Publicacions
Ma emá iques,
Vol
37
(1993),
443-463
.
A
bs ac
SOME
THEOREMS
OF
PHRAGMEN-LINDELOF
TYPE
FOR
NONLINEAR
PARTIAL
DIFFERENTIAL
EQUATIONS
RAMÓN
QUINTANILLA
The
p esen
pape
s udies
second
o de
pa ial
di e en ial
equa ions
in
wo
independen
a iables
o
he
o m
Di (pl
¡u,,
J'
-l
u,l
,
P21U,2
I
n-1
u,2)
=
0
.
We
ob ain
decay
es i-
ma es
o
he
solu ions
in
a
semi-in ini e
s ip
.
The
esul s
may
be
seen
as
heo ems
o
Ph agmen-Lindelo
ype
.
The
me hod
is
s ongly
based
on
he
ideas
o
Ho gan
and Payne
[5], [6],
[8]
.
1
.
In oduc ion
In
[5]
Ho gan
and
Payne
s udied he
asymp o ic beha iou
o
solu ions
o
he
equa ions
:
(P(-'C,
u,
VU)u,
a
),a
=
0,
on
he
semi-in ini e s ip
R
=
{(x1,
x2)/0
_<
x2
<_
h,
x1
>_
0}
unde
bounda y
condi ions
:
(1
.1)

u(x
1
,
0)
=
u(xi, h)
=
0,

x1
>
0,
(1
.2)

u
and
u,
a

+
0
(uni o mly
in
x2)
as
x
1
--,
oo,
(1
.3)

u(O,x2)
=
(x2),
0
<
x2
<
h,
whe e
he
p esc ibed
unc ion
is
su icien ly
smoo h
and
sa is ies
(0)
_
(h)
=
0
.
This
s udy
was
made
o
wo
kinds
o
unc ions
(case
1)

O
<
ml
:5p<
M
l
+
K,P(P
2
+
q2),
(case
2)

0
<
m2
<
P
-1
<
M2
+
K2P(P2
+
q2),
444

R
.
QUINTANILLA
whe e
p
=
au/Ox1
and q
=
8u/¿9x2
.
Ho gan
and
Payne
ob ained
expo-
nen ial
decay
o
he
solu ions
.
Recen ly
[8]
he
assump ion
(1
.2)
on
he
beha iou
o
he
solu ion
a
in ini y
has
been weakened and
many
o he s
geome ies
o
he
domains
ha e
been
conside ed
.
These
esul s
we e
mo i a ed
by
he
desi e
o
es ablish
e sions
o
Sain -Venan 's
p inciple
in
elas ici y
.
A
good
su ey
o
esea ch
on
his
p inciple
in
se e al
kind
o
p oblems
may
be
ound
in
[3],
[4]
.
Simila
me hods
a e
cu en ly
used
in
he
s udy
o
cons ained
elas ic
cylinde s
o
a iable
c oss
sec ion
[111,
[12]
.
Recen
esul s
using
ela ed
me hods
may
be
ound
in
[9],
[10]
.
Fu he mo e,
some
conside a ions
o
he
p-laplacian
equa ion
we e
gi en
in
[13]
and
o
nonlinea
ou h
o de
equa ions
in
[15]
.
Al e na i ely,
he
esul s
may
be
iewed
as
heo ems
o
Ph ag nen-
Lindelo
ype
[2]
.
Following
he
main
ideas
o
he
wo k
o
Ho ganand
Payne,
we
gene -
alize
he
esul s o a la ge class o
equa ions
.
F om
now
on,
R
will
be
a
s ip
lying
in
he
hal
plane
x1
>_
0
wi h
ixed
la e al
cu es
c(x1),c(x1)
o
x
1
>_
0
.
We
may
designa e
R
=
{(XI
,
X2)
E
118
2
/x1
>
0
and
c(x1)
<
x2
<_
¿(x1)}
.
We
will
suppose
ha
he
wid h
o
he
c oss
sec ion
L
z
=
{(x1,
x2)
E
R/x1
=
z}
is
bounded
abo e
by a
cons an
h
.
We
obse e
ha
he
Di ichle
homogeneous
bounda y
condi ion
a e
:
(l
.lb)

u(x1,
c(x1))
=
u(x1,
¿(x1))
=
0
o
all
x1
>
0,
and
he
end
condi ion
may
be
w i en
as
:
(1
.3b)

u(0,
x2)
=
(X2)
o
all
(0,
x2)
E
Lo,
whe e
he
p esc ibed
unc ion
sa is ies
(c(0))
=
(¿(O»=
0
.

