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

我们研究了数学反问题的特解。这些解称为复几何解，简称CGO解。已知我们可以成功地将线性复相函数的CGO解应用于许多问题。近年来，人们研究了具有非线性复相函数的新型CGO解。但这种新方法仅限于求解椭圆型方程的拉普拉斯方程。所以没有人理解一般情况下具有非线性相函数的CGO解的意义。在本研究项目中，我们研究了新的具有非线性相函数的CGO解，这种解可以适用于包括双曲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。利用这一估计，我们可以研究洛伦度规及其相关算子。这是关于双曲方程的新的反问题。

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.