Mountain pass theorem

The mountain pass theorem is an existence theorem from the calculus of variations, originally due to Antonio Ambrosetti and Paul Rabinowitz.[1] Given certain conditions on a function, the theorem demonstrates the existence of a saddle point. The theorem is unusual in that there are many other theorems regarding the existence of extrema, but few regarding saddle points.

Statement

The assumptions of the theorem are:

  • is a functional from a Hilbert space H to the reals,
  • and is Lipschitz continuous on bounded subsets of H,
  • satisfies the Palais–Smale compactness condition,
  • ,
  • there exist positive constants r and a such that if , and
  • there exists with such that .

If we define:

and:

then the conclusion of the theorem is that c is a critical value of I.

Visualization

The intuition behind the theorem is in the name "mountain pass." Consider I as describing elevation. Then we know two low spots in the landscape: the origin because , and a far-off spot v where . In between the two lies a range of mountains (at ) where the elevation is high (higher than a>0). In order to travel along a path g from the origin to v, we must pass over the mountains—that is, we must go up and then down. Since I is somewhat smooth, there must be a critical point somewhere in between. (Think along the lines of the mean-value theorem.) The mountain pass lies along the path that passes at the lowest elevation through the mountains. Note that this mountain pass is almost always a saddle point.

For a proof, see section 8.5 of Evans.

Weaker formulation

Let be Banach space. The assumptions of the theorem are:

  • and have a Gateaux derivative which is continuous when and are endowed with strong topology and weak* topology respectively.
  • There exists such that one can find certain with
.
  • satisfies weak Palais–Smale condition on .

In this case there is a critical point of satisfying . Moreover, if we define

then

For a proof, see section 5.5 of Aubin and Ekeland.

References

  1. Ambrosetti, Antonio; Rabinowitz, Paul H. (1973). "Dual variational methods in critical point theory and applications". Journal of Functional Analysis. 14 (4): 349–381. doi:10.1016/0022-1236(73)90051-7.

Further reading

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