In
his
pape
we
p o e
he
exponen ial
decay
o
classical
solu ions
o
he
equa ion
:
(1
.4)

DZ (pllpln
-1
p,P21gin
-1
q)
=0,

n
>
0,
whe e p
i
=
pi(x, u,
Vu)
in
he
semi-in ini y
s ip
R
subjec
o
he
bound-
a y
condi ions
(l
.lb),
(1
.3b)
and
unde
sui able
hypo hesis
a
in ini y
(H
.s)
which
is
he
na u al
ansla ion
o
ou
case
o
he
hypo hesis
used
in
[8]
.
Ou
esul s
may
be
seen as
analogues
o
Ph agmen-Lindelo
he-
o em
o
he
equa ions
and
domains
we
s udy
.
We
es ablish
ou
esul s
o
ou
clases
o
equa ions
depending
on
he
unc ions
p1
and
p2
.
The
main
esul
is
o
es ablish
an
exponen ial
decay
es ima e
o
he
ene gy
:
(1
.5)

E
s
(z)
=
R
z
IDI'(
+1)
(PlIPI"
1
+P2Igj
+1
)dA
(H
.0)
Type
I
Type
II
Type
III
Type
IV
SOME
THEOREM
OF
PHRAGMEN-LINDELOF
TYPE

445
con ained
in
he
subdomain
R
z
=
{(x1, x2)
E
1[8
2
/0
<
z
<
x1
<
oo}
n
R
p o ided
ha
he
o al
ene gy
E
S
(0)
is
ini e
o s
=
0,
1
.
Ne e heless
he
me hod
may
be
used
o
all
na u al
numbe s
s
.
Te
ensu e
ha
he
unc ion
Eo
is
well
de ined
we
need
o
impose
a
in ini y
he
condi ion
ha
he
solu ions
sa is y
:
lim
in
z
-
1
1

(P1
IPIn
+1
+
P2I
qI
n+1)dx2
=
0,
L
z
while
e
ensu e
ha
E
S
(s
>
0)
is
well de ined
we
suppose
ha
(H
.0)
is
sa is ied
oge he
wi h
he
condi ion
en
he
beha io
o
he
solu ions
a
in ini y
:
(H
.s)

lzm
in
z
-
1
I

Iul
s(n+1)
(Pl
lpln+1
+
P2I
qI n
+
1
)dx2
<
OO
.
~
We
may
obse e
ha
whene e p1 and
P2
a e
bounded
abo e
by a
con-
inuous
unc ion
o
u,
p and
q
hen
condi ion
(1
.2)
implies
(H
.s)
o
all
sEN
.
We
sepa a e
he
hypo heses
in o
he
ollowing
ou
ypes
:
0
<
m1
<
P2
Pi
1
,
and
0
<
m2
<
P2
<
M2
+
K2(P1IPIn+1
+
P2IgIn+1)
.
0
<
m1<
p=
1P2
1/n,
0
<
p21
<
(M2
+
K2(P1IpIn+1
+
P2Igjn+1))n
.
0<M2
<
piP2,
(1
.8)

0
<
p1
<
M1
+K1(Plipi
n+1
+P2IgI
n+1
)
.
0
<m2<p2P1
n
>
0
<
m1
<
p-
1
< M1+
K1(PlIpin+1
+
P2IgIn+1)
.
44
6

