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0. Define U1 C C U as in the previous proof and choose a smooth cutoff function ( satisfying
PROOF
{
( = l on U1 , spt ( () c U, O < ( < I.
Then
=
Since /k
.--..
-
fu div ((
( 1 - ()
· Dfk dx.
f in L 1 ( U), the first term in (*) converges to
-fu div (
0, we calculate c 0,
c,
1d(f')'rp dx = - 1d f 'rp' dx =
-
1 \rl< * f)rp' dx
= - 1b f(fl, * r.p) ' dx Thus fe I ( ! ) 'I dx < l i D /II (a, b) . d 3. Claim: f E U"'(a, b).
< I I D! I I (a, b).
'
Proof of Claim: Choose
{ !1 }�1
c
BV(a, b) n C= (a , b) so that
fJ --> f in L1 (a, b),
fJ --> f C a.e.
and For each y, z E (a, b) ,
1b 1!} 1 dx
__,
I I D JII (a, b).
fJ (z) = /j(Y) +
lz f} dx .
Averaging with respect to y E (a, b) , we obtain
1 /j (z)l < ib l fJ I dy + 1b If} I dx, and so
sup llfJ I I L�(a, ) < oo.
b n a.e., Since fJ --> f .e I I JII L�(a,b) < oo . J
5.10 Essential variation on lines
219
4. It follows from the claim that each point of approximate continuity of f is
a Lebesgue point and hence
f ' (t) -> f(t) 0 for each point of approximate continuity of f. Consequently, for each partition {a < t 1 < < t + 1 < !J}, with each I J a point of approximate continuity of f , as f ___.
·
·
·
,
"' \ f(tJ+ I ) - f( IJ ) l = lim � t -O j =l rn
"' \ f ' (tJ+I ) - f ' ( IJ ) l � j=l ut
< lim sup ( -o
1/J
\ (/' ) '\ dx
u
< \ \ D f \ \ (a, b) .
Thus ess V,�J < \ \ D f \ \ ( a, b) < 5.10.2
oo.
I
Essential variation on a.e. line
We next extend our analysis to BV functions on lR11 • Suppose f : R11 -> Ilt Then for k = I , . . . , n, set x' Xk +t , . . . X 11 ) E Rn - l and t E R, write NOTATION
=
(x 1 , • • . Xk - l .
,
Thus ess V,i' h means the essential variation of h as a function of t E (a, b) , for each fixed x' . LEMMA 1
Assume f E L/oc( R11 ) , k E { I , . . . , n } , x
'
,_.
- oo
oo.
Then the mapping
ess v: !k
is L,n- l _measurable. PROOF
According to Theorem I , for L, n- l a.e. x' E Rn- l ,
ess v: fk
=
\ \ D fk \ \ (a, b)
= sup
{ lb
}
fk(x' , t)cp'(t) dt \ cp E C� ( a, b), J cp \ < I .
BV Functions and Sets of Finite Perimeter
220
Let
be a countable, dense subset of C,� (a, b) n {I 'PI <
{ 'P:i }j 1
x' ,_.
1 }.
!, fk(:r', t) rpj (t) dt (L
is .C"- 1 -measurable for j
is .C" - 1 -measurable.
=
I , . . . and so
I
THEOREM 2
Assume f E L/oc (JRn ) . Then f E B Vioc (R" ) if and only if
L ess V�fk dx'
for each k
=
=
fl•
*
oo
I . . . , n, a < b, and compact set I< C R" - 1 • ,
PROOF I. First suppose f E
Let /'
<
B V.oc ( Rn ) . Choose k, a, b,
J{ as above. Se
f, as before. Then
1 If' - !I dx lim sup l J D /' 1 dx
lim £-0
£-0
=
0,
c
c
<
oc .
Thus for 'Hn-l a.e. x' E I<,
where
Hence ess v; !k < lim inf ess v; fie .�o
for 7-i"- 1 a.e. x' E I<
Then
S.ll
221
A criterion for finite perimeter
Thus Fatou's Lemma implies
r ess v,:· h dx'
JK
r ess v,:· Ji, dx ' , �o
JK lim inf 1 - dx ,�o c lim sup l i Df'l dx ,�o c
< lim inf =
<
Df' Dx k
<
oo .
2. Now suppose f E Lloc (R" ) and
l ess v�·/k dx'
for all k I , . . . , n, a < b, and compact I 'P I < I, and choose a, b, and k such that
< K
==
00
C R"- 1 • Fix
E
C,� ( R" ) ,
spt (r.p) C {x I a < Xk < b} . Then Theorem I implies
1
JRn
I n
dx <
1 ess va !k dx b
K
< ,
oo,
for K
=
{x'
E
Rn- l I (. . . Xk - l o t, Xk + t .
As this estimate holds for k
A
5.11
=
I, .
.
.
, n,
.
•
.
) E spt (r.p) for some t E R}.
f E BVioc ( Rn )
.
I
criterion for finite perimeter
We conclude this chapter by establishing a relatively simple criterion for a set E to have locally finite perimeter. NOTATION Write x E Rn as X t X E IR. The projection P : Rn
=
=
n
P(x) DEFINITION Set and x' E Rn - l .
=
(x' , t) , for x'
--->
Rn - l is
=
(x 1 1 . . . , Xn -d
E
Rn- l ,
x'
N(P I A, x' )
=
1-f ( A n p- 1 {x'}) for Borel sets A c Rn
BV Functions and Sets of Finite Perimete r
222
LEMMA I (i) The mapping
N(P \ A, x') is £ " - 1 -measurahle. (ii) JJR - ' N(P \ A, x') dx' < 7-l" - 1 ( A).
x'
,_.
..
PROOF Assertions (i) and (ii) follow as in the proof of Lemma 2, Section 3.4. 1 : see also the remark in Section 3.4. 1 . DEFINITIONS
Let
E
C
R" he C'-measurahle.
{
We defllle
( B (x, r ) I = x E R" \ lim £" r-0 rrt
E) == o
to be the measure theoretic interior of E and
{
( (x , r) n E) 0 = x E R" I lim £" B r-0 rn
==
o
} }
to be the measure theoretic exterior of E. REMARK Note a* E = R" - ( I U 0). Think of I as denoting the "inside" and 0 as denoting the "outside" of E. I LEMMA 2 (i) I, 0,
(ii)
and a* E are Borel measurable sets. C"( ( I - E) U (E - I)) = 0.
PROOF I. There exists a Borel set C c R" for all £"-measurable sets T. Thus
{
E such that C" ( C n T)
( (x r) n C ) = o I = x \ lim £" B , r-0
rn
}
=
C" ( T
-
E)
,
and so is Borel measurable. The proof for 0 is similar. 2. Assertion (ii) follows from Corollary 3 in Section 1 .7 . I .
I
Let E c R" be £" -measurable. Then E has locally finite perimeter if. and only THEOREM I
CRITERION FOR FINITE PERIMETER
if,
for each compact set K c R" .
5.11 A criterion for finite perimeter
223
!'ROOF
I. Assume tirst (*) holds, fix a > 0, and set
U = ( - a , a )" C R" . To sirnplify notation slightly, let us write = x' E R - 1 , t from Lernma I and hypothesis (*) { N( P I U n a. E, z) dz < 'H" - 1 (U n a.E) < oo. z
"
= :r,.
E JR.
( ** )
}fin -\
z E R" - 1
Define for each
Assurne
E C�(U) , I'PI < I , and then cornpute
J div ( e ) dx J UX�'Pn dx 1n -l [1 f' (t) ::,, ( t) dt] dz < fv ess v::ar dz
r di v ( rpen ) dx =
J
E
I
r.p
=
rt
I
=
where
z,
(* * *)
c Rn - t .
V = ( - a, a) n- t
2. For positive integers k and m, define these sets: G(k) H(k)
Note
=
=
{ a(n - 1 ) n < I { x E R C(B(x, r) n i)
} }
a(n - 1 ) 3 n < 0 E R r" for 0) x
rn for O < r
<
I X + se n E 0 for 0 < s < 3/m}, = G(k) n {x I X - se n E 0 for 0 < s < 3/m} , H+ (k, m) = H (k) n {x I X + sen E I for 0 < s < 3/m}, H- (k, m) = H(k) n {x I X - sen E I for 0 < s < 3/m} .
c+ (k, m) c- (k, m)
=
G(k) n {x
3 k
,
3. Claim #I: (k, m = l , 2, . . . ) .
Proof of Claim #I: For fixed k, m, write
U 00
c+ (k, m)
=
j= - oo
Gi ,
BV Functions and Sets of Finite Perimeter
224
where G1
=
G +(k m) n ,
{x
l
}
j- l < x11 < i . m
m
Assume z E ll�-"- 1 , 0 < 1· < min { l /k, 1 /m } , and B ( z 1·) n P(GJ) =/= 0. Then there exists a point b E GJ n p - 1 (B(z , T)) C G(k) such that ,
b, + ;
> sup{x n I x E Gj n p- I (IJ(x, r)) } .
