Variable Coefficient Fourier Analysis

变系数傅里叶分析

• 批准号: 2348996
• 负责人: Christopher Sogge
• 金额: $ 39.09万
• 依托单位: Johns Hopkins University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2024
• 资助国家: 美国
• 起止时间: 2024-07-01 至 2027-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2348996&HistoricalAwards=false
• 关键词: Variable; Coefficient; Fourier; Analysis

The PI will study several problems in Geometric Harmonic Analysis. The settings for these problems involve geometric manifolds of dimension two or more. Associated with a given manifold are fundamental objects called eigenfunctions. These are the fundamental modes of vibration of the manifold, and they are the higher dimensional analogs of the familiar trigonometric functions for the circle. Designers of musical instruments are well aware that the shape of, say, a drum or the soundboard of stringed instrument affects the basic tones that it omits, as well as the sound volume. Similar phenomena arise for manifolds, and the PI will study precisely how their shapes, such as how they are curved, affect the properties properties of eigenfunctions. Just as in music, one particularly expects different shapes and geometries to become more apparent in the behavior of the fundamental modes of vibration as the frequency becomes larger and larger. These eigenfunctions are solutions of a differential equation that is similar to the wave equation, and the PI will study similar problems involving it. The general theme is to study how solutions of wave equations are affected by their physical backgrounds, such as whether or not black holes are present or whether the background becomes very close to a vacuum near infinity. This project provides research training opportunities for graduate students.Among the specific problems the PI shall study, they wish to obtain improved estimates that measure the size and concentration of eigenfunctions. In order to do this, they will develop what is called ``global harmonic analysis’’, which is a mixture of classical harmonic analysis, microlocal analysis, and techniques from geometry. The basic estimates are Lp-estimates for eigenfunctions and quasimodes and related highly localized L2 estimates that are sensitive to concentration. The main questions center around how the geometry and the global dynamics of the geodesic flow affect the estimates and the kinds of functions that saturate them. The latter issue is closely related to the much-studied (but still not well-understood) questions of concentration, oscillation, and size properties of modes and quasimodes in spectral asymptotics. These questions are also naturally linked to the long-time properties of the solution operator for the wave equation, Schrodinger equation, and resolvent estimates coming from the metric Laplacian. High frequency solutions and obtaining sharp results under geometric assumptions are particularly interesting. They will also study functions that saturate the estimates in different ways depending on the sign of the sectional curvatures of the manifolds.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.

PI 将研究几何调和分析中的几个问题，这些问题的设置涉及与给定流形相关的基本对象，称为本征函数，它们是流形的基本振动模式。乐器设计者很清楚，鼓或弦乐器音板的形状会影响其省略的基本音调，流形也会出现类似的现象，PI 将精确研究它们的形状（例如它们的弯曲方式）如何影响本征函数的特性，就像在音乐中一样，人们特别期望不同的形状和几何形状。随着频率变得越来越大，这些特征函数是类似于波动方程的微分方程的解，PI 将研究涉及它的类似问题。到研究波动方程的解如何受到其物理背景的影响，例如黑洞是否存在或背景是否变得非常接近无穷大的真空。该项目为研究生提供了研究培训机会。 PI 将研究，他们希望获得改进的估计来测量本征函数的大小和浓度。为了做到这一点，他们将开发所谓的“全局调和分析”，它是经典调和分析和微局部分析的混合。 ，以及来自的技术基本估计是本征函数和准模态的 Lp 估计以及对浓度敏感的相关高度局域化 L2 估计，主要问题集中在测地流的几何和全局动力学如何影响估计以及函数类型。后一个问题与光谱中模式和准模式的浓度、振荡和尺寸特性密切相关。这些问题也自然地与波动方程、薛定谔方程的解算子的长期性质相关，并且求解来自度量拉普拉斯的估计并在几何假设下获得清晰的结果是特别有趣的。还将研究根据流形截面曲率的符号以不同方式使估计饱和的函数。该奖项是 NSF 的法定使命，并且通过使用基金会的智力评估进行评估被认为值得支持优点和更广泛的影响审查标准。