Mathematical Analysis of free boundary problems related to a variational problem

与变分问题相关的自由边界问题的数学分析

• 批准号: 09640170
• 负责人: OMATA Seiro
• 金额: $ 2.3万
• 依托单位: Kanazawa University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 1997
• 资助国家: 日本
• 起止时间: 1997 至 1998
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-09640170/
• 关键词: Free boundary problem; Variational problem; Nonlinear partial differential equations; Numerical Analysis; Minimizing methods; Superconductivity; Liquid crystals

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).

我们主要研究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（即自由边界是极小化器的一组奇点，并且能量集中在其上。因此，我们可以认为我们的目的是处理解奇点上的能量集中现象。在这个立场上，我们处理了以下类型的问题：（1）发展与最小化移动边界泛函相关的椭圆自由边界问题的正则理论，（2）开发一种通过最小化的数值方法 过程中，(3)开发了一种解决双曲自由边界问题的相关方法。对于问题(1)，在二维情况下，我们成功地展示了一些非线性情况下自由边界的规律性。对于(2)，我们处理了主要出现在超导现象中的Ginzburg-Landau泛函。在此，我们开发了一种针对抛物线和双曲问题的离散莫尔斯半流方法。对于（3），我们 在某些兼容性条件下构造了与双曲自由边界问题相关的强解。此外，我们开发了一个软件来高精度地解决这个问题。我们将这些结果总结为 7 篇论文（已发表或正在出版），我预印本（已提交）。

项目成果

A.Kodama: "Characterization of generalized complex ellipsoids in C^n-from the viewpoint of biholomorphic automorphisms" Geometric Complex Anal.363-369 (1996)

A.Kodama：“从双全纯自同构的角度来表征 C^n 中的广义复椭球体”Geometric Complex Anal.363-369 (1996)

T.Nagasawa K.Nakane S.Omata: "Hyperbolic Ginzburg Landau system" Nonliear Analysis. to appear.

T.Nagasawa K.Nakane S.Omata：“双曲 Ginzburg Landau 系统”非线性分析。

H.Imai,S.Omata,K.Nakane K.Kikuchi: "Numerical analysis of a free boundary problem governed by a hyperbolic equation" in Proc.Third China-Japan Seminar on Numerical Mathematics,Science Press Beijing New York. 214-221 (1998)

H.Imai，S.Omata，K.Nakane K.Kikuchi：“双曲方程控制的自由边界问题的数值分析”载于第三届中日数值数学研讨会论文集，科学出版社北京纽约。

S.Omata T.Okamura K.Nakane: "Numerical analysis for the discrete Morse semiflow related to the Ginzburg Landau funcional" Nonlinear Analysis. to appear.

S.Omata T.Okamura K.Nakane：“与 Ginzburg Landau 函数相关的离散莫尔斯半流的数值分析”非线性分析。

T.Nagasawa, K.Nakane and S.Omata: "Numerical computations for a hyperbolic Ginzburg-Landau system" in Proceedings of the Eighth International Colloquium on Differential Equations Plovdiv, Bulgaria, August. 18-23 (1997)

T.Nagasawa、K.Nakane 和 S.Omata：“双曲 Ginzburg-Landau 系统的数值计算”，第八届国际微分方程座谈会论文集，保加利亚普罗夫迪夫，8 月。