Thus, by the definition of c+ ( k, m ) , we have
Take the .C11-measure of each side above to calculate
� C- 1 (P(G1) n B(z, T)) < C'(O n B(b, 3T)) < a��:� I ) (3T)" ,
since b E G(k). Then .cn - 1 (P(G1) n B(z , T)) 2 < ltm sup - 3 r�o a (n - l ) 'fn - l .
for all
z
E Rn - l . This implies (j
=
0 ± 1 , ±2, . . . ).
and consequently c- 1 (P(G+ (k, m) ))
=
0.
Similar arguments imply
for all k, m. 4. Now suppose
U 00
z
E V-
P[G+ (k, m) uG- (k, m)uH + (k. m)uH- (/;. m)]
k ,m= l
and
N(P I u n a* E, z) <
(* * * *)
00.
Assume -a < t 1 < · < tm + l < a are points of approximate continuity of r . Notice that Jr(tj+ l ) - r ( tJ ) I =I= 0 i f and only if J r (tj+l ) - r (tj) l = I In the latter case we may, for definiteness. suppose ( z , t1 ) E I. ( : . ti + ' )
·
.
S.ll
225
A criterion for finite perimeter
of tJ + I must contain points s such that continuous at s. Consequently, ess v�" r = sup
(z, s) E
0 and r is approximately
"'
L: Jr ( tJ + d - r ( tj ) l
j=I
the supremum taken over points - a < t , < · · · < tm+ l < a such that ( z, tj ) E 0 U I and /" is approximately continuous at each tJ . 5. Claim #2: If (z, u ) E I and ( z, v) E 0, with 1L < v, there exists u < t < 11 such that ( z , t) E a* E. Proof of Claim #2: Suppose not; then observe that Ic
U G(k), 00
(z, t) E 0 U I 0c
k=l
for all
u
< t<
v.
We
00
U H(k),
k= l
and that the G(k), H(k) are increasing and closed. Hence there exists ko such that (z, u ) E G(ko) , (z, v) E H(ko). Now H(ko) n G ( ko) = 0, and so
Uo = sup{ t I (z, t) E G(ko) ,
t<
v} < v.
Set
vo
=
inf{t
I (z, t) E H(ko),
t>
Uo }-
Then
(z, vo) E H(ko),
(z, Uo) E G(ko), u
and
<
Uo
<
Vo
<
v,
{ (z, t) I Uo < t < vo} n [H(ko) U G(ko)] = 0.
Next, there exist
Uo < 81
< tl <
Vo
with (z, s!) E I, (z , tl ) E 0; this is a consequence of (* * * *). Arguing as above, we find k1 > ko and numbers u , , v1 such that Uo
<
U!
<
V!
<
Vo,
(z, v,) E H( k! ),
BV Functions and Sets of Finite Perimeter
226
and (z , t) !f. ll(kt ) U G(k 1 ) if u1 < t < 111 . Continuing, there exist and sequences { uJ }} 1 , { vJ }} 1 such that
kJ
-+ oo
< U I < . . . , Vo > 11 1 > 112 . . . , 1t i < vi for all j I , 2, . . . , (z, ui ) E G(kJ), (z, vJ) E 1/ ( kj ), (z, t ) rf. G(kJ ) U H (kj ) if uj < t < Vj . 'Uo
=
Choose
,lim
J -00
Then y
whence
,lim Vj .
J -00
00
(z, t) !f. U [G(ki ) U H(ki )],
=
j= l
I.1m sup
r�o
and
Uj < t <
cn (B(y, r) n E) a(n - I ) > - 3 n +l:--'rn ---'--
,C n (B(y, r) - E) a(n - 1 ) > hm sup - 3n + 1 rn r �o .
Thus y E 8.E. 6. Now, by Claim #2 , if z satisfies (* * * *), ess Vaa fZ < Card { t 1 - a < t < a, (z , t) E o E } = N(P 1 u n a. E , z) . .
Thus (* * *) implies
fv ess vaa r dz < fv
N(P I u n a.E, z)
dz <. 1{'' -l (U n a. E) < 00
and analogous inequalities hold for the other coordinate directions. According to Theorem 2 in Section 5. 10, E has locally finite perimeter. 7. The necessity of (*) was established in Theorem I in Section 5.8. I
6
Differentiability and Approximation by C 1 Functions
In this final chapter we examine more carefully the differentiability properties of BV, Sobolev, and Lipschitz functions. We will see that such functions are differentiable in various senses for C" a.e. point in R" , and as a consequence are equal to C 1 functions except on small sets. Section 6. 1 investigates differentiability .C" a.e. in certain U'-senses, and Section 6.2 extends these ideas to show functions in W l.1' for p > n are in fact .C" a.e . differentiable in the classical sense. Section 6.3 recounts the elementary properties of convex functions. In Section 6.4 we prove Aleksandrov 's Theorem, asserting a convex function is twice di fferentiable C' a.e. Whitney's Extension Theorem, ensuring the existence of C 1 extensions, is proved in Section 6.5 and is utilized in Section 6.6 to show approximation by C 1 functions.
6.1 6. 1.1
LP differentiability; Approximate differentiability L1
'
differentiability a.e. for
Assume f E
BV
BVioc (R" ) .
NOTATION We recall from Section 5 . 1 the notation
[D/] = [D/]ac + [Df],
=
C L Df + [Df], ,
where D f E L.'oc ( R" ; R" ) is the density of the absolutely continuous part [D f]ac of [D f] , and [D /], is the singular part.
227
Differentiability and Approximation by C1 Functions
228
We first demonstrate that near [/' a.e. point integral norm by a linear tangent mapping. THEOREM 1
Assume
f E BVioc(R11 ).
(i
lJ(x,r)
Then for C' a.e.
E R" ,
lf(y) - f(x) - Df(x) (x - y) J 1 • dy ·
PROOF I. en a.e. point
x E Rn
)
,.. I
( )
as
= o 1·
r --+
0.
satisfies these conditions:
i
r�O B(x,r) lf(y) - f(x) l
(a) lim
x
x, f can be approximated in an
dy = 0.
[ r�o JB(x,r) I DJ(y) - Df(x) l dy = 0.
(b) lim
n r�o J[Df],J (B(x, r))/r = 0.
(c) lim
2. Fix such a point x; we may as well assume x = 0. Choose r > 0 and let f' = fl• * f . Select y E B(r) and write g(t) = f'(ty). Then
g(l) = g(O) + that is,
[ g'(s) ds,
1' Df'(sy) · y ds = /'(0) + Di (O) · y + 1' [Df'( sy) - Df(O) ] · y ds.
f'(y) = /'(0) +
3. Choose any function average over B( r):
{
JB(r)
rp E C� (B(r))
with
I 'PI <
I , multiply by
and
rp (y)(f'(y) - /'(0) - Df(O) · y) dy
[' (1 = f' !_s ( =
Jo
Jo
Ja (r)
)
rp (y)[Df'( sy) - D/ (0)] · y dy ds
()
f rp z [Df'(z ) - D/(0) ] · dz s Ja(rs) z
) ds.
(*)
22J
6.1 £1' differentiability; Approximate differentiability
Now
g, (s )
Jl3(r.< ) cp C) /Jf' (z) dz - r !' ( ) d iv (cp c) ) .J - J f(z) div (cp (�s ) z) dz = J cp c) z d [ D f] s z) z z J cp ( Jl3(r.• ) cp ( ) Df(z) dz +
=
· z
-�
z
=
/J ( r· ,< )
-+
S
dz
n(•·•l
·
£3(•·.•)
=
Furthermore,
z
·
S
=
£3 ( 1'.< )
·
d [Df], .
S
J rs) J DJ' (z) l dz ; Jl3(rs) }Rr n Df!,(Z - y) f(y) dy
< ..:._ l + s" s
J g,(n s ) J
z
£3 (
,
rn
dz
{ f!,(z - y) d [D f] dz £3( rs) }Rn
f r < : r ,(z - Y ) d J I DJII dz f! }l3(rs) }Rn
=
S
.
S
=
rn { { f!,(z - y) dz dJ IDJII }Rn B(rs)
J
S
<
:n J{B(rs+<) J{B(rs)n B( ,<) dz di i D/1 1
S
f
y
min( (rst ' fn )( rs + ('t c < t'n
sn
<
C
for 0 <
f,
s < I.
Differentiability and Approximation by C1 Functions
230
4. Therefore, applying the Dominated Convergence Theorem to (*), we find
j cp(y)(f(y) - f(O) - Df(O) y)) dy Ja(r) ·
< =
Cr
1 o
1
�� �
[ I DJ (z ) - Df (O ) I dz d.� + C r t I [D J , ( ( rs )) 1s ' .fo Jn( •··•l
ds
o(r) as r -> 0.
