Variable Coefficient Fourier Analysis

变系数傅立叶分析

• 批准号: 9734866
• 负责人: Christopher Sogge
• 金额: $ 9.48万
• 依托单位: Johns Hopkins University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1998
• 资助国家: 美国
• 起止时间: 1998-07-01 至 2001-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9734866&HistoricalAwards=false
• 关键词: Variable; Coefficient; Fourier; Analysis

Proposal: DMS-9734866 Principal Investigator: Christopher D. Sogge Abstract: Sogge will work on various problems in Fourier analysis and nonlinear wave equations. He would like to concentrate mainly on two types of problems. First, he is interested in proving local and global existence theorems for nonlinear wave equations in various settings, including wave equations outside of convex obstacles. For this, the main analysis involves proving a suitable "Strichartz estimate." Such estimates in the situation where there is a boundary are much more delicate than in the case without boundary; however, some preliminary results have already been obtained in joint work with Hart Smith. Sogge is also very interested in studying maximal averages over curves and especially interested in trying to establish such estimates in the curved space setting. Sogge has been intrigued for some time by certain questions in nonlinear partial differential equations and Fourier analysis. Such issues arise naturally in many contexts. For instance, the equations of general relativity are just nonlinear wave equations. One of the main things Sogge would like to do is to show that such equations have solutions in the region outside of an obstacle. This might be thought of as a model for equations describing black holes, where the obstacle plays the role of the black hole. Sogge is also interested in basic problems in Fourier analysis. Since its introduction in the pioneering work of Fourier himself on problems of heat conduction, Fourier analysis has been the main tool to study differential equations, which constitute the formal language of physics.

提案：DMS-9734866主要研究者：Christopher D.娑葛 翻译后摘要：索格将工作在傅立叶分析和非线性波动方程的各种问题。他想主要集中讨论两类问题。首先，他有兴趣证明非线性波动方程在各种环境中的局部和全局存在定理，包括凸障碍物外的波动方程。 为此，主要的分析包括证明一个合适的“哈茨估计”。“在有边界的情况下的这种估计比在没有边界的情况下要微妙得多;然而，在与哈特·史密斯的联合工作中已经获得了一些初步结果。 索格也非常感兴趣的研究最大平均曲线，特别是有兴趣试图建立这样的估计在弯曲的空间设置。 Sogge对非线性偏微分方程和傅立叶分析中的某些问题感兴趣已有一段时间了。这些问题在许多情况下自然会出现。 例如，广义相对论的方程就是非线性波动方程。Sogge想做的主要事情之一是证明这样的方程在障碍物之外的区域有解。这可能被认为是描述黑洞的方程的模型，其中障碍物扮演黑洞的角色。Sogge也有兴趣在傅立叶分析的基本问题。自从傅立叶自己在热传导问题上的开创性工作中引入傅立叶分析以来，傅立叶分析一直是研究微分方程的主要工具，微分方程构成了物理学的形式语言。