Mathematical inverse problems based on analysis of complex geometrical optics solutions and the applications to science and engineering

基于复杂几何光学解分析的数学反问题及其在科学和工程中的应用

• 批准号: 21740107
• 负责人: TAKUWA Hideki
• 金额: $ 2.66万
• 依托单位: Doshisha University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Young Scientists (B)
• 财政年份: 2009
• 资助国家: 日本
• 起止时间: 2009 至 2012
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-21740107/
• 关键词: 逆問題; 複素幾何光学解; 擬リーマン幾何学; カーレマン評価式; カ―レマン評価; 弱擬凸性; 双曲型逆問題; カーレマン評価; 複素幾何光学

We have studied the special solutions to the mathematical inverse problems. These solutions are called complex geometrical solutions, in short, CGO solutions. It was known that we could succeed to apply CGO solutions with linear complex phase functions to many problems. Recently new CGO solutions with nonlinear complex phase functions have been studied. But this new approach was restricted to the problems about elliptic equations as Laplace equations. So no one has understood the meaning of CGO solutions with nonlinear phase functions in general cases. In this research program we have studied new CGO solutions with nonlinear phase functions which can be applicable to general equations including hyperbolic equations. More precisely, we can derive new nonlocal Carleman estimates. By using this estimate we can study Lorentian metric and operators associated it. This is the new inverse problem related to hyperbolic equations.

我们研究了数学反问题的特解。这些解被称为复杂几何解，简称CGO解。众所周知，我们可以成功地将具有线性复相函数的CGO解应用于许多问题。最近，人们研究了新的具有非线性复相函数的CGO解。但这种新方法仅限于关于椭圆型方程作为拉普拉斯方程的问题。因此，在一般情况下，没有人理解具有非线性相函数的CGO解的含义。在这个研究项目中，我们研究了新的具有非线性相函数的CGO解，它可以适用于包括双曲型方程在内的一般方程。更准确地说，我们可以得到新的非局部Carleman估计。通过使用这个估计，我们可以研究洛伦兹度量和与之相关的算子。这是与双曲型方程有关的新的反问题。