Take the supremum over all
cp as above to find
_[ lf(y) - f(O) - Df(O ) yJ dy o(r) as r -> 0. J a ( r)
(**)
=
·
Finally, observe from Theorem I (ii) in Section 5.6.1 that
5.
(
[ IJ(y) - f(O) - Df(O) . Yl .::. , dy JB ( r)
)
-;;n- 1
JJD(f - /(0) - Df(O) y)JJ(B( r)) C r••-t <
·
C J_[ J f(y) - /(0) - Df(O) · yJ dy B( r) o (r) as r -> 0, +
=
according to (**), (b), and (c). 6.1.2
u·
I
differentiability a.e. for
W 1 ·P ( I
<
p
<
n
)
We can improve the local approximation by tangent planes if function. THEOREM 2 Assume f E W1::' (lRn ) for
I
<
p<
Then for
[ Ja(x,r)
is a Sobolev
.en a.e. x E Rn ,
' / p )1 ( lf(y) - f(x) - Df(x) (y - x)IP· dy n.
f
·
PROOF I. .e n a.e. point
=
o( r)
as
r -> 0.
x E Rn satisfies (a) limr�o fa(x,r) lf(x) - j (y)JP dy 0. (b) lim r� o fa x,r ) J Df(x) - Df(y) JP dy 0. ( 2. Fix such a point x; we may as well assume x 0. Select .p E C� (B( r)) < I , where 1 /p + 1/q I. Then, in the previous proof, with J J cp JJ L• ( B ( r)) =
=
=
=
as
231
6.1 LP differentiability; Approximate differentiability
we calculate
1
JU(•• )
cp (y)(f (y) - f(O ) - D f( O) · y) dy
1 f 1 < r 1 (1
=
1
� S
0
0
) [ D f( z) - D/ (0) ] · z d:: ds z ( cp l3(rs) S
Jl3 ( rs)
cp C)'1 dz S
Since
i
l3(rs)
we obtain
cp (
z
-
S
)q
/l
) (f
dz =
l
<
J D f (z) - /Jf( O) I1' d::
U(r,< )
!,
B(r)
{Ja(r·) cp(y)(f(y) - / (0) - Df(O) y) dy ·
Taking the supremum over all functions
�r ( }{
=
a( n ) r"
,
o(r l -nfq ) as r -+ 0.
cp as above gives
Jf (y) - /(0 ) - Df(O ) y JP dy ·
B ( r)
I
J cp (yW dy <
)
1 /p
)
=
o(rl- n/ q ) ,
and so
(
_[ lf(y) - /( 0) - D /( 0 ) · yJ P dy JB(r)
)
1 /p =
o(r ) as r -+ 0.
3. Thus Theorem 2(ii) in Section 4.5 . 1 implies
(1
JB(r)
Jf(y) - /( 0 ) - Df(O) yJP• dy ·
1( +C({
< Cr
Ja(r)
)
I D J (y) - D/(O) JP dy
according to (*) and (b).
1 /p
J f(y) - /( 0) - Df (O) yJP dy
JB(r) = o(r) r -+ 0, as
)
1 /p •
I
·
)
l /p
1ll'
ds.
Differentiability and Approximation by C1 Functions
232
6.1.3
Ap proximate differentiability
DEFINITION Let f : R" --> R'" . We say x E Rn if there exists a linear mapping
/, : R"
such that
f is approximately differentiable at
-->
R"'
. lf(y) - f(x) - L(y - x) l = 0. ap lim v�x I Y - xl
(See Section 1 .7.2 for the definition of the approximate limit.) As proved below, such an L , if it exists, is unique. We write
NOTATION
ap
Df( x )
for L and call ap D f(x) the approximate derivative of f at x. THEOREM 3
An approximate derivative is unique and, in particular, ap D f = 0 en a.e. on {! 0}. =
PROOF
Suppose . f(y) - f(x) - L(y - x)l ap I 1m l y�x I Y - xl
and
Then for each
f >
0
. f(x) - L'(y - x) l ap I1m lf(y) = 0. y�x I Y - xl
0,
(
{
L (y- x) l (y) J if B n en r) (x, yl �2:1 . l lim r �o £n ( B (x, r)) .
and
=
> f
}) })
{
if(y) -f���-.;,f < y- x ) l > (' B e n r) (x, n y I . hm HO en( B (x, r)) If L =I= L' , set 6t' = I l L - L' l l = max I (L - L' )(z) l > 0
(
and consider then the sector
{
=0
(*)
0.
( **)
=
JzJ=l
S = Y I I (L - L')(y - x) l
>
�
II L - L' I I y - x l
}·
233
6.1 LP differentiability; Approximate differentiability
Note
e" ( B (x , r) n S) = e"(R(x, r)) -
a >
0
for all r > 0. But if y E S,
3 fiY
_
x' l
I l L - L' I I I Y - x l 2 < I (L - L' )(y - x)l < l f (y) - f (x) - L(y - x) l
=
+ l f(y) - f(.r ) -
L '(y - x) l
so that Sc
{Y 1 1 /(y) - f(x) - L(y - x) l } {y 1 1 J(y) - f(x)y -xL' (y - x)l } . >f
I y- xI
u
>f
I - I
Thus (*) and (**) imply
e" ( B (x, r ) n S) r�o en(B (x, r)) lim
a contradiction to (* * *). THEOREM 4 Assume f E BVioc (Rn ).
I
=
O
'
Then f is approximately differentiable en a.e.
REMARK (i) We show in addition that
ap DJ
=
Df
_cn a.e.,
the right-hand function defined in Section 5 . 1 . (ii) Since YV;� ( Rn ) C BVioc ( Rn ) (I < p < oo), we see that each Sobolev function is approximately differentiable en a.e. and its approximate deriva tive equals its weak derivative en a.e. I PROOF
Choose a point x E Rn such that
J lf(y) - f(x) - Df(x ) (y - x) l dy = o (r) as JB(x,r) en a.e. X will do according to Theorem I . ·
r ->
Suppose
f(y) - !(x) - Df (x) · (y - x)l . ap I1msup l y�x I Y - XI
>
()
>
0.
0;
(*)
Differentiability and Approximation by C1 Functions
234
Then there exist 1"J
-+
0 and 1 > 0 such that
C'' ( {y E B (x, rJ ) l l f(y) - f(x) - D f(x) · (y - x)l > 9Jy - xi }) > I > 0 . 0' ( n )rj' Hence there exists a > 0 such that
.C"( { y E B(x , r.;) - R(x, a rj ) l l f(y) - f( :r ) - Df(x) · (y - x)J > Oi y - xJ}) > 1 - -2 a(n)rj'
for j
=
I , 2, . . . . Since Jy - xl > a r1 for y E B (x, 1·J ) - B(x , m·J ) ,
.C"( {y E B (x, rJ ) I I f(y) - f(x) - Df(x) · (y - x) I > Bari }) > 1 - 2 a(n)rj'
( **)
for j = I , . . . . But by (*), the expression on the left-hand side of (**) is less than or equal to a contradiction to (**). Thus
o(r ) 9arJJ·
=
o( I ) as r1
0,
-+
. sup l f(y) - f(x) - Df(x) (y - x ) l ap I 1m = 0, X y�x IY - I ·
and so ap Df(x)
6.2
=
Differentiability a.e. for W 1 •P
Df(x).
I
(p > n)
Recall from Section 3. 1 the following definition: DEFINITION
A function f : R"
exists a linear mapping
-+
Rm
is differentiable at x E R" if there
such that f(y) - f(x) - L(x - y)J lim l y �x lx - Yl
=
O.
6.2 Differentiability a.e. for W i .P (p
NOTATION write
>
235
n)
If such a linear mapping
t
exists at x, it is clearly unique, and we
/) f(;r)
for
We call Df( .'"C ) the derii'Giil·e of f
L.
THEOREM 1 Let f E W1�11 ( R" ) for
at
J'.
some < p < oo. Then f is differentiable .C" a.e., and its derivative equals its weak derivative .C" a.e. 11
PROOF Since W1�00 ( lR" ) C W1�' ( R" ) , we may as well assume For .C" a.e. x E R" , we have
{
r�o Ja (x,r) J DJ(z) - Df(x) l lim
1' dz
=
n
<
p
<
oo.
(* )
0.
Choose such a point x, and write
g (y)
=
f(y) - f(x) - Df(x) · (y - x)
(y E B (x, r)).
Employing Morrey's estimate from Section 4.5.3, we deduce
Jg(y) - g(x) l for r = Jx - yJ. Since g (x)
=
<
Cr
I
Ja (x,r)
0 and Dg
lf(y) - f(x) - Df(x) · (y - x)l J y - xl according to (*).
