Taylor M.E. Partial Differential Equations III: Nonlinear Equations
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258 14. Nonlinear Elliptic Equations
then follows from Corollary 12.12 and Proposition 12.14 of Chap. 13, just as in
(12b.93). Hence Corollary 12B.5 is applicable. This result, established by [Wen],
was an important precursor to Proposition 12.13 of Chap.13.
13. Elliptic regularity IV (Krylov–Safonov estimates)
In this section we obtain estimates for solutions to second-order elliptic equations
of the form
(13.1) Lu D f; Lu D a
jk
.x/ @
j
@
k
u C b
j
.x/ @
j
u C c.x/u;
on a domain R
n
. We assume that a
jk
;b
j
,andc are real-valued and that
a
jk
2 L
1
./, with
(13.2) jj
2
a
jk
.x/
j
k
ƒjj
2
;
for certain ;ƒ 2 .0; 1/.Wedefine
(13.3) D D det .a
jk
/; D
D D
1=n
:
A. Alexandrov [Al] proved that if jbj=D
2 L
n
./ and c 0 on ,then
(13.4) u 2 C.
/ \ H
2;n
loc
./; Lu f on ;
implies
(13.5) sup
x2
u.x/ sup
y2@
u
C
.y/ C C kD
1
f k
L
n
./
;
where C D C
n; diam ;kb=D
k
L
n
. We will not make use of this and will
not include a proof, but we will establish the following result of I. Bakelman [B],
essentially a more precise version of (13.5) for the special case b
j
D c D 0
(under stronger regularity hypotheses on u). It is used in some proofs of (13.5)
(see [GT]).
To formulate this result, set
(13.6)
C
Dfy 2 W u.x/ u.y/ C p .x y/; 8 x 2 ;
for some p D p.y/ 2 R
n
g:
If u 2 C
1
./,theny belongs to
C
if and only if the graph of u lies everywhere
below its tangent plane at
y; u.y/
.Ifu 2 C
2
./,thenu is concave on
C
,that
is, .@
j
@
k
u/ 0 on
C
.
13. Elliptic regularity IV (Krylov–Safonov estimates) 259
Proposition 13.1. If u 2 C
2
./ \ C./, we have
(13.7) sup
x2
u.x/ sup
y2@
u.y/ C
d
nV
1=n
n
D
1
.a
jk
@
j
@
k
u/
L
n
.
C
/
;
where d D diam , and V
n
is the volume of the unit ball in R
n
.
To establish this, we use the matrix inequality
(13.8) .det A/.det B/
1
n
Tr AB
n
;
for positive, symmetric, n n matrices A and B. (See the exercise at the end of
this section for a proof.) Setting
(13.9) A DH.u/ D
@
j
@
k
u.x/
;BD
a
jk
.x/
;x2
C
;
where H.u/ is the Hessian matrix, as in (3.7a), we have
(13.10) jdet H.u/jD
1
1
n
a
jk
@
j
@
k
u
n
on
C
:
Thus Proposition 13.1 follows from
Lemma 13.2. Fo r u 2 C
2
./ \ C./, we have
(13.11) sup
x2
u.x/ sup
y2@
u.y/ C
d
V
1=n
n
Z
C
jdet H.u/j dx
1=n
:
Proof. Replacing u by u sup
@
u, it suffices to assume u 0 on @.Define
./ to be
S
y2
.y/,where
(13.12) .y/ Dfp 2 R
n
W u.x/ u.y/ C p .x y/; 8 x 2 g;
so .y/ ¤;,y 2
C
. Also, if u 2 C
1
./ (as we assume here),
(13.13) .y/ DfDu.y/g; for y 2
C
:
Thus the Lebesgue measure of ./ is given by
(13.14) L
n
./
D L
n
.
C
/
D L
n
Du.
