T R I T A - MAT - 1981 - 9 ,
Mathematics
A P P L I C A T I O N S OF V A R I A T I O N A L
I N E Q U A L I T I E S TO A
MOVING BOUNDARY PROBLEM FOR HELE SHAW FLOWS
by
Bjbrn
Gustafsson
T A B L E OF CONTENTS
I.
INTRODUCTION
a. General
introduction
b. A s u r v e y o f t h e
c.
II.
Some f u r t h e r
contents
bibliographical
FORMULATIONS OF T H E MOVING BOUNDARY
a . N o t a t i o n s and
b. A c l a s s i c a l
c.
CONDITION
preliminaries
formulation
(A)
A semi-weak f o r m u l a t i o n
(B)
d . T h e weak f o r m u l a t i o n
(C)
e . Two l i n e a r c o m p l e m e n t a r i t y
III.
notes
f.
Two v a r i a t i o n a l
g.
T h e moment i n e q u a l i t y
problems
inequalities
(D1) and
( E 1 ) and
(E2)
(F)
R E L A T I O N S BETWEEN T H E MOVING BOUNDARY C O N D I T I O N S , AND
E X I S T E N C E AND U N I C I T Y
RESULTS
a.
( A ) =» ( B )
( T h m 1)
b.
( B ) => ( C )
(Thm 2)
c.
( C ) => ( D )
( T h m 3 , Thm 4 )
d.
( D ) .» ( E )
(Thm 5)
e.
E x i s t e n c e , u n i q u e n e s s and r e g u l a r i t y
solutions
to
(E)
of
(Thm 6)
f.
( D ) •» ( C )
(Lemma 7 , Lemma 8 , Thm 9 )
g.
(C) * (F)
( T h m 10)
I V . SUMMARIZING R E S U L T S AND A P P L I C A T I O N S
a . Main t h e o r e m s
b. A d d i t i o n a l
c.
V.
(D2)
properties
Applications
REFERENCES
(Lemma 1 1 , Thm 1 2 , Thm 13)
(Lemma 1 4 , Thm 15)
t o q u a d r a t u r e domains
( T h m 16)
-1-
I.
INTRODUCTION
a. General
introduction
The aim o f the present paper is to prove a global
existence
and
u n i q u e n e s s t h e o r e m f o r a k i n d o f weak s o l u t i o n t o a m o v i n g b o u n d a r y
l e m (MBP) a r i s i n g
in two-dimensional
i n t r o d u c e d by S. R i c h a r d s o n i n
T h e o r e m 13 ( p . 6 9 ) .
[17]
H e l e Shaw f l o w s .
prob-
T h i s p r o b l e m was
. Our main r e s u l t s a r e summarized
We a l s o g i v e some a p p l i c a t i o n s
to quadrature
in
domains
(p.78f).
O u r MBP o c c u r s a s a s p e c i f i c e x a m p l e i n a c l a s s o f MBP:s w h i c h c a n
be d e s c r i b e d a s
follows:
L e t t h e r e t o e a c h r e g i o n D i n some s u i t a b l e c l a s s o f r e g i o n s
]R
n
be a s s o c i a t e d
( b y some r u l e t o be s p e c i f i e d )
f i n e d , h a r m o n i c and p o s i t i v e
in a neighbourhood in
3 D , t a k i n g t h e b o u n d a r y v a l u e z e r o on 3D.
main D = D Q
t>0
(it
to
the problem i t
such t h a t
i s assumed
3D
t
that
D of the
de-
boundary
T h e n , g i v e n an i n i t i a l
t o f i n d a f a m i l y o f d o m a i n s t -> D^
moves w i t h t h e v e l o c i t y
here
a f u n c t i o n g^
g r a d g^
in
-(grad
do-
for
9o )| p
t
has a c o n t i n u o u s
g
extension
3D ).
t
More g e n e r a l l y t h e r o l e o f t h e b o u n d a r y
b y some d i s t i n g u i s h e d s u b s e t S o f i t ,
3Dj>S
t
remains
3D a b o v e c a n be
so t h a t o n l y S
t
played
moves
while
fixed.
T h i s d e f i n e s t h e c l a s s o f MBP:s w h i c h we h a v e i n m i n d .
In order
s p e c i f y a s i n g l e p r o b l e m i n i t one has t o s p e c i f y t h r e e t h i n g s :
the
to
class
o f r e g i o n s D u n d e r c o n s i d e r a t i o n , t h e r u l e w h i c h a s s o c i a t e s g^ w i t h D a n d
the subset
S<=3D f o r e a c h D.
L e t us a t o n c e n o t i c e t h a t a f a m i l y t -* D
kind necessarily
vector field
We s h a l l
i s an i n c r e a s i n g f a m i l y
- g r a d g^
t
s o l v i n g a MBP o f t h e
( D ^ cz D^. f o r T < t )
since
p o i n t s p e r p e n d i c u l a r l y o u t f r o m D^ a t 3 D
g i v e t h r e e e x a m p l e s i n t h e c l a s s o f MBP:s j u s t
t h e t h i r d o f w h i c h i s t h e one w h i c h s h a l l
be t r e a t e d i n t h i s
above
the
t
(or
S^).
described,
paper.
-2-
Ex.
1:
D is
Let
or
]R
and l e t ,
i n t h e c l a s s o f bounded domains c o n t a i n i n g
K,
g^ be t h a t
function
9
K be a f i x e d c o m p a c t s e t i n ]R
i n D-^K w h i c h t a k e s t h e b o u n d a r y
_ M
on
K
° " 10
on
3D
and l e t
whenever
harmonic
values
,
S = 3D .
T h e MBP s o a r i s i n g o c c u r s
cathode, D^K
in electrochemistry,
i s an e l e c t r o l y t e
which gradually dissolves
i s an a n o d e , t h e s u r f a c e
n
under e l e c t r o l y s i s
to the r u l e described above.
g r a d g^ i s t h e e l e c t r i c
and R ^ D
Here
current
in which case K is a
so t h a t
D grows
-g^ is the e l e c t r i c
according
potential
(up to a p o s i t i v e f a c t o r ) .
of
and
See [ 7 ]
for
details.
2
Ex.
2: L e t
L
be a s u b s p a c e o f
R
3
(or
R ) of codimension one, l e t
be o n e o f t h e h a l f - s p a c e s
in which i t
L as b e i n g h o r i z o n t a l
H as b e i n g t h e l o w e r h a l f - s p a c e )
and
t h e c l a s s o f subdomains
D of
separates
H such t h a t
R
(or
3D = L U S
R )
(think
J
for
H
and
of
consider
some
sufficient-
l y smooth c u r v e ( o r s u r f a c e r e s p e c t i v e l y )
S i n H.
L e t h be a f i x e d
positive
f u n c t i o n on L s u f f i c i e n t l y s m a l l a t i n f i n i t y ( t y p i c a l l y h ( x ) =
_h 2
x
= ae
, a,b>0
if
L = R)
in D which takes the boundary
9
D
rh
on
L
Mo
on
S
T h i s model
ing water zone
of the l a t t e r
and d e f i n e
t o be t h a t h a r m o n i c
values
.
is suggested in [16] to d e s c r i b e the growth o f a
( t h e r e g i o n D) p e n e t r a t i n g a r o c k
is
i m p i n g e d on b y a h e a v y w a t e r j e t .
to the hydrostatic
portional
to the flow
propagat-
( H ) when t h e s u r f a c e
T h i s example is
i n s t a n c e o f p o r o u s medium f l o w g o v e r n e d by D a r c y ' s
tional
function
law.
p r e s s u r e o f t h e w a t e r and
g^ i s
- g r a d g^
proporis
pro-
velocity.