R
.
QUINTANILLA
We
obse e
ha
when
pI
=
p2
Types
I
and
III
become
iden ical
and
na u ally
gene alize
Case
1
in
[5]
.
We
can
say he
same
hing
abou
Types
II
and
IV
in
he
case
n
=
1
.
Then
ou
hypo heses
a e
a
na u al
gene aliza ion
o
Case
2 in
[5]
.
A
amily
o
unc ions
sa is ying
he
o egoing
condi ions
is
gi en
by
:
(1
.10)
pl
=
(1+ llpl
n+l
+ 2lgl
n+I
)yand
p2
=
(1+ ,Ipln+I+ 2lgln+1)x
whe e
l
and
2 a e
wo
posi i e
numbe s
.
I
0
<_
y
<
x
<_
y
+
1
hen
an
easy
calcula ion
p o es ha
hese
unc ions a e
o
he
Type
I
by
aking
m
l
=
m2
=
M2
=
1
and
K
2
=
max(x l,
x 2)
.
In
he
case
0
<_
x
<_
-ny,
he unc ions
(1 .10)
a e
o
Type
II
by
aking
ml
=
M
2
=
1
and
K
2
=
0
.
In
a
simila
way
ou
unc ions
a e
o
Type
III
i
0
<_
-y
<_
x/n by
aking
m2 = Ml
=
1
and
K
l
=
0
.
Fo
he
Type
IV
we
can
ake
m
l
= m2=
M
2
=
1
and
Kl
=
max( l, 2)
whene e
-z <
y
<
0
and
x
>
-y
.
In
Sec ion
2
we
s a e
wo
lemmas
which
a e
use ul
in
he
s udy
o
he
exponen ial
decay
o
he
ene gies
E
S
o
he
Type
I
o
IV
.
In
Sec ion
3
we
p o e
he
exponen ial
decay
o
E
S
o
all
he
Types
.
We
gi e
a
comple e
p oo
o
Type
I,
bu
p o ide
only
a
ske ch
o
he
o he h ee
ypes
.
We
also
gi e
in
Sec ion
4
a
heo em
o
decay
o
he
ene gies
wi hou
he
es ic ion
ha
he
c oss
sec ion
is
bounded
abo e,
bu he
hypo heses
on
asymp o ic
beha io
(H
.0)
and
(H
.s)
mus
be
modi ied
o
ano he
(H'
.s)
.
Sec ion
5
is
de o ed
o
ob ain
Ln+
l
-es ima es
in
he
whole
s ip
.
We
make
a
simila
s udy
in
Sec ion
6
bu
o
he
c oss-sec ion
.
In
Sec ion
7
we
ob ain
an
es ima e
o
Eo
o
he
Neumann
bounda y p oblem
.
Sec ion
8,
which
concludes
he
pape ,
summa ises
ou
esul s
by
lis ing
he
a ious
Types
and
measu es
o
which
exponen ial
decay
ha e been
es ablished
.
An
appendix
o
he
pape
desc ibes
a
me hod
o
es ima ing
an uppe
bound
o
he
o al
ene gies
E
S
(0)
in
e ms
o
he
end
condi ions
.
Acknowledgmen s
.
I
am
g a e ul
o
P o
.
C
.
O
.
Ho gan
and
P o
.
R
.
J
.
Knops
o
hei
commen s
on
an
ea lie
e sion
.
2
.
Two
p e ious
lemmas
In
his
sec ion
we
es ablish
wo
lemmas
abou
in eg a ion
which
a e
use ul
in
p o ing
he
esul s o
Sec ion
3
.
Fi s ,
we
gene alize
a
weigh ed
Poinca é
inequali y,
due
o
Ho gan
and Payne
[5,
p
.
314]
.
Lemma
2
.1
.
Le
p
:
[0,
h]
,
l[8+
-
{0}
be
a
con inuous unc ion
and
u
:
[0,
h]
->
R
a di e en iable
unc ion
such
ha u(0)
=
u(h)
=
0
.

Then
SOMETHEOREM
OF
PHRAGMEN-LINDELOF
TYPE

447
he e
exis
wo
cons an e
0 +1
and
B
such
ha
:
h

h
Plul +ldx
<_
/3 +1B +1
1
p- Idu/dxl +ldx,
0

0
whe e
Q +1
depends
on
and
B
=
h
pdx
.
P oo
.
We
conside
he
change
o
a iable
:
so
ha ,
d
=
pdx,
hence
we
see
:
=
x
p(e)da,
0
oh
PI
uI +l
dx
=
l
a
I4dl
+l
( )d
.
0

0
Now,
we
can
apply
Poinca é
inequali y
[1],
o
ob ain
:
o

I
UI
+ld
<
P +1B +1
J
B
I(du/d )I-+ld ,
whe e
Q +1
is
a
cons an ha
depends
on
"
+
1"
.
Upon
ecalling
ha
:
we
ob ain
he
desi ed
inequali y
.
F om
Lemma
2
.1
we
may
ob ain
an
inequali y
use ul
in
he
s udy
o
he
beha iou
o
E
s
:
ph
(2
.1)

PI
uI
( +1)(s+l)dx
<
0
whene e
u(0)
=
u(h)
=
0
.
This
inequali y
may
be
p o ed
by
applying
he o me
lemma
o
he
unc ion
us+1(s
+
1)
-l
.
Now,
we
ecall
he
ollowing
esul
which
is
p o ed
in
[5,
p
.
315]
.
Lemma
2
.2
.
Le
p
:
[0,
l)
->
ll8+
-
{0}
be
a
con inuous
unc ion
such
ha
:
Then
:
du/d
=
(du/dx)(dx/d )
=
p-1du/dx,
h
<
(s+1) +1(0 +l)-lB +1
p- luls( +1)I(du/dx)I'+ldx,
0
o
Zpdx<Az+B

whe e
A,B>0
.
J
zp
-l
dx>_
Á-Á
2
,
o allz>0
.
o

448
In
his
sec ion
we
s a e
and
p o e
some
heo ems
o
he
exponen ial
decay
o
he
ene gy
(1
.5)
o
each
o
he
ypes
conside ed
in
Sec ion
1
.
We
gi e a
comple e
p oo
only
in
he
case
o
Type
I
.
We
ske ch he
p oo
in
he
o he
cases
.
In
o de
o
p o e
he
heo ems,
we
need
o
in oduce
he
unc ion
:
(3
.3)
whe e
and
R
.
QUINTANILLA
3
.
Decay
o
he
ene gy
G,
(z)
=
s(n
+
1
1)
+
1

