2000 Fiscal Year Final Research Report Summary
Variational Problems on an infinite network and their applications
| Project/Area Number |
11640202
|
|---|
| Research Category |
Grant-in-Aid for Scientific Research (C)
|
| Allocation Type | Single-year Grants |
|---|
| Section | 一般 |
|---|
| Research Field |
Global analysis
|
|---|
| Research Institution | Shimane University |
|---|
Principal Investigator |
YAMASAKI Maretsugu Shimane Univ., Science and Engineering, Professor, 総合理工学部, 教授 (70032935)
|
|---|
| Co-Investigator(Kenkyū-buntansha) |
SUGIE Jitsuro Shimane Univ., Science and Engineering, Professor, 総合理工学部, 教授 (40196720)
FURUMOCHI Tetsuo Shimane Univ., Science and Engineering, Professor, 総合理工学部, 教授 (40039128)
AIKAWA Hiroaki Shimane Univ., Science and Engineering, Professor, 総合理工学部, 教授 (20137889)
|
|---|
| Project Period (FY) |
1999 – 2000
|
| Keywords | infinite network / Hardy's inequality / convex programs / duality theorem / discrete Laplacian / eigenvalue problem / Poincare-Sobolev' inequality / variational problem |
|---|
| Research Abstract |
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.
|
