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

点击翻译按钮获取中文摘要

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.