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

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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

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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

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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

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  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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