C
/
Z
C
jdet H.u/j dx:
260 14. Nonlinear Elliptic Equations
Thus it suffices to show that if u 2 C./ \ C
2
./ and u 0 on @,then
(13.15) sup
x2
u.x/
d
V
1=n
n
L
n
./
:
This is basically a comparison result. Assume sup u >0is attained at x
0
.Let
W
1
be the function on whose graph is the cone with apex at
x
0
; u.x
0
/
and
base @ f0g. Then, if
W
1
.y/ denotes the function (13.12) with u replaced by
W
1
,wehave
(13.16)
u
./
W
1
./:
Similarly, if W
2
is the function on B
d
.x
0
/ whose graph is the cone with apex at
x
0
; u.x
0
/
and base fx Wjx x
0
jDdgf0g,then
(13.17)
W
1
./
W
2
B
d
.x
0
/
:
Finally, the inequality
(13.18) sup W
2
d
V
1=n
n
L
n
W
2
.B
d
.x
0
//
is elementary, so we have (13.15), and hence Lemma 13.2 is proved.
We now make the assumption that
(13.19)
ƒ
;
jbj
2
;
jcj
;
and establish the following local maximum principle, following [GT].
Proposition 13.3. Let u 2 H
2;n
./; Lu f; f 2 L
n
./. Then, for any ball
B D B
2R
.y/ and any p 2 .0; n, we have
(13.20) sup
x2B
R
.y/
u.x/ C
8
<
:
1
Vol.B/
Z
B
.u
C
/
p
dx
1=p
C
R
kf k
L
n
.B/
9
=
;
;
where C D C.n;;R
2
;p/.
Proof. Translating and dilating, we can assume without loss of generality that
0 2 and B D B
1
.0/. We will also assume that u 2 C
2
./ \ H
2;n
./,since
if (13.20) is established in this case, the more general case follows by a simple
approximation argument.
13. Elliptic regularity IV (Krylov–Safonov estimates) 261
Given ˇ 1,define
(13.21) .x/ D
1 jxj
2
ˇ
; for jxj1:
Setting v D u on B,wehave
(13.22)
a
jk
@
j
@
k
v D a
jk
@
j
@
k
u C 2a
jk
.@
j
/.@
k
u/ C ua
jk
@
j
@
k
.f b
j
@
j
u cu/ C 2a
jk
.@
j
/.@
k
u/ C ua
jk
@
j
@
k
:
Let
C
v
be as in (13.6), but with u replaced by v,and replaced by B. Clearly,
u 0 on
C
v
.Wehave
(13.23) jDvj
v
1 jxj
on
C
v
;
so
(13.24)
jDujD
1
jDv uDj
1
v
1 jxj
C ujDj
2.1 C ˇ/
1=ˇ
u on
C
v
:
Hence
(13.25)
a
jk
@
j
@
k
v
n
16ˇ
2
C 2ˇ
ƒ
2=ˇ
C 2ˇjbj
1=ˇ
C c
o
v C f
C
2=ˇ
v C f;
on
C
v
,whereC D C.n;ˇ;;/. Of course, a
jk
@
j
@
k
v 0 on
C
v
.Ifˇ 2,we
have, upon applying Proposition 13.1 to v,
(13.26)
sup
B
v C
2=ˇ
v
C
L
n
.B/
C
1
kf k
L
n
.B/
C
1
sup
B
v
C
12=ˇ
.u
C
/
2=ˇ
L
n
.B/
C
1
kf k
L
n
.B/
:
Choose ˇ D 2n=p 2,sowehave
(13.27) sup
B
v C
1
sup
B
v
C
1p=n
ku
C
k
p=n
L
p
.B/
C
1
kf k
L
n
.B/
:
(Hereweallowp<1, in which case kk
L
p
is not a norm, but (13.27) is still
valid.) Using the elementary inequality
(13.28) a
1p=n
b
p=n
"a C "
n=p1
b; 8 " 2 .0; 1/;
262 14. Nonlinear Elliptic Equations
and taking a D sup
B
v
C
;bDku
C
k
L
p
.B/
,and" D 1=2C
1
,wehave(theR D 1
case of) (13.20), so Proposition 13.3 is proved.
Replacing u by u,wehaveanestimateonsup
B
R
.y/
.u/ when Lu f .