1(
=
C
=
o( l )
)
Df - Df(x) , this reads
1(
<
J DgJ P dz
1 /p
Ja( x,r)
I D!(z) - Df(x)J P dz
as y -> x
)
1 /P
As an application we have a new proof of Rademacher's Theorem, Sec tion 3. 1 .2: THEOREM 2
Let f Rn -> R be a locally Lipschitz function. Then f is differentiable en a.e :
PROOF
According to Theorem 5 in Section 4.2.3, f E wl�oo ( Rn ).
I
DifferentiabiliJy and Approximation by C 1 Functions
236
6.3
Convex fu nctions
DEFINITION
A function f : IR"
-----+
IR
is called convex if
- >..) y) < Af(x) + ( I - >.. ) f(y )
f(>..x + ( I
for all 0 < >.. < I , x, y E !R" . THEOREM I
Let f : IR"
-----+
----
IR
·
be convex.
(i) Then f is locally Lipschitz on IR" , and there exists a constant C, depending
only on n , such that
su�. lfl < C J
B(x , 2 )
JB(x,r)
and
lfl dy
i
ess sup I DJI < C lfl dy r B(x,r) B(x , D for each ball B (x, r ) C IR" . (ii) If, in addition, f E C2 (1Rn ) . then D2 f
>
0 on IR" ,
that is, D2 f is a nonnegative definite symmetric matrix 011 IR" . PROOF I. Suppose first that f E C2 (!Rn ) and is convex. Fix x E IR" . Then for each y E !Rn and ).. E (0, 1 ),
f(x + >.. ( y - x)) < f(x) Thus
f(x + >.. ( y
� x)) - f(x) < f(y)
Let >.. -----+ 0 to obtain
f(y)
+ >.. ( f(y) - f(x) ).
>
f(x)
>
f(z)
f(x ) .
+ Df(x) · (y - x)
for all x, y E !Rn . 2. Given now B (x, r ) C IR", we fix a point
f(y)
_
+ DJ (z)
z
·
E B (.r. r/2). Then (*) implies
(y - ;;) .
237
6.3 Convex functions
We integrate this inequality with respect to y over
f(z) < j
Jn(z, f )
B(z, r/2) to find
f (y) dy < C J
JD(x, r)
Next choose a smooth cutoff function
lfl dy.
( E C,� ( IR" ) satisfying
0 < ( < I, (= I
on
r ( = 0 on IR" - B(x, r) B(x, ), . 2
Now (*) implies
f (z) > f(y) + Df (y) · (z - y) . Multiply this inequality by
f (z)
j
D (x,r)
((y) dy >
j
((y) and integrate with respect to y over B(x, r):
D( x,r)
= r
f (y )((y) dy + {
}D ( x,r)
((y) DJ(y ) · (z - y) dy
f (y)[( (y) - div (((y)(z - y))] dy
}B(x ,r) > -c r 111 dy. }B(x,r)
This inequality implies
f (z) >
-C J
JB(x ,r)
lfl dy,
which estimate together with (**) proves
lf(z)l < C J
JB(x,r)
3. For
If I dy.
(* * *)
z as above, define
Sz =
{ Y l r4 < 1 y - xi < 2r , Df(z) · (y - z) > 21 1Df(z) I IY - zl
and observe
}
,
Differentiability and Approximation by C1 Functions
238
where C depends only on
n.
Use (*) to write 1'
f(z) + 1 Df(z)l 8 for all y E Sz. Integrating over Sz gives f(y)
I DJ(z)l <
>
�i
,.
D(r. :; )
lf( y ) - f(z)l dy .
This inequality and (* * *) complete the proof of assertion (i) for C2 convex functions f. 4. If f is merely convex, define f ' = 1/f * f , where � > 0 and TJ• is the standard mollifier. Claim: f' is convex. Proof of Claim: Fix x, y E IR", 0 < A < I . Then for each z E IR",
f(z - (Ax + ( I - A)y))
= f(A(z - x) + ( I - A)(z - y)) < Af(z - x) + ( I - A)f( z - y).
Multiply this estimate by TJ,(z)
f' (Ax + ( 1 - A)y)
0 and integrate over IR":
>
= r
}'U{n
f(z - (AX + ( I - A)y))1).(z) dz
< A r f(z - X) TJ. ( Z ) dz
}'U{n
+ ( I - A) { f(z - y)1),(z) dz
}JRn
= AJ '(x) + ( I - A) J' (y). 5. According to the estimate proved above for smooth convex functions, we have sup (lf'l + r i Df' l) < C J
B(x, ¥)
JB(x,r)
IJ ' I dy
for each ball B (x, r ) C !Rn . Letting � -----+ 0, we obtain in the limit the same estimates for f. This proves assertion (i). 6. To prove assertion (ii), recall from Taylor's Theorem
f(y)
= f(x) + Df(x ) · (y -x)+(y-x)T ·
This equality and (*) yield
(y ·- xf ·
1 ( 1 - s)D2f(x+s(y-x)) ds· (y-x) . 1
1 ( I - s)D2f (x + s(y - x)) ds (y 1
·
x
)
>
0
239
6.3 Convex functions
for all x, y E to compute:
IR" .
Thus, given any vector �, set
e Send t
-+
0 to prove
THEOREM 2
Let f : IR" such that
-+
y
· 1 1 ( I - s)D2f(x + stO
IR be convex.
==
J:
+ t � in ( * * * *) for t
ds � ·
>
0
> o.
Then there exist signed Radon measures
ILiJ
= f.J)'
( i, j = I , . . . , n ) for all
C1(IRn ).
Furthermore, the measures Jlii are nonnegative (i
PROOF I. Fix any vector � E !Rn , 1�1 I , � = (�1 , . . . , �n ). Let '7• be the standard mollifier. Write f' = TJ• * f . Then f' is smooth and convex, whence ==
D2f' Thus for all
Let
E -+
E
>
0.
C1(IRn ) with
0 to conclude
Then Corollary 1 in Section 1 .8 implies the existence of a Radon measure f.1� such that
for all
Differentiability and Ap("ibximation by C1 Functions
240
'
Thus
where
I
THEOREM 3 Let f : �n -----+
IR be convex.
Then
8J 8J -, . . . , E BVioc ( IR ). 8XI {)Xn n
PROOF
Let
V CC IR
n
,
E
C�(V, IR"), I 'P I < I . Then for k = l , . . .
<
L llik(V) < oo. i =l n
, n,
I
NOTATION In analogy with the notation introduced in Section 5 . 1 , let us write for a convex function f:
.. where I: : !Rn -----+ M n x n is I I D2 JII -measurable, with lEI = I I ID2 f l l a.e. We also write (i, j = l , . . . . n).
241
6.4 Second derivatives a. e. for convex functions
By Lebesgue's Decomposition Theorem, we may further set
- JLaijc + JL.ij• , JL ij -
where i f1aci < < e" '
But then
JL I.J .<
j_
e"
•
1111 --en L fij r-ae
(i j = l ,
82!
8x18x 1
...
n , . . . ,
),
82!
8x18X n
D2 f =
'
82!
82!
8xn 8Xn
8xn8x l 11 Il
,..... ac
• • •
,...11a.. 1cn
,...11nl a.. c . . . ,...11a..ncn
Thus [D2f] = [D2J] .c + [D2J) . = e n L D2J + [D2JJ.. , so that D2f E L/oc(JRn ; Mn x n ) is the density of the absolutely continuous part [D2 f] ac of [D2 f]
6.4
Second derivatives a.e. for convex functions
Next we show that a convex function is twice differentiable a.e. This asser tion is in the same spirit as Rademacher's Theorem, but is perhaps even more remarkable in that we have only "one-sided control" on the second derivatives.
Differentiability and Approximation by C 1 Functions
242
THEOREM I ALEKSANDROV'S THEOREM
Let f : IR" -----+ IR be convex. Then f has a second del'h•ative C" precisely. for C" a.e. x ,
a .l'.
More
I f(y) - j(1:) - Df(x) (y - x) - (y - x) T D2f(x) · (y - x) 2 ·
·
= o( IY - xl 2 ) PROOF I. e n a.e. point (a)
as y -----+ x.
X satisfies these conditions:
Df(x) exists and lim _[
r-;O JB(x,r) I DJ (y) - Df(x) l dy = 0.
2 2 J r-;0 JB(x,r) I D f(y) - D f(x)l dy = 0. lim I [D2 f] s l (B (x, r )) /rn = 0. r-;0
(b) lim (c)
2. Fix such a point x; we may as well assume x f' = TJ. * f. Fix y E B(r). By Taylor's Theorem,
f'(y)
= f' (O) + Df'(O)
·
y+
== 0.
Choose
[(1 - s)yT · D2f'(sy)
r > 0 and let
·
y ds.
Add and subtract ( l /2 )yT · D2 f(O) y: ·
f'(y)
= f' (O) + D f'(O) · y + � YT +
1 (1 - s)yT I
·
·
D2f(O) · y
[D2 f'(sy) - D2 f(O)J · y ds .