L

P1IPI
n-1
PI
UI
,
(n
+1
)udx2
.
z
o
he
di e gen e
heo em,
his
may
be
ew i en
as
:
Because
(3
.2)

G,
(z)
=
G,(zo)
+
o
all
z
>
zo
>
0
.
Di ec
di e en ia ion
gi es
:
Gs(z)
=
IUI
'(n+1)
(PlIPI'
+1
+P2IgIn+1
)dx2
.
L
.
Holde 's
inequali y
applied
o (3
.1)
leads
o
he inequali y
:
(3
.4)

IG,(z)I
<_
1
P1IuI
n+1

s(n+1)

+i

(s+1)(n+1)dxz]
n4-1
.
<
s(n
+
1)
+
1
[
p1I
pj

I
uI

dx2~
nn

~IL
.
Now,
we
s a e
he
ollowing
esul
.
Theo em
3
.1
.
Le
u
E
C
2
(R)nC
1
(RUóR)
be
a unc ion
ha
sa is ies
equa ion
(1 .4),
he
bounda y
condi ions
(1
.1b)
and
he
end
condi ions
(1
.3b),
(H
.0)
and
(H
.s)
.
We
suppose
ha p,
and
P2
sa is y
condi ions
o
Type
1(1
.6)
.
Then
o
all
z
>
0
we
ha e
:
(3
.5)

ES(z)
<
ES(0)eQle-
.alz,
x
uls(n+1)
(P1
IpI
n+1
+
P2IgI
n+1
)dA
a
L
I

Q

K2Eo(0)(n+
1)(s(n+
1)
+
1)mi
1
=

n
M2
h
2
(s
+
1)nnnn
(n+
1)(s(n+
1)
+
1)m
1
-
"
0
1+1
m2
1

M2h(s
+
1)nnn+~
nn1
On+1
m2
SOME
THEOREM
OF
PHRAGMEN-LINDELOF
TYPE

449
P oo
.
Le
G,(z)
be
de ined
by
(3
.1)
.
On
using
he
i s
inequali y
o
(1
.6)
in
he
inequali y
(3
.4)
we
ob ain
:
whe e
nn}1
lGs(z)l

s(n
+
l
l)
+
1
[1,
Pl lpln+l
luls(n+l)dx2]
n+l
x
x
~~
Pzlul
(S+l)(n+l)
dx2]n+
1
G
L
z
(s
+
1)B(z)

n+l

S
n+l

Pl
I
pl

I
ul
(

)
dx2]
n+1
x
(s(n
+
1)
+
1)mi
+1
~3n+i

L=
x

P2
nluls(n+1)
lgln+ldx
]
1
2
n+1
z
(s
+
1)nn+l
11

B(z)
x
(n
+
1)
(s(n
+
1)
+
1)mi
+1
~
3
n+i
m2
x~
lul
s(n+l)
(Pllpl
n+l
+P2[4l
n+l
)dx2,
whe e
B
(z)
=
Lz
P
2
(z,
x2)dx2
.
We
obse e
ha
he
second
inequali y
ollows
om
(2
.1)
and
he
hi d
one om
he
second
inequali y
in
(1
.6)
.
Thus
we
may
conclude
ha
:
(3
.6)

lG,(z)1
C
k,B(z)Gs(z),
k
-

(s
+
1)nn
+1
S
-

_1

1

(n
+
1)
(s(n
+
1)
+
')Mi
+1

+i
m2
and
B(z)
has
been
de ined
p e iously
.
F om
inequali y
(3
.6)
deduce
he
ollowing
wo
i s -o de di e en ial
inequali ies
:
(3
.7)

G,
5
(z)
<
k,B(z)Gs(z),
(3
.8)

-
GS(z)
<
ksB(z)Gs(z)
.
z
G,(z)
>
G
S(z*)
x
exp
CkS

d
l

7

.
£Z
i,,
P2
(77,x2)dx2
we
may
We
now
show
ha
G
S
(z)
G
0
o
all
z
>_
0
.
To
his
end,
le
us
suppose
ha
he e
exis s
z*
>
0
such
ha
G
s
(z*)
>
0
.
Then
in eg a ion
o
(3
.7)
gi es
:
45
0

R
.
QUINTANILLA
Bu
om
(3
.7)
and
he
second
inequali y
in
(1
.6),
we
ha e
:
J
=
IUI'(+
1
)(PlIPI"
1
+P2Igj'+
1
)dx2
>
ks
1
G
s
(z*)
(

x
M2h
+
K2
L>
(P1Ipin+1
+
P2IgIn+1)dx2)
(3
.9)

x exp
(k,,
-1
Iz

d l
.
M2h
+
K2
i

(p1Ipin+1
+
P21qln+1)dX2
and
since
condi ion
(H
.0) is
sa is ed,
we
ob ain
:
Z_
1
1

