Harmonic Analysis and PDE

该提案解决了物理学和工程学中出现的几个问题。我们描述了我们提出的问题之一。在工程问题中出现的一个基本情况是构建由两种密度不同的材料组成的膜。我们对所构造的膜提出了额外的要求，即它们还具有第一本征值尽可能最低的附加性质。这是膜具有最低色调的要求。现在出现的自然问题是：（1）这样的最佳配置存在吗？(2)膜之间的连接区域，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，即奇异集的Hausdorff维数是多少。(3)如果问题中存在对称性，它是否反映并施加了材料必须粘合以尊重这种对称性的条件。这个问题也适用于我们已经进行的数值模拟，部分数值模拟为我们提供了什么样的严格定理可以被证明的洞察力。 该项目实际上源于M向PI提出的问题。今井和我。东京电子通信研究所的大西和K。东京都立大学的仓田。其目的是设计具有两种不同密度的材料的复合材料，使得所得到的复合材料具有最低的色调。例如，天真地看待这个问题的一种方法是将两种材料拼凑在一起，使得所得的复合膜在所有可能的结构中以最低的可能方式振动，这些结构可以通过将两种给定材料粘合在一起形成.第一个问题是（1）是否总是存在一个可能的最优配置？(2)构成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)如果在膜的形状上有一些额外的对称性，它是否迫使材料以某种方式粘合在一起，从而使这种对称性得到尊重？这些都是基本的工程问题，出现在许多情况下，说振动是要保持在最低限度。计算机研究有所帮助，但最佳配置的描述是关于材料排列的严格定理。这是我们的项目中的许多问题之一，可以总结为：在复合膜的设计中确定最佳形状，以使振动最小化，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.