DiBenedetto E. Degenerate Parabolic Equations
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46
ill.
HOlder
continuity of solutions
of
degenerate
parabolic equations
Remark
3.1. The proof of Proposition
3.1
will show that indeed it
is
sufficient
to
work with
the
number w
and
the
cylinder Q
(aoRP,
R)
linked
by
(3.1)
essosc u <
w.
Q(aoRp,R) -
This fact
is
in
general not verifiable, for a given
box,
since its dimensions would
have to
be
intrinsically defined
in
terms
of
the essential oscillation of u within it.
Therefore the role of having introduced the cylinder Q
(RP-E,
2R)
and hav-
ing
assumed (2.2)
is
that (3.1) holds true for
the
constructed
box
Q
(aoRP,
R).
It
will
be
part of the proof of
Pf<)position
3.1
to show that at each step
the
cylinders
Q(n)
and
the
essential oscillation
of
u within
them
satisfy
the
intrinsic geometry
dictated
by (3.1).
To
begin the proof. inside Q (
aoRP
,
R)
consider subcylinders of smaller
size
constructed
as
follows. The number w being fixed. let
So
be
the smallest positive
integer
such that
(3.2)
w
-
<6
2
8
0 _
0,
where the number 6
0
is
introduced
in
(3.11) of
Chap.
II
in
the
derivation of the
local energy estimates. Then construct cylinders
(3.3)
[(0,
t) + Q (dR",
R)]
,
R
Figure
3.1
3.
The main proposition
47
These are contained inside Q
(aoRP,
R) if the number A
is
chosen larger that
2
80
and if t ranges over
_ {AP-2 _ (2
80
}P-2}
RP
< t <
o.
w
p
-
2
The structure of
the
proof
is
based
on
studying separately
two
cases. Either
we
can find a cylinder of the type of
[(0,
l)
+ Q
(dRP,
R)] where u
is
mostly large,
or such a cylinder cannot
be
found.
In either case
the
conclusion
is
that the essential
oscillation of u
in
a smaller cylinder about
(xo,
to)
decreases
in
a
way
that can
be
quantitatively measured.
In
the arguments to follow
we
assume (2.2)
is
in
force
and determine later the numbers
A, e and
eo.
Remark
3.2. For later
use
we
estimate the quantity
(
w )
-2
1!1!±.cl
G(w,
R}
==
'YRNIt
2
8
0
dr,
where
'Y
is
a constant depending only upon the data and
K.
is
defined
in
(3.5) of
Chap. II. From the defu\ition of
din
(3.3) it follows that
(3.4)
and
G(w
R}
< A
RNltw-
b
, _ I ,
where b = 2 +
(p
_
2)
p(1 +
K.}
r
Al
= A
2
+(P-2)P(1;,,)
•
Along the proof
we
will encounter quantities of the type
AiRNltw-b,
i =
1,2,
...
,t, where
Ai
are constants that can
be
determined a priori only
in
terms of
the data and are independent of
w
and
R.
We
may
assume without loss of generality
that they satisfy
(3.5)
Indeed if not,
we
would have
w:5
C
Reo
for the choices
C = max
A~/b
l~i9
t
and
and
the first iterative step of
the
Proposition would
be
trivial.
Remark
3.3. The proof below and (2.2) show that
the
numbers e
and
eo
can
be
taken
as
NK.
eo
=
b'
e =
(p-
2}eo.
In
the estimates to follow
we
denote
with
'Y
a generic positive constant that
can be calculated a priori depending only
upon
the data and that
may
be
different
in
different contexts.
48
ill.
HOlder
continuity of solutions
of
degenerate parabolic equations
3-(i). About the dependence on
lIulloo.n
T
We
will use the energy and logarithmic estimates
of
Propositions 3.1 and 3.2
of
Chap. II for the truncated functions
(u
-
k)±
over cylinders contained in
Q
(aoRP
,
R).
When working with
(u
-
k)
_ we will use the levels
for some
i
~
o.
These levels are admissible since
lI(u
-
k)-lIoo.Q(IJoRP.R)
::;;
6
0
•
When working with
(u
-
k)
+ we will take levels
for some
i
~
o.
These are also admissible since
II
(u - k)+
lloo.Q(lJoRP.R)
::;;
6
0
•
Let us fix 6
0
as
in (3.11)
of
Chap. II. Then, sincew::;;2I1ull
oo
.nT'
(3.2) holds
true
if
we choose So so large that
2
80
=
8~2I1ulloo.nT.
