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

研究了数学反问题的特解。这些解称为复几何解，简称CGO解。我们知道，我们可以成功地应用CGO解决方案与线性复相函数的许多问题。近年来，人们研究了具有非线性复相函数的CGO解。但这种新方法仅限于求解类似于拉普拉斯方程的椭圆型方程。因此，在一般情况下，没有人理解具有非线性相函数的CGO解的意义。在本研究计划中，我们研究了新的CGO解的非线性相函数，可以适用于一般方程，包括双曲型方程。更准确地说，我们可以得到新的非局部Carleman估计。利用这个估计，我们可以研究Lorentian度量及其相关的算子，这是与双曲型方程有关的新的反问题。

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.