Mathematical Sciences: Inverse Boundary Value Problems

对偏微分方程反问题数学理论的持续研究将得到该奖项的支持。这项工作的主要数学成分是所谓的狄利克雷到诺伊曼映射：给出偏微分方程的边界值，建立其解，然后给出其在边界处的法向导数。建立了边界值与法向导数之间的关系。其中一个主要问题涉及映射的可逆性和逆存在时的规律性。在物理世界中，测量法向导数——通量——往往比了解边界值或解在其定义区域内的值要容易得多。这种类型的问题发生在阻抗层析成像中，人们试图通过测量边界处的电压和电流来测量物体的内部电阻率。迄今为止的进展包括可逆性的各种必要条件，但没有对称假设的存在就没有好的充分条件。在大于2个维度上取得了更好的结果。二维层析问题和各向异性情况（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.