2
Ex.
3: Let
D range over a l l
o r i g i n , and l e t ,
r e s p e c t t o 0 £ D,
for
i.e.
such
bounded domains i n
D, g
Q
R
containing
be t h e G r e e n ' s f u n c t i o n
an
the
for D with
(L)
-3-
g^(z)
= - l o g |z| + h a r m o n i c
in
g (z)
=0
on 3D .
D
H e r e we h a v e i d e n t i f i e d
S = 3D i n t h i s
This
and used z = x + i y
t h i s means t h e f o l l o w i n g :
2
(corresponding
and m o d e r a t e r a t e .
Then the r e g i o n o f f l u i d
the mathematical
model
The f u n c t i o n g
Q
in
]R
fluid
is
the
grow according
to
described.
is here proportional
to the hydrostatic
pressure
in the d i r e c t i o n
of
perpen-
]R - p l a n e ) , a n d a c c o r d i n g t o t h e s o - c a l l e d H e l e Shaw
- g r a d g^ i s p r o p o r t i o n a l
to the average of i t
(that
At
vis-
injected at a constant
D will
( t h e p r e s s u r e t u r n s o u t t o be c o n s t a n t
2
to the
occupied
have high
Newtonian and i n c o m p r e s s i b l e .
to the o r i g i n
be
distance
i s t o be p a r t l y
t o t h e r e g i o n D) w h i c h m i g h t w e l l
point corresponding
dition
occurs
L e t two l a r g e s u r f a c e s
The space between the s u r f a c e s
b u t w h i c h i s s u p p o s e d t o be
equation
It
t o e a c h o t h e r a n d t o t h e ]R - p l a n e a n d a t a s m a l l
from each o t h e r .
dicular
variable.
origin.
parallel
the f l u i d
as
H e l e Shaw f l o w s w i t h f r e e b o u n d a r i e s a n d w i t h a
More p r e c i s e l y ,
cosity
J
i s t h e MBP t o w h i c h t h e p r e s e n t p a p e r i s d e v o t e d .
source at the
by a f l u i d
with
example.
in two-dimensional
lying
]R
D ,
to the f l u i d v e l o c i t y ,
o r more
correctly
across the gap. T h i s g i v e s the moving boundary
3D m o v e s w i t h t h e v e l o c i t y
- g r a d g^ ,
up t o a c o n s t a n t
confac-
tor).
The assumption that the f l u i d
harmonic f u n c t i o n
a logarithmic
(div ( -grad
singularity
is
i n c o m p r e s s i b l e i m p l i e s t h a t g^ i s a
gQ) = 0) and t h e i n j e c t i o n o f f l u i d
at the o r i g i n f o r
g^ .
Further,
g^ t a k e s a
c o n s t a n t v a l u e on 3D ( a c o n s t a n t w h i c h c a n be c h o s e n t o be z e r o )
the pressure outside the f l u i d
the pressure drop across
l y constant along
high).
is constant
be a p p r o x i m a t e -
p r e s u p p o s e s t h a t t h e c u r v a t u r e o f 3D i s n o t
T h i s m o t i v a t e s t h a t g^ e q u a l s t h e G r e e n ' s f u n c t i o n f o r
respect to the
origin.
because
( t h e a i r p r e s s u r e ) and because
3D due t o t h e s u r f a c e t e n s i o n w i l l
3D ( t h i s
gives
D with
too
-4-
F o r more d e t a i l s
and t e c h n i c a l
background to t h i s example,
see
[17] .
A s i d e f r o m t h i s H e l e Shaw i n t e r p r e t a t i o n
Ex.
3 (possibly with several
to describe
"migration of oil
in particular
Another
logarithmic
Chapter XV,
the choice f o r
singularities)
in hydrostatic
g^
in
has b e e n u s e d
environment".
See
[15],
§7.
i n t e r p r e t a t i o n o f the moving boundary
conditions in the
ex-
amples above i s t h a t t h e y d e s c r i b e a one-phase S t e f a n problem w i t h
zero
specific
tem-
heat ( i . e .
D is a melting
solid held at i t s melting
p e r a t u r e , t a k e n t o be z e r o , D i s t h e g r o w i n g l i q u i d p h a s e w i t h
g^ a n d t h e l i q u i d
a n d p . 26 f f
i s supposed to have z e r o s p e c i f i c
in the present
paper
(Ex.
3 a b o v e ) was i n t r o d u c e d
e q u a t i o n f o r t h e Riemann mapping
f r o m t h e u n i t d i s c o n t o t h e domain
(C
[3]
In t h a t paper the moving boundary c o n d i t i o n
f o r m u l a t e d as a d i f f e r e n t i a l
plane
See e . g .
paper.
T h e MBP t o be t r e a t e d i n t h i s
S. R i c h a r d s o n i n [ 1 7 ] .
heat).
temperature
and assuming t h a t D
t
(identifying
]R
is
function
with the
is simply connected). This
complex
differential
e q u a t i o n i s o f a q u i t e uncommon t y p e a n d no e x i s t e n c e o r u n i q u e n e s s
for solutions
uniqueness
however,
to i t
proof for
is given in [17].
solutions
motion
( t h e complex
e x i s t e n c e and
t o t h e same d i f f e r e n t i a l
t o have been g i v e n i n [ 2 1 ] .
Richardson also discovers
(A l o c a l
in
by
a.
equation
proof
partial
seems,
)
[17] a sequence o f simple c o n s t a n t s
moments, see p. 9 )
and uses them t o o b t a i n a f u n c t i o n a l
for his d i f f e r e n t i a l
of
equation
e q u a t i o n f o r t h e Riemann mapping
function.
I n t h e p r e s e n t p a p e r we r e - f o r m u l a t e
l e a d s us i n t o t h e t h e o r y o f v a r i a t i o n a l
t h e o r y we a r e a b l e t o g i v e a g l o b a l
a k i n d o f w e a k s o l u t i o n t o t h e MBP.
R i c h a r d s o n ' s MBP i n a w a y w h i c h
inequalities.
By u s i n g
this
e x i s t e n c e and u n i q u e n e s s p r o o f
for
-5-
The t h e o r y of v a r i a t i o n a l
ful
inequalities
i n h a n d l i n g MBP:s o f v a r i o u s t y p e s .
w o r k s b y G. D u v a u t a n d
inequality
CM. Elliott.
formulation for classical
h a s a l r e a d y p r o v e d t o be u s e -
M o s t r e l e v a n t f o r us a r e
In [5] Duvaut g i v e s a
type
outlines
(our Ex.
variational
inequality formulations
1 is t r e a t e d in [6] §4).
Elliott
solutions.
3) f r o m a p u r e l y mathematical
intended to p a r t l y
fill
t h i s gap.
paper by E l l i o t t - J a n o v s k y w i l l
ly overlap since E l l i o t t
tions
and n u m e r i c a l
3).
exist
s i m p l e t r e a t m e n t o f t h e H e l e Shaw MBP
point of view
The p r e s e n t paper
P r o b a b l y o u r p a p e r and t h e
is
announced
seems t o w o r k a l o n g s o m e w h a t d i f f e r e n t
questions
our
a l s o announces a paper
complement each o t h e r r a t h e r than
we a n d s i n c e t h e w o r k o f E l l i o t t
[6]
a n d o t h e r s t h e r e seems s o f a r t o
no c o m p l e t e , d e t a i l e d a n d r e l a t i v e l y
(Ex.
