| Project/Area Number |
11640221
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Global analysis
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| Research Institution | Shikoku University |
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Principal Investigator |
TAKEUCHI Hiroshi Shikoku University, Faculty of Management and Information Science, Professor, 経営情報学部, 教授 (20197271)
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| Co-Investigator(Kenkyū-buntansha) |
HARIMA Tadahito Shikoku University, Faculty of Management and Information Science, Associate Professor, 経営情報学部, 助教授 (30258313)
KATSUDA Atsushi Okayama University, Faculty of Science, Associate Professor, 理学部, 助教授 (60183779)
SAKAI Takashi Okayama University, Faculty of Science, Professor, 理学部, 教授 (70005809)
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| Project Period (FY) |
1999 – 2001
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| Project Status |
Completed (Fiscal Year 2001)
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| Budget Amount *help |
¥3,300,000 (Direct Cost: ¥3,300,000)
Fiscal Year 2001: ¥900,000 (Direct Cost: ¥900,000)
Fiscal Year 2000: ¥1,300,000 (Direct Cost: ¥1,300,000)
Fiscal Year 1999: ¥1,100,000 (Direct Cost: ¥1,100,000)
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| Keywords | P-harmonic map / P-Laplace operator / graph / spectrum / 無限グラフ / グリーン核 / リーマン多様体 / P-harmonic morphism / P-ラプラシアン / 第1固有値 / スペクトラム |
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| Research Abstract |
Let u : M → N be a smooth map between Riemannian manifolds and p a real number 1 < p < ∞. We call u a p-harmonic map if it is a critical point of the p-energy functional ∫_M | du |^pdx. In the case of p = 2, it becomes the usual harmonic map. When N is a real number, the map u becomes the p-harmonic function and it is the solution of Δ_pu =div(|∇u|^<p-2>∇u) = 0. When M is the n-dimensional sphere S^n and p is equal to the dimension of M (dim M = n = p), we can get the existence of n-harmonic maps from S^n to N. This is the generalization of the results of Sacks-Uhlenbeck, which is the case of n = p = 2.
Let N be a real number. For the p-Laplacian Δ_p, we define the first eigenvalue of the p-Laplacian as the least real number λ for which the equation Δ_pu = -λ|u|^<p-2>u has a nontrivial solution u. Before, we had several estimates for them on Riemannian manifolds, such as the Faber-Krahn type inequality, the Cheeger type inqulity, and the Cheng type inequality. We get a discrete analogue in this project term, that is, we define the p-Laplacian on graphs and get the Cheeger type inequality and the Brooks type inequality. Let G_1 = (V_1, E_1) and G_2 = (V_2, E_2) be two graphs and φ : V_1 → V_2 an onto mapping. The map φ is said to be a p-harmonic morphism of G_1 to G_2 if for any p-harmonic function f at y = φ(x) ∈ V_2, the composition φ^* f = f ο φ is p-harmonic function at x ∈ V_1. We show the p-harmonic morphism is equivalent to the horizontally conformal.
Next we consider the solution of p-Laplace equations which coincide with Green kernels in the case of p = 2 and give some estimates.
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