DiBenedetto E. Degenerate Parabolic Equations
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xii
Contents
III. HOlder continuity
of
solutions
of
degenerate
parabolic equations
§
1.
The regularity theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
..
41
§2. Preliminaries
......................................
43
§3. The main proposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
..
44
§4.
The
first alternative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
..
49
§5. The fllSt alternative continued . . . . . . . . . . . . . . . . . . . . . . . . .
..
52
§6. The first alternative concluded
......................
'"
.
..
55
§7. The second
alternative.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
..
58
§8. The second alternative continued . . . . . . . . . . . . . . . . . . . . . . .
..
62
§9. The second alternative
concluded.
. . . . . . . . . . . . . . . . . . . . . .
..
64
§lO. Proof
of
Proposition 3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
..
68
§11. Regularity up
to
t = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
..
69
§12. Regularity up to
ST. Dirichlet data
.......................
72
§13. Regularity
at
ST. Variational data . . . . . . . . . . . . . . . . . . . . . .
..
74
§14. Remarks on stability
.................................
74
§15. Bibliographical
notes.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
..
75
IV.
Holder continuity
of
solutions
of
singular
parabolic equations
§ I. Singular equations and the regularity theorems . . . . . . . . . . . . . .
..
77
§2.
The
main proposition
.............................:..
79
§3. Preliminaries
......................................
81
§4.
Rescaled iterations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
..
84
§5. The first alternative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
..
88
§6.
Proof
of
Lemma 5.1. Integral inequalities. . . . . . . . . . . . . . . . . .
..
92
§7. An auxiliary proposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
..
95
§8.
Proof
of
Proposition 7.1 when (7.6) holds
...................
97
§9. Removing the assumption (6.1)
..........................
101
§ 10. The second alternative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
..
102
§11. The second alternative concluded
.........................
106
§ 12. Proof
of
the main proposition
...........................
109
§13. Boundary regularity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
..
110
§14. Miscellaneous
remarks.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
..
114
§15. Bibliographical notes
...............................
" 116
v.
Boundedness
of
weak solutions
§
1.
Introduction.......................................
117
§2. Quasilinear parabolic
equations.
. . . . . . . . . . . . . . . . . . . . . . .
..
118
§3. Sup-bounds
.....................................
" 120
§4. Homogeneous structures. The degenerate case
p > 2
...........
122
§5. Homogeneous structures. The singular case 1
< p < 2
..........
125
§6. Energy estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
..
128
Contents xiii
§7.
Local iterative inequalities
.............................
131
§8. Local iterative inequalities
(P>
max
{
1;
J~2})
............. 134
§9.
Global iterative inequalities. . . . . . . . . . . . . . . . . . . . . . . . . . .
..
135
§1O.
Homogeneous structures and 1
<p$max
{
1;
J~2}
............ 137
§11. Proof
of
Theorems
3.1
and 3.2
...........................
138
§12. Proof
of
Theorem 4.1
.................................
140
§13. Proof
of
Theorem 4.2
.................................
142
§14. Proof
of
Theorem 4.3
.................................
143
§15. Proof
of
Theorem 4.5
.................................
144
§16. Proof
of
Theorems 5.1 and 5.2
...........................
147
§17. Natural growth conditions
..............................
149
§18. Bibliographical notes
.................................
155
VI.
Harnack estimates:
the
case
p>2
§1. Introduction
.......................................
156
§2. The intrinsic Harnack inequality
..........................
157
§3.
Local comparison functions
.............................
159
§4. Proof
of
Theorem
2.1
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
..
163
§5.
Proof
of
Theorem 2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
..
167
§6.
Global versus local estimates
............................
169
§7.
Global Harnack estimates
.......
".
. . . . . . . . . . . . . . . . . . . .
..
171
§8.
Compactly supported initial data
.........................
172
§9. Proof
of
Proposition 8.1
.....
. . . . . . . . . . . . . . . . . . . . . . . .
..
174
§ 10. Proof
of
Proposition 8.1 continued . . . . . . . . . . . . . . . . . . . . . .
