Variational Problems on an infinite network and their applications

无限网络上的变分问题及其应用

• 批准号: 11640202
• 负责人: YAMASAKI Maretsugu
• 金额: $ 2.3万
• 依托单位: Shimane University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 1999
• 资助国家: 日本
• 起止时间: 1999 至 2000
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-11640202/
• 关键词: infinite network; Hardy's inequality; convex programs; duality theorem; discrete Laplacian; eigenvalue problem; Poincare-Sobolev' inequality; variational problem; 変分法; 離散倉持関数; 倉持ポテンシャル; Hardyの不等式

1. Inequalities on networks have played important roles in the theory of networks. We study several famous inequalities on networks such as Wirtinger's inequality, Hardy's inequality, Poincare-Sobolev's inequality and the strong isoperimetric inequality, etc. These inequalities are closely related to the smallest eigenvalue of weighted discrete Laplacian. We discuss some relations between these inequalities and the potential-theoretic magnitude of the ideal boundary of an infinite network.2. We estimate the smallest eigenvalue of a weighted discrete Laplacian by using a discrete Kuramochi potential with some numerical experiments.3. We give a dual characterization for the smallest eigenvalue of a weighted Laplacian by using an optimal solution of a variational problem on a network.4. We study the conjugate duality for optimization problems on an infinite network. In contrast to earlier approach we do not employ Hilbert or Banach space methods. As an application we obtain generalizations of some basic inverse relations from discrete potential theory.5. We attempt to obtain more useful information from the output of the verification method proposed by Alefeld, Chen and Potra. We use the Farkas lemma to check the nonexistence of solutions of linear complementarity problems.

1. 网络不等式在网络理论中发挥了重要作用。我们研究了几个著名的网络不等式，如Wirtinger不等式、Hardy不等式、Poincare-Sobolev不等式和强等周不等式等。这些不等式与加权离散拉普拉斯算子的最小特征值密切相关。我们讨论了这些不等式与无限网络理想边界的势论大小之间的一些关系。2．通过数值实验，利用离散Kuramochi势估计了加权离散拉普拉斯算子的最小特征值。 3．我们通过使用网络上变分问题的最优解给出了加权拉普拉斯算子的最小特征值的双重表征。 4．我们研究无限网络上优化问题的共轭对偶性。与早期的方法相反，我们不采用希尔伯特或巴纳赫空间方法。作为应用，我们从离散势理论中得到了一些基本反关系的推广。 5．我们试图从 Alefeld、Chen 和 Potra 提出的验证方法的输出中获取更多有用的信息。我们使用法卡斯引理来检查线性互补问题解的不存在性。

项目成果

村上温、山崎稀嗣: "Inequlities on infinite networks"Mem. Fac. Sci. Shimane Univ. Ser.B: Mathematical Science. 33(to appear). (2000)

Atsushi Murakami，Maretsugu Yamazaki：“无限网络上的不等式”Sci.Shimane Univ.B：数学科学（2000 年）。

W.Oettli and M.Yamasaki: "Duality theorems on an infinite network"Optimization. Vol.48. 1-15 (2000)

W.Oettli 和 M.Yamasaki：“无限网络上的对偶定理”优化。

村上温、山崎稀嗣: "Discrete Kuramochi function"Proc. 3rd. Intern. Conference on Difference Equation and Applications. 313-322 (1999)

Atsushi Murakami，Maretsugu Yamazaki：“离散 Kuramochi 函数”Proc。第 3 期差分方程和应用会议。

A.Murakami and M.Yamasaki: "Discrete Kuramochi function"Proc.3rd.Intern.Conference on Difference Equation and Applications. 313-322 (1999)

A.Murakami 和 M.Yamasaki：“离散 Kuramochi 函数”Proc.3rd.Intern.Conference on Difference Equation and Applications。

X.Chen, Y.Shogenji and M.Yamasaki: "Verification for existence of solutions of linear complementarity problems"Linear Algebra and Its Applications. Vol.324. 15-26 (2001)

X.Chen、Y.Shogenji 和 M.Yamasaki：“线性互补问题解的存在性验证”线性代数及其应用。