Harmonic Analysis and PDE

该提案解决了《物理与工程》中出现的几个问题。我们描述了我们提出的问题之一。工程问题中出现的一种基本情况是由两种不同密度的材料组成的膜。我们对所构造的膜提出了附加要求，即它们还具有第一本征值尽可能低的附加性质。这是对膜具有最低色调的要求。现在出现的自然问题是：(1)这种最优构型存在吗？(2)膜之间的连接区，即所谓的自由边界是光滑的吗？如果不光滑，奇点的性质是什么，即奇点集的Hausdorff维度是多少。(3)如果问题中存在对称性，它是否反映并强加了这样的条件：材料必须如此粘合以尊重这种对称性。这个问题也适用于数值模拟，我们也进行了数值模拟，在某种程度上，数值让我们洞察到可以证明什么样的严格定理。该项目实际上源于M向PI提出的问题。东京电子通信研究所的Imai和I.Ohnishi以及东京都大学的K.Kurata。其目的是设计具有两种不同密度的材料的复合材料，从而得到具有最低色调的复合材料。例如，天真地看待这个问题的一种方法是将两种材料拼接在一起，这样得到的复合膜以尽可能低的方式振动，而不是通过将两种给定的材料粘合在一起而形成的所有可能的构型。第一个问题是(1)总是有一个最佳的构型可能吗？(2)构成薄膜的两种材料之间的界面是光滑的还是一定有尖锐的尖峰？(3)如果薄膜的形状有一些额外的对称性，它是否迫使材料以一种尊重这种对称性的方式粘合在一起？这些都是基本的工程问题，在许多情况下都会出现，比如振动要保持在最低限度。计算机研究是有帮助的，但确切地说，对最佳构型的描述，是关于材料排列的严格定理。这是我们项目中的许多问题之一，可以概括为：在复合膜的设计中确定最佳形状以最小化振动，如果可能的话，确定一个明确的配方(算法)，告诉我们如何实现这种最佳配置。

The proposal addresses several questions that arise in problemsin Physics and Engineering. We describe one of the problems thatwe have proposed. A basic situation that arises in Engineering problems is to construct membranes composed out of two materials with different densities. We impose additional requirements on the membranesconstructed, that is they also have the additional property that the first eigenvalue be lowest possible. This is the requirement that the membrane have the lowest tone. Natural questions that now arise are (1) Does such anoptimal configuration exist?(2) Is the junction region between themembranes, the so-called free boundary smooth? If not smooth whatis the nature of the singularities, i.e. what is the Hausdorff dimensionof the singular set. (3) If there is symmetry in the problem does it reflectand impose conditions that the materials have to be so glued as to respect this symmetry. This problem is also amenable to numerical simulationwhich we have also carried out, and in part the numerics have provided us with insight into what sort of rigorous theorems can be proved. The project really stems from questions proposed to the PI byM. Imai and I. Ohnishi of the Institute of Electro-Communicationsin Tokyo and K. Kurata of the Tokyo Metropolitan University. Theobject is to design composites of materials with two different densitiessuch that the resulting composite has the lowest tone. For example one way to naively view this problem is to patch together two materials insuch a way that the resulting composite membrane vibrates in theslowest possible way out of all possible configurations thatcan be formed by gluing together the two given materials. The firstquestion is (1) Is there always an optimal configuration possible?(2) Is the interface between the two materials making up themembrane smooth or necessarily are there sharp spikes? (3) Ifthere is some additional symmetry in the shape of the membranedoes it force that the materials be glued together in a waysuch that this symmetry is to be respected? These are basicEngineering questions that arise in many situations where say vibrationsare to be kept to a minimum. Computer studies help, but what nailsdown with certainty, the description of the optimal configuration,are rigorous theorems about the arrangement of the materials. This isone of many problems in our project which can be summarized by sayingthat it is the determination of optimal shapes in the designof composite membranes so as to minimize vibrations and if possibledetermine a explicit recipe(algorithm) that will tell us howto achieve this optimal configuration.