液晶の数理解析と変分問題の研究

液晶的数学分析和变分问题的研究

• 批准号: 06740106
• 负责人: 三沢 正史
• 金额: $ 0.58万
• 依托单位: Shinshu University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Encouragement of Young Scientists (A)
• 财政年份: 1994
• 资助国家: 日本
• 起止时间: 1994 至 无数据
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-06740106/
• 关键词: Ericksenモデル; Ginzburg-Landauモデル; 変分問題; 調和写像; p-調和写像; 熱型勾配流; 退化放物型偏微分方程式

1 数学の理論研究.液晶の電磁気学的平衡状態を記述するEricksenモデルの数理解析は調和写像型変分問題に帰着される.調和写像型変分問題に対する熱型勾配流を与える非線形放物型偏微分方程式を近似する後退差分型変分汎関数の最小化関数の族について次の結果を得た.(1)後退差分型変分汎関数のEuler-Lagrange方程式である後退差分偏微分方程式の解の族に対して、解の正則性を示す上で基本的役割を果たすHarnack不等式が近似に関して一様に成り立つ(Nonlinear Worldに掲載予定).粘性のある液晶の電磁気学的平衡状態を記述するEricksenモデル及び超電導の理論におけるGinzburg-Landauモデルの数理解析はp-調和写像型変分問題に帰着される(p>1).p-調和写像型変分汎関数を近似する退化型変分汎関数の変分問題に対応した熱型勾配流を記述する非線形退化放物型偏微分方程式の解の構成について次の結果を得た.(2)強解のクラスに対するa-prioriな評価の構成.特にあるweightを伴ったエネルギーの単調性を示す不等式が成り立つことを証明した.(京都大学数理解析研究所講究録掲載予定).(3)(3)の結果を基礎にして,時間大域的な弱解の構成とその解が(空間変数に関しての)一階微分とともに部分的にヘルダー連続であることを証明することに成功した(Courant研究所のF.H.Lin教授に評価され,投稿先を検討中).(4)有界な弱解のクラスに対するa-prioriな評価の構成(Rice Univ.のR.Hardt教授に評価され,投稿先を検討中).解の最良の部分的滑らかさ(解の不連続点(特異点)の集合の大きさがハウスドルフ測度の意味でどこまで小さくなるか)については考察中である.2.数値シミュレーション.液晶の変分問題(調和写像型変分問題)に対応した熱型勾配流を近似する後退差分型変分汎関数の最小化関数の族(離散的勾配流)による数値シミュレーションについては次のことを行なった.(1)空間次元が1の場合について離散的勾配流の時間発展の数値実験を数値解析用ソフト(MATHEMATICA)を使って行なった.空間次元が高次元の場合に離散的勾配流の時間発展の数値シミュレーションを行なうにはメモリ増設が必要である(現有メモリは10MB,スムーズに作動させるためには40MBが必要).また,そのプログラム作成は今後の課題である.

1 the study of mathematical theory. The equilibrium state of liquid crystal electromagnetism Ericksen analysis and image analysis problems are very important. The equation of partial differential equation of non-linear type is similar to that of non-linear partial differential equation. (1) backward differential equation, Euler-Lagrange equation, backward differential equation, backward differential equation. The solution of the positive property shows that the basic service cutting results are similar to the Harnack inequality, which is determined by the Nonlinear World inequality. The equilibrium state of viscous liquid crystal electromagnetism A brief account of the equilibrium of Ericksen Electro-Magnetics understanding of the number of viscous liquid Crystal Electro-Magnetics and p&gt (p&gt) 1) the partial differential equation of the non-linear degenerate type partial differential equation is solved by solving the partial differential equation of the non-linear degenerate object type. (2) the solution of the partial differential equation of the non-linear degenerate object type has been successfully solved. (2) the solution of the partial differential equation of the non-linear degenerate object type has been successfully solved. (2) the solution of the partial differential equation of the non-linear degenerate object type has been successfully solved. (2) the solution of the partial differential equation of the non-linear degenerate object type has been successfully solved. (2) the solution of the partial differential equation of the non-linear degenerate object type has been successfully solved. (2) the solution of the partial differential equation of the non-linear degenerate object type. In this case, the inequality is expressed in the form of an inequality, which is different from that of the weight. (Institute of Mathematical Analysis, Kyoto University). (3) (3) the results show that the weak solution of a large area of time is converted into a part of the differential equation part of the differential equation solution (space number analysis). (professor F.H.Lin of the Institute of Courant studies shows that it is successful. (4) there is a bounded weak solution. (4) there is a weak solution to the problem. (4) the contribution should be completed in the first place. (4) there is a bounded weak solution (Rice Univ). (4) the contribution is divided into two parts. Professor R.Hardt is in the middle of submitting a contribution. To solve the best part of the slippery part (solution of the best part of the slippery part. 2. I'm going to make a lot of noise. The liquid crystal distribution problem (image and image distribution problem) is similar to that of the differential distribution flow. The differential distribution model minimizes the number of users (scattered distribution flow). (1) the number of spatial parameters is different. (1) the spatial dimension 1 matches the dispersion of the distribution flow in the time domain. Several parsers use "MATHEMATICA" to make sure that the lines are not valid. Empty-dimensional and high-dimensional mapping flows are distributed in time, and the number of applications is not available. The operation is necessary for 40MB. (currently available for 10MB, it is necessary for users to perform such operations. In the future, the problem will be solved in the future.

项目成果

Masashi Misawa: "Pointwise estimates for solutions to nonlinear difference differential equations of elliptic-parabolic type" Nonlinear World. (掲載予定).

Masashi Misawa：“椭圆抛物型非线性差分微分方程解的逐点估计”《非线性世界》（待出版）。