Mountain pass theorem
The mountain pass theorem is an existence theorem from the calculus of variations. 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.
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:
then the conclusion of the theorem is that c is a critical value of I.
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.
Let be Banach space. The assumptions of the theorem are:
- and have a Gâteaux 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
For a proof, see section 5.5 of Aubin and Ekeland.
- Jabri, Youssef (2003). The Mountain Pass Theorem, Variants, Generalizations and Some Applications (Encyclopedia of Mathematics and its Applications). Cambridge University Press. ISBN 0-521-82721-3.
- Evans, Lawrence C. (1998). Partial Differential Equations. Providence, Rhode Island: American Mathematical Society. ISBN 0-8218-0772-2.
- Aubin, Jean-Pierre; Ivar Ekeland (2006). Applied Nonlinear Analysis. Dover Books. ISBN 0-486-45324-3.
- Bisgard, James (2015). "Mountain Passes and Saddle Points". SIAM Review. 57 (2): 275–292. doi:10.1137/140963510.