Harmonic Analysis and PDE

The proposal addresses several questions that arise in problems in Physics and Engineering. We describe one of the problems that we 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 membranes constructed, 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 an optimal configuration exist?(2) Is the junction region between the membranes, the so-called free boundary smooth? If not smooth what is the nature of the singularities, i.e. what is the Hausdorff dimension of the singular set. (3) If there is symmetry in the problem does it reflect and impose conditions that the materials have to be so glued as to respect this symmetry. This problem is also amenable to numerical simulation which 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 by M. Imai and I. Ohnishi of the Institute of Electro-Communications in Tokyo and K. Kurata of the Tokyo Metropolitan University. The object is to design composites of materials with two different densities such that the resulting composite has the lowest tone. For example one way to naively view this problem is to patch together two materials in such a way that the resulting composite membrane vibrates in the slowest possible way out of all possible configurations that can be formed by gluing together the two given materials. The first question is (1) Is there always an optimal configuration possible? (2) Is the interface between the two materials making up the membrane smooth or necessarily are there sharp spikes? (3) If there is some additional symmetry in the shape of the membrane does it force that the materials be glued together in a way such that this symmetry is to be respected? These are basic Engineering questions that arise in many situations where say vibrations are to be kept to a minimum. Computer studies help, but what nails down with certainty, the description of the optimal configuration, are rigorous theorems about the arrangement of the materials. This is one of many problems in our project which can be summarized by saying that it is the determination of optimal shapes in the design of composite membranes so as to minimize vibrations and if possible determine a explicit recipe(algorithm) that will tell us how to achieve this optimal configuration.