Asymptotic Expansions, Inverse Problems and Homogenization of Boundary Values

数学科学：边界条件的渐近展开式、反问题和均匀化。我们将使用渐近展开式技术来解决两类问题。首先是从边界测量中精确检测小的不均匀性的位置和形状。关于非均匀性大小的解的渐近展开式将在它们还不知道的地方发展。将引入一种新的程序，它使用这些展开来查找有关孔的信息。该程序将测试不同类型的方程和形状的非均匀性。第二个问题涉及方程边值的均匀化，其中介质具有潜在的周期结构。渐近分析有助于找到极限方程或有效方程。这种分析需要考察边界层函数，它是具有周期或几乎周期边界条件的半空间上的解。材料科学和电磁学中出现的许多方程都涉及一些小参数。例如，这个参数可以表示飞机机翼内部一个小缺陷的直径。我们将从数学上分析这种缺陷对电位的影响。从这个分析中，我们将开发和测试一种新的方法来找到缺陷的大小和位置。实际上，这种方法只需要在物体的外部感应电流并测量由此产生的电势。我们还将在微观水平上分析两种材料混合在一起的宏观行为。在这种情况下，小参数表示微观尺度的大小。我们将利用我们的分析更准确地模拟波在混合介质中的传播，用于石油勘探和材料科学。

NSF Award Abstract - DMS-0072511Mathematical Sciences: Asymptotic Expansions, Inverse Problems, and Homogenization of Boundary ConditionsAbstract0072511 MoskowWe will use the technique of asymptotic expansions to solve two types of problems. The first is the accurate detection of the location and shapes of small inhomogeneities from boundary measurements. Asymptotic expansions of the solution with respect to the size of the inhomogeneity will be developed where they are not yet known. A new procedure will be introduced which uses these expansions to find information about the holes. The procedure will be tested for different types of equations and shapes of inhomogeneities. The second problem involves the homogenization of boundary values for equations where the medium has an underlying periodic structure. Asymptotic analysis will aid in finding the limiting or effective equations. This analysis requires examining boundary layer functions which are solutions on a half space with periodic or almost periodic boundary conditions. Many equations that arise from material science and electromagnetics involve some small parameter. This parameter could represent, for example, the diameter of a small imperfection inside an airplane wing. We will analyze mathematically the effects of such imperfections on electric potentials. From this analysis we will develop and test numerically a new method to find the sizes and locations of imperfections. In practice, this method would require inducing electrical currents and measuring the resulting electric potential only on the exterior of the object. We will also analyze the macroscopic behavior of two materials mixed together at the microscopic level. The small parameter in this case represents the size of the microscopic scale. We will use our analysis to model more accurately the propagation of waves through mixed media, for use in oil exploration and materials science.