In
f o r problems of
t o g e t h e r w i t h V . J a n o v s k y on o u r H e l e Shaw p r o b l e m ( E x .
Despite the works of E l l i o t t
variational
t w o - p h a s e S t e f a n p r o b l e m s * / and
i n d i c a t e s e x i s t e n c e and u n i q u e n e s s p r o o f s f o r t h e i r
Elliott
the
complete-
lines
seems t o be m o r e d i r e c t e d t o w a r d s
w h i l e the p r e s e n t work i s c o m p l e t e l y
than
applicatheo-
retical .
Acknowledgements:
I am v e r y much i n d e b t e d t o P r o f e s s o r H a r o l d S . S h a p i r o
h a v i n g p r o p o s e d t h e p r o b l e m a n d f o r t h e many l o n g a n d v a l u a b l e
s i o n s on i t w i t h o u t w h i c h I w o u l d p e r h a p s n o t h a v e b e e n a b l e t o
t h e p r o j e c t t o an e n d .
excellent
I a l s o wish to thank Gunhild Melin f o r
and r a p i d t y p i n g o f t h i s
for
discuscarry
her
manuscript.
/Here " c l a s s i c a l " stands f o r the r e q u i r e m e n t t h a t both phases s h a l l have
s t r i c t l y p o s i t i v e s p e c i f i c heat.
When o u r p r o b l e m i s i n t e r p r e t e d as a
S t e f a n p r o b l e m t h e s p e c i f i c h e a t i s z e r o ( c f . p . 4 a n d 26 f f ) .
/ R e c e n t l y t h e r e h a s a p p e a r e d t h e w o r k [19] o f M. S a k a i w h i c h c o n t a i n s
a d e t a i l e d t h e o r e t i c a l s t u d y o f t h e H e l e Shaw p r o b l e m .
S e e p . 13 f f .
-6-
b. A s u r v e y o f t h e
This
tion
contents
paper c o n s i s t s
is Section
In Section
of four sections of which the present
introduc-
I.
I I we f o r m u l a t e a n u m b e r o f p r o p e r t i e s , ( A ) t o ( F ) , o f o n e
2
parameter f a m i l i e s
t ->
o f domains i n
express our moving boundary
The f i r s t
classical
of these,
sense, i.e.
This formulation
boundary
of i t .
9D
w h i c h i n v a r i o u s ways a r e
( A ) , expresses that t + D
that
to
condition.
9D^
s o l v e s o u r MBP i n a
t
moves w i t h t h e v e l o c i t y
is not v e r y f l e x i b l e
t o be r a t h e r s m o o t h a n d
t
]R
because i t
- g r a d g^
requires the
also involves a
.
t
parametrization
Thus, t h i s formulation cannot cover occasions at which
changes
connectivity.
(B)
i s a weakened and somewhat more f l e x i b l e way o f f o r m u l a t i n g
moving boundary c o n d i t i o n .
I t does not r e q u i r e
p a r a m e t r i z a t i o n o f 9D^, a l t h o u g h i t
(directly
involves a contour
We t h e n a r r i v e a t o u r c o n c e p t o f weak s o l u t i o n ,
freed ourselves
ther,
from all
regularity
( C ) makes no e x p l i c i t
requirements
is.
integral
(C).
I n ( C ) we h a v e
<>
\
1
=
Here
A
g
D
I t can r a t h e r e a s i l y
+
2
6
f u n c t i o n o f D^, g ^
i s t h e Green's f u n c -
with respect to the o r i g i n , extended to a l l
D ,
t
is nothing else than
See p . 2 6 f f .
by
and 6Q i s t h e D i r a c measure a t t h e
T h e s e f u n c t i o n s a r e c o n s i d e r e d as d i s t r i b u t i o n s
Also
distribution
^ 0
t o be z e r o o u t s i d e
language.
be
equation
is the c h a r a c t e r i s t i c
tion for
it
t
Fur-
function.
s e e n t h a t t h e m o v i n g b o u n d a r y c o n d i t i o n can be e x p r e s s e d i n
l a n g u a g e by t h e
a
along
f o r the boundary.
r e f e r e n c e t o the Green's
L e t us i n d i c a t e m o r e c l o s e l y w h a t ( C )
at least)
the
the condition
on
]R^.
(B) formulated w i t h i n
( 1 ) c a n be g i v e n a d i r e c t p h y s i c a l
defining
origin.
Actually,
(1)
distribution
interpretation.
-7-
*
By i n t e g r a t i n g
(1) with respect to
t
and
putting
t
one
< >
3
D
X
T
obtains
" \
Thus,
=
(1)
A
u
(5)
u^ = 0
t
> 0
+
2
7
T
of
(2),
#
0
6
( 3 ) m u s t be i n c r e a s i n g
(for t > 0 )
outside
Now ( C )
tribution
u
on
]R
( 3 ) combined w i t h ( 4 ) and ( 5 ) , i . e .
which s a t i s f i e s
(2) of
seen t h a t these c o n d i t i o n s
of
must a l s o
> 0 and a s o l u t ~
satisfy
we s a y
that
i s a weak s o l u t i o n i f f o r e a c h t > 0 t h e r e e x i s t s a d i s 7
the coupling
solutions
t
n
T
is e s s e n t i a l l y
t
u
Since g
and
D
t -> D . ( f o r t > 0 )
~
replace
t
i s e q u i v a l e n t t o ( 2 ) and ( 3 ) t o g e t h e r .
t i o n t -*
(4)
t
u
u
t
(3)-(5).
The c o n d i t i o n s
to the Green's
( 4 ) and
(5)
f u n c t i o n , and i t w i l l
be
are s t r o n g enough t o guarantee uniqueness
for
(given DQ) .
(3) - (5)
Observe the remarkable f e a t u r e of the transformation leading from
to ( 3 ) - (5) of having
on t ;
l i b e r a t e d the problem f r o m any e s s e n t i a l
t h e c o n c e p t o f weak s o l u t i o n
meaning f o r each f i x e d
particular
value of
value of
t
i s a c o n c e p t w h i c h has an
dependence
independent
t , a n d a weak s o l u t i o n c a n a l w a y s be f o u n d f o r
w i t h o u t b o t h e r i n g about t h e s o l u t i o n f o r any
t .
/ T h e s u b s c r i p t t i n u^. i s u s e d t o i n d i c a t e t h a t u^. i s a f u n c t i o n
o n ] R w h i c h d e p e n d s on t ( i . e . t i s a p a r a m e t e r ) . We n e v e r u s e
s u b s c r i p t s to denote p a r t i a l d e r i v a t i v e s in t h i s paper.
2
(1)
any
other
-8-
In Sections
I I a n d I I I we w a n t t o w o r k
2
on a b o u n d e d d o m a i n Q c= IR
condition
(C)
( a
sufficiently
is not formulated e x a c t l y
t o make some m i n o r m o d i f i c a t i o n s
is
of
that the D :s
it.
c o n s i d e r a t i o n and t h i s
It
t-set
is also required that
Next to c o n d i t i o n
Xj)
problems
( C ) come c o n d i t i o n s
( L C P : s ) , one f o r each
( 5 ) by r e p l a c i n g
X
from
< 1 .
D
t
U
T
• (Au^ + A ^ )
t
X
- 1 2 7 r f 6
"UQ
D
+
0
(7)
°
=
(D2)
t
*/
([0,T],
t
under
0 < T < °°).
t.
express
complemen-
In these problems the
explicit
(D1) is o b t a i n e d from
(3) by t h e i n e q u a l i t y
(3)-
resulting
such
that
= 0
is the solution -
t
v
in
s m a l l ) , we h a v e
of so-called l i n e a r
t.
