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I cam across the Mountain Pass Theorem, mentioned for example at http://en.wikipedia.org/wiki/Mountain_pass_theorem. In (very) loose terms, it somewhat reminds me of Rolle's theorem. Trying to understand it better in the infinte-dimensional setting, I came across a multiple variables Mountain Pass Theorem in finite dimensional spaces, due to Courant. In this case the critical value $c$ is attained for a certain curve $g$ (using Wikipedia's notation).

What surprises me is that in the Mountain Pass Theorem proper the critical value $c$ is defined as an $\inf \max$. While I am trying to work out the details, I am sure it would have been defined as a $\min \max$ if it were possible.

I cannot get around this point. The critical value $c$ is never attained by any curve $g$, and yet is a stationary point of the functional. the theorem proves there is a saddle, and yet in general no curve crosses it at the lowest height possible?? This totally defies my understanding. Has anybody got a word of wisdom? Thanks

Gerry Myerson
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An aedonist
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1 Answers1

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The infinte-dimensional Mountain Pass Theorem states like this (and the finite-dimensional is very similar):

The Mountain Pass Theorem. Let X be a Banach space, $ I:X \rightarrow \mathbb{R}$ a $C^1$-functional satisfying the Palais-Smale condition and $I[0]=0$. Suppose \begin{align*} &(A) \quad \; \; \; \exists \; \rho,\alpha > 0 \; \text{such that} \; I|_{\partial B_{\rho}} \geq \alpha,\\& (B) \quad \; \;\exists \; e \in X\backslash B_{\rho} \; \text{with} \; I[e] \leq 0, \end{align*} where $B_{\rho}$ denotes the ball of radius $\rho$ around 0. \ then $I$ has a critical value $c\geq \alpha$ and $c$ is characterized by \begin{equation}c=\inf_{\gamma \in \Gamma} \max_{u\in \gamma([0,1])} I[u],\end{equation} where \begin{equation}\Gamma=\{ \gamma \in C([0,1],X)\; | \; \gamma(0)=0, \gamma(1)=e \} \end{equation}.

As you see the maximum is taken over a compact set (and thus it is attained), while the set $\Gamma$ is not guranteed to have a minimum, thus the use of infimum instead.

Kamir Raman
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