Harmonic Analysis and PDE

摘要：本文提出了pde领域的几个问题。这些问题大多属于非线性微分方程的子领域，也有一些属于线性微分方程的子领域。非线性问题自然地从物理和几何以及物理和几何的相互作用中产生。线性问题也与几何和物理有关。本文重点讨论了微分方程解的各种定性特征、解的水平线、特征函数的节点线、解的光滑性等。我们选择这些问题的部分原因是我们认为它们中的许多与经典谐波分析有很自然的联系，事实上，我们已经开始使用经典谐波分析的技术来解决它们。这在提案正文中有说明。在我们的建议中，物理学中出现的一些问题与涡旋现象有关。在这里，我们继续由以前的拨款资助的工作。我们研究的涡是由流体在二维球体上和平面上的流动而产生的涡。利用点涡分布的连续极限，为求解规定的高斯曲率方程提供了一种新的方法，同时对保形不变偏微分方程也有了深刻的认识。还提出了源于薛定谔方程和几何及其对椭圆算子谱的影响的问题。我们从拟议的研究计划中挑出一个重要组成部分。在今天的环境中，研究所谓的智能材料，主要是复合材料已经变得很重要。由于它们的重量轻，它们是首选的材料，因为它们的强度和重量轻。本文将复合材料的振动特性作为一个问题来研究。特别是应该如何使用复合材料来建造物体，以尽量减少它们的基本振动特性。直观地说，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，一个问题是，如果我们需要构建一个对称的物体，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，这一定不能最小化可能由振动引起的应力，我们需要以一种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问题，而且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.