Mountain pass theorem

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:

• ${\displaystyle I}$ is a functional from a Hilbert space H to the reals,
• ${\displaystyle I\in C^{1}(H,\mathbb {R} )}$ and ${\displaystyle I'}$ is Lipschitz continuous on bounded subsets of H,
• ${\displaystyle I}$ satisfies the Palais–Smale compactness condition,
• ${\displaystyle I[0]=0}$,
• there exist positive constants r and a such that ${\displaystyle I[u]\geq a}$ if ${\displaystyle \Vert u\Vert =r}$, and
• there exists ${\displaystyle v\in H}$ with ${\displaystyle \Vert v\Vert >r}$ such that ${\displaystyle I[v]\leq 0}$.

If we define:

${\displaystyle \Gamma =\{\mathbf {g} \in C([0,1];H)\,\vert \,\mathbf {g} (0)=0,\mathbf {g} (1)=v\}}$

and:

${\displaystyle c=\inf _{\mathbf {g} \in \Gamma }\max _{0\leq t\leq 1}I[\mathbf {g} (t)],}$

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 ${\displaystyle I[0]=0}$, and a far-off spot v where ${\displaystyle I[v]\leq 0}$. In between the two lies a range of mountains (at ${\displaystyle \Vert u\Vert =r}$) 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 ${\displaystyle X}$ be Banach space. The assumptions of the theorem are:

• ${\displaystyle \Phi \in C(X,\mathbf {R} )}$ and have a Gateaux derivative ${\displaystyle \Phi '\colon X\to X^{*}}$ which is continuous when ${\displaystyle X}$ and ${\displaystyle X^{*}}$ are endowed with strong topology and weak* topology respectively.
• There exists ${\displaystyle r>0}$ such that one can find certain ${\displaystyle \|x'\|>r}$ with

${\displaystyle \max \,(\Phi (0),\Phi (x'))<\inf \limits _{\|x\|=r}\Phi (x)=:m(r)}$.

• ${\displaystyle \Phi }$ satisfies weak Palais–Smale condition on ${\displaystyle \{x\in X\mid m(r)\leq \Phi (x)\}}$.

In this case there is a critical point ${\displaystyle {\overline {x}}\in X}$ of ${\displaystyle \Phi }$ satisfying ${\displaystyle m(r)\leq \Phi ({\overline {x}})}$. Moreover, if we define

${\displaystyle \Gamma =\{c\in C([0,1],X)\mid c\,(0)=0,\,c\,(1)=x'\}}$

then

${\displaystyle \Phi ({\overline {x}})=\inf _{c\,\in \,\Gamma }\max _{0\leq t\leq 1}\Phi (c\,(t)).}$

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

References

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 reading

• 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.