Having chosen
So
this way. (3.2) is verified when working within
any
subdomain
of
D
T
•
The a priori knowledge
of
the norm
lIulloo.n
T
is required through the number
So.
If
the lower order terms b(x, t,
u,
Du)
in (1.1) satisfy
(A~)
of
§5
of
Chap. II,
then. as remarked there. the energy and logarithmic inequalities hold true for the
truncated functions
(u
-
k)±
with
no
restriction
on
the levels k. Thus in such a
case
So
can be taken to be one and no a priori knowledge
of
lIulloo.nT
is needed.
The numbers
A and
Ai
introduced in (3.5) will be chosen to be larger than
2
80
• In the proof below we will choose them
of
the type
and
A
. -
2;0+h,
.- ,
i =
0,1,2,
...
,
where
hi
~
0 will be independent
of
lIulloo.nT.
We
have just remarked that
if
the
lower order terms b(x, t,
u,
Du)
satisfy
(A~)
of§5
of
Chap. II, then
So
can be taken
to be one.
We
conclude that for equations with such a structure, the numbers
Ai
can be determined a priori only in terms
of
the data and independent
of
the norm
lIulloo.nT·
4.
The first alternative
49
4. The first alternative
LEMMA
4.1.
There
exists a
number
Vo
E (0, 1) independent 0/
w,
R, A
such
that
if/or
some
cylinder
of
the
type
[(0,
t) + Q
(dRP, R)]
I(X,t) E
[(O,t)
+Q(dRP,R)] lu(x,t} < IL- +
2~0
1::5
VolQ
(dRP,
R)
I,
then
(4.1)
u(x,t)
>
IL-
+
2S~+1
a.e.
(x,t)
E
[(O,t)
+Q(d(f)",f)]·
PROOF:
Fix
a cylinder for which the assumption of the lemma holds.
Up
to a
translation
we
may
assume that (0, t) = (0,
O),
and
we
may work within cylinders
Q(dPP,p) , O<p::5R. Let
R R
Rn
="2
+
2n+1'
n =
0,1,2,
...
,
construct
the
family of nested cylinders Q
(dR~,
Rn)
and
let
(n
be
a piecewise
smooth cutoff function
in
Q (
dR~,
Rn)
such that
(4.2)
We
will
use
the
energy inequalities
of
Proposition
3.1
of Chap
II.
written over
the
cylinders Q
(d~,
Rn),
for the functions (u - k
n
)-.
where
forn=O,
1,2,
...
,
k = -
~
W
n
IL
+
28
0
+1
+ 2
80
+1+
n
'
In
this setting. (3.8) of Chap.
II
takes the
form
(4.3) esssup
J(u
- k
n
):
(!(x,
t)dx +
jr
flD
(u -
knL
(nlPdxdr
-dR~<t<O
J ,
KRn
Q(dR~,Rn)
::5
7:
{ J J (u -
kn)~
dxdr +
~
J J (u - k
n
):
dXdr}
Q(dR~,Rn)
Q(dR~,Rn)
,,(1+,,)
+7 { J
IAkn'RJr)l~dr}
r
-dR~
We
will
show
that
as
n
-+
00
50
DI.
HOlder
continuity of
solutions
of
degenerate
parabolic
equations
Ilx[(U-k
n
L
>O]dxdr-+O.
Q(dR!:.Rn)
Since k
n
'\.
koo
=
J,L-
+
2.::'+1'
this would imply that
thereby proving the lemma.
We
observe that
and estimate above the first two tenns on the right hand side
of
(4.3) by
7
~:
(2:
0
)2
(2:
0
r-
2
II
X
[(u
- knL >
0]
dxdr
Q(d~.Rn)
+7:
(2:J
2
~
II
X
[(u
- knL >
0]
dxdr
Q(dR~,R,,)
:57:
(2:J"
Ilx[(U-knL
>O]dxdr,
Q(d~.R,,)
where we have used the defmition (3.3)
of
d. Combining these remarks in (4.3)
and dividing through by d, we arrive at
(4.4)
esssup
I(u
-
kn)~
(:(x,
t)
dx
+
-d
1
I
liD
(u - knL
(nl"dxdr
-dR~<t<O
11
KRn
Q(d~.R,,)
:57~:
(2:J"
~
Ilx[(U-knL
>O]dxdr
Q(d~.Rn)
{
0
}~
(
W
),,-2
~
1 I
1:
+7
2Bo
d,.
~dR~IAkn.R,,(r)l·dr
In (4.4) we introduce the change
of
time-variable z =
tid
which transfonns
Q
(d~,
Rn)into
Qn
==
Q(~,Rn)
==
KR"
x{-~,O}.
