Critical points of lipschitz functions on smooth manifolds
Copyright policy )Critical points of lipschitz functions on smooth manifolds
The author extends the main results of Lyusternik-Shnirel'man theory to the case of Lipschitz mappings between \(C^ 2\)-manifolds. Let U be open in a Banach space E and \(f: U\to {\mathbb{R}}^ a \)Lipschitz mapping. For \(x\in U\), \(v\in E\) let \(f^ 0(x,v):= \limsup_{u\to x,t\to 0+}t^{- 1}[f(u+tv)-f(u)]\). Then the Clarke differential of f at x is defined to be the set \(\partial f(x)\) of all \(\ell \in E'\) such that \(\leq f^ 0(x,v)\) for \(v\in E\). A typical result is: Let M be a paracompact complete \(C^ 2\)-Banach manifold and \(f: M\to {\mathbb{R}}^ a \)Lipschitz function which satisfies the Palais-Smale condition. Let \(m\leq n\), \(- \infty
category of the manifold, Lyusternik-Shnirel'man category of a space, topological complexity à la Farber, topological robotics (topological aspects), Palais-Smale condition, Abstract critical point theory (Morse theory, Lyusternik-Shnirel'man theory, etc.) in infinite-dimensional spaces, generalized Clark derivative, level sets, Critical points and critical submanifolds in differential topology, Cobordism and concordance in topological manifolds, \(\mathrm{O}\)- and \(\mathrm{SO}\)-cobordism, Lyusternik-Shnirel'man theory, Topology of the Euclidean \(n\)-space, \(n\)-manifolds (\(4 \leq n \leq \infty\)), critical points of Lipschitz functions on smooth manifolds, Lipschitz mappings
category of the manifold, Lyusternik-Shnirel'man category of a space, topological complexity à la Farber, topological robotics (topological aspects), Palais-Smale condition, Abstract critical point theory (Morse theory, Lyusternik-Shnirel'man theory, etc.) in infinite-dimensional spaces, generalized Clark derivative, level sets, Critical points and critical submanifolds in differential topology, Cobordism and concordance in topological manifolds, \(\mathrm{O}\)- and \(\mathrm{SO}\)-cobordism, Lyusternik-Shnirel'man theory, Topology of the Euclidean \(n\)-space, \(n\)-manifolds (\(4 \leq n \leq \infty\)), critical points of Lipschitz functions on smooth manifolds, Lipschitz mappings
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