Mountain Pass Theorem

The mountain pass theorem is an existence theorem from the calculus of variations, originally due to Antonio Ambrosetti and Paul Rabinowitz. 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,
* $I\in C^1(H,\mathbb)$ and $I'$ is Lipschitz continuous on bounded subsets of ''H'',
* $I$ satisfies the Palais–Smale compactness condition,
* I 0,
* there exist positive constants ''r'' and ''a'' such that I geq a if $\Vert u\Vert =r$, and
* there exists $v\in H$ with $\Vert v\Vert >r$ such that I leq 0.
If we define:
:$\Gamma=\$
and:
:c=\inf_\max_ I mathbf(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, and a far-off spot ''v'' where I 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
:$\max\,(\Phi(0),\Phi(x'))<\inf\limits_\Phi(x)=:m(r)$.
* $\Phi$ satisfies weak Palais–Smale condition on $\$.

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
:$\Gamma=\$
then
:$\Phi(\overline x)=\inf_\max_\Phi(c\,(t)).$

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

References

Further reading

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* {{cite book , first=Robert C. , last=McOwen , title=Partial Differential Equations: Methods and Applications , location=Upper Saddle River, NJ , publisher=Prentice Hall , year=1996 , isbn=0-13-121880-8 , pages=206–208 , chapter=Mountain Passes and Saddle Points , chapter-url=https://www.google.com/books/edition/_/TuNHsNC1Yf0C?hl=en&gbpv=1&pg=PA206

Mathematical analysis
Calculus of variations
Theorems in analysis