en>SmackBot: remove Erik9bot category,outdated, tag and general fixes

2009-12-16T22:49:55Z

remove Erik9bot category,outdated, tag and general fixes

New page

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 point (mathematics)|critical point]]s, in particular [[saddle point]]s. The Palais-Smale condition is a condition on the [[functional (mathematics)|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 [[proper map]]s: functions which do not take unbounded sets into bounded sets. In the calculus of variations, where one is typically interested in infinite-dimensional [[function space]]s, the condition is necessary because some extra notion of [[compactness]] beyond simple boundedness is needed. See, for example, the proof of the [[mountain pass theorem]] in section 8.5 of Evans.

== Strong formulation ==

A continuously [[Fréchet derivative|Fréchet differentiable]] [[functional (mathematics)|functional]] <math>I\in C^1(H,\mathbb{R})</math> from a [[Hilbert space]] ''H'' to the [[real number|reals]] satisfies the Palais-Smale condition if every [[sequence]] <math>\{u_k\}_{k=1}^\infty\subset H</math> such that:
* <math>\{I[u_k]\}_{k=1}^\infty</math> is bounded, and
* <math>I'[u_k]\rightarrow 0</math> in ''H''
has a convergent subsequence in ''H''.

== Weak formulation ==
Let ''X'' be a [[Banach space]] and <math>\Phi\colon X\to\mathbf R</math> be a [[Gâteaux derivative|Gâteaux differentiable]] functional. The functional <math>\Phi</math> is said to satisfy the '''weak Palais-Smale condition''' if for each sequence <math>\{x_n\}\subset X</math> such that
* <math>\sup |\Phi(x_n)|<\infty</math>,
* <math>\lim\Phi'(x_n)=0</math> in <math>X^*</math>,
* <math>\Phi(x_n)\neq0</math> for all <math>n\in\mathbf N</math>,

there exists a critical point <math>\overline x\in X</math> of <math>\Phi</math> with
:<math>\liminf\Phi(x_n)\le\Phi(\overline x)\le\limsup\Phi(x_n).</math>

== References ==

* {{cite book | first=Lawrence C. | last=Evans | title=Partial Differential Equations | publisher=American Mathematical Society | location=Providence, Rhode Island | year=1998 | isbn=0-8218-0772-2}}

{{DEFAULTSORT:Palais-Smale compactness condition}}
[[Category:Calculus of variations]]

Connectedness locus - Revision history

en>SmackBot: remove Erik9bot category,outdated, tag and general fixes