Mathematical Analysis of free boundary problems related to a variational problem

我们主要研究了一个与变分问题相关的自由边界问题。由于我们的问题有一个特点，即自由边界是一个极小点的奇异点集合，并且能量集中在它上面。因此，我们可以认为我们的目的是在解的奇点上处理能量集中现象。在这种观点下，我们处理了以下类型的问题：(1)发展了与移动边界最小化泛函相关的椭圆自由边界问题的正则性理论；(2)通过最小化过程发展了一种数值方法；(3)发展了一种与解决双曲自由边界问题相关的方法。对于问题(1)，在二维情况下，我们成功地证明了某些非线性情况下自由边界的正则性。对于(2)，我们处理了主要出现在超导现象中的金兹堡-朗道泛函。在此，我们开发了一种基于离散莫尔斯半流的抛物线型和双曲型问题的求解方法。对于(3)，我们构造了一些相容条件下双曲型自由边界问题的强解。此外，我们还开发了一个软件来解决这个问题，精度很高。我们将这些结果总结成7篇论文（已发表或已出版）和1篇预印本（已提交）。

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).