Taylor M.E. Partial Differential Equations III: Nonlinear Equations
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198 14. Nonlinear Elliptic Equations
We will obtain such an estimate in terms of the Sobolev constants .
j
/ and C
j
,
defined below. Ingredients for the analysis include the following two lemmas, the
first being a standard Sobolev inequality.
Lemma 9.1. Fo r v 2 H
1
.
j
/; n=.n 2/,
(9.10) kv
k
2
L
2
.
j
/
.
j
/
krvk
2
L
2
.
j
/
Ckvk
2
L
2
.
j
/
:
The next lemma follows from (9.6)ifwetake D 1 on
j C1
, tending roughly
linearly to 0 on @
j
.
Lemma 9.2. If v>0is a subsolution of L, then, with C
j
D C.
j
;
j C1
/,
(9.11) krvk
L
2
.
j C1
/
C
j
kvk
L
2
.
j
/
:
Under the geometrical conditions indicated above on
j
, we can assume
(9.12) .
j
/
0
;C
j
C.j
2
C 1/:
Putting together the two lemmas, we see that when v satisfies (9.8),
(9.13)
kv
k
2
L
2
.
j C1
/
.
j C1
/
h
C
2
j
kvk
2
L
2
.
j
/
Ckvk
2
L
2
.
j C1
/
i
0
.C
2
j
C 1/kvk
2
L
2
.
j
/
:
Fix 2 .1; n=.n 2/.Now,ifv satisfies (9.8), so does
(9.14) v
j
D v
j
;
by (9.4). Note that v
j C1
D v
j
.Nowlet
(9.15) N
j
Dkvk
L
2
j
.
j
/
Dkv
j
k
1=
j
L
2
.
j
/
;
so
(9.16) kvk
L
1
.O/
lim sup
j !1
N
j
:
If we apply (9.13)tov
j
,wehave
(9.17) kv
j C1
k
2
L
2
.
j C1
/
0
.C
2
j
C 1/kv
j
k
2
L
2
.
j
/
:
Note that the left side is equal to N
2
j C1
j C1
, and the norm on the right is equal to
N
2
j C1
j
. Thus (9.17) is equivalent to
9. Elliptic regularity III (DeGiorgi–Nash–Moser theory) 199
(9.18) N
2
j C1
h
0
.C
2
j
C 1/
i
1=
j C1
N
2
j
:
By (9.12), C
2
j
C 1 C
0
.j
4
C 1/,so
(9.19)
lim sup
j !1
N
2
j
1
Y
j D0
h
0
C
0
.j
4
C 1/
i
1=
j C1
N
2
0
.
0
C
0
/
1=.1/
2
4
exp
1
X
j D0
j 1
log.j
4
C 1/
3
5
N
2
0
K
2
N
2
0
;
for finite K. This gives Moser’s sup-norm estimate:
Theorem 9.3. If v>0is a subsolution of L,then
(9.20) kvk
L
1
.O/
Kkvk
L
2
.
0
/
;
where K D K.
0
;C
0
;n/.
H¨older continuity of a solution to Lu D 0 will be obtained as a consequence
of the following “Harnack inequality.” Let B
Dfx Wjxj <g.
Proposition 9.4. Let u 0 be a solution of Lu D 0 in B
2r
.Pickc
0
2 .0; 1/.
Suppose
(9.21) measfx 2 B
r
W u.x/ 1g >c
1
0
r
n
:
Then there is a constant c>0such that
(9.22) u.x/ > c
1
in B
r=2
:
This will be established by examining v D f.u/ with
(9.23) f.u/ D maxflog.u C "/; 0g;
where " is chosen in .0; 1/. Note that f is convex, so v is a subsolution. Our first
goal will be to estimate the L
2
.B
r
/-norm of rv. Once this is done, Theorem 9.3
will be applied to estimate v from above (hence u from below) on B
r=2
.
We begin with a variant of (9.5), obtained by taking w D
2
f
0
.u/ in (9.3).
