Asymptotic Expansions, Inverse Problems and Homogenization of Boundary Values

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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.