Iuls(n+1)
(P1
Ipln
+l
+
P2I
gI
n
+
1
)dx2
>
~
k
-1
G
s
(z*)
>

exp(ks
l
e
-1
K2
l
ln(M2h
+
cK2 7)
I
z*)
_
z(M2h
+
K2ez)
ks
l
G,5(z
* )

M2h
+
eK2z
-

]eK2ks
z(M2h
+
K2ez)
M2h
+
eK2z*
o
all
e
small
and
o
all
z
>
z(e)
>
0
.
I
hen
ollows
ha
:
(3 .10)

z-1~
luls(n+1)(PlIpin+1
+
P21ql
n+1
)dX2
>
L
z
-1
>
ks
l
Gs(z
*
)[M2h
+
eK2z]
eksK2
_
2
[M2h
+
eK2z*]
ek
.9K
2.
On
le ing
z
-->
oo,
we
obse e
ha
o
e
suicien ly
small
he
igh -hand
side
ends
o
in ini y,
which
con adic s
condi ion
(H
.s)
.
Hence
we ha e
p o ed
ha
G,
(z)
<_
0
o
all
z
>_
0
.
On
ecalling
inequali y
(3
.8),
we
ha e
:
(3
.11)

E
S
(z)
<
E
S
(0)
exp
(-
ks
1

oz
B(97)-ld77/
J
0
The
uppe
bound
o
P2
in
(1
.6)
gi es
he
inequali y
:
while
om
lemma
2
.2,
we
conclude
:
z
Á,,
p
2
dA
<
M
2
hz
+
K2Eo(0),
o
B(

z
_
K2Eo(0)
z

l)

d~
>
M
2
h

MZh2
Thus,
lle
desi ed
es íma e
is
es ablished
.
(3
.12)
whe e
and
whe e
and
SOMETHEOREM
OF
PHRAGMEN-LINDELOF
TYPE

451
Theo em
3
.2
.
Le
uE
C
2
(R)nCI(RUOR)
be
a
unc ion
ha
sa is ies
equa ion
(
1
.
4),
he
bounda y
condi ions
(1
.1b)
and
he
end
condi ions
(1
.3b),
(H
.0)
and
(H
.s)
.
We
suppose
ha
p
l
and
p2
sa is y
condi ions
o Type
11(1
.7)
.
Then
o
all
z
>
0
we
ha e
:
ES
(
z
)
<
ES
(0)eQ2 e-UZZ,
K2Eo(0)(n
+
1)(s(n
+
1)
+
1)mi
+1 13n+
Q2
=

i
M2
h
2
(s+1)n
n
n+1
'
a2
-
(n
+
1)
(s(n
+
1)
+
1)M1
+
1
~n+i
.
M2h(s
+
1)nn+1
Ske ch
o
P oo
.:
Rom
inequali y
(3
.4)
we
ob ain
a
simila
o (3
.6)
bu
wi h
B
(z)
=
L,
p2
l
/
n
(z,
x2)dx2
and
~_n_-
k,
=

(s
+
1)n
+l
(n
+
1)
(s(n
+
1)
+
1)m
i+
1
Qn+i
Theo em
3
.3
.
Le
uE
C
2
(R)nc
1
(RUOR)
be
a unc ion
ha
sa is ies
equa ion
(
1
.4
+),
he
bounda y
condi ions
(1 .1b)
and
he
end
condi ions
(1
.3b),
(H
.0)
and
(H
.s)
.
We
suppose
ha
pl
and
p2
sa is y
condi ions
o
Type
111(1
.8)
.
Then
o
all
z
>
0
we
ha e
:
(3 .13)

E
.(z)
<
ES(0)eQ3e-13Z,
Q3
=
K,Eo(0)(n+
1)(s(n+
1)
+
1)m2+1
On+i
M1
h
2
(s
+
1)n
nn+1
1 1
as
-
(n
+
1)(s(n
+
1)
+
l)mz
+1
~n+i
.
M
i
h(s
+
1)nn+1
inequali y
Ske ch
o
P oo
.
:
As
be o e, inequali y
(3
.4)
leads
o
an
inequali y
sim-
ila
o
(3
.6)
bu
wi h
B
(z)
=
Lz
p
l
(z,x2)dx
2
and
ks
=

