Numerical and Analytical Studies of Boundary Value Problems for PDE's. Direct and Inverse Problems

9704575 Vogelius The research funded by this grant will focus on analytical and numerical studies of partial differential equations. There will be an emphasis on direct as well as on inverse problems. As far as direct problems are concerned, the effective boundary layer behavior encountered in connection with PDE's with rapidly oscillating coefficients will be investigated. PDE's of this form are used to model the behavior of composite materials. Other ``homogenization' problems for composite materials, for example, certain problems associated with the relationship between microscopic and macroscopic failure (including debonding of closely spaced fibers and delamination) will be studied. Concerning inverse problems, those to be investigated include 1) inverse 'coefficient' problems for linear elliptic equations with spatially varying coefficients and 2) inverse 'source' problems for semilinear elliptic equations. The data used for identification in both cases consists of overdetermined (Dirichlet or Neumann) boundary data. The analytical component of this work concerns such questions as identifiability and continuous dependence. For the inverse 'coefficient' problem, the investigation of the identification of cracks, inhomogeneities and corrosion damage will be continued and a study related to the ``imaging' of thin films will be initiated; for the inverse 'source problem,' connections with the well known Schiffer (or Pompeiu) conjecture will be investigated with particular emphasis on the relation between the ``smoothness' of the domain and the ability to identify. The numerical work will be devoted to the design of effective reconstruction methods, which to the largest extent possible rely on structural information about the solutions of the underlying PDE. The research funded by this grant will focus on analytical and numerical problems related to continuum mechanics. The research on direct problems has applications to several impor tant practical problems, for instance the assessment of the strength and potential failure of composite materials (including fracture, debonding of fibers and delamination). Special emphasis is put on the understanding of the relationship between microscopic and macroscopic behavior. The research on inverse problems has immediate applications to 1) medical impedance imaging, 2) nondestructive testing of mechanical parts (and sensors) as well as 3) the interpretation of magnetic diagnostics for Tokamak (fusion) devices. Part of this research is concerned with determining the sufficiency of the proposed boundary data for the various identifications (proving uniqueness and continuous dependence results). Another part of this research involves the design of effective algorithms, for instance for the detection and location of cracks and inhomogeneities in metal components as well as for the determination of the level of oxidation of thin films (gas sensors) using real experimental data. There will be an active involvement of post-doctoral researchers as well as graduate students and hopefully even some advanced undergraduate students in various aspects of the research.