Setting also
v(·,z) = u(·,zd) and (n(-,z) = (n(·,zd),
the inequality (4.4) can be written more concisely as
4.
The
first alternative
51
where
we
have
set
o
and
IAnl
= J IAn(z)ldz.
-Rl:
Since
(11
-
knL
<n
vanishes
on
the lateral boundary of Qn,
by
Corollary
3.1
of
Chap.
I
we
have
The
left
hand
side
of (4.5)
is
estimated
below
by
II
(11
-
knL
1I:,Qn+l
~
Ik
n
-
k,.+1
IPIAn+1I
~
2P(~+2)
(;:0
r I
A
n+1l·
Combining
these
estimates gives
(4.6)
IA
11+m:;
IAn+d
~
-y4
np
n
RP
~
+~4·P
(2~.r'
dol':"'
1A.lm
tllA.(Z)lidz}
·
Divide
by
IQn+11
and
introduce
the
quantities
Using
also
the
fact
that. by virtue of
Remark
3.2
and
(3.5)
RNIC
(!:!...)
-2
d
P
(1:"! < 1
2
80
- ,
we
obtain
from
(4.6)
in
dimensionless fonn
52
m.
H6lder
continuity of
solutions
of degenerate
parabolic
equations
y.
1 < "'4
np
{y'l+~
+
Y.~Zl+"}
n+
_I
n n n ,
'tin = 0,1,2,
....
Next by the .embedding
of
Proposition 3.3
of
Chap. I
Therefore
Z <
",4
np
{y. +
Zl+"}
n+l
_
Inn
,
'tin =
0,
1,2,
....
From Lemma 4.2
of
Chap. I it follows that Y
n
and Zn tend to zero as n -+
00,
provided
where 8
0
=
min
{
N;"p;
It}.
5. The frrst alternative continued
Suppose the assumptions
of
Lemma
4.1
are verified for some box
of
the
type
[(0,
l)
+ Q
(dRP,
R)].
We
will exploit the fact that at the time level
(
W
)2-
P
(R)P
-8
= t -
2Bo
"2'
the
functionx-+u(x,
-8)
is strictly above the level
",-
+
2.::'+1'
in
the
cube K
R
/
2
•
To simplify the symbolism let us set p =
~
and construct the cylinder
Q(8,p)
==
Kpx(-8,0),
p=
R/2.
The length 8
of
such a cylinder satisfies
(5.1)
- < - < - 2
P
•
(
W
)2-P
8
(W)2-P
2-
0
-
pP
- A
The next lemma asserts that, owing to (4.1), the set where u(·, t) is close to
I.e
,
within the smaller cube
KR/4'
can be made arbitrarily small for all time levels
-8~t~0.
S.
The fllSt alternative continued
S3
LEMMA 5.1.
For
every
number
Vi
E (0,1),
there
exists a positive integer
81.
depending
only
upon
the
data
and
independent
of
w, R,
such
that
'lit E
(-9,0).
PROOF:
Consider the logarithmic estimates
of
Proposition 3.2
of
Chap. II, written
over the cylinder
Q
(0,
p),
for
(u-k)_,
As a number c in the definition
of
W,
we take
w
c -
-=--7":'"":"-
- 2
s
o+
i
+
n
'
where n is
to
be chosen. Thus we
take
where
n>
1,
Hi; = esssup
(u
-
(JL-
+ 2
W+
i
))
::;
2
w+
i
.
Q(fJ.p)
So
_
80.
For
t=
-0,
by virtue
of(4.1)
we have
(u
-
(JL-
+
2.~+dL
=0,
and therefore
W(x,
-0)
= 0,
These remarks in (3.14)
of
Chap. II yield
(5.2) I w
2
(x, t)(P(x)dx
::;
; II
W!W
u
!2-
P
dxdr
Kp Q(fJ.p)
.ell±!!l
+
~(2":'+'
r'
~
+
In
H,
(2<.:,+0>
n
(lIA',p(T)
I i
dT)
· ,
where A;.i·) is defined in (3.4)
of
Chap. II and x
-+
(x)
is a piecewise smooth
cutoff function in
Kp
that equals one on
Kp/2
and such that
ID(I::;
4/
p.
Next
W::;ln(
~
)
=nln2
~
and
! !
2-P
! _ W
!P-2
(W
)P-2
WU
=
Hk
- (u - k)_ + 2
so
+l+
n
::;
2
80
•
Therefore, in also view
of
(5.1), the first term on the right hand side
of
(5.2) is
estimated above by
54
m.