(s
+
1)n-
++1
(n
+
1)(s(n
+
1)
+
1)m2+1
Qn+i
45
8

R
.
QUINTANILLA
sec ion
is
o
ske ch
simila
esul s
o
Neumann
condi ions,
whene e
he
c oss-
sec ion
is
he
in e al
[0,h]
.
A
deep
s udy
o his
p oblem
o
he
case
p1
=
P2
and
n =
1
may
be
ound
in
[10]
.
Thus
we
examine
he
equa ions
(1
.4)
subjec
o
he
condi ions
a
in ini y
(H
.0),
and
he
end
condi ion
:
and
he
bounda y
condi ions
:
implies
:
(7
.2)
inequali y
o
Eo
:
Pl¡PI'
-l
p
=
9(x2)
on x1
=
0,
q(x1,
0)
=
q(xl,
l)
=
0
o
all

xl
>
0
.
We
also
make
an
assump ion
on
he
end
condi ion
(1
.3)
which
is
ela ed
o
he condi ion
ha
he
load be
sel -equilib a ed
o
an
elas ic
ma e ial
(see
among
o he s
[3],
[7])
:
(7
.1)

I
LO

=
0
.
Lo
On
using
he
di e gen e
heo em,
we
see
ha
he o me
assump ion
Plipl
n-l
pdx2
=
0
o
all
x1
>
0
.
Lx
Now,
we
ake
(xl,
x2)
=
u(x1,
x2)
-
ü(x1),
whe e
,
will
be
explici ly
de ined
la e
.
We
ema k
ha
q(u)
=
q( )
.
Because
o
(7
.2)
we
ha e
:
(7
.3)

Eo(z)
= -

Plipl
n-1
p dx2
.
L
x
Di ec
di e en ia ion o
(1
.5),
wi h
s
=
0,
yields
:
(7
.4)

Eó(z)
=
-
I
Lz
(Pl
IPI+1
+
P2IgIn+1)dx2
.
In a
simila
way
o
he de i a ion
o (3
.11),
we
ob ain
he
ollowing
(7
.5)
Eó(z)+e¿(z)Eo(z)
<-
Lx
P2Igln+1dx2+0
.n+1(z)
Áz
p1 n+ldx2
.

SOMETHEOREM
OF
PHRAGMEN-LINDELOF
TYPE

459
Now,
we
may
ob ain
simila
es ima es
o
(3
.5),
(3
.12),
(3
.13)
and
(3
.14)
wi h
s
=
0
by
aking
espec i ely
:
L,
UP2dx2
ú(X1)
=

o
he
Type
I,
L
.P2dx2
ic(xl)=
L
z
uP2
i/ndx
2
o
he
Type
II,
L
z
P2
1/ndx2
L,
upldx2
ú(X1)
=

o
he
Type
III,
L,
pidx2
i=
upi
ldx2
ú(X
1
)
=

1

o
he
TypeIV
.
L~
Pi
dx
2
8
.
Summa y
To
conclude
we
summa ize
b ie ly
he
cases
whe e
we
ha e
es ablished
exponen ial
decay
.
In
Sec ion
3,
we
ha e
p o ed
he
exponen ial
decay
o
he
ene gies
E,
s
o
all
he
Types
(I
o
IV)
whene e (H
.0)
and
(H
.s)
a e
sa is ied
and
he
c oss-Sec ion
is
bounded
.
In
Sec ion
5,
we
ha e
p o ed
ha
i
(H
.0)
is
sa is ied
and
he
c oss
Sec ion
is
bounded,
hen
he e
is
exponen ial
decay
o
J(z)
o
he unc ions
o
Type
I
.
We
may
say
he
same
hing
o
many
unc ions
o
Type
III
.
We
ha e
also
es ablished
he
exponen ial
decay
o
J(z)
o
solu ions
o
he
equa ions
con aining
he
unc ions
o
Type
II,
111
and
IV
whene e (H
.0)
and (H
.s)
a e
sa is ied
and
he
c oss-sec ion
is
bounded
.
In
Sec ion
6,
we
ha e
p o ed
ha
i
(H
.0)
is
sa is ied
and
he
c oss-
sec ion
is
bounded,
we
may
conclude
he
exponen ial
decay
o
H(z)
o
unc ions
o
Type
1
.
We
ha e
also
p o ed
he
exponen ial
decay
o
H(z)
o
he
solu ions
o
he
equa ions
ha ing
unc ions
o
Type
IV whene e
(H
.0)
and (H
.s)
a e
sa is ied
and
he
c oss-sec ion
is
bounded
.
Some
commen s
on
he
beha io
o
he
unc ion
L(z)
a e
also
s a ed
.
In
o de
o ob ain
a
good
es ima e
o
he
decay
o
J(z)
and
H(z)
we
may
use
inequali ies
(5
.1),
(5
.2),
(6
.1)
and
(6
.2)
.
No e
howe e
ha
we
ha e
no
ound
es ima es
o
he alues
o
he
k
i
in
(5
.2)
no
when
we
es ima e
H(z)
in (6
.2)
.
Ne e heless,
i
should
be
possible
o
ob ain
hese
alues
.
In
Sec ion
4,
we
ha e ex ended
he
exponen ial
decay
esul s
o
he
ene gy
o
an
enla ged
caass
o
domains
.
46
0