The identity (for smooth f )is
(9.24)
Z
2
f
00
jruj
2
dV C 2
Z
h f
0
ru; r i dV D.Lu;
2
f
0
/:
200 14. Nonlinear Elliptic Equations
This vanishes if Lu D 0. Applying Cauchy’s inequality to the second integral, we
obtain
(9.25)
Z
2
h
f
00
.u/ ı
2
f
0
.u/
2
i
jruj
2
dV
1
ı
2
Z
jr j
2
dV:
Now the function f.u/ in (9.23) has the property that
(9.26) h De
f
is a convex functionI
indeed, in this case h.u/ D maxf.u C "/; 1g. Thus
(9.27) f
00
.f
0
/
2
D e
f
h
00
0:
Thus f
00
.u/jruj
2
f
0
.u/
2
jruj
2
Djrvj
2
if v D f.u/.Takingı
2
D 1=2 in
(9.25), we obtain
(9.28)
Z
2
jrvj
2
dV 4
Z
jr j
2
dV;
after one overcomes the minor problem that f
0
has a jump discontinuity. If we
pick to D 1 on B
r
and go linearly to 0 on @B
2r
, we obtain the estimate
(9.29)
Z
B
r
jrvj
2
dV Cr
n2
;
for v D f.u/, given that Lu D 0 and that (9.26) holds.
Now the hypothesis (9.21) implies that v vanishes on a subset of B
r
of measure
>c
1
0
r
n
. Hence there is an elementary estimate of the form
(9.30) r
n
Z
B
r
v
2
dV Cr
2n
Z
B
r
jrvj
2
dV;
which is bounded from above by (9.29). Now Theorem 9.3, together with a simple
scaling argument, gives
(9.31) v.x/
2
Cr
n
Z
B
r
v
2
dV C
2
1
;x2 B
r=2
;
so
(9.32) u C" e
C
1
; for x 2 B
r=2
;
for all " 2 .0; 1/.Taking" ! 0, we have the proof of Proposition 9.4.
9. Elliptic regularity III (DeGiorgi–Nash–Moser theory) 201
We remark that Moser obtained a stronger Harnack inequality in [Mo3], by a
more elaborate argument. In that work, the hypothesis (9.21) is weakened to
(9.21a) sup
B
r
u.x/ 1:
To deduce the H¨older continuity of a solution to Lu D 0 given Proposition 9.4
is fairly simple. Following [Mo2], who followed DeGiorgi, we have from (9.20)
a bound
(9.33) ju.x/jK
on any compact subset O of
0
,givenu 2 H
1
.
0
/; Lu D 0.Fixx
0
2 O,such
that B
.x
0
/ O, and, for r ,let
(9.34) !.r/ D sup
B
r
u.x/ inf
B
r
u.x/;
where B
r
D B
r
.x
0
/. Clearly, !./ 2K. Adding a constant to u, we can assume
(9.35) sup
B
u.x/ Dinf
B
u.x/ D
1
2
!./ D M:
Then u
C
D 1 C u=M and u
D 1 u=M are also annihilated by L.Theyare
both 0 and at least one of them satisfies the hypothesis (9.21), with r D =2.
If, for example, u
C
does, then Proposition 9.4 implies
(9.36) u
C
.x/ > c
1
in B
=4
;
so
(9.37) M
1
1
c
u.x/ M in B
=4
:
Hence
(9.38) !.=4/
1
1
2c
!./;
which gives H¨older continuity:
(9.39) !.r/ !./
r
˛
;˛Dlog
4
1
1
2c
:
We state the result formally.
202 14. Nonlinear Elliptic Equations
Theorem 9.5. If u 2 H
1
.
0
/ solves Lu D 0, then for every compact O in
0
,
there is an estimate
(9.40) kuk
C
˛
.O/
C kuk
L
2
.
0
/
:
It will be convenient to replace (9.40) by an estimate involving Morrey spaces,
which are discussed in Appendix A at the end of this chapter. We claim that under
the hypotheses of Theorem 9.5,
(9.41) ru
ˇ
ˇ
O
2 M
p
2
;pD
n
1 ˛
;
where the Morrey space M
p
2
consists of functions f satisfying the q D 2 case
of (A.2). The property (9.41) is stronger than (9.40), by Morrey’s lemma (Lemma
A.1). To see (9.41), if B
R
is a ball of radius R centered at y; B
2R
,thenlet
c D u.y/ and replace u by u.x/ c in (9.6), to get
1
2
Z
2
jruj
2
dV 2
Z
ju.x/ cj
2
jr j
2
dV:
Taking D 1 on B
R
, going linearly to 0 on @B
2R
,gives
(9.42)
Z
B
R
jruj
2
dV CR
n2C2˛
;
as needed to have (9.41).