(3)
> 0
Here ^
*t
forced
( D 1 ) i s t h e p r o b l e m o f f i n d i n g u ^ , d e f i n e d on Q, w i t h
Thus
b o u n d a r y v a l u e z e r o on SFI
u
t
0
(D1) and (D2) w h i c h
has d i s a p p e a r e d .
the occurrence of D
>
for all
r
for all
t
o c c u r r e n c e o f t h e domains
r
compact in
t h e m o v i n g b o u n d a r y c o n d i t i o n as a s e r i e s
tarity
>
h a s t o be a b o u n d e d s e t
u €Hg(fi)
Therefore,
T h u s , the Dirac measure in
are r e l a t i v e l y
t
large disc).
(HUtt))
a s a b o v e b u t we h a v e b e e n
r e p l a c e d by an a p p r o x i m a t i o n t o i t
to require
(6)
in a Sobolev space
= u
t
i s t h e same as
+ ^
essentially
in
Q,
on
dQ
7
.
of
(D1) but e x p r e s s e d i n terms o f t h e
function
instead:
t
1
6Q i n
that
( 7 ) has t o be s m o o t h e d o u t .
u eH ! (fi),
t
(
)
Then I ^ e H g ^ )
and i t
is
required
-9-
Av
< 0
t
(8)
(v -* )
t
.Av
t
=
t
Each o f the LCP:s
tional
inequalities,
Find ( f o r each t )
(9)
/ V(u-u ) • Vu
t
and ( E 2 )
t
(D1) and (D2) i s e q u i v a l e n t t o a s e r i e s o f
( E 1 ) and ( E 2 ) r e s p e c t i v e l y .
u GHg(ft)
> 0
for all
t
v e HQ(Q)
u £ Hj(ft) with
is
< 0 and
t
Au + A ^
t
< 0
,
t
> 0 for
is
F i n d v G H g ( f 2 ) such t h a t v
(10)
such t h a t Au^ + A ^
t
(E1)
w i t h v > ip
t
> ^
and / v ( v - v )
• Vv
t
t o o u r MBP i s t h e
t
t e n c e o f an i n f i n i t e n u m b e r o f s i m p l e c o n s t a n t s o f m o t i o n f o r
the complex
c
n< t>
D
/ z
all
t
One v e r y n i c e p r o p e r t y o f s o l u t i o n s t -> D
(11)
varia-
it,
exisnamely
moments
n
(z = x + iy)
dx dy
f o r n = 1 , 2 , 3 , . . . . T h e z e r o t h o r d e r moment C g ( D )
t
increases l i n e a r l y with t.
= |D^| = a r e a o f D^
F o r t h e c a s e t h a t t h e D^ a r e
simply
con-
nected the set
( 1 1 ) o f c o n s t a n t s o f m o t i o n may a l s o be e x p e c t e d t o be
complete,
g i v e n Dp s i m p l y c o n n e c t e d , a map
i.e.
suitable
regularity
requirement)
property
that the c (D^)
n
Our f i n a l
remain
formulation
(F)
some
s h o u l d be c o m p l e t e l y d e t e r m i n e d b y
the
constant.
( F ) o f the moving boundary c o n d i t i o n i s a
s t r e n g t h e n e d f o r m o f t h i s moment p r o p e r t y .
satisfies
t -* D^. ( s a t i s f y i n g
i f f o r each t > 0
N a m e l y , we s a y t h a t t -»• D^
-10-
j CP
(12)
>
27RT - C P ( 0 )
for every
+
/ CP
s u b h a r m o n i c f u n c t i o n TP i n D - T o s e e w h y ( 1 2 ) s t r e n g t h e n s ( 1 1 )
T
observe that a natural
(13)
/ CP = 27Tt - C P ( 0 )
D
D
T h e moment i n e q u a l i t y
Section
ing
CP i n D
T
(CP = z
yields
1 1
(11) ) .
( 1 2 ) t u r n s o u t t o be s u f f i c i e n t l y
strong
to
to ( C ) .
III
consists
precise relationships
ships
that
0
harmonic f u n c t i o n
be e q u i v a l e n t
( 1 1 ) is
+ / CP
t
for every
generalization of
essentially
o f a number o f theorems g i v i n g
between the c o n d i t i o n s
(and the f l o w o f Section
III
( A ) to ( F ) . These
the
relation-
i n g e n e r a l ) a r e as s h o w n i n t h e
follow-
diagramme.
Thm
(A)
=>
Thm
1
2
(B)
Thm
(0
Thm
Thm
3
==>
Thm 9
10
(E1)
Thm
Lemma 8
4-
Thm 4
5
(E2)l
(D2)
(F)
5
(D1)
hm
+
7
Minimal
properties
and
monotonicity
properties
(Thm 7
with cor )
Have u n i q u e
solutions
with
regularity
properties
(Thm 6)
s
V
H e r e an
arrow
in the sense of
, say
(C)
Thm 3
= ^ > ( D 1 ) , means t h a t a s o l u t i o n t + D .
( C ) i n a s i m p l e w a y (made p r e c i s e
vides a solution t + u
t
of ( D 1 ) .
in Theorem 3 ) p r o -
-11-
As t h e diagramme shows t h e c o n d i t i o n s
while
tion
(A)
III
implies
( B ) and ( B )
implies
(C) to (F)
(C) to ( F ) .
are a l l
equivalent
The main goal o f
i s t o show t h e e x i s t e n c e and u n i q u e n e s s o f s o l u t i o n s t o
(weak s o l u t i o n s ) .
This result
i s s t a t e d as C o r o l l a r y
( A ) => ( B ) => ( C ) , s h o w i n g t h a t a c l a s s i c a l
together with the fact
solution
t h a t a weak s o l u t i o n
9.1.
(C)
The
steps
i s a l s o a weak
solution,
is unique, serve to motivate
c o n c e p t o f weak s o l u t i o n a n d a l s o p r o v e s u n i q u e n e s s f o r c l a s s i c a l
The hardest step
or
(C)(Theorem 9).
( u ^ and v^ r e s p e c t i v e l y )
while
i n v o l v e s showing that a c e r t a i n f u n c t i o n
function of a set
This
ties
of
(Theorem
III
(C) concerns sets
(namely
-Av^.)
(D )
is a
Its proof requires regularity
(D1) and (D2)
possessed by them ( L e m m a
Section
(D1) and (D2) c o n c e r n
D^).
properties
proper-
corollaries).
c o n c l u d e s by showing t h e e q u i v a l e n c e between ( C ) and
ft
<= ] R
2
( a d i s c ) ) . T h i s h a s f o r c e d us t o s m o o t h o u t t h e
of the Green's function
space).
(F)
10).
I n S e c t i o n I I I we w o r k w i t h i n a S o b o l e v s p a c e ( H Q ( ^ ) on a f i x e d
domain
step
characteristic
( T h e o r e m 6 ) and a l s o m o n o t o n i c i t y
7 with
(D1)
func-
this
t
( n a m e l y t h e s e t w h i c h i s t o be t h e c o m p l e m e n t o f
i s t h e c o n t e n t o f Lemma 8 .
of the solutions
Since
the
solutions.
in t h e diagramme i s t o c a r r y o v e r a s o l u t i o n o f
(D2) t o a s o l u t i o n o f
tions only
Sec-
Section
(in order that
it
shall
belong to that
IV begins by showing t h a t the s o l u t i o n s
depend i n any e s s e n t i a l
w a y on t h e c h o i c e o f
Q
bounded
singularity
Sobolev
o f ( C ) do n o t
( a s l o n g as i t
l y l a r g e ) and a r e n o t a f f e c t e d by t h e s m o o t h i n g o f t h e G r e e n ' s
is
sufficient-
function
(Lemma 1 1 ) .
