Inverse Boundary Value Problems in Partial Differential Equations

PI建议研究通过电磁波探测的非侵入性检测方法背后的数学理论。这种方法的目标是通过只进行表面测量而不钻孔或采集样本来确定物体的内部成分。这一理论在乳腺恶性肿瘤的早期检测中具有潜在的应用价值，并可能带来更经济的石油勘探技术。这个想法是在物体的表面施加电磁波，并通过表面测量分析其与内部材料的相互作用。这个项目的目的之一是证明当只在表面的一部分上进行表面测量时，仍然可以恢复材料的内部成分。PI建议从偏微分方程组(PDE)反边值问题的角度来研究非侵入性的检测方法，即解决边值问题并测量解在边界处的法向导数。问题是，人们能否从这些测量中恢复偏微分方程的系数。PI建议对电导率方程和Pauli-Dirac系统解决这个问题。PI建议对偏微分方程组使用变分收敛方法和Carleman估计。

The PI proposes to study the mathematical theory behind non-invasive methods of detection via probing by electromagnetic waves. The goal of such methods is to determine the interior composition of an object by making only surface measurements and without drilling holes or taking samples. The theory has potential applications in the early detection of malignant breast tumors and can lead to more economical oil exploration techniques. The idea is to apply an electromagnetic wave on the surface of the object and analyze its interaction with the interior material via surface measurements. One of the objectives of this project is to show that when the surface measurements are taken on only a part of the surface one can still recover the interior composition of the material.The PI proposes to study non-invasive methods of detection from the point of view of inverse boundary value problems for partial differential equations (PDEs), where one solves a boundary value problem and measures the normal derivative of the solution at the boundary. The question is whether one can recover the coefficients of the PDE from these measurements. The PI proposes to address this problem for the conductivity equation and the Pauli-Dirac system. The PI proposes to use variational convergence methods and Carleman estimates for systems of PDEs.