Thus, when Lu D f and the hypotheses of Proposition 13.3 hold, we have
(13.29) sup
B
R
.y/
jujC
8
<
:
1
Vo l .B/
Z
B
juj
p
dx
1=p
C
R
kf k
L
n
.B/
9
=
;
:
Next we establish a “weak Harnack inequality” of [KrS], which will lead to
results on H¨older continuity of solutions of Lu D f . This result will also be
applied directly in the next section, to results on solutions to certain completely
nonlinear equations.
Proposition 13.4. Assume u 2 H
2;n
./; Lu f in ; f 2 L
n
./, and u 0
on a ball B D B
2R
.y/ .Then
(13.30)
0
B
@
1
Vol.B
R
/
Z
B
R
u
p
dx
1
C
A
1=p
C
inf
B
R
u C
R
kf k
L
n
.B/
;
for some positive p D p.n;; R
2
/ and C D C.n;;R
2
/.
As before, there is no loss of generality in assuming B D B
1
.0/. Also, replac-
ing L and f by
1
L and
1
f , we can assume D 1.
To begin the proof, take ">0and set
(13.31)
u D u C " Ckf k
L
n
.B/
;wD log
1
u
;
v D w; g D
f
u
;
where is given by (13.21). Note that w is large (positive) where
u is small. We
have
(13.32)
a
jk
@
j
@
k
v Da
jk
@
j
@
k
w 2a
jk
.@
j
/.@
k
w/ wa
jk
@
j
@
k
a
jk
.@
j
w/.@
k
w/ C b
j
@
j
w CjcjCg
2a
jk
.@
j
/.@
k
w/ wa
jk
@
j
@
k
2
a
jk
.@
j
/.@
k
/ wa
jk
@
j
@
k
C .jbj
2
CjcjCg/;
where the last inequality is obtained via Cauchy’s inequality, applied to the inner
product hV;W iDV
j
a
jk
W
k
.
13. Elliptic regularity IV (Krylov–Safonov estimates) 263
Now the form of implies that a
jk
@
j
@
k
0 provided 2.ˇ 1/a
jk
x
j
x
k
C
a
jj
jxj
2
a
jj
, and hence
(13.33) 2ˇjxj
2
nƒ H) a
jk
@
j
@
k
0:
Thus, if ˛ 2 .0; 1/,then
(13.34) ˇ
n
2˛
; jxj˛ H) a
jk
@
j
@
k
0:
Hence, on the set B
C
Dfx 2 B W w.x/ > 0g,wehave
(13.35)
a
jk
@
j
@
k
v 4ˇ
2
1 jxj
2
ˇ2
jxj
2
C v
B
˛
sup
B
˛
a
jk
@
j
@
k
!
C
jbj
2
CjcjCg
4ˇ
2
ƒ Cjbj
2
CjcjCg C
2nˇƒ
1 ˛
2
v
B
˛
:
Note that kgk
L
n
.B/
1. Thus Proposition 13.1 yields
(13.36) sup
B
v C
1 Ckv
C
k
L
n
.B
˛
/
;
with C D C.n;˛;;/.
Note that if u satisfies the hypotheses of Proposition 13.4 and t 2 .0; 1/,then
u=t satisfies L.u=t/ f=t, and the analogue of w in (13.31)isw k,where
k D log.1=t/. The function g in (13.31) is unchanged, and, working through
(13.32)–(13.36), we obtain the following extension of (13.36):
(13.37) sup
B
.w k/ C
1 Ck.w k/
C
k
L
n
.B
˛
/
; 8 k 2 R;
with constants independent of k.
The next stage in the proof of Proposition 13.4 will involve a decomposition
into cubes of the sort used for Calderon–Zygmund estimates in 5 of Chap. 13.
To set up some notation, given y 2 R
n
;R>0,letQ
R
.y/ denote the open cube
centered at y, of edge 2R:
(13.38) Q
R
.y/ Dfx 2 R
n
Wjx
j
y
j
j <R; 1 j ng:
If ˛<1=
p
n,thenQ
˛
D Q
˛
.0/ B.