..
177
§
11.
Proof
of
Proposition
8.1
concluded . . . . . . . . . . . . . . . . . . . . . .
..
179
§12. The Cauchy problem with compactly supported initial data
........
180
§13. Bibliographical notes
.................................
183
VII.
Harnack estimates
and
extinction profile for
singular equations
§
1.
The Harnack inequality
...............................
184
§2. Extinction in finite time (bounded domains)
..................
188
§3. Extinction in finite time (in
RN)
.........................
191
§4.
An integral Harnack inequality for all 1 < p < 2
...............
193
§5.
Sup-estimatesfor
J~l
<p<2
..........................
198
§6. Local subsolutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
..
199
§7.
Time expansion
of
positivity
............................
203
§8.
Space-time configurations . . . . . . . . . . . . . . . . . . . . . . . . . . . .
..
204
§9.
Proof
of
the Harnack inequality
..........................
206
§1O.
Proof
of
Theorem 1.2
.................................
211
§11. Bibliographical notes
.................................
214
xiv
Contents
VIII. Degenerate
and
singular parabolic systems
§
1.
Introduction.......................................
215
§2. Boundedness
of
weak solutions . . . . . . . . . . . . . . . . . . . . . . . .
..
218
§3. Weak differentiability
of
IDul2j1
Du
and energy estimates for IDul 223
§4.
Boundedness
of
IDul.
Qualitative estimates
..
,
..............
231
§5. Quantitative sup-bounds
of
IDul . . . . . . . . . . . . . . . . . . . . . . . . . 238
§6. General structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243
§7. Bibliographical notes
.................................
244
IX.
Parabolic p-systems:
HOlder
continuity of
Du
§
1.
The main theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245
§2. Estimating the oscillation
of
Du
. . . . . . . . . . . . . . . . . . . . . . . . . 248
§3.
HOlder
continuity
of
Du
(the case p >
2)
..................
251
§4.
HOlder
continuity
of
Du
(the case 1 < p < 2 )
................
256
§S.
Some algebraic Lemmas
..............................
258
§6. Linear parabolic systems with constant coefficients . . . . . . . . . . . . . 263
§7.
The perturbation lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
..
268
§S.
Proof
of
Proposition 1.1-(i)
.............................
275
§9. Proof
of
Proposition 1.1-(ii)
............................
278
§IO. Proof
of
Proposition 1.1-(iii) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282
§
11.
Proof
of
Proposition
1.1
concluded . . . . . . . . . . . . . . . . . . . . . .
..
284
§12. Proof
of
Proposition
1..2-(i)
.............................
286
§13. Proof
of
Proposition 1.2 concluded. . . . . . . . . . . . . . . . . . . . . . . . 288
§14. General structures
...................................
291
§15. Bibliographical notes
.................................
291
X.
Parabolic p-systems: boundary regularity
§
1.
Introduction.......................................
292
§2. Flattening the boundary
..
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294
§3.
An iteration lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297
§4.
Comparing w and v (the case p >
2)
....................
299
§5. Estimating the local average
of
IDwl (the case p >
2)
..........
304
§6.
Estimating the local averages
of
w (the case p >
2)
...........
305
§7.
comparingwandv(thecasemax{1;;~2}<p<2)
.........
309
§8. Estimating the local average
of
IDwl
.....................
313
§9. Bibliographical notes
.................................
315
XI.
Non-negative solutions
in
E
T
•
The
case p> 2
§1. Introduction
.......................................
316
§2. Behaviour
of
non-negative solutions as
Ixl
--+
00
and as t
""
0
....
317
§3. Proof
of
(2.4)
......................................
319
§4.
Initial traces
.......................................
322
Contents
xv
§5. Estimating
IDul
p
-
1
in
ET
............................
323
§6.
Uniqueness for data in LtoARN)
........................
326
§7. Solving the Cauchy problem
...........................
, 330
§8.
Bibliographical notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
..