So far we have dealt with the homogeneous equation, Lu D 0. We now turn to
regularity for solutions to a nonhomogeneous equation. We will follow a method
of Morrey, and Morrey spaces will play a very important role in this analysis. We
take L as in (9.1), with a
jk
measurable, satisfying
(9.43) 0<
0
jj
2
X
a
jk
.x/
j
k
1
jj
2
;
while for simplicity we assume b; b
1
2 Lip./.WeconsideraPDE
(9.44) Lu D f:
It is clear that, for u 2 H
1
0
./,
(9.45) .Lu; u/ C
X
k@
j
uk
2
L
2
;
so we have an isomorphism
(9.46) L W H
1
0
./
! H
1
./:
9. Elliptic regularity III (DeGiorgi–Nash–Moser theory) 203
Thus, for any f 2 H
1
./,(9.44) has a unique solution u 2 H
1
0
./. One can
write such f as
(9.47) f D
X
@
j
g
j
;g
j
2 L
2
./:
The solution u 2 H
1
0
./ then satisfies
(9.48) kuk
2
H
1
./
C
X
kg
j
k
2
L
2
:
Here C depends on ;
0
;
1
,andb 2 Lip./.
One can also consider the boundary problem
(9.49) Lv D 0 on ; v D w on @;
given w 2 H
1
./, where the latter condition means v w 2 H
1
0
./. Indeed,
setting v D u C w, the equation for u is Lu DLw; u 2 H
1
0
./. Thus (9.49)is
uniquely solvable, with an estimate
(9.50) krvk
L
2
./
C krwk
L
2
./
;
where C has a dependence as in (9.48).
Our present goal is to give Morrey’s proof of the following local regularity
result.
Theorem 9.6. Suppose u 2 H
1
./ solves (9.44), with f D
P
@
j
g
j
;g
j
2
M
q
2
./; q > n, that is,
(9.51)
Z
B
r
jg
j
j
2
dV K
2
1
r
R
n2C2
;D 1
n
q
2 .0; 1/:
Assume L is of the form (9.1), where the coefficients a
jk
satisfy (9.43) and
b; b
1
2 Lip./.LetO , and assume <
0
D ˛, for which Theorem
9.5 holds. Then u 2 C
.O/; more precisely, ru 2 M
q
2
.O/, that is,
(9.52)
Z
B
r
jruj
2
dV K
2
2
r
R
n2C2
:
Morrey established this by using (9.48), (9.50), and an elegant dilation argu-
ment, in concert with Theorem 9.5. For this, suppose B
R
D B
R
.y/ for each
y 2 O. We can write u D U C H on B
R
,where
(9.53)
LU D
X
@
j
g
j
on B
R
;U2 H
1
0
.B
R
/;
LH D 0 on B
R
;Hu 2 H
1
0
.B
R
/;
204 14. Nonlinear Elliptic Equations
and we have
(9.54) krU k
L
2
.B
R
/
C
1
kgk
L
2
.B
R
/
; krH k
L
2
.B
R
/
C
2
kruk
L
2
.B
R
/
;
where kgk
2
L
2
D
P
kg
j
k
2
L
2
.Letusset
(9.55) kF k
r
DkF k
L
2
.B
r
/
:
Also let .g
j
;R/be the best constant K
1
for which (9.51) is valid for 0<r R.
If g
.x/ D g.x/, note that
.g
;
1
S/ D
n=2
.g; S/:
Now define
(9.56)
'.r/ D sup
˚
krU k
rS
W U 2 H
1
0
.B
S
/; LU D
X
@
j
g
j
; on B
S
;
.g
j
;S/ 1; 0 < S R
:
Let us denote by '
S
.r/ the sup in (9.56) with S fixed, in .0; R.Then'
S
.r/
coincides with '
R
.r/, with L replaced by the dilated operator, coming from the
dilation taking B
S
to B
R
. More precisely, the dilated operator is
(9.57) L
S
D b
S
@
j
a
jk
S
b
1
S
@
k
;
with
a
jk
S
.x/ D a
jk
.SR
1
x/; b
S
.x/ D b.SR
1
x/;
assuming 0 has been arranged to be the center of B
R
. To see this, note that if
D S=R, U
.x/ D
1
U.x/,andg
j
.x/ D g
j
.x/,then
(9.58) LU D
X
@
j
g
j
” L
S
U
D
X
@
j
g
j
:
Also, rU
.x/ D .rU /.x/,sokrU
k
S=
D
n=2
krU k
S
.