P
We a r e t h e n a b l e t o s u m m a r i z e much o f t h e w o r k i n S e c t i o n
m u l a t i n g a k i n d o f Main T h e o r e m - T h e o r e m 13.
initial
states that this
t -*
defined for all
solution satisfies
i
I I I be
for-
I t s t a t e s t h a t g i v e n an
d o m a i n Dg ( a n o p e n b o u n d e d s e t c o n t a i n i n g t h e o r i g i n )
u n i q u e weak s o l u t i o n
1
there
is a
t £ [0,«>) o f o u r MBP. I t
t h e moment i n e q u a l i t y a n d t h a t i t
also
has
-12-
those monotonicity
properties which i t
D
t
t
increases with
served for all
and t h a t
i s supposed t o h a v e , namely
inclusions
between i n i t i a l
that
domains a r e
pre-
t > 0 .
T h e n a t u r e o f o u r MBP i s s u c h t h a t o n e e x p e c t s
and n i c e r w i t h i n c r e a s i n g
t.
For example, f o r a l l
nicer
9D^. i s
expected
t>0,
to consist of a n a l y t i c
curves
t -» oo
should approach that of a c i r c u l a r
t h e shape o f
(even i f
t o become
3DQ d o e s n o t ) a n d a s y m p t o t i c a l l y
disc.
as
We a r e
n o t a b l e t o p r o v e a n y c o m p l e t e r e s u l t o f t h a t s o r t b u t we a t l e a s t make
some s t e p s
sists
in that d i r e c t i o n .
of analytic
T h u s i n T h e o r e m 15 we p r o v e t h a t
curves f o r t > 0 under the assumptions that
n i t e l y c o n n e c t e d and t h a t i t c o n t a i n s
Dg c o m p a c t l y
l a r g e o r t h e a s s u m p t i o n t h a t 3Dg i s a n a l y t i c
This
domains
The f i r s t
i s p r e s e r v e d by s o l u t i o n s
asserts
c.
result states
is
to so-called
sufficientthe
theorem).
quadrature
o f o u r MBP ( T h e o r e m 1 6 ) , a n d t h e s e c o n d
bibliographical
fi-
t h a t t h e p r o p e r t y o f b e i n g a QD
t h e e x i s t e n c e o f QD:s o f c e r t a i n k i n d s
Some f u r t h e r
t
(Remark 1 a f t e r
paper c o n c l u d e s w i t h two a p p l i c a t i o n s
(QD:s).
is
( i . e . t h a t DQ <= D^.).
T h e l a s t a s s u m p t i o n c a n be r e p l a c e d b y t h e a s s u m p t i o n t h a t
ly
9D^. c o n -
(Corollary
result
16.1).
notes
T r e a t m e n t s o f MBP:s o f k i n d s s i m i l a r t o o u r s by t h e methods o f
tional
inequalities
article
space
[11] and
a r e f o u n d i n [ 5 ] and
[6],
See a l s o §3 o f t h e
C h a p t e r V I I I o f t h e r e c e n t book [ 1 2 ]
variasurvey
(MBP:s i n one
dimension).
In the context
of Stefan problems t h e r e
"weak s o l u t i o n " , d i s c u s s e d e . g .
lution
i n [ 8 ] and
i s an e s t a b l i s h e d n o t i o n
[3].
T h i s c o n c e p t o f weak
i s n o t t h e same a s t h a t u s e d i n t h e p r e s e n t
In [3] Crowley proves u n i c i t y
[ 8 ] and [ 3 ] )
t o MBP:s o f o u r t y p e
paper.
f o r weak s o l u t i o n s
(in particular
of
( i n the sense
that of Ex.
of
1, p . 2 ) .
so-
-13-
I n t h e p r e s e n t paper two k i n d s o f LCP:s
kinds of v a r i a t i o n a l
side.
Our c o n c e p t o f
the problems
(E2)
inequalities
( ( E 1 ) and ( E 2 ) ) a r e t r e a t e d s i d e
weak s o l u t i o n
(D1) and ( E 1 ) .
( ( D 1 ) and ( D 2 ) ) and two
(C)
i s most n a t u r a l l y
(E1).
For example the v a r i a t i o n a l
inequalities
of solutions
of variational
and [ 2 ] ) c o n c e r n t h o s e o f t h e t y p e
(among o t h e r t h i n g s )
subject of
[19]
[6]
on e x i s t e n c e ,
(e.g.
unicity
[12],
[13]
(E2).
(preprint)
[ 1 9 ] o f M. S a k a i
o u r H e l e Shaw MBP i s t r e a t e d .
The main
k i n d s , a n d t h e r e s u l t s on t h e MBP a r e
of this.
and
a r r i v e d at in [ 5 ] ,
i s t h e c o n s t r u c t i o n and i n v e s t i g a t i o n o f q u a d r a t u r e
mains of r a t h e r general
as a p p l i c a t i o n s
(D2)
and
inequalities
R e c e n t l y t h e r e has a p p e a r e d t h e paper
in which
with
t y p e than (D1)
and [11] a r e o f the k i n d ( E 2 ) , and most l i t e r a t u r e
and r e g u l a r i t y
connected
The main r e a s o n f o r a l s o i n v o l v i n g
i s t h a t t h e s e p r o b l e m s a r e o f more c o n v e n t i o n a l
by
do-
obtained
These r e s u l t s are s i m i l a r to (or s l i g h t l y
strong-
e r t h a n ) t h o s e o f t h e p r e s e n t p a p e r , w h i l e t h e methods used a r e q u i t e
dif-
ferent.
L e t us i n d i c a t e r a t h e r b r i e f l y w h a t S a k a i has d o n e on t h e H e l e
Shaw
problem.
1) S a k a i f o r m u l a t e s a c o n c e p t o f s o l u t i o n o f t h e H e l e Shaw f l o w s . * /
concept
is s i m i l a r to (but weaker than) our concept of c l a s s i c a l
in the sense that the core of
it
to our
(p. 22).
however,
t>0
equation
(iii)
a global
and i t
of
(A)
one ( i . e .
(essentially)
Sakai's concept of solution
i s a l w a y s t o be d e f i n e d f o r
and t o change c o n n e c t i v i t y .
Sakai's concept of solution is very complicated.
that a solution
it.
a solution
solution
in this
sense always e x i s t s
(Uniqueness of s o l u t i o n s
but
he
is
is,
all
i s f o r m u l a t e d i n a way t h a t a l l o w s t h e domain t o d e v e l o p
t a i n kinds of s i n g u l a r i t i e s
of this
i s an e q u a t i o n e q u i v a l e n t
This
cer-
As a c o n s e q u e n c e
Sakai
not
i s , however, proved; c f . 4
able
expects
to
prove
below.)
* / W h e n e v e r t h e t e r m " s o l u t i o n " ( o f H e l e Shaw f l o w s ) a p p e a r s on t h e f o l l o w ing pages,
i t i s u n d e r s t o o d t o mean s o l u t i o n i n t h i s s t r i c t s e n s e o f
Sakai.
The e x a c t d e f i n i t i o n o f t h i s concept o f s o l u t i o n i s , however,
t o o c o m p l i c a t e d t o be r e p r o d u c e d h e r e .
