Fourier Analysis on Bounded and Exterior Domains

马修·布莱尔的这项数学研究项目涉及傅立叶分析及其在偏微分方程中的应用。这里特别感兴趣的是波动方程和薛定谔方程的Strichartz时空可积界和局部光滑型。该建议还涉及紧致黎曼流形上拉普拉斯的本征函数的密切相关的估计。这些问题实际上源于经典傅里叶分析中的问题，例如Stein-Tomas限制定理，相关技术涉及到密切相关的振荡积分估计。当这些方程在欧几里德空间甚至无边界的黎曼流形上被提出时，这些不等式的基本公式已经存在，但对于它们对初边值问题的有效性却知之甚少。在这里，人们考虑一个区域，或者更一般地，一个带边界的黎曼流形，施加齐次边界条件对能量流动有显著的影响，有时甚至抑制扩散。因此，这些估计的证明通常涉及微局部或相空间分析，这为理解这一现象提供了途径。对于光滑的边界，Blair和他的合作者已经取得了巨大的成功，他们将解实现为波包的叠加，然后证明了由此产生的振荡积分产生了期望的不等式。在有角域上，这些估计的有效性更是鲜为人知，但Blair和他的合作者已经得到了关于多边形域的一些结果。Matthew Blair的这个数学研究项目是在傅立叶分析领域：这是数学的一个分支，在数学和物理理论的发展中发挥着重要作用。本项目所进行的研究为研究光波和声波提供了数学基础。傅立叶分析继续在加深我们对模拟这一行为的方程的理解方面发挥着重要作用。特别是，这些研究对硬边界表面的存在如何影响波的发展产生了进一步的洞察。例如，如果一个人在礼堂里听交响乐，听到的声音会受到声波从墙上反射的方式的影响。从这个意义上说，了解大厅的形状如何影响其声学是很重要的。虽然这当然是一个经典的问题，但关于这些相互作用如何影响色散特性，还有更多的需要理解。此外，这项工作对于分析由光纤和水波引起的密切相关的非线性方程是重要的，在理解和限制可能发生的各种类型的不稳定性方面有很多工作要做。

This mathematics research project by Matthew Blair deals with Fourier analysis and its applications to partial differential equations. Of particular interest here are space-time integrability bounds of Strichartz and local smoothing type for the wave and Schrödinger equations. The proposal also concerns closely related estimates on eigenfunctions of the Laplacian on a compact Riemannian manifold. These problems actually stem from those in classical Fourier analysis, such as the Stein-Tomas restriction theorem, and the relevant techniques involve closely related oscillatory integral estimates. While basic formulations of these inequalities exist for these equations when they are posed over Euclidean space and even Riemannian manifolds without boundary, less is known about their validity for initial boundary value problems. Here one considers a domain or more generally a Riemannian manifold with boundary, and the imposition of homogeneous boundary conditions affects the flow of energy in significant ways, sometimes even inhibiting dispersion. Consequently, the proofs of these estimates typically involve microlocal or phase space analysis, which provide avenues for understanding this phenomena. For smooth boundaries, Blair and his collaborators have achieved a great deal of success by realizing solutions as a superposition of wave packets, then showing that the resulting oscillatory integrals yield the desired inequalities. On domains with corners, even less is known about the validity of these estimates, but Blair and his collaborators have been able to obtain some results for polygonal domains.This mathematics research project by Matthew Blair is in the area of Fourier analysis: this is a branch of mathematics that plays an important role in the development of mathematical and physical theories. The research pursued in this project provides the mathematical foundation for the study of light and sound waves. Fourier analysis continues to play a significant role in deepening our understanding of the equations which model this behavior. In particular, these investigations yield further insight as to how the presence of a hard boundary surface influences the development of waves. For example, if one listens to the symphony in an auditorium, the sounds heard are affected by the manner in which the acoustic waves reflect off the walls. In this sense, it can be important to understand how the shape of the hall influences its acoustics. While this is of course a classical problem, there is more to be understood in terms of how these interactions influence dispersive properties. Moreover, this line of work is important in the analysis of closely related nonlinear equations arising from fiber optics and water waves, where there is much to be done in understanding and limiting the various types of instabilities which can occur.