Palais-Smale compactness condition

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The Palais-Smale compactness condition is a necessary condition for some theorems of the calculus of variations.

The condition is necessary because the calculus of variations studies function spaces that are infinite dimensional — 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.

[edit] Strong formulation

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

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

is precompact in H.

[edit] Weak formulation

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