-14-
2)
S a k a i f o r m u l a t e s a c o n c e p t o f weak s o l u t i o n w h i c h i s as
(with our
notations):
G i v e n an a r b i t r a r y
bounded domain D ,
n
u
f o r each t > 0
(i)
follows
i s a domain c o n t a i n i n g
{Do.}
i s a weak s o l u t i o n
t>0
Dg s u c h t h a t
if
z
/ cpda + t ' t p ( O ) < / cpda
0
D
" t
D
for all
function
cp
w h i c h a r e s u b h a r m o n i c a n d i n t e g r a b l e on D^,
(ii)
has f i n i t e
area,
(iii)
is minimal w i t h the p r o p e r t i e s
(i)
and
(ii).
T h i s c o n c e p t o f w e a k s o l u t i o n i s s i m i l a r t o o u r moment i n e q u a l i t y
In f a c t ,
the essential
test class
difference
t h a n we h a v e .
is that
A further
f i n i t e a r e a o f the domains ( ( i i ) )
Sakai proves
cessarily
(iii)
domain s a t i s f y i n g
such t h a t
It follows
in [19]) that a
Theorem 3.7)
(i)
if
and ( i i )
satisfying
(i)
that there
(i.e.
However,
to ( i i i )
(i)
and ( i i )
t
then
t i o n a l s o i n our sense
(i.e.
s u i t a b l e R and r ) .
i s u n i q u e up t o n u l l - s e t s
satisfies
(F)
weak s o l u t i o n s a r e i d e n t i c a l
t
in
interval
the
solu-
[0,T]
in our
T h e r e f o r e the two concepts
modulo n u l l - s e t s .
k i n d o f answer t o a q u e s t i o n posed by Sakai
(i)
a n d s o i s a weak
( C ) on e a c h f i n i t e
9.1).
minimal)
D <=G).
On t h e o t h e r h a n d , a weak s o l u t i o n
(Corollary
Sakai
satisfying
f r o m w h a t has b e e n s a i d a b o v e t h a t a weak s o l u t i o n
o u r moment i n e q u a l i t y
ne-
point.
i s a minimum ( n o t o n l y
t h e r e i s a domain D
larger
requires
t h e r e i s no d i f f e r e n c e a t t h a t
G also satisfies
sense o f Sakai s a t i s f i e s
and f o r
is t h a t Sakai o n l y
i t c a n be l o o k e d u p o n a s a k i n d o f n o r m a l i s a t i o n .
(essentially
and ( i i )
difference
Sakai has a somewhat
w h i l e we r e q u i r e b o u n d e d n e s s .
i s bounded, so i n e f f e c t
As r e g a r d s
proves
( T h e o r e m 6.4
in ( i )
(F).
Incidentally
inequality
techniques.
of
this gives a
(p.113 in [19]) concerning
r e l a t i o n b e t w e e n h i s c o n c e p t o f weak s o l u t i o n a n d t h a t one o b t a i n e d b y
variational
sense
the
-15-
3)
S a k a i s h o w s e x i s t e n c e a n d u n i q u e n e s s f o r weak s o l u t i o n s . T h e
i s p r o v e d by s i m p l y c o n s t r u c t i n g
This construction
is rather
t h e domains
"by h a n d " ( m o r e o r
(i)
and ( i i )
due t o t h e r e q u i r e m e n t
t u r e domains
for
under
(iii).
of
4)
2) a b o v e .
(Actually
domains
Uniqueness
the
having
is then
trivial
Sakai proves e x i s t e n c e o f
( s u b h a r m o n i c , harmonic and a n a l y t i c )
w a y , a n d t h e w e a k s o l u t i o n s o f H e l e Shaw f l o w s a r e j u s t
Sakai proves t h a t a s o l u t i o n
i s a l s o a weak s o l u t i o n
Since Sakai's concept of solution
of classical
(Proposition
is stronger
i s b e t t e r than our Theorems
solutions.
Existence of solutions
S a k a i has a p a r t i a l
known a p r i o r i
is a solution
Sakai
result
to f u l f i l
if
(certain
(Corollary
solution
actually
13.4).
is a solution
i s bounded by a n a l y t i c
then, for t > 0 , every
3D^ i s a n a l y t i c
except at certain
" c o r n e r s " and ends o f s l i t s ) .
of these exceptional
although
curves.
non-
t>0
d e g e n e r a t e component o f
points
uniqueness
c e r t a i n of the requirements f o r a solution
{D.}
z
points
exceptional
The union o v e r a l l
i s a s e t w h i c h i s a t most c o u n t a b l y
t>0
infinite
13.3.)
Finally,
Sakai proves t h a t f o r
large
t
a s o l u t i o n o f H e l e Shaw f l o w s
i s a s i m p l y c o n n e c t e d domain bounded by a s i m p l e a n a l y t i c c u r v e and
it asymptotically
a s t + °° c o n v e r g e s t o a d i s c
mann m a p p i n g f u n c t i o n f r o m
the unit disc
13.5).
that
in the sense t h a t the
Rie-
( s c a l e d in the proper way)
con-
v e r g e s u n i f o r m l y on t h e c l o s u r e o f t h a t d i s c t o t h e i d e n t i t y map
sition
priori)
1 and 2 .
i s , however, never proved
shows t h a t a s o l u t i o n e s s e n t i a l l y
More p r e c i s e l y ,
(a
i n t h a t d i r e c t i o n a s s e r t i n g t h a t a weak
(Proposition
13.1
concept
T h e weak s o l u t i o n b e i n g u n i q u e , t h e above r e s u l t a l s o shows
6)
in
applications
is weaker than our
s o l u t i o n a n d h i s c o n c e p t o f weak s o l u t i o n
than that of o u r s , t h i s r e s u l t
5)
va-
this.)
in [19]).
for
quadra-
r a t h e r g e n e r a l c l a s s e s o f p o s i t i v e measures and f o r
rious classes of test functions
this
less).
l e n g t h y and i s done i n such a way t h a t
d o m a i n o b t a i n e d c a n b e p r o v e d t o be t h e m i n i m u m o f a l l
the p r o p e r t i e s
existence
(Propo-
-16-
F O R M U L A T I O N S OF T H E MOVING BOUNDARY
a. Notations
F
2
= I
:
R
D
identify
s u c h as z and
= t h e open d i s c
=
D(0;r)
=
D(0;1)
in
= {x £ F
: a < x < b}
[a,b]
= {x€
:
will
2
F
F
at the
with center
be t h e u s a g e
con
of
]R .
a
and r a d i u s
r.
a<x<b}
the o r i g i n .
(Actually
ft
will
a l w a y s be a d i s c
in
F
centered
origin.)
3D = t h e b o u n d a r y o f
D in
F ,
means
:
D in
D C ft ( i f
D C F
D i n ft ,
fi
D - D U 3D = t h e c l o s u r e o f
if
2
3 D = 3Df"lft = t h e b o u n d a r y o f
fi
in
(C w h e n e v e r
g e n e r a l l y denote our "universe", a f i x e d boundary region
containing
D E C
w i t h the complex plane
£ for variables
(a,b)
ft
F
Often the o n l y consequence o f t h i s w i l l
letters
D
preliminaries
we w i l l
venient.
D(a;r)
and
CONDITION
ft
F
2
( D <= F
2
D <= Q
if
2
)
i s open and b o u n d e d ,
DEFT)
2
da = dx dy
= element o f area measure in
F
(will
integrals)
|D| = a r e a o f
(also
D
Izl
if
D <= F
= yx +y
2
Xp = t h e c h a r a c t e r i s t i c
if
D C F
2
(
<D
X n
(z)
2
if
2
z = x + iy £ £ )
function of
D
1
for
z £ D
0
for
z € D
•
o f t e n be o m i t t e d
in
-17-
6 = 6Q = t h e D i r a c m e a s u r e w i t h r e s p e c t t o t h e o r i g i n
p o i n t mass a t
VU
.
