Mathematical Analysis of free boundary problems related to a variational problem

主要研究了一个与变分问题相关的自由边界问题。由于问题的自由边界是极小元的奇点集，能量集中在自由边界上，所以我们的目的是处理解的奇点上的能量集中现象。在此基础上，我们研究了以下几类问题：（1）发展了与极小化泛函活动边界相关的椭圆型自由边界问题的正则性理论;（2）发展了一种基于极小化过程的数值方法;（3）发展了一种求解双曲型自由边界问题的方法。对于（2），我们处理了主要出现在超导现象中的Ginzburg-Landau泛函。在这方面，我们发展了一种方法，由于离散莫尔斯半流抛物和双曲问题。对于（3），在某些相容条件下，我们构造了一个与双曲型自由边值问题有关的强解.此外，我们开发了一个软件来解决这个问题，具有良好的精度。我们将这些结果总结为7篇论文（发表或出版），我预印（提交）。

We mainly investigated a free boundary problem related to a variational problem. Since our problem has a feature (hat the free boundary is a set of singular points of a minimizer and the energy concentrate on it. So, we can cosider that our purpose is on treating the energy concentration phenomena on the singularity of solutions. In this stand point of view, we treated the following type of problems :(1)Develop Regularity theory of elliptic free boundary problem related to minimizing functional with moving boundary,(2)Develop a Numerical method via a minimization process,(3)Develop a method related to solve a hyperbolic free boundary problem.For problem (1), in 2-dimensional case, we successfully showed regularity of free boundary on some nonlinear case. For (2), we treated the Ginzburg-Landau functional which mainly appear in superconducting phenomema. In this, we developed a method due to discrete Morse semiflow for parabolic and hyperbolic problems. For (3), we constucted a strong solutions related to hyperbolic free boundary problems under some compatibility conditions. Moreover we developed a software to solve this with good accuracy. We summed up these results into 7 papers (appeared or in press) and I preprint (submitted).