Mathematical Sciences: Inverse Boundary Value Problems

关于逆的数学理论的继续研究 偏微分方程中的问题将得到支持， 这个奖。 这项工作的主要数学成分是 所谓的Dirichlet到Neumann映射： 偏微分方程，解为 建立，其在边界处的法向导数为 给出 边界值与 建立了正规导数。 其中一个主要问题是 关于地图的可逆性和规则性， 当它存在的时候， 在现实世界中， 测量法向导数-通量-比测量 我知道很多关于边界值或解的值 在其定义范围内。 这类问题发生在 阻抗断层成像，其中一个人试图测量内部 通过测量电压和电流， 边界 迄今取得的进展包括各种必要条件 的可逆性，但没有很好的充分条件， 不存在对称性假设。 更好 已经在大于2的维度上实现了结果。 的 二维层析成像问题和各向异性情况（其 可能发生在地球物理学中）是重要的挑战。

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.