The cube decomposition we will use in the proof of Lemma 13.5 below can
be described in general as follows. Let Q
0
be a cube in R
n
,let' 0 be an
element of L
1
.Q
0
/, and suppose
R
Q
0
'dx tL
n
.Q
0
/; t 2 .0; 1/. Bisecting
the edges of Q
0
, we subdivide it into 2
n
subcubes. Those subcubes that satisfy
264 14. Nonlinear Elliptic Equations
R
Q
'dx tL
n
.Q/ are similarly subdivided, and this process is repeated
indefinitely. Let F denote the set of subcubes so obtained that satisfy
Z
Q
'dx>tL
n
.Q/I
we do not further subdivide these cubes. For each Q 2 F, denote by
e
Q the
subcube whose subdivision gives Q.SinceL
n
.
e
Q/=L
n
.Q/ D 2
n
, we see that
(13.39) t<
1
L
n
.Q/
Z
Q
'dx 2
n
t; 8 Q 2 F:
Also, setting F D
S
Q2F
Q and G D Q
0
n F ,wehave
(13.40) ' t; a.e. in G:
This subdivision was also done in the proof of Lemma 5.5 in Chap. 13. Let us
also set
e
F D
S
Q2F
e
Q;sinceQ 2 F )
e
Q … F ,wehave
(13.41)
Z
e
F
'dx tL
n
.
e
F/:
In particular, when ' is the characteristic function
of a measurable subset
of Q
0
, of measure t L
n
.Q
0
/, we deduce from (13.40)–(13.41)that
(13.42) L
n
./ D L
n
. \
e
F/ tL
n
.
e
F/:
We have the following measure-theoretic result:
Lemma 13.5. Let Q
0
be a cube in R
n
;w2 L
1
.Q
0
/, and, for k 2 R,set
(13.43)
k
Dfx 2 Q
0
W w.x/ kg:
Suppose there are positive constants ı<1and C such that
(13.44) sup
Q
0
\Q
3r
.z/
.w k/ C
whenever k and Q D Q
r
.z/ Q
0
satisfy
(13.45) L
n
.
k
\ Q/ ıL
n
.Q/:
13. Elliptic regularity IV (Krylov–Safonov estimates) 265
Then, for all k 2 R,
(13.46) sup
Q
0
.w k/ C
1 C
log
L
n
.
k
/=L
n
.Q
0
/
log ı
!
:
Proof. We show by induction that
(13.47) sup
Q
0
.w k/ mC;
for any m 2 Z
C
and k 2 R such that L
n
.
k
/ ı
m
L
n
.Q
0
/.Thisistrueby
hypothesis if m D 1. Suppose that it holds for m D M 2 Z
C
and that L
n
.
k
/
ı
M C1
L
n
.Q
0
/.Define
e
k
by
(13.48)
e
k
D
[
˚
Q
3r
.z/ \ Q
0
W L
n
Q
r
.z/ \
k
ı L
n
Q
r
.z/
:
Applying the estimate (13.42), with t D ı, we see that either
e
k
D Q
0
or
(13.49) L
n
.
e
k
/ ı
1
L
n
.
k
/ ı
M
vol.Q
0
/;
and hence, replacing k by k C C , we obtain
(13.50) sup
Q
0
.w k/ .M C 1/C;
which verifies (13.47)form D M C 1.
Now, the estimate (13.46) follows by choosing m appropriately, and the lemma
is proved.
