Approximation and regularization of Lipschitz functions: Convergence of the gradients

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 $\lim _{n\to +\infty } \|f_n-f\|_{\infty }+ \|\nabla f_n-\nabla f\|_{\infty }=0$. 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 \begin{eqnarray*} &&\lim _{n\to +\infty } \|f_n-f\|_{\infty }=0,\\ &&\lim _{n\to +\infty } d_{Haus}(\operatorname {graph}(\nabla f_n), \operatorname {graph}(\partial f))=0. \end{eqnarray*} 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.