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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)


Approximation and regularization of Lipschitz functions: Convergence of the gradients

Authors: Marc-Olivier Czarnecki and Ludovic Rifford
Journal: Trans. Amer. Math. Soc. 358 (2006), 4467-4520
MSC (2000): Primary 49J45, 49J52, 57R12
Published electronically: May 9, 2006
MathSciNet review: 2231385
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Abstract: We examine the possible extensions to the Lipschitzian setting of the classical result on $ C^1$-convergence: first (approximation), if a sequence $ (f_n)$ of functions of class $ C^1$ from $ \mathbb{R}^N$ to $ \mathbb{R}$ converges uniformly to a function $ f$ of class $ C^1$, then the gradient of $ f$ is a limit of gradients of $ f_n$ in the sense that $ \operatorname{graph}(\nabla f)\subset \liminf_{n\to +\infty} \operatorname{graph}(\nabla f_n)$; second (regularization), the functions $ (f_n)$ can be chosen to be of class $ C^{\infty}$ and $ C^1$-converging to $ f$ in the sense that $ \lim_{n\to +\infty} \Vert f_n-f\Vert _{\infty}+ \Vert\nabla f_n-\nabla f\Vert _{\infty}=0$. In other words, the space of $ C^{\infty}$ functions is dense in the space of $ C^1$ functions endowed with the $ C^1$ pseudo-norm.

We first deepen the properties of Warga's counterexample (1981) for the extension of the approximation part to the Lipschitzian setting. This part cannot be extended, even if one restricts the approximation schemes to the classical convolution and the Lasry-Lions regularization. We thus make more precise various results in the literature on the convergence of subdifferentials.

We then show that the regularization part can be extended to the Lipschitzian setting, namely if $ f:\mathbb{R}^N \rightarrow {\mathbb{R}}$ is a locally Lipschitz function, we build a sequence of smooth functions $ (f_n)_{n \in \mathbb{N}}$ such that

    $\displaystyle \lim_{n\to +\infty} \Vert f_n-f\Vert _{\infty}=0,$  
    $\displaystyle \lim_{n\to +\infty} d_{Haus}(\operatorname{graph}(\nabla f_n), \operatorname{graph}(\partial f))=0.$  

In other words, the space of $ C^{\infty}$ functions is dense in the space of locally Lipschitz functions endowed with an appropriate Lipschitz pseudo-distance. Up to now, Rockafellar and Wets (1998) have shown that the convolution procedure permits us to have the equality $ \limsup_{n\to +\infty} \operatorname{graph}(\nabla f_n) =\operatorname{graph}(\partial f)$, which cannot provide the exactness of our result.

As a consequence, we obtain a similar result on the regularization of epi-Lipschitz sets. With both functional and set parts, we improve previous results in the literature on the regularization of functions and sets.

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Additional Information

Marc-Olivier Czarnecki
Affiliation: Institut de Mathematiques et Modelisation de Montpellier UMR 5030 CNRS, Université Montpellier 2, place Eugène Bataillon, 34095 Montpellier Cedex 5, France

Ludovic Rifford
Affiliation: Institut Girard Desargues, Université Claude Bernard Lyon I, 43 Bd du 11 Novembre 1918, 69622 Villeurbanne Cedex, France
Address at time of publication: Departement de Mathématiques (Bâtiment 425), Université de Paris-Sud, 91405 Orsay, France

PII: S 0002-9947(06)04103-1
Keywords: Lipschitz functions, epi-Lipschitz sets, approximation, regularization, gradient convergence, normal cone
Received by editor(s): January 28, 2003
Received by editor(s) in revised form: August 28, 2004
Published electronically: May 9, 2006
Article copyright: © Copyright 2006 American Mathematical Society