333
XII. Non-negative solutions in E
T
.
The case 1 <p<2
§l. Introduction
.......................................
334
§2. Weak solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
..
337
§3. Estimating
IDul
...................................
340
§4. The weak Harnack inequality and initial traces . . . . . . . . . . . . . . . . 344
§5. The uniqueness theorem
...............................
346
§6.
An auxiliary proposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
..
350
§7. Proof
of
the uniqueness theorem. . . . . . . . . . . . . . . . . . . . . . . .
..
362
§8. Solving the Cauchy problem
...........................
, 362
§9. Compactness
in the space variables. . . . . . . . . . . . . . . . . . . . . . . . 363
§ 10. Compactness in the t variable . . . . . . . . . . . . . . . . . . . . . . . . .
..
366
§1l. More on the time-compactness
..........................
370
§12. The limiting process
.................................
371
§13. Bounded solutions. A counterexample
......................
376
§ 14. Bibliographical notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
..
379
Bibliography
......................................
381
I
Notation and function spaces
1.
Some notation
Let
il
be a bounded domain in
RN
of
boundary
oil
and for 0 < T <
00
let
ilT
denote the cylindrical domain
ilx
(0,
TJ.
Also let,
ST
==
oil
x
[0,
TJ,
r
==
STU(ilx
{O})
denote the lateral boundary and the parabolic boundary
of
ilT
respectively.
If
il
is a sphere
of
radius P > 0 centered at some
Xo
E R
N
,
we denote it by
Bp(xo)
==
{Ix
-
xol
<
p},
and
if
Xo
coincides with the origin, we let
Bp(O)
==
Bp.
The boundary
oil
will be assumed to satisfy the property
of
positive geomet-
ric density, i.e.,
{
there exists
0:*
E
(0,1)
and
Po
> 0 such that
VXo
E
oil,
(1.1) for every ball Bp(x
o
) centered at
Xo
and radius p 5,
Po
I SliBp(x
o
)I 5,
(1
- o:*)IBp(xo)l,
where
1171
denotes the Lebesgue measure
of
a measurable set E.
At times it will be necessary to assume that
oil
is
of
class
C1,A
for some
A E (0, 1). That is, there exist a positive number
Po
such that for all
Xo
E
oil,
the
portion
of
oil
within the ball Bpo(x
o
) can be represented, in a local system
of
coordinates, as the graph
of
a
C1,A
function
¢(x
o
)
such that
¢(x
o
)
(x
o
)
=
O.
We
set
(1.2)
2 I. Notation
and
function
spaces
Here
for
a smooth function ¢ defined on a compact subset
IC
of
Rft, for some
positive integer
n
(1.3)
[¢h
IC
==
sup
1¢(x)1
+
sup
I¢(x) -
¢~Y)I.
'zEIC
(Z,II)EIC Ix -
YI
The
boundary
aa
is
piecewise
smooth
if it satisfies (1.1) and
is
the
union
of finitely
many
portions of
(N
-1)
-dimensional hypersurfaces of class Cl,.x.
If
a a
is
piecewise smooth,
we
say that a certain quantity,
say
C or
'Y,
depends
upon
the
structure
of
aa
if it can
be
calculated apriori only
in
terms of the numbers
o·
,Po
in
the definition (1.1), the number of
the
components making
up
aa
and
the
III
.
1111
+.\
norm
of each
of
the
components
making
up
aa.
If
1 E
Lq(a),
1
~
q
~
00,
denote
by
II/lIq,n
the
Lq(a)-norm of
I.
The
function
1
is
in
Lfoc(a) if
II/lIq,lC
is
finite for all compact subsets
IC
of
a.
Let
q,
r
~
1.
A function 1 defined and measurable
in
aT
belongs
to
if
11/11
••
.,,,, '"
(l
([
IfI'dz)
f
dT)
1 <
00.
