Variational Methods for Nonlinear Eigenvalue Problems
Variational Methods for Nonlinear Eigenvalue Problems
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DOI：
10.1007/978-3-642-10940-9_4
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发表时间：
2009
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影响因子：
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通讯作者：
P. Rabinowitz
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文献类型：
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作者：
P. Rabinowitz
这些讲座的目的是介绍非线性特征值问题的变分方法，包括在抽象的背景下和应用于非线性偏微分方程时。几种不同的情况将被处理。我们的研究始于“关于Lust temk-Schnirelmann类型的定理”。由Ljusternik[1]得到的最简单的结果是：如果f是ℝn上的偶连续可微的实值函数，则至少有n对不同的临界点(即f‘(X)=λx，λ=(f’(X)，x)的点)。这个定理作为更一般情况的原型，其中一个人在流形(通常是“球状”)上有一个实值函数(通常是偶数)，并使用与流形相关的拓扑不变量来获得该泛函所具有的临界点的数目的下界。莫尔斯理论处理了类似的问题，事实上，S有许多共同之处。然而，Ljusternik-Schnirelmann型定理对光滑性的要求没有Morse理论严格(C_1而不是C_2)，临界点不一定是非退化的。这些事实结合在一起，使得Ljusternik-Schnirelmann理论更适用于非线性偏微分方程。另一方面，当可以使用莫尔斯理论时，它提供了关于临界点及其类型的更详细的信息(例如，见[2])。
The goal of these lectures is to present an introduction to variational methods for nonlinear eigenvalue problems both in an abstract setting and as applied to nonlinear partial differential equations. Several different situations will be treated. Our study begins with “Theorems on Ljustemik-Schnirelmann type”. The simplest such result, which is due to Ljusternik [1] states: If f is an even continuously differentiable real valued function on ℝn, thenpossesses at least n distinct pairs of critical points (i.e. points at which f′(x) = λx, λ = (f′(x),x)). This theorem serves as a prototype for more general situations where one has a real valued function (usually even) on a manifold (usually “spherelike”) and uses topological invariants associated with the manifold to obtain lower bounds for the number of critical points the functional possesses. Morse theory treats similar questions and indeed there are many ideás in common. However smoothness requirements for theorems of Ljusternik-Schnirelmann type are less stringent (C1rather than C2) than in Morse theory and critical points need not be nondegenerate. These facts combine to make the Ljusternik-Schnirelmann theory more applicable to nonlinear partial differential equations. On the other hand when it can be used, Morse theory gives more detailed information on critical points and their types (see e.g. [2]).