Returning to the estimation of the functions defined in (13.31), we see that
(13.36) implies
(13.51) sup
B
v C
1 Ckv
C
k
L
n
.Q
˛
/
C
1 C
vol.Q
C
˛
/
1=n
sup
B
v
C
;
where Q
˛
D Q
˛
.0/, as stated below (13.38), and
Q
C
˛
Dfx 2 Q
˛
W v.x/ > 0gDfx 2 Q
˛
W u.x/ < 1g:
Hence, if C is the constant in (13.36),
(13.52)
vol.Q
C
˛
/
vol.Q
˛
/
1
4˛C
n
D H) sup
B
v 2C:
Now choose ˛ D 1=3n,andtake D .4˛C /
n
,asin(13.52). Using the coor-
dinate change x 7! ˛.x z/=r, we obtain for any cube Q D Q
r
.z/ such that
B
3nr
.z/ B, the implication
266 14. Nonlinear Elliptic Equations
(13.53)
vol.Q
C
/
vol.Q/
H) sup
Q
3r
.z/
w C.n;;/:
With ˛ and as specified above, take ı D 1 ; Q
0
D Q
˛
.0/, and note that
the estimate (13.53) holds also when w is replaced by w k,andQ
C
is replaced
by the set fx 2 Q W w.x/ k>0g, as a consequence of (13.37). Let
(13.54) .t/ D L
n
fx 2 Q
0
W u.x/ > tg
:
Setting k D log 1=t, we have from Lemma 13.5 the estimate
(13.55) .t/ C
inf
Q
0
t
1
u
; 8 t>0;
where C D C.n;;/; D .n;;/. Replacing the cube Q
0
by the inscribed
ball B
˛
.0/; ˛ D 1=3n, and using the identity
(13.56)
Z
Q
0
.u/
p
dx D p
Z
1
0
t
p1
.t/ dt;
we have
(13.57)
Z
B
˛
.u/
p
dx C
inf
B
˛
u
p
; for p D
2
:
The inequality (13.30) then follows by letting " ! 0 if we use a covering
argument to extend (13.57) to arbitrary ˛<1(especially, ˛ D 1=2) and use the
coordinate transformation x 7! .x y/=2R. Thus Proposition 13.4 is established.
Putting together (13.29)and(13.30), we have the following.
Corollary 13.6. Assume u 2 H
2;n
./; Lu D f on ; f 2 L
n
./, and u 0
on a ball B D B
4R
.y/ .Then
(13.58) sup
B
R
.y/
u.x/ C
1
inf
B
2R
.y/
u C
R
kf k
L
n
.B
4R
/
;
for some C
1
D C
1
.n;;R
2
/. In particular, if u 0 on ,
(13.59) Lu D 0 H) sup
B
R
.y/
u.x/ C
1
inf
B
2R
.y/
u.x/:
We can use this to establish H¨older estimates on solutions to Lu D f . We will
actually apply Corollary 13.6 to L
1
D a
jk
@
j
@
k
Cb
j
@
j
,soL
1
u D f
1
D f cu.
Suppose that
13. Elliptic regularity IV (Krylov–Safonov estimates) 267
(13.60) a D inf
B
4R
.y/
u sup
B
4R
.y/
u D b:
Then v D .ua/=.b a/ is 0 on B
4R
.y/,andL
1
v D f
1
=.b a/, so Corollary
13.6 yields
(13.61) sup
B
R
.y/
u a
b a
C
1
inf
B
2R
.y/
u a
b a
C
R
1
b a
kf cuk
L
n
.B
4R
/
:
Without loss of generality, we can assume C
1
1. Now given this, one of the
following two cases must hold:
(i) C
1
inf
B
2R
.y/
u a
b a
1
2
sup
B
R
.y/
u a
b a
;
(ii) C
1
inf
B
2R
.y/
u a
b a
<
1
2
sup
B
R
.y/
u a
b a
:
If case (i) holds, then either
sup
B
R
.y/
u a
b a
1
2
or inf
B
2R
.y/
u a
b a
1
4C
1
;
and hence (since we are assuming C
1
1)
(13.62) (i) H) osc
B
R
.y/
u
1
1
4C
1
osc
B
4R
.y/
u:
If case (ii) holds, then
sup
B
R
.y/
u a
b a
2R
1
b a
kf cuk
L
n
.B
4R
/
;
so
(13.63) (ii) H) osc
B
R
.y/
u
2R
kf cuk
L
n
.B
4R
/
;
which is bounded by C
2
R in view of the sup-norm estimate (13.29).Consequently,
under the hypotheses of Corollary 13.6, we have
(13.64) osc
B
R
.y/
u max
C
2
R;
1
1
C
1
osc
B
4R
.y/
u
;