Finite dimensional "Mountain Pass Lemma" - MathOverflow

Finite dimensional "Mountain Pass Lemma"

2012-08-30T15:06:31Z

Question Does anyone know of a good reference which I can cite for the finite dimensional version of Mountain Pass Lemma?

Motivation I am writing a paper and found myself using the following result:

Let $f$ be a proper smooth real-valued function on $\mathbf{R}^3$ such that $f(0) = 0$, $f|_{B_1(0)} \geq 0$, $f|_{\partial B_1(0)} \geq 1$ and $\exists p \in {\partial B_2(0)}$ such that $f(p) = 0$. Then $\exists q\in \mathbf{R}^3 \setminus B_1(0)$ such that $f'(q) = 0$ and $f(q) \geq 1$.

For the time being I referred to Ambrosetti and Rabinowitz's JFA article for the mountain pass lemma, but citing a Banach space version for a finite-dimensional Euclidean space application gives me a funny feeling. (Also, if feels like such a result could in principle be found in not-so-advanced undergraduate textbooks...)

Answer by Liviu Nicolaescu for Finite dimensional "Mountain Pass Lemma"

2012-08-30T16:02:47Z

My book An Invitation to Morse Theory, 2nd Edition, Springer Verlag 2011 describes the finite dimensional Mountain Pass Lemma in Example 2.53. There I work on a compact manifold, but the compactness of the manifold can be substituted by a properness assumption on the function. In the same section I explain a more general Min-Max principle (Thm. 2.51) and in Example 2.53 I explain how this implies the Mountain Pass Lemma.

Answer by Igor Rivin for Finite dimensional "Mountain Pass Lemma"

2012-08-30T20:13:46Z

This seems to be in L Evans's PDE book, section 8.5

Answer by Willie Wong for Finite dimensional "Mountain Pass Lemma"

2012-09-03T07:27:20Z

For historical interest: A friend pointed me to the book

Youssef Jabri, The Mountain Pass Theorem: Variants, Generalizations and Some Applications, CUP

which asserts that one of the earliest known published version of the finite dimensional mountain pass theorem was due to

Richard Courant, Dirichlet's Principle, Conformal Mapping, and Minimal Surfaces, Interscience

published originally in 1950. The version stated and proven by Courant does not, technically speaking, imply the result I stated in the question text (the points $0$ and $p$ are assumed to be local minima of the function $f$). But a simple modification of the deformation lemma (for example, as in Liviu's book that he mentioned) would do.

Answer by Chris Gerig for Finite dimensional "Mountain Pass Lemma"

2013-04-19T02:33:37Z

I have stumbled across Richard Palais' (co-author Chuu-lian Terng) Critical Point Theory and Submanifold Geometry (Springer Lecture Notes in Math 1353). This is an awesome book!

The "Mountain Pass Lemma" for finite-dimensional manifolds is presented as Theorem 9.2.7 (pg189).