Smooth approximation of Lipschitz functions on Riemannian manifolds

By D. Azagra, J. Ferrera, Fernando López-Mesas, Y. Rangel. 2006

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Abstract We show that for every Lipschitz function f defined on a separable Riemannian manifold M (possibly of infinite dimension), for every continuous e : M → ( 0 , + ∞ ) , and for every positive number r > 0 , there exists a C ∞ smooth Lipschitz function g : M → R such that | f ( p ) − g ( p ) | ⩽ e ( p ) for every p ∈ M and Lip ( g ) ⩽ Lip ( f ) + r . Consequently, every separable Riemannian manifold is uniformly bumpable. We also present some applications of this result, such as a general version for separable Riemannian manifolds of Deville–Godefroy–Zizler's smooth variational principle.

Subfields: Applied Mathematics

View Paper: https://www.semanticscholar.org/paper/7064185516f6f0248ffa7bb53379f15590fc9fb3