Mountain pass theorem

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

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 Gâteaux 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.

References

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.

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