Approximation and regularization of Lipschitz functions: Convergence of the gradients
Authors: Czarnecki Marc-Olivier , Rifford Ludovic 
Published: June 2006

We examine the possible extensions to the Lipschitzian setting of the classical result on

C
1

C^1

-convergence: first (approximation), if a sequence

(

f
n

)

(f_n)

of functions of class

C
1

C^1

from

R

N

\mathbb {R}^N

to

R

\mathbb {R}

converges uniformly to a function

f
f

of class

C
1

C^1

, then the gradient of

f
f

is a limit of gradients of

f
n

f_n

in the sense that

graph
⁡
(
∇
f
)
⊂

lim inf

n
→
+
∞

graph
⁡
(
∇

f
n

)

\operatorname {graph}(\nabla f)\subset \liminf _{n\to +\infty } \operatorname {graph}(\nabla f_n)

; second (regularization), the functions

(

f
n

)

(f_n)

can be chosen to be of class

C

∞

C^{\infty }

and

C
1

C^1

-converging to

f
f

in the sense that
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="limit Underscript n right-arrow plus normal infinity Endscripts double-vertical-bar f Subscript n Baseline minus f double-vertical-bar Subscript normal infinity Baseline plus double-vertical-bar nabla f Subscript n Baseline minus nabla f double-vertical-bar Subscript normal infinity Baseline equals 0">

lim

n
→
+
∞

‖

f
n

−
f

‖

∞

+
‖
∇

f
n

−
∇
f

‖

∞

=
0

\lim _{n\to +\infty } \|f_n-f\|_{\infty }+ \|\nabla f_n-\nabla f\|_{\infty }=0

. In other words, the space of

C

∞

C^{\infty }

functions is dense in the space of

C
1

C^1

functions endowed with the

C
1

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
:

R

N

→

R

f:\mathbb {R}^N \rightarrow {\mathbb R}

is a locally Lipschitz function, we build a sequence of smooth functions

(

f
n

)

n
∈

N

(f_n)_{n \in \mathbb {N}}

such that
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row 1st Column Blank 2nd Column a m p semicolon 3rd Column a m p semicolon limit Underscript n right-arrow plus normal infinity Endscripts double-vertical-bar f Subscript n Baseline minus f double-vertical-bar Subscript normal infinity Baseline equals 0 comma 2nd Row 1st Column Blank 2nd Column a m p semicolon 3rd Column a m p semicolon limit Underscript n right-arrow plus normal infinity Endscripts d Subscript upper H a u s Baseline left-parenthesis g r a p h left-parenthesis nabla f Subscript n Baseline right-parenthesis comma g r a p h left-parenthesis partial-differential f right-parenthesis right-parenthesis equals 0 period EndLayout">

a
m
p
;

a
m
p
;

lim

n
→
+
∞

‖

f
n

−
f

‖

∞

=
0
,

a
m
p
;

a
m
p
;

lim

n
→
+
∞

d

H
a
u
s

(
graph
⁡
(
∇

f
n

)
,
graph
⁡
(
∂
f
)
)
=
0.

\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

∞

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

lim sup

n
→
+
∞

graph
⁡
(
∇

f
n

)
=
graph
⁡
(
∂
f
)

\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.