Mountain pass theorem

The mountain pass theorem is an existence theorem from the calculus of variations, originally due to Antonio Ambrosetti and Paul Rabinowitz.{{cite journal|title=Dual variational methods in critical point theory and applications|journal=Journal of Functional Analysis|doi=10.1016/0022-1236(73)90051-7|volume=14|issue=4|pages=349–381|year=1973|last1=Ambrosetti|first1=Antonio|last2=Rabinowitz|first2=Paul H.|doi-access=}}{{cite book |first=Paul H. |last=Rabinowitz |chapter=The Mountain Pass Theorem: Theme and Variations |pages=237–271 |title=Differential Equations: Proceedings of the 1st Latin American School of Differential Equations |editor1-first=D. G. |editor1-last=de Figueiredo |editor2-first=C. S. |editor2-last=Hönig |publisher=Springer |location=New York |series=Lecture Notes in Mathematics |volume=957 |year=1982 |isbn=0-387-11951-5 }} 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 $\{x\in X\mid m(r)\le\Phi(x)\}$ .

In this case there is a critical point $\overline x\in X$ 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

: $\Phi(\overline x)=\inf_{c\,\in\,\Gamma}\max_{0\le t\le 1}\Phi(c\,(t)).$

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

Mountain pass theorem

Statement

Visualization

Weaker formulation

References

Further reading