R
.
QUINTANILLA
Sec ion
7
is
de o ed
o
ob aining
exponen ial
decay
o
Eo
o
he
Neu-
mann
p oblem,
whene e
he
c oss
sec ion
is
cons an ,
o
he
Types
I
o
IV
.
The
o al
ene gies
:
Appendix
A
:
To al
ene gy
bounds
(A
.1)

ES
(0)
=

Iuls(n
+1)
(PlIPIn
+1
+P21gjn+
1
)dA,
R
con ained
in
he
s ip
appea s
in
he
es ima es
.
We
now
ob ain
uppe
bounds
o
E,,
(0)
in
e ms
o
he
end
da a
o
unc ions
o
Type
I
.
Simila
es ima es
may
be
ob ained
o
he
o he ypes
.
We
will
conside
ha
R
is
de e mined
by
wo
s aigh
lines
:
c(x
1
)
=
clxl
and
c(xl)
=
C2x1
+
h
.
Fo
an
a bi a y
C
l
unc ion
O(xl,
x2),
de ined
on R,
Holde 's
inequal-
i y
yields
:
(A
.2)
(A
.5)
R
(P1IPIn
-lp0,1
+P2Igjn
-
lq0,2
)IulsnosdA
<
<
ES(0)~+i
~~
(Pl10,1
In+1
+P210,2In+1)101-9(71+1)dAl
+1
.
R
I
0
is
chosen
o
sa is y
(l
.lb),
(1
.2)
and
(1
.3),
he
di e gence
heo em
shows
ha
he
le
hand
e m
o
(A
.2)
is
:
(A
.3)

1

1
,~

e(u)P11PI
n-1PI
ul
s(n+1)udx2
=
E,5(0)
.
s(n+)+
La
Thus,
we
ob ain
:
(A
.4)

E
S
(0)
<
R
(
.110,1
In+1
+P2I0,2
I
n
+l)IOIs(n
+1)
dA
.
Fo
he unc ions
o
Type
I
we
ha e om
(A
.4)
:
ES(o)
<_
J

Pz(ml
1
10,1
In+1
+
10,2
In+
l
)I0Is(n
+1)
dA<
R
<
_~(M2+K2(P1p
n+1
+P2g
n+l
))(m1
1
10,1
I
n+1
+10,2
In+l)101s(n+1)dA
<
R
I
R(ml
llo,1
In+1
+10,2
I
n+l
)10Is(n+
1
)dA+
+K2
mR
(m
l
110,1
In+1
+
10,2
I
n
+
l
)IOIs(n
+1)
&(O)
.
Thus,
on
se ing
F(0)
=
1
-
K2maXR(mi
110,1
In+l
+
10,2
In+1)
>
0
,
we
ob ain
Le
us
ake
0
=

(52-clxl)h

e
-"Ixl
.
Easy
calcula ions
lead
o
:
(
(c2
-
el)x1+h
)
and
On
supposing
ha
Thus
:
Eo(0)
<
SOMETHEOREM
OF
PHRAGMEN-LINDELOF
TYPE

46)1
Eo(
0
)
<
F(0)
-'
M2

(m1
1
10,1
1'+'
+
10,2
n
+
1
)dA
.
IR
10,1
1
<-
e-751
[yI
I
+
I 'l

CA

],
(C2
-
CI)Xl
+
h
0,2
=
e
-751
'

(x2
-
cixl)h

h
(
(C2-
Cl)xl+h)
(c2
-
Cl)xl+h
K2
1
>
[
o
,
aX(I
'
(X2)I
n+1
+mi
1
[71 (X2)I
+C2I
. (X2)1]
n+1
),
we
ha e
an
uppe
bound
:
Eo(0)
<
M2
¡'
R
1-(-+1)-Y-1
(Mi
-1[yL I
+Iili

c2h

]n
+1
+
I
lI
n+1
(
h

)
n
+
l
dA
<

(~2-~l)xl+h

(c2-cl)xl+h
1-K2
max[
o
h](I
'
(5z)In
+l
+TY l
1
[ -
Y¡ (
5
2)~-2L
' (
5
2)Il
n+1
)
M2

h

c2-cl

h

1

i
n+l

i
n+l
<
h
[(n+1)
-y
+
(n+1)
2
.
7
2

( 0
(ml

[71
l
+
C2l
I

+
I
I

)dX2)
1-
K2max[o,hl(I '(X2)In+l+mil[-Y1 (X2)I+c21 '(X2)I]n+1)

,
which
a e
inse ion
in o
inequali y
(A
.5),
yields
he
es ima e
Es
(0)
<
K
2
O'aX
s(n+1)
(I '(X2)In+1+
+mi
1
[Y
1
(X2)I
+
C21 ~(X2)IIn+l)Eo(0)+
M2
h(n
+
1)7(1
+
s(n
+
1))
+
_

(C2
-
cl)
+
h
[

(n
+
1)
2
7
2
(1
+
s(n
+
1))2

]X
X
(
h
1
.
I ls(n
+1)
(mi
l
[7l l
+C2l ll]n+l
+
I l
in
+l
)dX2
0
462

R
.
QUINTANILLA
Re e ences
1
.

