Some Questions in Inverse Problems and the Mixed Problem for Laplace's Equation in Lipschitz Domains

本课题将研究偏微分方程中的几个问题，这些问题会导致谐波分析中一些有趣的问题。第一组问题与Sylvester和Uhlmann研究的反二次问题有关。我的主要兴趣是考虑先验的规则假设，这似乎是研究这个问题所必需的。我提出了一种技术来建立一个唯一性定理在维度3和更大的系数，只有一个导数。此外，我将考虑二维问题，其中这个定理是已知的。在二维中，我建议将结果从方程扩展到系统。第二组问题与拉普拉斯方程的混合问题有关。这里的目标是为这些问题的解决方案获得最优的规律性结果。实例表明，正结果强烈地依赖于域的几何形状和Dirichlet和Neumann数据所在的集合。反电导率问题是通过在边界处进行电测量来确定物体内部物理性质问题的数学公式。这一问题及其相关问题在医学成像和材料无损评价中具有重要的实际意义。本项目提出的理论研究可能对如何改进这些问题的实际实施有所启发。拉普拉斯方程的混合问题模拟了确定固体内部温度的问题，其中部分边界是绝热的。我的研究重点是了解区域的几何形状如何影响我们解决这个问题的能力。

This project will study several questions in partial differentialequations which lead to interesting questions in harmonicanalysis. The first set of questions are related to the inverseconductivity problem as studied by Sylvester and Uhlmann. My maininterest is considering the a priori regularity assumption which seemto be necessary to study this problem. I propose a technique toestablish a uniqueness theorem in dimensions 3 and larger forcoefficients which only have one derivative. In addition, I willconsider the two-dimensional problem, where such a theorem isknown. In two dimensions, I propose to extend the result fromequations to systems. The second set of questions are related to themixed problem for Laplace's equation. The goal here is to obtainoptimal regularity results for solutions to these problems. Examplesindicate that the positive result depend strongly on the geometry ofthe domain and the sets where Dirichlet and Neumann data areposed.The inverse conductivity problem is a mathematical formulation of theproblem of determining the interior physical properties of an objectby making electrical measurements at the boundary. This and relatedproblems are of practical importance in medical imaging and in thenondestructive evaluation of materials. The theoretical investigationsproposed in this project may shed some light on how to improvepractical implementation of these problems. The mixed problem forLaplace's equation models the problem of determining the temperaturein the interior of a solid where part of the boundary is insulated.My research is focused on understanding how the geometry of theregion of the region affects our ability to solve this problem.