Palais-Smale compactness condition

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' [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
* sup |Phi(x_n)|,
* limPhi'(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

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