A NONLINEAR EIGENVALUE PROBLEM
A NONLINEAR EIGENVALUE PROBLEM
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非线性特征值问题
DOI：
10.1142/9789812811066_0005
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发表时间：
2004
影响因子：
11.5
通讯作者：
P. Lindqvist
中科院分区：
文献类型：
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作者：
P. Lindqvist
2000年6月我在帕多瓦数学分析学院的讲座记录在这些笔记中。它们是我1994年10月在Jyvaskyla的b[37]讲座的更新和扩展版本。特别地，包括了最近令人兴奋的渐近情况的发展，这被称为∞-特征值问题。我要感谢帕多瓦大学提供的资金支持。我特别感谢马西莫·兰扎·德·克里斯托福里斯的友好协助。我感谢Harald Hanche-Olsen在排版的最后调整方面的帮助。这几节课讲的是一个非线性特征值问题它被认为是线性情况的正确推广。到目前为止，我已经在四大洲做过关于这个主题的演讲，我坚持这个看似非常特殊的问题有两个原因。首先，人们可以在没有任何光谱理论知识的情况下研究有趣的问题。第二，据我所知，有许多悬而未决的问题很容易说明。较高的特征值是“神秘的”。线性特征值问题的主要例子是在r中给定的有界区域内找到边值为零的方程∆u+λu = 0的所有非平凡解。这是Dirichlet边值问题。（在Neumann边值问题中，法向导数在边界处为零。）不用说，这已经被推广到许多方面：黎曼曲面和流形，方程∆u + λu + V u = 0与一个潜在的V，更一般的微分算子比拉普拉斯，等等。
My lectures at the Minicorsi di Analisi Matematica at Padova in June 2000 are written up in these notes1. They are an updated and extended version of my lectures [37] at Jyvaskyla in October 1994. In particular, an account of the exciting recent development of the asymptotic case is included, which is called the ∞-eigenvalue problem. I wish to thank the University of Padova for financial support. I am especially grateful to Massimo Lanza de Cristoforis for his kind assistance. I thank Harald Hanche-Olsen for his kind help with final adjustments of the typesetting. These lectures are about a nonlinear eigenvalue problem that has a serious claim to be the right generalization of the linear case. By now I have lectured on four continents about this theme and my reason for sticking to this seemingly very peculiar problem is twofold. First, one can study the interesting questions without any previous knowledge of spectral theory. Second, to the best of my knowledge there are many open problems easy to state. The higher eigenvalues are “mysterious”. The leading example of a linear eigenvalue problem is to find all nontrivial solutions of the equation ∆u+λu = 0 with boundary values zero in a given bounded domain in R. This is the Dirichlet boundary value problem. (In the Neumann boundary value problem the normal derivative is zero at the boundary.) Needless to say, this has been generalized in numerous ways: to Riemann surfaces and manifolds, to equations ∆u + λu + V u = 0 with a potential V , to more general differential operators than the Laplacian, and so on.