Palais–Smale compactness condition

The Palais–Smale compactness condition, named after Richard Palais and Stephen Smale, is a hypothesis for some theorems of the calculus of variations. It is useful for guaranteeing the existence of certain kinds of critical points, in particular saddle points. The Palais-Smale condition is a condition on the functional that one is trying to extremize.

In finite-dimensional spaces, the Palais–Smale condition for a continuously differentiable real-valued function is satisfied automatically for

A continuously Fréchet differentiable functional ${\displaystyle I\in C^{1}(H,\mathbb {R} )}$ from a Hilbert space H to the reals satisfies the Palais–Smale condition if every sequence ${\displaystyle \{u_{k}\}_{k=1}^{\infty }\subset H}$ such that:

• ${\displaystyle \{I[u_{k}]\}_{k=1}^{\infty }}$ is bounded, and
• ${\displaystyle I'[u_{k}]\rightarrow 0}$ in H

has a convergent subsequence in H.

Weak formulation

Let X be a Banach space and ${\displaystyle \Phi \colon X\to \mathbf {R} }$ be a Gateaux differentiable functional. The functional ${\displaystyle \Phi }$ is said to satisfy the weak Palais–Smale condition if for each sequence ${\displaystyle \{x_{n}\}\subset X}$ such that

• ${\displaystyle \sup |\Phi (x_{n})|<\infty }$,
• ${\displaystyle \lim \Phi '(x_{n})=0}$ in ${\displaystyle X^{*}}$,
• ${\displaystyle \Phi (x_{n})\neq 0}$ for all ${\displaystyle n\in \mathbf {N} }$,

there exists a critical point ${\displaystyle {\overline {x}}\in X}$ of ${\displaystyle \Phi }$ with

${\displaystyle \liminf \Phi (x_{n})\leq \Phi ({\overline {x}})\leq \limsup \Phi (x_{n}).}$

References

• ISBN 0-8218-0772-2
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• S2CID 122094186
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• Palais, R. S.; Smale, S. (1964). "A generalized Morse theory". Bulletin of the American Mathematical Society. 70: 165–172.
  doi:10.1090/S0002-9904-1964-11062-4
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