Also 1 E Lf::c(a
T
),
iff
or every compact subset
IC
of a and every subinterval
[tl'
t2J
C
(0,
TJ
Whenever q = r
we
set
Lq,r(aT)
==
Lq(a
T
), Lf::c(aT)
==
Lfoc(aT) and
II/lIq,q;nT
==
II/lIq,nT' These definitions
are
extended in the obvious
way
when
either q or r are infinity.
If
1 E
CI(a
T
),
we
denote
by
DI
==
(I",,,
1"'2''''
,lZ:N)
the
gradient of 1
with
respect
to
the space variables
only.
The
spaces
WI,p(a)
and
W!,p(a),
p~
I,
are
defined
by
W
l,p
(a)
is
the completion of
Coo
(a)
under
the
norm
Ilvllwl,p(n)
==
IIvllp,n
+ IIDvllp,n, v E
Coo(a)nLP(a).
W~,p(a)
is
the
completion
ofC~(a)
under
the
norm
IIvllw~,p(n)
==
IIDvllp,n,
v E
C~(a).
Equivalently
WI
,p
(a)
is
the Banach space
of
functions v E V
(a)
whose
gener-
alised derivatives
v""
belong
to
LP(a)
for all
i=
1,2,
...
,
N.
A function v E
Lfoc
(a)
is
in W,!;:
(a)
if
for
every compact subset
IC
c
a,
v E
WI,P(K).
2.
Basic
facts
about
WI.pCo)
and
W~·P(O)
3
We
let WI,oo(n) denote
the
space of functions v E
LOO(n)
whose
disttibu-
tional derivatives
v
z
•
are
in
LOO(n),
for
i = 1,2,
...
, N. The
space
W,~:o(n)
is
defined
analogously.
2.
Basic
facts
about W
1
,P(il)
and
W~'P(il)
We
collect
here
a
few
facts
that
will
be
of frequent
use
in
what
follows.
The
first
is
about
the
Gagliardo-Nirenberg multiplicative embedding
inequality.
THEOREM 2.1. Let v E w;,"(n), p
~
1.
For
every fixed
number
s
~
I
there
exists a constant C
depending
only
upon
N, p and s
such
that
(2.1)
where
Q E
[0,
1],
p,
q
~
1,
are
linked
by
(2.2)
and their
admissible
range
is:
(2.2-i)
q
N=I,
qE[S,OO],
QE
[O'P+S~_I)];
(2.2-ii)
if 1
~
p < N, a E
[0,
I]
and
qE
[s,
:!p]
if
s~
:!p'
qE[:!p'S]
if
S~:!p;
(2.2-iii)
if p
~
N >
1,
q E
[s,
00)
and
oE
[0,
NP+~:-N»).
COROLLARY
2.1.
LetvE
W;,"(n),
and
assume
pE
[I,N).
There
exists
a
con-
stant
'Y
depending
only
upon
N, and p,
such
that
(2.1)'
where
PROOF:
We
may
take
Q = 1
and
S = 1
in
(2.2-ii).
Np
q=--.
N-p
H
8n
is
piecewise
smooth,
functions v
in
WI,"(n)
are
defined
up
to
8n
via
their traces.
We
will
denote
by
vlan
the
trace
on
8n
of a
function
v E
Wl,"(n)
.
4 I. Notation
and
function spaces
THEOREM
2.2. Assume that
an
is piecewise smooth. There exists a constant C
depending only upon N, p and the structure
of
an
such that
(2.3)
where
(2.3-i)
(2.3-ii)
[
1
(N
-I)P]
qE
'N
'
-p
q E [1,00),
if 1 < p < N,
if
p=
N.
If
an
is piecewise smooth, the space
W;,"(n)
can
be
defined equivalently
as
the
set of functions v E
Wl,"(n)
whose
trace on
an
is zero.
Remark
2.1.
The
embedding inequalities of Theorem 2.1 and Corollary 2.1 con-
tinue
to
hold
for
functions v
in
Wl,p(n),
not necessarily vanishing
on
an
in
the
sense
of the traces, provided
we
assume
further that
an
is
piecewise smooth and
that
(2.4)
f
v(x)dx
=
o.
n
In such a case the constant C depends
upon
S,p,
q,
a,
N
and
the structure of
an.
