Singularities in Oscillatory Integrals and Inverse Problems

本项目涉及几个不同的问题，其共同的主题是控制、估计或重构振荡积分算子、傅里叶积分算子和反问题中出现的奇异性，第一个问题涉及建立具有一般齐次多项式相位函数的三维振荡积分算子的最佳衰减率。在第二，Calderon问题的标量conductivitieswith跳跃不连续性在两个维度将进行分析，与奇点（适当解释）的边界数据产生的信息thelocations，并可能的大小，跳跃的电导率在内部。衰减Radon变换的识别问题的线性化版本将分析使用技术从微观局部和谐波分析在第三个问题。最后，一个线性化的双曲反问题涉及到研究与三维焦散线的存在所产生的特定奇异几何相关的傅里叶积分算子的组成。这些问题的进展将有助于理解偏微分方程，用于求解它们的算子及其解的行为。许多自然定律都可以用偏微分方程来表示，这些方程控制着感兴趣的物理量，比如电磁场强度或声波施加的压力。在这个项目中开发的技术适用于管理各种波传播的方程，并基于几何观点来理解存在的奇异性。 最后三个问题有可能帮助设计或改进无损检测、医学成像和地震勘探系统。

ABSTRACT The project involves work on several distinct problems with the commonthemes of controlling, estimating or reconstructing singularities that arise inoscillatory integral operators, Fourier integral operators and inverse problems.The first problem concerns establishing optimal decay rates for oscillatoryintegral operators in three dimensions having generic homogeneous polynomialphase functions. In the second, the Calderon problem for scalar conductivitieswith jump discontinuities in two dimensions will be analyzed, with singularities(suitably interpreted) of boundary data yielding information about thelocations, and possibly sizes, of the jumps of the conductivity in the interior. A linearized version of the identification problem for the attenuated Radontransform will be analyzed using techniques from micro-local and harmonic analysisin the third problem. Finally, a linearized hyperbolic inverse problem involvesstudying the composition of Fourier integral operators associated to specificsingular geometries arising from the presence of caustics occurring genericallyin three dimensions.Progress on these problems will add to the understanding of partial differentialequations, the operators, which are used to solve them, and thebehavior of their solutions. Many laws of nature are expressed as partial differentialequations, which govern physical quantities of interest, such as electromagneticfield strength or pressure exerted by acoustic waves. The techniques to be developedin this project are applicable to equations that govern various kinds of wavepropagation and are based on a geometric point of view in understanding thesingularities, which are present. The last three problems have the potential toassist in the design or refinement of systems for nondestructive testing, medical imaging, and seismic exploration.