Mountain pass theorem

Last updated December 24, 2024

The mountain pass theorem is an existence theorem from the calculus of variations, originally due to Antonio Ambrosetti and Paul Rabinowitz.^[1]^[2] 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,
$I\in C^{1}(H,\mathbb {R} )$ 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 =\{\mathbf {g} \in C([0,1];H)\,\vert \,\mathbf {g} (0)=0,\mathbf {g} (1)=v\}

and:

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 $I[0]=0$ , 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:

$\Phi \in C(X,\mathbf {R} )$ 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 $\|x'\|>r$ with

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

.

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

In this case there is a critical point ${\overline {x}}\in X$ of $\Phi$ satisfying $m(r)\leq \Phi ({\overline {x}})$ . 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\leq t\leq 1}\Phi (c\,(t)).

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

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References

↑ 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.
↑ Rabinowitz, Paul H. (1982). "The Mountain Pass Theorem: Theme and Variations". In de Figueiredo, D. G.; Hönig, C. S. (eds.). Differential Equations: Proceedings of the 1st Latin American School of Differential Equations. Lecture Notes in Mathematics. Vol. 957. New York: Springer. pp. 237–271. ISBN 0-387-11951-5.