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Results tagged with critical-point-theory
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user 6101
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votes
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Critical points in Hilbert space
Especially in Calculus of Variations and Mechanics, a submanifold $Y$ of a manifold $X$ is usually called "a natural constraint" for a functional $f$ on $X$, if the special circumstance that you are c …
- 60.6k
4
votes
Minimizing sequence $\implies$ Palais–Smale sequence
In fact it's just the MVT to $DF$:
$$\|DF(x_n)-DF(y_n)\|\le \|x_n-y_n\|\sup\|D^2F\|=O(\|x_n-y_n\|)=o(1),$$ so $\|DF(x_n)\|=o(1)$ too.
$$*$$
Note that if $D^2f$ is not bounded, it is not true, and (a …
- 60.6k