Mountain pass theorem

Summary

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.

StatementEdit

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.

VisualizationEdit

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 formulationEdit

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.

ReferencesEdit

  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 readingEdit

  • Aubin, Jean-Pierre; Ekeland, Ivar (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.
  • Evans, Lawrence C. (1998). Partial Differential Equations. Providence, Rhode Island: American Mathematical Society. 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.
  • Mawhin, Jean; Willem, Michel (1989). "The Mountain Pass Theorem and Periodic Solutions of Superlinear Convex Autonomous Hamiltonian Systems". Critical Point Theory and Hamiltonian Systems. New York: Springer-Verlag. pp. 92–97. 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.