：它询问是否可以使用有限的控制（例如，通过边界数据）将系统（例如，由pde建模）驱动到首选状态。另一个感兴趣的领域是逆问题，它询问是否可以通过仅进行有限的测量（例如，解的边界值）来确定系统（在数学上，PDE）。鉴于波在物理学中的普遍存在，波的控制和逆问题都与科学和工程中的重要问题密切相关。

A wide variety of phenomena in science, economics, and engineering are mathematically modelled by partial differential equations, or PDEs. Wave equations form an important subclass of PDEs and are found within many fundamental equations of physics, such as the Maxwell equations (electromagnetics), Yang-Mills equations (particle physics), Einstein field equations (gravitation), and Euler equations (fluid dynamics). In particular, the wave structure hidden within the Einstein equations led to the prediction of gravitational waves a century before their experimental detection in 2016.A basic problem in PDEs, and for wave equations in particular, is to find a unique solution given appropriate data. This can be interpreted as being able to "predict the future" given initial conditions for a system. On the other hand, in settings where the equation may not always be solved, it remains pertinent to ask whether solutions, if they exist, remain unique; this is the problem of unique continuation. Intuitively, this question asks whether there is a one-to-one correspondence between initial data and solutions.There is now significant literature surrounding the theory of unique continuation. Modern developments began with the work of Carleman in 1939 and continued with breakthroughs by Calderón, Hörmander, Tataru, and many others. The main analytic technique is a class of weighted inequalities now known as Carleman estimates.My recent contributions in this direction revolve around unique continuation properties for linear and nonlinear waves; this also forms the backbone of my proposed research. The focus is on studying "degenerate" settings, for which the classical theory fails to apply. Another essential goal is the development of robust geometric techniques that apply to a wide variety of curved settings.The bulk of the proposed research programme deals with applying the results and techniques of this unique continuation theory toward other problems. In the past decades, there have been numerous connections between unique continuation and other aspects of PDEs and physics. For instance, unique continuation results for wave equations have been recently applied in relativity toward symmetry extension and rigidity results. Furthermore, Carleman estimates, in particular for wave equations, have been an invaluable tool for studying inverse and control theory problems in the context of PDEs and differential geometry.One ongoing research project, a major component of the proposed programme, is to apply unique continuation techniques to study holographic principles in theoretical physics. This is largely motivated by the AdS/CFT correspondence, which roughly posits a correspondence between gravitational dynamics in Anti-de Sitter spacetime and conformal field theories on its boundary. While this idea has been tremendously influential in theoretical physics, there has been scant progress in terms of rigorous mathematical formulations and results. This project ultimately aims to develop mathematical underpinnings to these physical ideas. A first major step in this endeavour, in the context of classical relativity, is to formulate and prove a correspondence statement as a unique continuation problem for the Einstein field equations.Another major aspect of my research is to apply these techniques, novel Carleman estimates in particular, toward other problems for geometric PDEs. One example is the question of controllability: this asks whether one can drive a system (say, modelled by PDEs) to a preferred state using limited controls (for instance, through boundary data). Another area of interest is inverse problems, which asks whether one can determine a system (mathematically, a PDE) by making only limited measurements (for instance, the boundary values of solutions). Given the prevalence of waves in physics, both control and inverse problems for waves are well-connected to important questions in science and engineering.