H
.
BREZIS,
"Analyse
onc ionnelle,"
Masson,
Pa is,
1983
.
2
.
J
.
B
.
CONWAY,
"Func ions o
one
complex
a iable,"
G
.T
.M
.,
Sp inge -Ve lag,
New
Yo k,
Second
Edi ion,
1978
.
3
.

C
.
O
.
HORCAN
AND
J
.
K
.
KNOWLES,
"Recen
de elopmen s
con-
ce ning
Sain -
Venan 's
P inciple,"
Ad ances
in
Applied
Mechanics
23,
(Ed
.
J
.
W
.
Hu chinson
and
T
.
Y
.
Wu),
Academic
P ess,
New
Yo k,
1983,
pp
.
179-269
.
4
.

C
.
O
.
HORCAN,
Recen de elopmen s
conce ning
Sain -Venan 's
P inciple
:
An
upda e,
Appl
.
Mechanics Re iews
42
(1989),
295-303
.
5
.

C
.
O
.
HORCAN
AND
L
.
E
.
PAYNE,
Decay
es ima es
o
second-o de
quasilinea
pa ial
di e en ial
equa ions,
Ad
.
i
n
Applied
Ma h
.
5
(1984),309-332
.
6
.

C
.
O
.
HORCAN
AND
L
.
E
.
PAYNE,
Decay
es ima es
o
a
class
o
second-o de
quasilinea
equa ions
in
h ee
dimensions,
A ch
.
Ra
.
Mech
.
Anal
.
86
(1984),
279-289
.
7
.

R
.
J
.
KNOPS
AND
L
.
E
.
PAYNE,
A
Sain -Venan
P inciple
o
non-
linea
elas ici y,
A ch
.
Ra
.
Mech
.
Anal
.
81
(1983),
1-12
.
8
.

C
.
O
.
HORCAN
AND
L
.
E
.
PAYNE,
Decay
Es ima es
o
a
class
o
nonlinea
bounda y
alue
p oblems
in
wo
dimensions,
S
.LA
.M
.
Jou
.
Ma h
.
Anal
.
2
0
(1989),
782-788
.
9
.

C
.
O
.
HORCAN
AND
L
.
E
.
PAYNE,
On
he
asymp o ic beha io
o
solu ion
o
inhomogeneous
second-o de
quasilinea
pa ial
di e en-
ial
equa ions,
Qua
.
Appl
.
Ma hema ics
XLVII
(1989),
753-771
.
10
.
C
.
O
.
HORCAN
AND
L
.
E
.
PAYNE,
On
Sain -Venan 's
P inciple
in
ini e
an i-plane
shea
:
An
ene gy
app oach,
A ch
.
Ra
.
Mech
.
Anal
.
10
9
(1990),
107-137
.
11
.
J
.
N
.
FLAVIN,
R
.
J
.
KNOPS
AND
L
.
E
.
PAYNE,
Decay
es ima es
o
cons ained
elas ic
cylinde
o
a iable
c oss sec ion,
Qua
.
Appl
.
Ma hema ics
XLVII
(1989),
325-350
.
12
.
R
.
J
.
KNOPS,
S
.
RIONERO
AM)
L
.
E
.
PAYNE,
Sain -
Venan 's
p incipie
on
unbounded
egions,
P oc
.
Roy
.
Soc
.
Edinbu gh
115A
(1990),319-336
.
13
.

J
.
N
.
FLAVIN,
R
.
J
.
KNOPS
AND
L
.
E
.
PAYNE,
Asymp o ic
beha -
io
o
solu ion
o
semi-linea
ellip ic
equa ions
on
he
hal -cylinde ,
Jou
.
Appl
.
Ma h
.
Phys
.
(ZAMP)
43
(1992),
405-421
.
SOME
THEOREM
OF
PHRAGMEN-LINDELOF
TYPE

463
14
.
V
.
G
.
MAZ'JA,
"Sobole
spaces,"
Sp inge -Ve lag,
Be lin,
1985
.
15
.
C
.
O
.
HORCAN
ANDL
.
E
.
PAYNE,
A
Sain -
Venan
p inciple
o
a
heo y
o
nonlinea plane
elas ici y,
Qua
.
Appl
.
Ma hema ics
L
(1992),641
-
675
.
Depa amen
de
Ma emá ica
Aplicada
II
Uni e si a Poli écnica
de
Ca alunya
Te assa,
Ba celona
SPAIN
Rebu
el
2
de
Juliol
de
1993