Harmonic Analysis and PDE

调和分析和偏微分方程

• 批准号: 0200628
• 负责人: Sagun Chanillo
• 金额: $ 10.26万
• 依托单位: Rutgers University New Brunswick
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2002
• 资助国家: 美国
• 起止时间: 2002-06-01 至 2007-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0200628&HistoricalAwards=false
• 关键词: Harmonic; Analysis; PDE

PI: Sagun Chanillo, Rutgers UniversityDMS-0200628Abstract:In this proposal several problems are proposed, all in the field ofPDE. Most of these problems are in the sub-field of non-linearPDE, and some in linear PDE. The non-linear problemsarise naturally from Physics andGeometry and the interaction of Physics and Geometry. The linear problems are also tied in withGeometry and Physics. Thereis a strong focus in this proposal on various qualitativefeatures of solutions to PDE, their level lines, nodal linesof eigenfunctions, smoothness of solutions and so on. We have selected the problems in part because we view many of them as having very natural connections with classical Harmonic analysis, and in fact we have made some start on solving them using techniques from classical Harmonic analysis. This is described in the body of the proposal. Some of the problems that arise from Physics in our proposal are connected with the phenomena ofvortices. Here we continue work that was funded by previous grants.The vortices we study are those that arise from fluid flow ontwo-dimensional spheres and on the plane. Taking the continuum limit of the point vortex distribution leads us to a new technique for solving the prescribed Gauss curvature equation which also gives a tremendous insight into conformally invariant PDE's. Problems are also posedthat stem from the Schrodinger equation and Geometry and itsinfluence on the spectrum of elliptic operators.We single out an important component of the proposed researchproposal. In todays environment it has become important tostudy so-called smart materials, more principally composites.Because of their lightness they are preferred materialsto use for their strength and lightness. In this proposalwe study as one problem the vibrational characteristicsof composites. In particular how should one build objects usingcomposite materials so as to minimize their basic vibrational characteristics.Intuitively the larger the natural frequency with whichan object vibrates, the more it is susceptible to stressand breakage. For example one question one can ask is, if we need to build a symmetric object like a washer does itmean the composite has to be arranged symmetrically respectingthe symmetry of the washer? We find that in our research for example thatthis necessarily does not minimize the stresses that can be caused by vibrationand we need to arrange the composite in a non-symmetric wayto construct the washer. This is not the only problem we study in ourproposal but also problems in Differential Geometry and Physics.

PI：Sagun Chanillo，Rutgers UniversityDMS-0200628摘要：在这个建议中提出了几个问题，都在PDE领域。这些问题大多属于非线性偏微分方程的子领域，也有一些属于线性偏微分方程的子领域。非线性问题是物理学与几何学以及物理学与几何学相互作用的产物。线性问题也与几何和物理有关.有一个强烈的重点在这个建议的各种qualitativefeatures的解决方案，PDE，他们的水平线，nodal linesof eigenfunctions，光滑的解决方案等，我们选择的问题，部分原因是因为我们认为他们中的许多人有非常自然的联系，与经典的谐波分析，事实上，我们已经取得了一些开始解决他们使用的技术，从经典的谐波分析。提案正文对此作了说明。在我们的建议中，物理学中出现的一些问题与旋涡现象有关。在这里，我们继续以前赠款的工作。我们研究的旋涡是那些从二维球体和平面上的流体流动中产生的。以连续极限的点涡分布导致我们的一个新的技术来解决规定的高斯曲率方程，这也给了一个巨大的洞察共形不变偏微分方程的。本文还提出了薛定谔方程和几何学中的一些问题及其对椭圆算子谱的影响，并指出了本文研究的一个重要组成部分.在今天的环境中，研究所谓的智能材料，尤其是复合材料，已经变得很重要。由于它们的轻质，它们因其强度和轻质而成为首选材料。在这个命题中，我们把复合材料的振动特性作为一个问题来研究。特别是如何使用复合材料建造物体，以使其基本振动特性最小化。直觉上，物体振动的固有频率越大，它就越容易受到应力和断裂的影响。例如，一个问题是，如果我们需要建造一个对称的物体，比如一个垫圈，这是否意味着复合材料必须对称地排列，尊重垫圈的对称性？我们发现，在我们的研究中，例如，这并不一定能最小化可能由振动引起的应力，我们需要以非对称的方式排列复合材料来构造垫圈.这不仅是我们在我们的建议中研究的问题，而且也是微分几何和物理学中的问题。