3. Fix any function
1, multiply the equation above
j
=
t ( l - s)
�
(
·
)
J
6.4 Second derivatives a.e. for convex functions
243
as E -----+ 0
Furthermore, as in Section 6. 1 . 1 , we may calculate
r D2TJ. (Z - y)f(y) dy dz }'U{n { TJ. (z - y) d [ D2f ] dz }'U{n
·--
min ((rs )n , En )(r s + Et < -C S nEn for O < E, s < l by (**) ·
[
�
]
j cp(y) f(y) - f(O) - D f(O) · y - yr · D2 f(O) y dy JB(r) < Cr2
1 o
1
r JB(rs)
·
ID2 f( z) - D2 f( O) I dz ds + Cr2 by (**) with
x =
0.
�
1 I [D2 J] . l (rs)) ds r (s ) o
1
DifferentiabiliJy and Approximation by C1 Functions
244
Take the supremum over all
'fJ
as above to obtain
{ jh(y) j dy = o(r2 ) as r -----+ 0 Jn(r·)
)
for I T 2 D f(O) · y. y - y 2 5. Claim #/: There exists a constant C such that C (r > 0). ��p j Dhj < r j jhj dy + Cr Jn ( r ) B(r/2)
h(y) = f(y) - f(O) - Df(O)
·
·
Proof of Claim #1: Let A = j D 2 f(O) j. Then g = h + (A/ 2) jyj2 is convex: apply Theorem I from Section 6.3. 6. Claim #2: supB(r/2) jhj = o(r2 ) as r -----+ 0. Proof of Claim #2; Fix 0 < E , T) < I , T) i / n < 1 /2. Then C' { z E B (r) j jh(z) j > tr2 } <
{ jhj dz � Er }B(r)
= o(r n ) as r -----+ 0, < TJ C'( B(r))
by (* * * * )
for 0 < r < r0 = r0(t , TJ) ·
Thus for each y E B (r/2) there exists z E B (r) such that
and for if not,
C' {z E B(r) l lh(z) l > Er2 } > C' ( B (y, u ) ) = a(n) w" = TJ C'( B ( r)). Consequently,
jh(y) j < jh (z) j + jh(y) - h (z) l .
< tr2 + sup j Dh j u
B(r) < tr2 + C1J 1 1n r2 = 2tr2 ,
provided we fix T) such that CT) 1 /n
= t
by Claim #1 and (* * * * ) and then choose 0 <
r
< r0•
245
6.5 Whitney's Extension Theorem
7. Accordi ng to Claim #2,
sup f (y) - f (O) - Df (O) B(r/2) This proves (*) for x
6.5
I
0.
=
·
y -
�2 YT D2f (O) y ·
as r
·
--+ 0.
Whitney's Extension Theorem
We next identify conditions ensuring the existence of a C 1 extension f of a given function f defined on a closed subset C of IR" . Let C C IR" be a closed set and assume f : C -+ IR, d : C --+ IR" are given functions. NOTATION
\x)·(y - x J R( y , x) = J(yJ-J(xJ-d lx- y (ii) Let K C C be compact, and set (i)
pK(b )
THEOREM I
=
(x, y E C, x f= y).
sup { I R(y, x)l l 0 < lx - vi < b, x, y E K} .
WHITNEY'S EXTENSION THEOREM
Assume J, d are continuous, andfor each compact set K
C C,
pK(b) --+ 0 as b --+ 0.
Then there exists a function f
:
!Rn
--+ IR such that
(i) f is C 1 • (ii) f = f, Df = d on C. -
-
PROOF I. The proof will be a kind of "C 1 -version" of the proof of the Extension Theorem presented in Section 1 .2. Let U = !Rn - C; U is open. Define
I min{ l , dist(x, C)} . 20 B y Vitali's Covering Theorem, there exists a countable set { xi } } 1 that r(x)
=
U B(xj , 5r(xj )) 00
U
=
j=l
C U such
Differentiability and Approximatidn by C1 Functions
246
and the balls { B (x j , r( xi ))} j 1 are disjoint. For each x E U, define
Sx
=
{xj I B(x, IOr(x)) n IJ(xj , I Or(J:j )) f= 0}.
2. Claim #I: Card (Sx) < ( 1 29 ) " and 1 /3 < r(;r) /r(J:j) < 3 if Xj E Sr . Proof of Claim #I: 1f x1 E S,, then I I lr(x) - r(xJ ) I < 1 x - xi l < ( 1 0 (r(J·) + r(xj))) 20 20 I = (r(x) + r(x1)). 2 Hence In addition, we have
l x - xi l + r(xj ) < I O(r(x) + r(xj )) + r(xj ) = IOr(x) + l l r ( :z· j) < 43r(x ); consequently,
B(xJ, r(xj))
C B(x, 43r(x)).
As the balls {B (xJ , r(xJ ) ) }) 1 are disjoint and r(xJ ) Card ( Sx)a(n) whence
>
r(x) j3,
( r�) ) n < a(n)(43 r(x)t
Card (Sx) < ( 1 29 t. 3. Now choose f-l : IR -+ IR such that f-l
E
0 < f-l < 1 ,
C00 ,
For each j
=
/-l(t )
=
I for t < I .
1 , . . . , define
(x Then
f-l(t)
E iR" ).
=
0 for t
>
2.
6.5 Whitney's Extension Theorem
247
Also c c1 . < If Xj E Sx r ( xJ ) r (x)
I Duj (x)l <
·
and
u1
=
0 on B(x. IOr(J: )) if Xj
Define 00
u (x) Since Uj
=
=
L u1 (x )
(x E IR" ).
j= l
0 on B (x, IOr(x)) if Xj
=
L
u1 (y) if y E B(x, I Or(x)).
Xj ESx
By Claim # I , Card (Sx) < ( 1 29 t ; this fact and (**) imply u E C 00 (U),
I Du (x) l < Now for each j
=
u > I on U
Cz r (x)
(x E U).
I , . . . , define VJ (x)
_ u1 (x)
=
(x E U).
u(x)
Notice
Dvj
=
Dui 0"
uJ· Du u2
Thus 00
2:: v1 (x)
=
1
j= l
00
L: Dvj (x) j=l
I Dvj (x)l <
=
0
(x E U)
c3 r (x )
The functions { Vj }j 1 are thus a smooth partition of unity in U.
Differentiability and Approximation by C1 F�nctions
248
4.
Now for each j
=
I , . . . , choo!>e any point sj E C such that
Finally, define f : IR" ----> IR this way: -
if X E C
f(x) f(x)
=
00
L vJ (x)[f ( sj ) + d( sJ ) · (x - sJ)] j=l
if x E U.
Observe
J E c= (U ) and
D J(x)
=
L { [f(sJ ) + d(sj) · (x - Sj )]Dvj (x ) + Vj (x) d(sj)}
(x E U )
Xj E Sx
5. Claim #2: Df(a)
=
d(a) for all a E C.
Proof of Claim #2: Fix a E C and let K = C n B(a, I ) ; K is compact. Define
IJ(x) - J(a) - d(a) · (x - a)l
=
=
lf(x) - f(a ) - d (a) (x - a)l ·
I R(x, a)l l x - al
<
ld(x) - d(a)l <
249
6.5 Whitney's Extension Theorem
Now suppose x E U, l :r - al < 1 /6. We calculate
l f(x) - f(a) - d(a) · (x - a)l = lf(x) - f(a) - d(a) · (x - a)l < L lvi (x) [f(sj ) - f( a ) + d(sJ ) · (x - sj) - d(a) · (:r - a) ] I Xj E Sx
<
L
Vj (x)lf(sj ) - f (a) + d(sj ) · ( a - Sj )l
x1 E S,
+
Vj (x)l (d(sj ) - d(a)) · (x - a)l .
L x1 E Sx
Now lx - al < 1 /6 implies r(x) < ( 1 /20) I x - al - Thus for Xj E
Sx .
I a - s1l < I a - Xjl + lx1 - sj l < 2 la - xj l < 2 (lx - al + lx - Xj l) < 2 (lx - al + I O(r(x) + r(xJ ) )) < 2 (lx - al + 40r(x)) < 6lx - al . Hence the calculation above and Claim #1 show
-
-
lf(x) - f(a) - d(a) · (x - a) l < C
l f(x) - f(a) - d(a) · (x - a) l = o(lx - al ) as x --+ a.
Thus Df(a) exists and equals d(a) . 6. Claim #3: J E C 1 (!Rn ).
Proof of Claim #3: Fix a E C, x E
!Rn,
lx - al < l j6. If x E C, then
I Df(x) - Dj(a)l = ld(x) - d(a) l <
If x E U, choose b E C such that
lx - bl = dist(x, C) .
Differentiability and Approximation by C 1 Functions
250
Thus
I Df(x) - Dj(a) l
=
I Df(x) - d (a)l
< I Df(x) - rl(b)l + ld(b) - rl(a)l .
