Boundedness of Laplacian eigenfunctions on manifolds of infinite volume
| Autor: | Patrícia Klaser, Miriam Telichevesky, Leonardo Prange Bonorino |
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| Rok vydání: | 2014 |
| Předmět: |
Mathematics - Differential Geometry
Statistics and Probability Pure mathematics Dimension (graph theory) Context (language use) 01 natural sciences 58J50 58J05 Mathematics - Analysis of PDEs 0103 physical sciences FOS: Mathematics 0101 mathematics Mathematics 010102 general mathematics Hadamard manifold Eigenfunction Riemannian manifold Differential Geometry (math.DG) Bounded function 010307 mathematical physics Geometry and Topology Mathematics::Differential Geometry Statistics Probability and Uncertainty Isoperimetric inequality Laplace operator Analysis Analysis of PDEs (math.AP) |
| DOI: | 10.48550/arxiv.1403.5552 |
| Popis: | In a Hadamard manifold $M$, it is proved that if $u$ is a $\lambda$-eigenfunction of the Laplacian that belongs to $L^p(M)$ for some $p \ge 2$, then $u$ is bounded and $\|u\|_{\infty} \le C \|u\|_p,$ where $C$ depends only on $p$, $\lambda$ and on the dimension of $M$. This result is obtained in the more general context of a complete Riemannian manifold endowed with an isoperimetric function $H$ satisfying some integrability condition. In this case, the constant $C$ depends on $p,\lambda$ and $H.$ Comment: 13 pages; change in the title; change in the abstract; improvement in the introduction; added some details in the proof of Theorem 2.2 |
| Databáze: | OpenAIRE |
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