Mountain pass theorem


The mountain pass theorem is an existence theorem from the calculus of variations, originally due to Antonio Ambrosetti and Paul Rabinowitz. 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:
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 where the elevation is high. 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. 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:
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.