Mitsuhiro Shishikura

2013-07-28T18:29:42Z

50.0.136.106: link preprint of annals paper

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 [[extremum|extrema]], but few regarding saddle points.

== Theorem statement ==
The assumptions of the theorem are:
* ''I'' is a [[functional (mathematics)|functional]] from a [[Hilbert space]] ''H'' to the [[real number|reals]],
* <math>I\in C^1(H,\mathbb{R})</math> and <math>I'</math> is [[Lipschitz continuous]] on bounded subsets of ''H'',
* ''I'' satisfies the [[Palais-Smale compactness condition]],
* <math>I[0]=0</math>,
* there exist positive constants ''r'' and ''a'' such that <math>I[u]\geq a</math> if <math>\Vert u\Vert =r</math>, and
* there exists <math>v\in H</math> with <math>\Vert v\Vert >r</math> such that <math>I[v]\leq 0</math>.
If we define:
:<math>\Gamma=\{\mathbf{g}\in C([0,1];H)\,\vert\,\mathbf{g}(0)=0,\mathbf{g}(1)=v\}</math>
and:
:<math>c=\inf_{\mathbf{g}\in\Gamma}\max_{0\leq t\leq 1} I[\mathbf{g}(t)],</math>
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 <math>I[0]=0</math>, and a far-off spot ''v'' where <math>I[v]\leq 0</math>. In between the two lies a range of mountains (at <math>\Vert u\Vert =r</math>) 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 <math>X</math> be [[Banach space]]. The assumptions of the theorem are:
* <math>\Phi\in C(X,\mathbf R)</math> and have a [[Gâteaux derivative]] <math>\Phi'\colon X\to X^*</math> which is continuous when <math>X</math> and <math>X^*</math> are endowed with [[strong topology]] and [[weak* topology]] respectively.
* There exists <math>r>0</math> such that one can find certain <math>\|x'\|>r</math> with
:<math>\max\,(\Phi(0),\Phi(x'))<\inf\limits_{\|x\|=r}\Phi(x)=:m(r)</math>.
* <math>\Phi</math> satisfies weak [[Palais-Smale condition]] on <math>\{x\in X\mid m(r)\le\Phi(x)\}</math>.

In this case there is a [[critical point (mathematics)|critical point]] <math>\overline x\in X</math> of <math>\Phi</math> satisfying <math>m(r)\le\Phi(\overline x)</math>. Moreover if we define
:<math>\Gamma=\{c\in C([0,1],X)\mid c\,(0)=0,\,c\,(1)=x'\}</math>
then
:<math>\Phi(\overline x)=\inf_{c\,\in\,\Gamma}\max_{0\le t\le 1}\Phi(c\,(t)).</math>

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

== References ==
* {{cite book | first=Youssef | last=Jabri | title=The Mountain Pass Theorem, Variants, Generalizations and Some Applications (Encyclopedia of Mathematics and its Applications) | publisher=Cambridge University Press | year=2003 | isbn=0-521-82721-3}}
* {{cite book | first=Lawrence C. | last=Evans | title=Partial Differential Equations | publisher=American Mathematical Society | location=Providence, Rhode Island | year=1998 | isbn=0-8218-0772-2}}
* {{cite book | first=Jean-Pierre | last=Aubin | coauthors=Ivar Ekeland | title=Applied Nonlinear Analysis | publisher=Dover Books | year=2006 | isbn=0-486-45324-3}}

[[Category:Mathematical analysis]]
[[Category:Calculus of variations]]
[[Category:Theorems in analysis]]

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Mitsuhiro Shishikura