Mountain pass theorem explained

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.

Statement

The assumptions of the theorem are:

I is a functional from a Hilbert space H to the reals,I\in C ^{1(H,R)} and I' is Lipschitz continuous on bounded subsets of H,I satisfies the Palais–Smale compactness condition,I[ 0 ] =0 ,

• there exist positive constants r and a such that

I[u]\geq a if \Vert u\Vert = r , and

• there exists

v\in H with \Vert v\Vert >r such that I[v]\leq 0 .If we define:\Gamma = \{ g \in C([ 0 , 1 ];H)\vert g ( 0 ) =0 , g ( 1 ) = v\} and:c =inf _{g\in\Gamma} max _{0\leq} I[ g (t)], 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

I[ 0 ] =0 , and a far-off spot v where I[v]\leq 0 . In between the two lies a range of mountains (at \Vert u\Vert = r ) 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

Let

X be Banach space. The assumptions of the theorem are:\Phi\in C(X, R) and have a Gateaux derivative \Phi'\colon X\to X ^{*} which is continuous when X and X ^{*} are endowed with strong topology and weak* topology respectively.

• There exists

r> 0 such that one can find certain \|x'\|>r withmax (\Phi( 0 ),\Phi(x'))< inf \limits _{\|x\|=r} \Phi(x) = :m(r) .\Phi satisfies weak Palais–Smale condition on \{x\in X\mid m(r)\le\Phi(x)\} .\overline x\in X of \Phi satisfying m(r)\le\Phi(\overline x) . Moreover, if we define\Gamma = \{c\in C([ 0 , 1 ],X)\mid c( 0 ) =0 ,c( 1 ) = x'\} then\Phi(\overline x) =inf _{c\in\Gamma} max _{0\le} \Phi(c(t)).

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

Further reading

• Book: Jean-Pierre . Aubin . Ivar . Ekeland . Ivar Ekeland . Applied Nonlinear Analysis . Dover Books . 2006 . 0-486-45324-3.
• Bisgard. James. Mountain Passes and Saddle Points. SIAM Review. 2015. 57. 2. 275–292. 10.1137/140963510.
• Book: Evans, Lawrence C. . Lawrence C. Evans . Partial Differential Equations . American Mathematical Society . Providence, Rhode Island . 1998 . 0-8218-0772-2.
• Book: Jabri, Youssef . The Mountain Pass Theorem, Variants, Generalizations and Some Applications . Encyclopedia of Mathematics and its Applications . Cambridge University Press . 2003 . 0-521-82721-3 . registration .
• Book: Jean . Mawhin . Jean Mawhin . Michel . Willem . Critical Point Theory and Hamiltonian Systems . New York . Springer-Verlag . 1989 . 0-387-96908-X . The Mountain Pass Theorem and Periodic Solutions of Superlinear Convex Autonomous Hamiltonian Systems . 92–97 . https://books.google.com/books?id=w6bTBwAAQBAJ&pg=PA92 .
• Book: McOwen, Robert C. . Partial Differential Equations: Methods and Applications . Upper Saddle River, NJ . Prentice Hall . 1996 . 0-13-121880-8 . 206–208 . Mountain Passes and Saddle Points . https://books.google.com/books?id=TuNHsNC1Yf0C&pg=PA206 .

Notes and References

1. Dual variational methods in critical point theory and applications. Journal of Functional Analysis. 10.1016/0022-1236(73)90051-7. 14. 4. 349–381. 1973. Ambrosetti. Antonio. Rabinowitz. Paul H..