G
R
A
D
U
g^(z^)
-
R
(the
n
unit
R )
n
| )
i s t h e G r e e n ' s f u n c t i o n f o r t h e domain D
(=-log|z-^|
=0
on
3D
g (z)
= g (z,0)
C (ft)
: the
D
0 e
in
order
C (ft)
A
|z-c|
ft
<
+
o f whose d e r i v a t e s
R which are l o c a l l y
(0 < A < 1 ) , i . e .
J J J U ) - " ( G ) |
z,;eK
z *£
u :
all
of
(m = 0 , 1 , 2 , . . . , o o ) .
e x i s t and a r e c o n t i n u o u s
with exponent
SUp
z)
s p a c e o f f u n c t i o n s ft -»• R
: t h e space o f f u n c t i o n s
a
D,
0 e D)
(linear)
< m
in
as a f u n c t i o n o f
(if
D
m
+ harmonic
which
Holder
continuous
satisfy
co
a
f o r e a c h c o m p a c t K <= ft .
C ' (ft)
m
a
: those functions
C (ft)
i n C (ft) whose d e r i v a t e s
of order
m
*/
such t h a t
!R„
K
1I
,r
2
the class of simply connected regions '
(i.e.
3D
is a Jordan curve of
admits a twice continuously
: t h e c l a s s o f open s e t s
D C R
: t h e c l a s s o f open bounded s e t s
Thus
to
(m-0,1,..., 0 < A < 1 ) .
a
S :
m belong
51 =
U
?L
R,r>0
with
2
D c= R
,
r
""/region = d o m a i n = o p e n s u b s e t o f
R
2
with
0 £ D
and
class
differentiate
.
K
D c R
o
B
r
parametrization).
C C D C C F .
with
0 £ D.
K
(R,r > 0).
-18-
= C " ( f i ) : t h e (1 i n e a r )
differentiate
V
(ft)
f u n c t i o n s ft -»• ]R
functions
on ft ( i n f i n i t e l y
w i t h compact
: the space o f r e a l - v a l u e d d i s t r i b u t i o n s
H '' (ft)
m
space o f t e s t
support).
on
ft
.
: t h e Banach space o f r e a l - v a l u e d d i s t r i b u t i o n s
:)
whose d i s t r i b u t i o n d e r i v a t i v e s
(m > 0 , 1 < p < » ) ,
_
m
H 'P(ft)
m
'
.
11 9
n,k>0
n+k<m
P
of order
3
n + k
n + k
\P(ft)
of the
n
D
n
K
w
k
i
t
n
u
ni,
n
e
l
Hj(ft)
: t h e c l o s u r e o f c"(fl)
= Hj' (ft)
2
P
^ )
(m
>
0,
H ' (ft)
.
1
<
p
in
1
2
1
We w i l l
Hg(ft)
the
p r i m a r i l y work
will
inner
o°).
<
K
.
2
form
u
H (ft) = H ' (ft)
m
L^(ft)
u 11
3 x V
n,k*0 3 x 3 y
n+k<m
m
belong to
of
normed by
: the space o f d i s t r i b u t i o n s
1
< m
on ft , a l l
-1
in t h e Sobolev space Hg(ft) and
a l w a y s be c o n s i d e r e d as a r e a l
Hilbert
H
(ft).
space, equipped
with
product
<-> = / - - - = / ^ l M f
This bilinear
Prop.23.4).
n o r m ||u ||^ ^
•
f o r m i s an i n n e r p r o d u c t
The norm i t
defined
induces,
above.
because
||u || = / ( u , u ) ,
ft
i s bounded
([20],
is equivalent
to
the
-19-
For
(1)
ue£>'(ft),
tpe$)(ft)
<u,cp> = <(p,u) = u(cp)
The f o l l o w i n g
(i)
o
we
set
(value of
u
facts are well-known
v e L (ft)
i f and o n l y i f
on cp).
(and easy t o
verify):
t h e map £ > ( f t ) £ c p - » (cp,v> i s c o n t i n u o u s
in
t h e L (ft) t o p o l o g y , and i n t h a t c a s e t h e ( u n i q u e ) c o n t i n u o u s e x t e n s i o n
2
i t t o L (ft) i s g i v e n by
(2)
(u,v)
= / u • v
ft
(u,v£L (ft))
(when v € H
i f and o n l y
i f £ > ( f t ) 9 c p - > <<p,v)
is continuous
The (unique) continuous e x t e n s i o n of
(ft)) d e f i n e s a b i l i n e a r
(also denoted ( . , . )
the dual
.
2
1
(ii)
v£H
(ft)
1
HfUft) t o p o l o g y .
-1
([20],
Prop. 23.1)
in
t h i s map t o
1
1
p a i r i n g between Hg(ft)
) , b y means o f w h i c h H Vft)
s p a c e o f H^(ft)
and
H
(u,v)
-1
namely
Thus - A
H^(ft)
identifies
meaning.
(4)
(u,v)
u£Hg(ft)
and
Despite that
vGH
u and
s p a c e ) and
( f t ) t h e p r o d u c t u*v
it will
of
the dual
of
H" (ft).
1
for
v
has i n g e n e r a l
(in similarity with
integral
( t h e p a i r i n g d e f i n e d under
/u«v = < u , v )
If
(ft)
-1
v e n i e n t t o use the symbolic
for
two r e p r e s e n t a t i o n s
( i n being a H i l b e r t
1
If
onto H
(u,v€Hj(fl))
( [ 2 0 ] , Thm 2 3 . 1 ) .
Q
with
that
= - (u,Av)
H (ft),
n
.
T h e L a p l a c i a n o p e r a t o r A i s an i s o m o r p h i s m o f H g ( f t )
(3)
the
1
H (ft)
(ft)
c a n be i d e n t i f i e d
1
with the property
of
ueHj(ft),
/u*v
(ii)
veH" (ft)
1
as
(2))
s o m e t i m e s be c o n -
an a l t e r n a t i v e
above).
in (4) to agree with Ju«v
ft
.
.
t h e n one wants t h e s y m b o l i c
This
notation
Thus,
i n ( 4 ) h a p p e n t o be s u c h t h a t t h e i n t e g r a l
i n the sense o f measure t h e o r y ,
no
is actually
/u«v
^
the case, at least
exists
integral
in
the
-20-
f o i l owing two
situations:
(i)
vel_ (ft)
(ii)
u
(then u - v G L
2
is a continuous
(in H" (ft))
u e HJ(«) C L (ft))
(ft) s i n c e
1
function
2
( i n H g ( f t ) ) and
These two cases w i l l
cover our
needs.
Using the above kind of n o t a t i o n formula
/VU'Vv = - /u-Av
(u,v£Hj(fi))
the elements of
u > 0
if
in
D c
ft
H (ft), H
will
u
tinuous function
On t h e o t h e r h a n d t h e
and such s e n t e n c e s w i l l
i s known t o have a r e p r e s e n t a t i o n
(necessarily
u n i q u e ) and s h a l l
ing in terms of t h a t continuous
as ( 6 ) and ( 7 ) when
D
ueHj(ft),
ft)
function.
is non-open w i l l
i n f o r m a t i o n about
(in
be
considered
be i n t e r p r e t e d
sentence
u
v£H~ (ft)
1
i f and o n l y
if
be u s e d o n l y
i f and o n l y
if
then have i t s usual
Similarly,
inequalities
meansuch
be u s e d o n l y when we h a v e a d d i -
the f o l l o w i n g are
(tp,v) > 0
the
in the form of a con-
and t h e i r meaning w i l l
(u,tp) > 0
if
d e p e n d on t h e
context.
true,
f o r a l l tp£H
( f t ) w i t h tp > 0
and
v > 0
in
D
distribution
u > 0
(ft),...
