Variable Coefficient Fourier Analysis

建议：DMS-9734866首席研究员：克里斯托弗·D·索格摘要：索格将致力于傅立叶分析和非线性波动方程中的各种问题。他想主要集中在两类问题上。首先，他感兴趣的是在各种环境下证明非线性波动方程的局部和整体存在定理，包括凸障碍外的波动方程。为此，主要的分析包括证明一个合适的“Strichartz估计”。在有边界的情况下，这样的估计比在没有边界的情况下要精细得多；然而，在与Hart Smith的联合工作中已经获得了一些初步结果。Sogge对研究曲线上的最大平均也非常感兴趣，尤其对在曲线空间环境下建立这样的估计感兴趣。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经有一段时间了。在许多情况下，这样的问题自然会出现。例如，广义相对论方程就是非线性波动方程。Sogge想要做的主要事情之一是证明这样的方程在障碍之外的区域中有解。这可能被认为是描述黑洞的方程的一个模型，其中障碍物扮演着黑洞的角色。索格还对傅立叶分析中的基本问题感兴趣。自从傅里叶分析被引入到他自己在热传导问题上的开创性工作中以来，它一直是研究微分方程的主要工具，而微分方程构成了物理学的形式语言。

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.