Since
< lb - xl + lx - a l < 2 lx - al,
l b - al we have
l d(b) - d(a)l
<
We thus must estimate:
I Df(x) - d(b) l
=
L
[f(s1 ) + d(sj) · (x - sj )]Dv1 (x)
L
[- f(b) + f(sj ) + d(sj ) (b - SJ )] Dv1 (x)
Xj ES,
<
·
XjES,
L
+
[(d(sj) - d(b)) (x - b)] Dv1 (x) ·
Xj ES,
< c
"
+ +
L
c
r(x)
L
Xj ESx
Xj ES,
Now
lx - bl and therefore
r(x)
=
< lx - al <
6
.
1 1 < - lx - bl - 20 120
1 < 3r (x) -< 40 r (xJ ) ·
1
1 < 20
·
.
6.6 Approximation by C1 functions
251
Hence
r (xJ )
=
Accordingly, if x1 E S" ,
I . 20 1 rJ - sJ I
lb - si l < l b - x l + l:r - J:J I + lxJ - sil < 201· (:1: ) + IO( r (.z: ) + r(.r1 )) + 20 r(x1 ) < 1 20r (x)
=
6l x - bl < 6 l x - a l .
Consequently (* * * *) implies
IDJ(x) - d(b)l < C
I D}(x) - Df(a)l < C
6.6
Approximation by
C1
I
functions
We now make use of Whitney's Extension Theorem to show that if f is a Lipschitz, BV, or Sobolev function, then f actually equals a C1 function J, _ except on a small set. In addition, Df = D f, except on a small set. 6.6.1
Approximation of Lipschitz functions
THEOREM 1
Suppose f : !Rn IR is Lipschitz continuous. Then for each E a C 1 function J : !Rn IR such that ----+
>
0, there exists
----+
C {x l f(x) "I f(x) or D}(x) "I Df(x)} < E.
In addition, sup I D fl < C Lip (f) JRn
for some constant C depending only on
n.
By Rademacher's Theorem, f is differentiable on a set A C !Rn , with cn(IRn A) = 0. Using Lusin 's Theorem, we see there exists a closed set B c A such that Df I s is continuous and cn(IRn - B) < Ej2 . Set PROOF
-
d(x)
=
Df(x)
DifferentiabiliJy and Approximation by C1 Functions
252
and
R( y, x )
_
=
f(y) - f(x) - d(x) (y - x) I x - yI ·
(x f= y).
Define also
TJ�.: (x) = sup { I R(y, x)l l y E B, 0 < lx - Yl < 1 /k}. Then
---
T)k(x) -+ 0 as k -+ oo , for all x E B.
By Egoroff's Theorem, there exists a closed set C C B such that
T)k
-+
0
uniformly on compact subsets of C,
and
C(B - C ) <
�.
This implies hypothesis (*) of Whitney's Extension Theorem. The stated estimate on supJRn I Dfl follows from the construction of f in the proof in Section 6.5, since supc ldl < Lip (f) and thus
I R I , I�PI < c Lip (! ) . 6.6.2
Approximation of
BV
I
functions
THEOREM 2 f:._et f E BV (IRn ) .
Then for each f : !Rn -+ IR such that
�
>
0, there exists a Lipschitz function
C{x I f(x) f= f(x) } < e. PROOF
I. Define for .\
R>..
=
>
0
{x
E
I I DJ I I !Rn I
��(x, r)) < .\ for all
r
>
}
0 .
2. Claim #1 : cn (IRn - R>.. ) < "'<'�,w i i DJI I (IRn ). Proof of Claim #1: According to Vitali's Covering Theorem, there exist disjoint balls { B (x ;, r; ) }f" 1 such that !Rn - RA
C U B (x; , 5r; ) 00
i=l
253
6.6 Approximation by C1 functions
and
Thus
5"a( n) >-. ( IR" ) . < I>:' 5"a(n) ) R. D/II I I C" ( IR" < >. 00
i =l 3. Claim #2: There exists a constant C, depending only on lf(x ) - f ( y)l < C.\l x - Yl
n,
such that
for C" a.e. x, y E R>-..
Proof of Claim #2: Let x E R>-. , r > 0. By Poincare's inequality, Theorem I (ii) in Section 5.6. 1 ,
��
x, )) ( C ( Dfl I I < C>.r. If - (f) x,rl d y < r Ja(x,r)
1
Thus, in particular,
IU )x,rfzk+' - U ) x,r;z• I <
1
JB(x,rj2k+')
r
If - U) x,r;z• I dy
Since
f (x ) for en a.e. X E R).. '
==
rlim -;0 (f) x,r
00
lf(x ) - (f)x ,rl < L I U)x,rf2'+' - U)x,r/2" 1 < C.\r. k=l
Now for x, y E R>-. , x f= y, set r = lx - Yl· Then
l (f)x,r - (f)y,rl <
1
JB(x,r)nB(y,r)
({
I U)x,r - f(z)l + lf(z) - (f)y,r l dz
)
lf(z) - (f)y,rl dz lf(z) - U)x,rl dz +J JB(y,r) JB(x,r) < C>.r.
Differentiability and Approximation by C1 Functions
254
We combine the inequalities above to estimate lf ( x) - f(y)j < C>..r
=
C>..!:r - yj
a.e. x, y E R>. . 4. In view of Claim #2, there exists a Lipschitz mapping f : R>. -+ IR such that f = f C' a.e. on R>.. Now recall Theorem I in Section 3. 1 and extend f to a Lipschitz mapping f : IR" -+ R I for
C"
COROLLARY I
Let f E BV(IR" ) . Then for each E > 0 thete exists a C 1 function f : IR" such that
-+
IR
C {x f ( x) "I f(x) or Df ( x ) "I Df(x)} < E.
I
PROOF
According to Theorems I and
2, there exists f E C 1 (1Rn ) such that
C ( { f "I ! } ) < E. Furthermore, Df( x)
.en a.e. on { ! 6.6.3
=
=
Df ( x)
!} according to Theorem 4 in Section 6. 1. ,
I
Ap proximation of Sobolev functions
THEOREM 3 Let f E W 1 ·P(!Rn )
for some 1 < p < oo. Then for each E > 0 there exists a Lipschitz function f : !Rn -+ IR such that C {x I f(x) "I f(x) } < E
\
and
I. Write g = If! + ! Dfj, and define for >.. > 0
PROOF
R>.
=
{
x E !Rn I r
JB(x,r)
g dy < >.. for all
r
>
o}.
6,6 Approximation by C 1 functions
255
Proof of Claim #1: By Vitali's Covering Theorem, there exist disjoint balls { IJ(x; , r; )}?" 1 such that IR" -
U B(x;, 5r;) 00
R>. c
; == 1
and
(i = l , . . . ) .
g dy > A J JB(x;,r, ) Hence I
A < .Cn (B r (x ; , ; )) <
_cn(B(x;, ) ) I
1';
and so
1 1
I
dy + .C"(B(x , )) g ; B( x; , r , ) n{ g> f l 1';
B(x,,r, )n{g>A/2)
B(x,r, )n{g<>./2}
g dy
A 2
g dy + -
a(n)r� < 2 r
). J
1
B(x,,r; ) n { g > A/2}
g dy
(i = 1 , . . . ) .
Using (*) therefore, we see
C(!Rn - R), ) < sna(n) L � 00
< 2 - 5"
i= l
1 < 2 ' 5" ( A J >.
-
1'
dy g {g> >./2} { {g > A/2}
<£ r ).P
=
o
J
{ IJI+ID II > A / 2}
( ;P ) as A -+
3. Claim #2: There exists a constant
lf (x)l < >., for .C" a.e.
x,
y E R>.
.
gP dy
)
-;; I
(C({ g > A/2})) 1 - �
I Dt i P + IJIP dy
oo.
C, depending only on n , such that
lf (x) - f (y) l < CA l x - Yl
Proof of Claim #2: This is almost exactly like the proof of Claim #2 in the proof of Theorem 2.
256
Differentiability and Approximation by C1 Functions
In view of Claim #2 we may extend f using Theorem l in Section 3.1 to a Lipschitz mapping f : IR" ----+ lR, with 4.
Lip ( ]) < CA,
Ill < A,
5. Claim #3: II ! - fl l w '·P IR" ( ) Proof of Claim #3: Since f
=
=
J=f
t::• a.e. on R >. .
o( l ) as A ----+ oo.
J on R>. , we have
f I f - 11 '' dx = f f dx }Rn }Rn - R >. I - 1 1''
r}JRn - ' IJI P dx + CAI'.c" (IR" - RA ) R
o ( l ) as A ----> oo,
according to Claim # l . Similarly, Df
=
DJ £" a.e. on R\ and so
rhn I DJ - D]I P dx < c hr " - ' I DJI P dx + CAPC( IR" - RA ) R = o( l ) as A --> oo.
