Mathematical Sciences: Inverse Boundary Value Problems

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Continuing research on the mathematical theory of inverse problems in partial differential equations will be supported by this award. The principal mathematical ingredient of the work is the so-called Dirichlet to Neumann map: The boundary values of a partial differential equation are given, the solution is established and its normal derivative at the boundary is then given. The relationship between the boundary values and the normal derivative is established. One of the primary issues concerns the invertibility of the map and the regularity of the inverse when it does exist. In the physical world it is often much easier to measure the normal derivative - flux - than to know much about the boundary values or the values of the solution within its region of definition. This type of problem occurs in impedance tomography where one attempts to measure internal resistivity in a body from measurements of voltage and current at the boundary. Progress to date includes a variety of necessary conditions for invertibility, but no good sufficient conditions are available without the presence of symmetry assumptions. Better results have been achieved in dimensions greater than two. The two dimensional tomography problem and anisotropic cases (which can occur in geophysics) are important challenges.