1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

9. WEAK SOLUTION FORMULATION 367

(the first derivative) such that

If (y) - f ( x) - D f ( x). (y - x) - ~ (y - x) T. D^2 f ( x). (y - x) I

= o (IY -x1^2 ) as y---+ x.

We also recall the following related notion (see p. 167 of [139] for ex-

ample).

DEFINITION 7.116 (Locally bounded variation). Let U be an open set

in Rn. A function f E Lfoc (U) has locally bounded variation if for every

open set U1 cc U,

sup {fu

1

f div¢ dx : ¢ E C~ (U1; Rn), 1¢1 :::; 1} < oo,

where the lower index c in C~ indicates having compact support. In this

case we write f E BVioc (U).

Clearly if f E C^1 (U), then f E BVioc (U) since

{ f div¢dx = - { 'Vf · ¢dx:::; meas(U1) · sup l'Vfl < oo

lu 1 lu1 suppefi

for all U1 CC U. On the other hand, as a partial converse, note that as

remarked on p. 166 at the beginning of Chapter 5 in [139], " ... a BV function

is 'measure theoretically C^1. '"
The following lemma is well known; see Theorem l(i) in §6.3 on p. 236
of [139] for the proof of (i), Aleksandrov's Theorem in §6.4 on p. 242 of
[139] for the proof of (ii), and Theorem 3 in §6.3 on pp. 240-241 of [139]
for the proof of (iii). Let B (r) c Rn denote the ball of radius r centered at
0.

LEMMA 7.117 (Regularity properties of convex functions). Let f: B (r)
---+ R be a convex function. Then

(i) f is locally Lipschitz,

(ii) (Aleksandrov's Theorem) f has second derivative D^2 f (x) for a.e.

x EB (r),

(iii) gfi has locally bounded variation for each i, and

D^2 f E Lfoc (B (r); Mnxn).

REMARK 7.118. Note that it follows easily from the convexity off that
D^2 f (x) ::'.'.: 0 when it exists.

We consider the approximation of continuous functions, in particular
convex functions, by smooth functions via the convolution with mollifiers.
A standard mollifier is T/ : Rn ---+ R defined by

T/ (x) ~ { ae^1 l(lxl

2

-^1 ) if lxl < 1,

0 if lxl ::'.'.: 1,