Finite dimensional "Mountain Pass Lemma" - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T05:46:40Z http://mathoverflow.net/feeds/question/105951 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/105951/finite-dimensional-mountain-pass-lemma Finite dimensional "Mountain Pass Lemma" Willie Wong 2012-08-30T15:06:31Z 2013-04-19T02:33:37Z <p><strong>Question</strong> Does anyone know of a good reference which I can cite for the <em>finite</em> dimensional version of Mountain Pass Lemma?</p> <p><strong>Motivation</strong> I am writing a paper and found myself using the following result:</p> <blockquote> <p>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 $f(q) \geq 1$. </p> </blockquote> <p>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...)</p> http://mathoverflow.net/questions/105951/finite-dimensional-mountain-pass-lemma/105963#105963 Answer by Liviu Nicolaescu for Finite dimensional "Mountain Pass Lemma" Liviu Nicolaescu 2012-08-30T16:02:47Z 2012-08-30T16:02:47Z <p>My book <em>An Invitation to Morse Theory</em>, 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.</p> http://mathoverflow.net/questions/105951/finite-dimensional-mountain-pass-lemma/105988#105988 Answer by Igor Rivin for Finite dimensional "Mountain Pass Lemma" Igor Rivin 2012-08-30T20:13:46Z 2012-08-30T20:13:46Z <p>This seems to be in L Evans's PDE book, section 8.5</p> http://mathoverflow.net/questions/105951/finite-dimensional-mountain-pass-lemma/106224#106224 Answer by Willie Wong for Finite dimensional "Mountain Pass Lemma" Willie Wong 2012-09-03T07:27:20Z 2012-09-03T07:27:20Z <p>For historical interest: A friend pointed me to the book </p> <ul> <li>Youssef Jabri, <a href="http://books.google.ch/books/about/The_Mountain_Pass_Theorem.html?id=uGWaffeFbroC&amp;redir_esc=y" rel="nofollow"><em>The Mountain Pass Theorem: Variants, Generalizations and Some Applications</em></a>, CUP</li> </ul> <p>which asserts that one of the earliest known published version of the finite dimensional mountain pass theorem was due to </p> <ul> <li>Richard Courant, <em>Dirichlet's Principle, Conformal Mapping, and Minimal Surfaces</em>, Interscience</li> </ul> <p>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. </p> http://mathoverflow.net/questions/105951/finite-dimensional-mountain-pass-lemma/128037#128037 Answer by Chris Gerig for Finite dimensional "Mountain Pass Lemma" Chris Gerig 2013-04-19T02:33:37Z 2013-04-19T02:33:37Z <p>I have stumbled across Richard Palais' (co-author Chuu-lian Terng) <strong>Critical Point Theory and Submanifold Geometry</strong> (Springer Lecture Notes in Math 1353). This is an awesome book!</p> <p>The "Mountain Pass Lemma" for finite-dimensional manifolds is presented as <strong>Theorem 9.2.7 (pg189)</strong>.</p>