Now for this family L
S
, one has a uniform bound on C in (9.48); hence '.r/ is
finite for r 2 .0; 1. We also note that the bounds in (9.40)and(9.42) are uniformly
valid for this family of operators. Theorem 9.6 will be proved when we show that
(9.59) '.r/ Ar
n=21C
:
In fact, this will give the estimate (9.52) with u replaced by U ; meanwhile such
an estimate with u replaced by H is a consequence of (9.42). Let H satisfy (9.42)
with ˛ D
0
.Wetake<
0
.
Pick S 2 .0; R and pick g
j
satisfying (9.51), with R replaced by S and K
1
by
K. Write the U of (9.53)asU D U
S
CH
S
on B
S
,whereU
S
2 H
1
0
.B
S
/; LU
S
D
LU D
P
@
j
g
j
on B
S
.Clearly,(9.51) implies
9. Elliptic regularity III (DeGiorgi–Nash–Moser theory) 205
(9.60)
Z
B
r
jg
j
j
2
dV K
2
S
R
n2C2
r
S
n2C2
:
Thus, as in (9.54) (and recalling the definition of '), we have
(9.61)
krU
S
k
S
A
1
K
S
R
n=21C
;
krH
S
k
S
A
2
krU k
S
A
2
K'
S
R
:
Now, suppose 0<r<S<R. Then, applying (9.42)toH
S
,wehave
(9.62)
krU k
r
krU
S
k
r
CkrH
S
k
r
K
S
R
n=21C
'
r
S
C A
3
K'
S
R
r
S
n=21C
0
:
Therefore, setting s D r=R; t D S=R, we have the inequality
(9.63) '.s/ t
n=21C
'
s
t
C A
3
'.t/
s
t
n=21C
0
;
valid for 0<s<t 1. Since it is clear that '.r/ is monotone and finite on .0; 1,
it is an elementary exercise to deduce from (9.63)that'.r/ satisfies an estimate
of the form (9.59), as long as <
0
. This proves Theorem 9.6.
Now that we have interior regularity estimates for the nonhomogeneous prob-
lem, we will be able to use a few simple tricks to establish regularity up to the
boundary for solutions to the Dirichlet problem
(9.64) Lu D
X
@
j
g
j
; u D f on @;
where L has the form (9.1),
is compact with smooth boundary, f 2 Lip.@/,
and g
j
2 L
q
./, with q>n.First,extendf to f 2 Lip./.Thenu D v C f ,
where v solves
(9.65) Lv D
X
@
j
h
j
;vD 0 on @;
where
(9.66) @
j
h
j
D @
j
g
j
b
1
@
j
a
jk
b@
k
f
:
We will assume b 2 Lip.
/;thenh
j
can be chosen in L
q
also.
The class of equations (9.65) is invariant under smooth changes of variables
(indeed, invariant under Lipschitz homeomorphisms with Lipschitz inverses, hav-
ing the further property of preserving volume up to a factor in Lip.
/). Thus
206 14. Nonlinear Elliptic Equations
make a change of variables to flatten out the boundary (locally), so we consider a
solution v 2 H
1
to (9.65)inx
n
>0; jxjR. We can even arrange that b D 1.
Now extend v to negative x
n
,tobeodd under the reflection x
n
7!x
n
.Alsoex-
tend a
jk
.x/ to be even when j; k < n or j D k D n, and odd when j or k D n
(but not both). Extend h
j
to be odd for j<nand even for j D n. With these
extensions, we continue to have (9.65) holding, this time in the ball jxjR. Thus
interior regularity applies to this extension of v, yielding H¨older continuity. The
following is hence proved.
