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2017 About the Degenerate Spectrum of the Tension Field for Mappings into a Symmetric Riemannian Manifold
Moussa Kourouma
Afr. Diaspora J. Math. (N.S.) 20(2): 45-68 (2017).

Abstract

Let $(M,g)$ and $(N,h)$ be compact Riemannian manifolds, where $(N,h)$ is symmetric, $v\in W^{1,2}((M,g),(N,h))$, and $\tau $ is the tension field for mappings from $(M,g)$ into $(N,h)$. We consider the nonlinear eigenvalue problem $\tau (u)-\lambda \exp _{u}^{-1}v=0$, for $u$ $\in W^{1,2}(M,N)$ such that $u_{\left\vert \partial M\right. }=v_{\left\vert \partial M\right.}$, and $\lambda \in \mathbb{R}$. We prove, under some assumptions, that the set of all $\lambda $, such that there exists a solution $(u,\lambda )$ of this problem and a non trivial Jacobi field $V$ along $u$, is contained in $\mathbb{R}_{+}$, is countable, and has no accumulation point in $\mathbb{R}$. This result generalizes a well known one about the spectrum of the Laplace-Beltrami operator $\Delta $ for functions from $(M,g)$ into $\mathbb{R}$.

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Moussa Kourouma. "About the Degenerate Spectrum of the Tension Field for Mappings into a Symmetric Riemannian Manifold." Afr. Diaspora J. Math. (N.S.) 20 (2) 45 - 68, 2017.

Information

Published: 2017
First available in Project Euclid: 17 May 2017

zbMATH: 1378.58013
MathSciNet: MR3652656

Subjects:
Primary: 35D , 35J , 49R50 , 58C , 58E , 58J

Keywords: bifurcation , eigenmapping , eigenvalue , Jacobi field , Laplace-Beltrami operator , Riemannian manifold , symmetry , tension field

Rights: Copyright © 2017 Mathematical Research Publishers

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Vol.20 • No. 2 • 2017
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