However it does
not
depend on the size of
n,
i.e.,
it
does
not
change under dilations
ofn.
Let
k
be
any
real
number
and
for
a function v E
Wl,p(n)
consider
the
trun-
cations of
v given
by
(2.5) (v - k)+
==
max{(v - k) ;
O},
(v -
k)_
==
max {
-(v
-
k)
;
O}.
LEMMA 2.1.
LetvEWl,P(n).
Thenforall
kER,
(v=Fk)± E
Wl'''(n).
Assume
in
addition that the trace
of
v on
an
is essentially bounded and
IIvlloo,an
~
ko, for some
ko
>
o.
Then for all
k~ko,
(v -
k)±
EW;,p(n).
COROLLARY
2.2.
Let
Vi
EWl,p(n),i=I,2,
...
,nEN.
Then
w
==
min {Vb
V2,
...
, v
n
}
E
Wl,"(n).
PROOF:
Assume
first
n=2.
Then
'.
{
.}
VI -
(V2
-
VI)+
V2
-
(Vl
-
V2)+
mm Vb
V2
= 2 + 2 .
The general case
is
proved
by
induction.
If
v
is
a continuous function defined
in
n
and
k < l
is
a pair of real numbers,
we
set
2.
Basic
facts
about
W1;P({})
and
W.,';p({})
5
r>~
-
{x E n I
vex)
> l},
(2.6)
[v
<
k]
-
{x E
nlv(x)
<
k},
[k
< v <
lj
-
{x
E n I k <
vex)
<
l}
.
LEMMA
2.2. Let v E WI,I(Bp(X
o
)) nC(Bp(xo)) for
some
p > 0 and
some
Xo
E
RN
and
let
k and I
be
any
pair
of
real
numbers
such
that k <
I.
There
exists a constant
"1
depending
only
upon
N, p and
independent
of
k,
l, v, x
o
,
p,
such
that
(2.7)
N+1
I
(l-
k)1
[v
>
1]1
~
"11
[~<
kjl
IDvldx.
[k<v<IJ
Remark
2.2. The conclusion
of
the lemma continues
to
hold for functions v E
WI,I
(n)nC(
n)
provided n is
convex.
We
will
use
it
in
the case n
is
a hemisphere
or a cube.
Remark
2.3. The continuity
is
not necessary to
the
conclusion of Lemma
2.2.
The
function
v has been assumed to
be
continuous to give
an
unambiguous meaning
to
the
definitions (2.6).
If
v
is
only
in
Wl,l
(n),
one could
fix
an
arbitrary repre-
sentative out of the equivalence class
v
say
v and define (2.6) accordingly. The
conclusion of
the
Lemma
is
independent of the choice of
v.
2-(i). Poincare-type inequalities
Inequality (2.7)
is
due
to
De
Giorgi
[33]
and
it
is
a particular case
of
a
more
general
Poincare-type inequality. The embedding (2.1)' of Corollary
2.1
gives a majorisa-
tion of the
Lq(n)-norm
of u solely
in
terms
of
the
LP(n)-norm
of
its gradient.
This
is
possible because one knows that u vanishes on an
in
the sense of
the
traces.
A Poincare-type inequality bounds some integral
norm
of a function u E Wl,p
(n)
in
terms
only
of some integral norm
of
its gradient, provided some information
is
available on
the
set where u vanishes.
PROPOSITION
2.1. Let n
be
a bounded convex
set
in
RN
and
let
I{)
E C
(1i)
satisfy
(2.8)
{
0
~
I{)
~
1, "Ix E
n,
the
sets [I{) >
k]
are
convex,
Vk
E (0,1).
Let v E WI,p(n),
p~
1,
and
assume
that
the
set
e
==
[v
=
Ojn[I{)
=
1]
has
positive
measure.
There
exists a constant C
depending
only
upon
Nand
p and
independent
of
v and I{),
such
that
(2.9)