Mountain pass theorem
The mountain pass theorem is an
Statement
The assumptions of the theorem are:
- is a functional from a Hilbert space H to the reals,
- and is Lipschitz continuouson 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
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 weak* topologyrespectively.
- 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
Further reading
- Aubin, Jean-Pierre; ISBN 0-486-45324-3.
- Bisgard, James (2015). "Mountain Passes and Saddle Points". SIAM Review. 57 (2): 275–292. .
- ISBN 0-8218-0772-2.
- 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.
- ISBN 0-387-96908-X.
- McOwen, Robert C. (1996). "Mountain Passes and Saddle Points". Partial Differential Equations: Methods and Applications. Upper Saddle River, NJ: Prentice Hall. pp. 206–208. ISBN 0-13-121880-8.