关于一类椭圆型方程扩散系数的唯一性研究

On the Uniqueness of Diffusion Coefficients for a Class of Elliptic Equations

研究了一类椭圆方程扩散系数的唯一性问题,这类问题在很多应用科学领域都有很重要的意义。基于吉洪诺夫正则化方法,把原来的问题转化一个优化问题。运用极值原理和正问题的能量估计对该问题进行了仔细分析,并对泛函作出了深入的研究,然后运用弗雷歇偏导理论证明了q的唯一性。此问题的主要难点:证明q的唯一性时有很多方法,但是在本文中,仅仅是用了弗雷歇偏导的性质就得到了这个结论,这就是本文所做的主要贡献。

This paper investigates a uniqueness problem of the diffusion coefficient in an elliptic equation, which has important applica- tion in a large field of applied science. Based on Tikhonov regularization method, the original problem is transformed to an optimization problem. The uniqueness of q was proved by carefully analyzing the cost functional and using the priori estimates of the forward problem and the extremum principle. The main difficulty of this problem is that the Frechet derivative properties were only used to get the u- niqueness, although there are a lot of ways to proof the uniqueness of q, which was considered the main contribution of the paper.