2004), “Approximation and Regularization of Lipschitz Functions

Abstract. We examine the possible extensions to the Lipschitzian setting of the classical result on C1-convergence: first (approximation), if a sequence (fn) of functions of class C1 from RN to R converges uniformly to a function f of class C1, then the gradient of f is a limit of gradients of fn in the sense that graph(∇f) ⊂ lim infn→+ ∞ graph(∇fn); second (regularization), the functions (fn) can be chosen to be of class C ∞ and C1-converging to f in the sense that limn→+ ∞ ‖fn − f‖ ∞ + ‖∇fn −∇f‖ ∞ = 0. In other words, the space of C∞ functions is dense in the space of C1 functions endowed with the C1 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 Lips-chitzian setting, namely if f: RN → R is a locally Lipschitz function, we build a sequence of smooth functions (fn)n∈N such that lim n→+∞ ‖fn − f‖ ∞ = 0, lim n→+∞ dHaus(graph(∇fn), graph(∂f)) = 0. In other words, the space of C ∞ functions is dense in the space of locally Lips-chitz 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 lim supn→+ ∞ graph(∇fn) = graph(∂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. 1