Mountain pass theorem

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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 extremums, but few regarding saddle points.

1 Theorem statement
2 Visualization
3 Weaker formulation
4 References

[edit] 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.

[edit] 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.

[edit] 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

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

$Φ$ 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 $Φ$ 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.

[edit] References

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.

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Categories: Mathematical analysis | Calculus of variations