Palais-Smale compactness condition

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.

Strong formulation

A functional "I" from a Hilbert space "H" to the reals satisfies the Palais-Smale condition if $Iin\; C^1(H,mathbb\{R\})$, and if every sequence $\{u\_k\}\_\{k=1\}^inftysubset\; H$ such that:
* $\{I\; [u\_k]\; \}\_\{k=1\}^infty$ is bounded, and
* $I\text{'}\; [u\_k]\; ightarrow\; 0$ in "H"is precompact in "H".

Weak formulation

Let $X$ be a Banach space and $Phicolon\; X\; omathbf\; R$ be a Gateaux differentiable functional. The functional $Phi$ is said to satisfy the weak Palais-Smale condition if for each sequence $\{x\_n\}subset\; X$ such that
*
* $limPhi\text{'}(x\_n)=0$ in $X^*$,
* $Phi(x\_n)\; eq0$ for all $ninmathbf\; N$,

there exists a critical point $overline\; xin\; X$ of $Phi$ with:$liminfPhi(x\_n)lePhi(overline\; x)lelimsupPhi(x\_n).$

References

*