COROLLARY 2 Let f E W 1 · P(IR" ) for
C 1 junction
J
:
IR"
---->
I
some l < p < oo. Then for each lR such that
Ln { x I f (x)
=I J (x)
or Df (x)
> 0, there exists a
€
=I D J(x)} <
and
PROOF
The assertion follows from Theorems l and 3.
I
€
Bibliography
[FA] [F] [F-Z] [F-R] [GA] [G-T] [G] [H] [L] (M] [MO] [MY] (R] (S] [ST]
Falconer, Fractal Geometry. Wiley, New York, 1 990. H. Federer, Geometric Measure Theory. Springer-Verlag, New York, 1969. H. Federer and W. Ziemer, "The Lebesgue set of a function whose distribution derivatives are p-th power summable," Indiana U. Math. J., 22, (1972), 1 39- 158. W. Fleming and R. Rishel, An integral formula for the total gradient variation, Arch. Math. ll, 21 8-222 (1960). F. R. Gantmacher, The Theory of Matrices, Vol. I. Chelsea Publishing Co., New York, 1977. D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, (2nd edn.). Springer-Verlag, New York, 1983. E. Giusti, Minimal Surfaces and Functions of Bounded Variation. Birkhauser, Boston, 1984. R. Hardt, An Introduction to Geometric Measure Theory. Lecture notes, Melbourne University, 1979. F.-C. Liu, "A Lusin property of Sobolev functions," Indiana U. Math. J. 26, 645-65 1 ( 1977). V. Maz'ja, Sobolev Spaces. Springer-Verlag, New York, 1985. F. Morgan, Geometric Measure Theory: A Beginners Guide. Academic Press, Boston; 1988. C. B. Morrey, Jr., Multiple Integrals in the Calculus of Variations. Springer-Verlag, New York, 1966. Ju. G. Reshetnjak, "Generalized derivatives and differentiability almost everywhere," Math USSR-Sbornik 4, 293-302 (1968). L. Simon, Lectures on Geometric Measure Theory. Centre for Mathe matical Analysis, Australian National University, 1984. E. Stein, Singular Integrals and Differentiability Properties of Func tions. Princeton University Press, Princeton, NJ, 1970. K.
257
258
BIBLIOGRAPHY
[Z]
W. Ziemer, Weakly Differentia!J/e Functions. Springer-Verlag. New York, 1 989.
Chapters I through 3 are based primarily on Federer ! F l. as well as Simon [S] and Hardt [H). Chapter 1 : Essentially this entire chapter closely follows [Fl. In Section 1 . 1 , Theorems I and 2 are [F, Sections 2.1 .3, 2. 1 .5]. Lemma I is [ F, Section 2.2.2) and Theorem 4 is [F, Section 2.2.5] . Theorems 5 and 6 nre from [ F, Section 2.3.2) and Theorem 7 is [F, Section 2.3.3]. See also [S, Chapter I ] and [H]. ln Section 1.2, Theorem I is a simplilied version of [F, Sections 3. 1 . 1 3, 3. 1 . 14]. Lusin's and Egoroff's theorems are in [F, Sections 2.3.5, 2.3.7] . The general theory of integration may be found in [F, Section 2.4): see in particular [F, Sections 2.4.6, 2.4.7, 2.4.9) for Fatou's Lemma, the Monotone Convergence Theorem, and the Dominated Convergence Theorem. Our treatment of product measure and Fubini's Theorem is taken directly from [F, Section 2.6). We relied heavily on [S] for Vitali's Covering Theorem and [H) for Besicov itch's Covering Theorem: see also [F, Sections 2.9. 12. 2.9. 1 3]. The differentia tion thoery in Section 1.6 is based on [H). [S], and [F, Section 2.9). Approximate limits and approximate continuity are in [F, Sections 2.9. 12, 2.9. 1 3). We took the proof of the Riesz Representation Theorem from [S, Sections 4. 1, 4.2] (cf. [F, Section 2.5)). A. Damlamian showed us the proof of Corollary I in Section 1.8. See [G, Appendix A] for Theorem 2 in Section 1 .9. Chapter 2: Again, our primary source is [F), especially [F, Section 2. 10]. Steiner symmetrization may be found in [F, Sections 2. 10.30, 2. 10.31]. We closely follow [H) for the proof of the lsodiametric Inequality, but incorporated a simplification due to Tam (communicated to us by R. Hardt). The proof 'Hn t:_ n is from [H) and [S, Sections 2.3-2.6]. We used [S, Section 3] for the density theorems in Section 2.3. Falconer [FA] and Morgan [MO] provide nice introductions to Hausdorff measure. Chapter 3: The primary reference is [F, Chapters l and 3]. Theorem l in Section 3.1 is [S, Section 5.1). The proof of Rademacher's Theorem, which we took from [S, Section 5.2], is due to Morrey (cf. [MY. p. 65)). Corollary l in Section 3. 1 is [F, Section 3.2.8). The discussion of linear maps and Jacobians in Section 3.2 is strongly based on [H) . S. Antman helped us with the proof of the Polar Decomposition Theo rem, and A. Damlamian provided the calculations for the Binet-Cauchy formula: see also Gantmacher [GA, pp. 9-12, 276-278). The proof of the Area Formula in Section 3.3, originating with [F, Sec tions 3.2.2-3.2.5], follows Hardt's exposition in [H). Our proof in Section 3.4 of the Coarea Formula also closely follows [H). and is in tum based on [F, Sections 3.2.8-3.2. 1 3). Chapter 4: Our main sources for Sobolev functions were Gilbarg and Trudinger [G-T, Chapter 7) and Federer and Ziemer [F-Z]. =
8/BUOGRAPHY
259
See [G-T, Sections 7.2 and 7.3] for mollification and local approximation by smooth functions. Theorem 2 in Section 4.2 is from [G-T, Section 7.6] and Theorem 3 is based on [G-T, proof of Theorem 7.25]. The product and chain rules are in [G-T, Section 7.41. See also [G-T, Section 7. 12] for extensions. Sobolev inequalities of various sorts are in !G-T, Section 7.7]. Lemma I in Section 4.5 is a variant of [G-T, Lemma 7. 16]. Compactness assertions are in [G-T, Section 7. 1 0]. We follow [F-Z] (cf. also Ziemer (Z] and Maz'ja [M]) in our treatment of capacity. Theorems 1-4 in Section 4.7 are from [F-Z], as are all the results in Section 4.8. Much more information is available in [Z] and (M]. Chapter 5: We principally used Giusti [G] and Federer [F, Section 4.5] for B V theory (cf. also [S, Section 6 ]). The Structure Theorem is stated, for in stance, in [S, Section 6 . 1 ]. The Lower Semicontinuity Theorem in Section 5.2 is [G, Section 1.9], and the Local Approximation Theorem is [G, Theorem l . l 7]. (This result is due to Anzellotti and Giaquinta). The compactness assertion in Theorem 4 of Section 5.2 follows [G, Theorem 1 . 19]. The discussion of traces in Section 5.3 follows [G, Chapter 2]. Our treatment of extensions in Section 5.4 is an elaboration of [G, Remark 2. 1 3]. The Coarea Formula for B V functions, due to Fleming and Rishel [F-R], is proved as in [G, Theorem 1.23]. For the lsoperi metric Inequalities, consult [G, Theorem 1.28 and Corollary 1 .29]. The Remark in Section 5.6 is related to [F, Section 4.5.9(18)]. Theorem 3 in Section 5.6 is due to Fleming; we followed [F-Z]. The results in Sections 5.7 and 5.8 on the reduced and measure-theoretic boundaries are from [G, Chapters 3 and 4] : these assertions were originally established by De Giorgi. [F, Section 4.5.9] presents a long list of properties of BV functions, from which we extracted the theory set forth in Section 5.9. Essential variation occurs in [F, Section 4.5 . 1 0] and the criterion for finite perimeter described in Section 5.1 1 is [F, Section 4.5. l l ]. Chapter 6: The principal sources for this chapter are Federer {F], Liu [L], Reshetnjak [R], and Stein [S]. Our treatment of £P-differentiability utilizes ideas from [ST, Section 8.1]. Approximate differentiability is discussed in [F, Sec tions 3.1 .2-3. 1 .5]. D. Adams showed us the proof of Theorem 1 in Section 6.2. We followed [R] for the proof of Aleksandrov's Theorem in Section 6.4, and we took Whitney's Extension Theorem from [F, Sections 3. l . l 3-3. l . l4]. The approximation of Lipschitz functions by C 1 functions is from [S, Section 5.3]: see also [F, Section 3 . l . l 5]. We used [L] for the approximation of Sobolev functions. We refer the reader to the sources listed in the B ibliography for complete citations of original papers, historical comments, etc.
