Mountain pass theorem

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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.

Theorem statement[edit]

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[edit]

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[edit]

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

then

For a proof, see section 5.5 of Aubin and Ekeland.

References[edit]

  • 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.