Itilder continuity of
solutions
of
degenerate
parabolic
equations
; ffl[lll[lul2-PdXdT S
'Yn;
(2~JP-2IKp/21
S 'YnAP-2IKp/21,
Q(9,p)
where
l'
and A are constants depending only
upon
the data. and A
has
to
be
deter-
mined
later.
The
second tenn
is
estimated
by
using
the
conditions (3.6). (3.7) of
Chap.
II.
linking the parameters r,
q,
K..
This gives
Ei!±cl
~(2'.:,+or'(l
+ In H;
(2
•.
:'+0
n
(lIA'-"<T)I~
dT)
.
(
w
)-2
(W)-Ei!±cl(P-2)
S'Y
n
2
so
+1+
n
A r R
NIt
IK
p
/
2
1.
The number n
will
be
detennined shortly. depending only
upon
the data and
inde-
pendent of w,
p.
Therefore
by
virtue of Remark 3.2
and
(3.5)
we
may
estimate
(
w
)-2
(~)_P(1:")(P_2)
Nit
<
n 2
Bo
+l+
n
A R _
1.
Combining these remarks
into
(5.2) yields
(5.3)
f 1[12(x,t)dx S
'Y
nAP
-
2
I
Kp/2
I ,
Kp/2
where
we
have used the fact that (== 1 on K
p/2.
The
integral
in
(5.3)
is
estimated
below
by
extending the integration
to
the smaller set
{x
E K
p
/ 2
1
u(x,
t) <
,.,.-
+
2Bo~l+n}
,
t E
(-0,0).
On
such a set
and
since
the
right hand side of this inequality
is
a decreasing function of Hi; •
we
have
1[12
~
In2(
20:+1
) = (n _
1)21n2
2.
2°o+
n
Putting this
into
(5.3) gives that for all t E
(-0,0)
(5.4)
Ix
E K
p
/
2
I
u
(x,
t)
<,.,.-
+
2Bo~l+n
I S
'Y
AP
-
2
(n
~
1)2I
K
p/21·
To
prove
the
lemma
we
have only
to
choose n sufficiently
large.
6. The
fmt
alternative concluded
55
6. The first alternative concluded
The infonnation
in
Lemma
5.1
will
imply that u
is
strictly bounded
away
from
p.-
in
a smaller cylinder.
To
make
this precise. consider the
box
Q(O,~)
=
Kp/2X(-0,0),
where
(J
satisfies
the
bounds
in
(5.1).
P = R/2,
LEMMA 6.1.
The
numbers
V1
E (0,1) and
81
» 1
can
be
chosen
a priori
depen-
dent only
upon
the
data and independent
of
w and R.
so
that
w
u(x, t) >
p.-
+ 2
81
+1'
a.e.
(x, t) E Q
((J,~)
.
PROOF:
We
will
use
the local energy estimates
of
Proposition
3.1
of
Chap.
II
in
the following setting.
Let
n = 0,1,2,
...
,
construct the cylinders
Q
((J,
Pn)
and let x
-+
(n (x)
be
a piecewise smooth cutoff
function
in
KPn
that equals one
on
KPn+1
and such that ID(n
1:$
2
n
+
3
/
p.
Write
the
inequalities (3.8) of
Chap.
II
over Q
(0,
Pn)
for
the
functions (u - k
n
)_.
where
w w
k
n
= p.- + 2
81
+
1
+ 2
s
l+
Hn
'
and observe that. owing
to
the
conclusion of Lemma 4.1.
(u -
knL
(x,
-0)
=
0,
IrIx
E
KPn,
n =
0,1,2,
....
With
these choices. the energy inequalities yield
(6.1) sup
j(u
-
kn)~
'~dx
+
,),-1
r
riD
(u -
knL
(niP
dxdr
-8<t<0
JJ
KPn
Q(8,Pn)
£lli!!.!.
::;
')'~P
jj(u
-
kn)~
dxdr
+"(
{JIAkn,pJr)lidr}
r
Q(8,Pn)
-8
The
first tenn
on
the left hand
side
is
estimated
below.
for
all t E
(-0,0).
by
j(u
-
kn)~
'~dx
~
(2~1)
2-p
j(U
-
kn)~
'~dx
KPn KPn
> 2-P
(2
81
)P-2
~
j(u
-k
)P
I"Pdx
>
~
j(u
-k
)P
I"Pdx
- A
pP
n -
'>n
-
pP
n -
'>n
,
KPn
KPn