Notation
A
Vector and set notation
IR"
n-dimensional real Euclidean space
e;
(0, . . . , l ,. . . ,0), with l in the ith slot a typical point in !Rn
lxl
x·y XT . A . y
(xr + xi + · · · + x�)1 XtYI + X2Y2 + · ' + Xn Yn ·
bilinear form :L�i=t a;1 x;yi, where and A = ((a;1)) is an n x n matrix
B (x, r)
{ Y E !Rn I lx - yl x, radius r
B (r) U( x, r)
B(O, r)
C (x, r, h)
< r}
x, y
E
!Rn
= closed ball with center
{y E !Rn I lx - Yl < r} = open ball with center x, radius r { y E !Rn I IY' - x' l < r, IY - X l < h} = open n n cylinder with center x, radius r, height 2h •
a (s)
7r2
r
(� + l )
(O < s < oo)
a(n)
volume of the unit ball in !Rn
Q(x, r)
{ y E !Rn I lxi - Yil < r, i = l , . . . , n } = open cube with center x, side length 2 r open sets, usually in !Rn
u,
v, w
V cc u K
V is compactly contained in U; i.e., V is compact and V c U compact set, usually in !Rn
261
Notation
262
indicator function of the set
E
closure of
E
interior of
[,'
E
Sa(E)
Steiner symmetrization of a set E ; Section 2.3
8E
topological boundary of E
8* E
reduced boundary of E; Section 5.7.1
8. E
measure theoretic boundary of E; Section 5.8
1 1 8 £11
perimeter measure of E; Section 5.1
B
Functional notation
i f df.L or (f)E
. I { f df.L = average of f on E with respect JL (E) E to the measure JL
J
(/)x,r
r f dx Js(x,r)
spt(J)
support of f
r. r r f IE
max (j, O), max ( -f, O)
f or Ef Tf
an extension of f ; cf. Sections 1 .2, 3. l . l , 4.4, 5.4, 6.5 trace of f; Sections 4.3, 5.3
Dj
derivative of f
[Dj ]
(vector-valued) measure for gradient of f E BV; Section 5.1 absolutely continuous, singular parts of [Df] ; Section 5.1 approximate derivative; Section 6.1.3
[Df]ac , [Df]s ap Dj
precise representative of f; Section l .7.l
f restricted to the set E
Jf Lip( f)
Jacobian of f ; Section 3.2.2
D2j
Hessian matrix of f
Lipschitz constant of f ; Sections 2.4. 1, 3. l . l
263
C Function spaces
[ [) 2 /I
(matrix-valued) measure for Hessian of convex f ; Section 6.3 absolutely continuous, singular parts of [ D 2 f]; Section 6.3 graph of f over the set A; Section 2.4.2
G (f , A)
C
Function spaces
U C(U) C(U) C"' ( U) Let
c
IR" be an open set.
U ----> lR I f continuous } {! C( U) I f uniformly continuous} lR I f is k-times continuously {! U ----> differentiable} {! Ck (U) I f is uniformly continuous on U for ]a] < k} functions in C(U), C(U), etc. with compact sup port (! functions f U ----> !Rm , f , Jm ) , with fi C( U), C(U), etc. for I, . . . {! u ----> lR I (fu 1 J I P dxr < 00, {! :
E
:
E
Cc(U), Cc(U), etc.
ocr
E
1,
=
:
i
j2 , • • •
=
, m
I
LP(U)
:
f Lebesgue measurable }
Loo(U) Lfoc( U) LP (U; JL)
{!
:
U ----> lR I
(I
< p < oo)
ess sup I! < oo, u
I
f Lebesgue measurable } {! : lR f E for each open V C C U}
U----> I LP(V) {! u ----> lR I (i I J J P dJ1 ) < oo, f JL-measurable } ( l < < oo ) {! U ----> lR I f is JL-measurable, JL- ess sup I J I < oo} I
p
:
p
:
u
Sobolev space; Section 4. 1
Notation
264
{f : IR"
KP
IR 1 J > o, J E
u· ,
DJ E
LP};
Section 4.7 space of functions of bounded variation; Sec tion 5.1
BV(U)
.en
__.
Measures and capacity
D
n-dimensional Lebesgue measure
'H.{;
approximate s-dimensional Hausdorff measure; Section 2. 1 s-dimensional Hausdorff measures; Section 2. 1
'H. "
Hausdorff dimension; Section 2.1 p-capacity; Section 4.7. 1
Other notation
E
restricted to the set A; Section l . l . l
p, L A
tL
p, L j
(signed) measure with density f with respect to p,; § 1 .3 derivative of v with respect to p,; Section 1 .6. 1 v is absolutely continuous with respect to p,;
v J.. p,
§ 1.6.2 v and p, are mutually singular: Section 1 .6.2
ap limx f
approximate limit: Section 1 .7.2
y-
ap lim sup f , ap lim inf f y -+ x
s 0
y -+ x
approximate lim su p, approximate lim inf; Sec tion 1.7.2 weak convergence; Section 1 .9 symmetric linear mapping; Section 3.2.1 orthogonal linear mapping; Section 3.2. 1
L*
adjoint of L ; Section 3.2. 1
[L]
Jacobian of linear mapping L ; Section 3.2.1
A(m, n)
{A : { 1, . . . , } n
Section 3.2. 1
---->
{ l . . . . , m} I A
increasing};
265
E Other notation
I\
projection associated with tion 3.2. 1 molliliers; Section 4.2. 1
p*
..!!J!_ = I
11
JL,
.!
'
u+
A
H
'
1r
-J
A
E
A(m, n); Sec
Sobolev conjugate of p; Section 4.5. 1
hyperplane, half spaces; Section 5.7.2 approximate lim sup, lim inf for B V function; Section 5.9 set of "measure theoretic jumps" for B V function; Section 5.9 essential variation; Section 5 . I 0
Index
Aleksandrov's Theorem, 242 approximate continuity, 47 approximate limit, 46
derivative, 37 Differentiation Theorem for Radon Measures, 40
Area Formula, 96
Domi nated Convergence Theorem, 20
Besicovitch's Covering Theorem, 30 Binet-Cauchy Formula, 89
Egoroff's Theorem, 16
blow-up, 1 99 Borel measure, 4 regular, 5 Borel set, 4 bounded variation, 1 66 B Y function, 1 66 approximation by Lipschitz functions, 252 approximation by smooth functions,
172
essential variation, 2 1 6
Fatou's Lemma, 1 9 finite perimeter, 167 Fubini's Theorem, 22
Gagli ardo-Nirenberg-Sobolev inequality,
138 Gauss--Green Theorem, 209
extension, 1 83 trace, 177 weak approximation of derivative, 1 7 5
Hausdorff dimension, 65 Hausdorff measure, 6 1
capacity, 147 Caratheodory 's Criterion, 9
integrable function, 1 8
chain rule, 1 30
Isodiametric Inequality, 69
change of variables, 99, 1 1 7 Coarea Formula, 1 1 2 for B V functions, 185
Isoperimetric Inequality, 1 90 .
convex function, 236
Jacobian, 88, 9 1
densities, 7 1
Lebesgue Decomposition Theorem, 42
267
Index
268
Lebesgue measure, 26
a-finite function,
Lebesgue-Besicovitch Differentiation Theorem, 43
a-finite set, 4
Lipschitz function, 79 approximation by C 1 functions, 25 1 Lusin 's Theorem, 1 5
II
simple function, 1 7
integrable, 1 7 Sobolev function, 1 2 1 . approximation by C 1 functions, 256 approximation by smooth functions,
Monotone Convergence Theorem, 20
1 22, 1 25, 1 27 extension. 1 35 fine propert ies, 1 60 trace, 1 33 Sobolev inequality for BV functions, 1 89 Sobolev space, 1 2 1 compact imbedding, 1 44 Steiner symmetrization, 67
Morrey 's inequality, 1 43
Structure Theorem
measurable function, measurable set, 2 measure,
I
II
·
regular, 4 . measure theoretic boundary, 208 measure theoretic interior, 45 molli fier, 1 22
multiplicity function, 93
for BV functions. 1 67 for sets of finite perimeter, 205 summable function, 1 8
perimeter, 1 7 0 perimeter measure, 1 7 0 Poincare 's inequality, 1 4 1 for B V functions, 1 89 polar decomposition, 87
trace of a B V function, 1 77 of a Sobolev function , 1 3 1
precise representative, 46, 160 product measure, 22 product rule, 1 29
variation measure, 49, 1 70 lower semicontinuity, 1 7 2 Vitali's Covering Theorem, 27
quasicontinuity, 1 60 weak compactness in LP, 57
Rademacher's Theorem, 8 1 Radon measure, 5
weak compactness of measures , 55
reduced boundary, 194
weak convergence of measures, 54
Relative Isoperimetric Inequality, 1 90 Riesz Representation Theorem, 49
weak convergence in LP, 57
weak derivative, 1 20 Whitney's Extension Theorem, 245