Theorem 9.7. Let u 2 H
1
./ solve the PDE
(9.67)
X
b
1
@
j
a
jk
b@
k
u
D
X
@
j
g
j
on ; u D f on @:
Assume g
j
2 L
q
./ with q>nD dim , and f 2 Lip.@/. Assume that
b; b
1
2 Lip./ and that .a
jk
/ is measurable and satisfies the uniform ellipticity
condition (9.43).ThenuhasaH
¨
older estimate
(9.68) kuk
C
./
C
1
X
kg
j
k
L
q
./
Ckf k
Lip.@/
:
More precisely, if D 1 n=q 2 .0; 1/ is sufficiently small, then ru belongs to
the Morrey space M
q
2
./, and
(9.69) kruk
M
q
2
./
C
2
X
kg
j
k
L
q
./
Ckf k
Lip.@/
:
In these estimates, C
j
D C
j
.;
1
;
2
;b/.
So far in this section we have looked at differential operators of the form
(9.1)inwhich.a
jk
/ is symmetric, but unlike the nondivergence case, where
a
jk
.x/ @
j
@
k
u D a
kj
.x/ @
j
@
k
u, nonsymmetric cases do arise; we will see an ex-
ample in 15. Thus we briefly describe the extension of the analysis of (9.1)to
(9.70) Lu D b
1
@
j
Œa
jk
C !
jk
b @
k
u
:
We make the same hypotheses on a
jk
.x/ and b.x/ as before, and we assume
.!
jk
/ is antisymmetric and bounded:
(9.71) !
jk
.x/ D!
kj
.x/; !
jk
2 L
1
./:
We thus have both a positive symmetric form and an antisymmetric form defined
at almost all x 2 :
(9.72) hV; W iDV
j
a
jk
.x/W
k
;ŒV;WD V
j
!
jk
.x/W
k
:
9. Elliptic regularity III (DeGiorgi–Nash–Moser theory) 207
We use the subscript L
2
to indicate the integrated quantities:
(9.73) hv; wi
L
2
D
Z
hv; wi dV; Œv;w
L
2
D
Z
Œv; w dV:
Then, in place of (9.3), we have
(9.74) .Lu;w/ Dhru; rwi
L
2
Œru; rw
L
2
:
The formula (9.4) remains valid, with jruj
2
Dhru; rui, as before. Instead of
(9.5), we have
(9.75)
Z
2
jruj
2
dV D2h ru; ur i
L
2
2Πru; ur
L
2
Z
2
gu dV;
when Lu D g on and 2 C
1
0
./. This leads to a minor change in (9.6):
(9.76)
1
2
Z
2
jruj
2
dV .2 C C
0
/
Z
juj
2
jr j
2
dV
Z
2
gu dV;
where C
0
is determined by the operator norm of .!
jk
/, relative to the inner prod-
uct h ; i.
From here, the proofs of Lemmas 9.1 and 9.2, and that of Theorem 9.3,go
through without essential change, so we have the sup-norm estimate (9.20). In the
proof of the Harnack inequality, (9.24) is replaced by
(9.77)
Z
2
f
00
jruj
2
dV C 2h f
0
ru; r i
L
2
C 2Πf
0
ru; r
L
2
D.Lu;
2
f
0
/:
Hence (9.25) still works if you replace the factor 1=ı
2
by .1 C C
1
/=ı
2
,where
again C
1
is estimated by the size of .!
jk
/. Thus Proposition 9.4 extends to our
present case, and hence so does the key regularity result, Theorem 9.5.Letus
record what has been noted so far:
Proposition 9.8. Assume Lu has the form (9.70), where .a
jk
/ and b satisfy the
hypotheses of Theorem 9.5, and .!
jk
/ satisfies (9.71). If u 2 H
1
.
0
/ solves
Lu D 0, then, for every compact O
0
, there is an estimate
(9.78) kuk
C
˛
.O/
C kuk
L
2
.
0
/
:
The Morrey space estimates go through as before, and the analysis of (9.64)is
also easily modified to incorporate the change in L. Thus we have the following:
Proposition 9.9. The boundary regularity of Theorem 9.7 extends to the opera-
tors L of the form (9.70), under the hypothesis (9.71)on.!
jk
/.