Mountain pass theorem

Template:Short description 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,
• $I\in C^1(H,ℝ)$ and $I^{\prime }$ 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 $I[u]\ge a$ if $\Vert u\Vert =r$, and
• there exists $v\in H$ with $\Vert v\Vert >r$ such that $I[v]\le 0$.

If we define:

$\mathrm{\Gamma }=\{𝐠\in C([0,1];H)|𝐠(0)=0,𝐠(1)=v\}$

and:

$c=\underset{𝐠\in \mathrm{\Gamma }}{inf}\underset{0\le t\le 1}{max}I[𝐠(t)],$

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 $I[v]\le 0$. 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:

• $\mathrm{\Phi }\in C(X,𝐑)$ and have a Gateaux derivative ${\mathrm{\Phi }}^{\prime }:X\to X^{\ast }$ which is continuous when $X$ and $X^{\ast }$ are endowed with strong topology and weak* topology respectively.
• There exists $r>0$ such that one can find certain $\Vert x^{\prime }\Vert >r$ with

$max(\mathrm{\Phi }(0),\mathrm{\Phi }(x^{\prime }))<\underset{\Vert x\Vert =r}{inf}\mathrm{\Phi }(x)=:m(r)$.

• $\mathrm{\Phi }$ satisfies weak Palais–Smale condition on $\{x\in X\mid m(r)\le \mathrm{\Phi }(x)\}$.

In this case there is a critical point $\bar{x}\in X$ of $\mathrm{\Phi }$ satisfying $m(r)\le \mathrm{\Phi }(\bar{x})$. Moreover, if we define

$\mathrm{\Gamma }=\{c\in C([0,1],X)\mid c(0)=0,c(1)=x^{\prime }\}$

then

$\mathrm{\Phi }(\bar{x})=\underset{c\in \mathrm{\Gamma }}{inf}\underset{0\le t\le 1}{max}\mathrm{\Phi }(c(t)).$

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

Further reading

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