Harmonic Analysis and PDE

Pi：Sagun Chanillo，罗格斯大学DMS-0200628摘要：在这项提案中提出了几个问题，都是在PDE领域。这些问题大多在非线性偏微分方程组的子域中，也有一些在线性偏微分方程组中。非线性问题是物理与几何、物理与几何相互作用的产物。线性问题也与几何和物理联系在一起。文中着重讨论了偏微分方程解的各种性质、水平线、特征函数的节点线、解的光滑性等。我们之所以选择这些问题，部分是因为我们认为其中许多问题与经典调和分析有着非常自然的联系，事实上，我们已经开始使用经典调和分析中的技术来解决它们。这一点在提案的正文中有描述。在我们的建议中，从物理学中出现的一些问题与涡旋现象有关。在这里，我们继续由前人资助的工作。我们研究的涡旋是由二维球体和平面上的流体流动引起的。利用点涡旋分布的连续极限，我们得到了一种新的方法来求解规定的高斯曲率方程，这也使我们对共形不变偏微分方程解有了很大的了解。我们还提出了源于薛定谔方程和几何的问题及其对椭圆算符频谱的影响。我们挑出了所提出的研究建议的一个重要组成部分。在当今的环境中，研究所谓的智能材料变得越来越重要，更主要的是复合材料。由于它们的轻质，它们因其强度和轻量化而成为首选材料。在这个建议中，我们把复合材料的振动特性作为一个问题来研究。尤其是如何使用复合材料来建造物体，以使其基本振动特性最小化。物体振动的固有频率越大，它就越容易受到应力和破坏。例如，人们可以问的一个问题是，如果我们需要建立一个对称的物体，就像垫圈一样，这是否意味着复合材料必须对称排列，而不是考虑垫圈的对称性？我们发现，例如，在我们的研究中，这不一定会使振动造成的应力最小化，我们需要以非对称的方式排列复合材料来构造垫圈。这不仅是我们在提案中研究的唯一问题，也是微分几何和物理中的问题。

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.