Q
i s o p e n , a l w a y s has i t s meaning and s h a l l
has no meaning i n g e n e r a l
If
-1
D ,
in
tional
form
T h u s s t a t e m e n t s such as
the sense o f d i s t r i b u t i o n s .
u > 0
(3) takes the
.
1
as d i s t r i b u t i o n s .
i s a Radon m e a s u r e
.
1
Primarily
v
;
1
for all tp£H (ft)
n
with
tp > 0 .
-21-
Some a b b r e v i a t i o n s
sometimes
MBP
moving boundary
LCP
linear complementarity
QD
quadrature
used:
problem
problem
domain
Formulas a r e numbered s u b s e q u e n t l y w i t h i n each s e c t i o n
When a f o r m u l a
i s r e f e r r e d t o f r o m a n o t h e r s e c t i o n i t s number i s
by t h e number o f t h e s e c t i o n t o w h i c h i t b e l o n g s
12 o f s e c t i o n
III).
b. A c l a s s i c a l
f o r m u l a t i o n o f t h e moving boundary
We s h a l l
O G D and such t h a t
1 et
(g^(z)
D
imply
a map ( a , b ) 3
connected
in
D,
to
D
g^ = 0
3D).
condition
D
t
regions,
Dc]R
2
2
g* X
C .
with respect to
on
For
the
The assumpVg^ have
c o n t a i n i n g 0 £ ]R .
T h e n we s a y
(A) or is a classical
there exists
(twice continuously
that
£S
s o l u t i o n o f our moving
boundary
2
problem i f
For
D = DU3D.
be an o p e n i n t e r v a l
t -»
be
case.
( a s i d e f r o m t h e e x i s t e n c e o f g^) t h a t gp and
( a , b ) c= F
satisfies
condition
3D i s a J o r d a n c u r v e o f c l a s s
= -log|z| +harmonic
continuous extensions
Let
= formula
i s a l s o a weak s o l u t i o n .
simply connected regions
denote the Green's f u n c t i o n f o r
t i o n s on
(III.12)
-Vg^ in order to
i s o b v i o u s how t o e x t e n d i t t o t h e m u l t i p l y
such t h a t
origin
preceded
we f o r m u l a t e t h i s c o n d i t i o n o n l y f o r s i m p l y c o n n e c t e d
Let S denote the class of a l l
DeS
solution
IV).
condition
3D o f D m o v e s w i t h t h e v e l o c i t y
able to prove l a t e r that a c l a s s i c a l
but i t
(e.g.
g i v e an e x a m p l e o f a p r e c i s e f o r m u l a t i o n o f t h e
that the boundary
simplicity
( I to
a map
z; : ]R / Z
differentiate)
such
* ( a , b ) -»• ]R
that
2
of class
C
-22-
(i)
c ( s , t ) £ 3D
(ii)
c(*,t)
for all
t
: 1R/Z
2
-* 3D.
s,t
is a
( o f c l a s s C ) f o r each
(10)
(iii)
9c(
aV
Comment:
3D^.
(i)
p
(g(s,t))
and ( i i )
te(a,b)
,
f o r a l l s . t .
say t h a t f o r each f i x e d
t , ?(*,t)
parametrizes
T h e p a r a m e t e r v a r i a b l e s ( i n w h i c h z; has p e r i o d 1) n u m b e r s t h e
ticles
- Vg
n
u
g
=-Vg
t}
diffeomorphism
D
on 3 D ,
and ( i i i )
t
(c(s,t)).
t
_
to D .
Here V g
s a y s t h a t e a c h s u c h p o i n t moves w i t h t h e
velocity
is the continuous extension of the gradient
n
u
parof
t
t
c. A semi-weak f o r m u l a t i o n o f the moving boundary
condition
N e x t we g i v e a f o r m u l a t i o n o f t h e m o v i n g b o u n d a r y c o n d i t i o n
s e r v e s as a
l i n k between the c l a s s i c a l
formulation.
containing
(B)
L e t S be as on p . 21
the o r i g i n .
t h e map ( a , b ) 3 t D
satisfies
be an o p e n
interval
that
£ S
t
(B) o r i s a semiweak s o l u t i o n
o f our moving boundary
oo 2
~~~~~~~~~~~~~~
f o r each
tp £ C ( F ) , / t p d x d y
is a continuously d i f f e r e n D
tiable function of
t
d
3 D
t
with
d
t
(a,b) c F
weak
condition
problem i f ,
D
f o r m u l a t i o n and t h e f i n a l
and l e t
T h e n we s a y
which
\
t
Comment:
denotes the d e r i v a t e in the d i r e c t i o n of the outward
—
—
3n
o f 3D. , ds d e n o t e s t h e a r c - l e n g t h m e a s u r e on 3D. .
normal
-23-
To see t h a t
does,
just
(B) expresses
integrate
smal1). This
t h e same m o v i n g b o u n d a r y c o n d i t i o n as
(11) w i t h r e s p e c t to time from
t
to
(A)
; t + fit ( 5 t > 0
gives
D
9
/ cp • - g j j t ds • fit + 0 ( 6 1 T )
3 g
/
cp -
/ cp = -
t 5t
D
D
+
t
9 D
expressing
that
t
the width of the s t r i p
"
from 9D
t
to 9 D
, is
t + g t
D
D ^ ^ . ^ D^, t h a t
• fit + <27(6t ) .
(A) wants i t
to
This
is precisely
condition
i s t h e f o r m u l a t i o n we a r e g o i n g t o w o r k w i t h a n d f o r w h i c h we
shall
p r o v e e x i s t e n c e and u n i c i t y
of solutions.
We g i v e t h e
first
and d i s c u s s
Let r , R , T > 0 ,
let
ft
i t afterwards.
^ denote the class of a l l
,r
T h e n we s a y t h a t
D
K
t h e map [ 0 , T ]
satisfies
(C) or
3 t -» D
°t
" °0
X
=
satisfies
u^ > 0
and
+
t
o p e n s e t s D C ]R
eft
2 7 T t
for all
' W
t£[0,T]
X ] D
r
2
with
Q = B
formulation
and
D
let
R
D C C D C C D .
r
K
D
R > r
i s a weak s o l u t i o n
boundary problem i f
X
what
be.
d . T h e weak f o r m u l a t i o n o f t h e moving b o u n d a r y
This
distance
2
T
-
the moving boundary c o n d i t i o n
is the
(with parameters r , R , T )
of our
moving
the function u = Uj.£Hg(ft) defined
by
-24-
Discussion:
—
—
XQ
First,
observe that equation
_^
" 2iTt •
- XQ
Hj(ft) + H" (ft)
| Xj)
e
H
(ft)
(13) r e a l l y d e f i n e s
and
A i s an
u.
t
since
isomorphism
.
1
T h e l e f t member o f ( 1 5 ) c a n e i t h e r be i n t e r p r e t e d as t h e d u a l i t y
1
-1
1
-1
p a i r i n g b e t w e e n H Q and H
( u ^ G H g , 1 - X ^ £ H ) o r a s an o r d i n a r y
?
t
Lebesgue i n t e g r a l ( 1 - X
» cf. p . 1 9 f ) .
Since 1 - X
(
that
t
t "
t h e i n t e g r a n d i n (15) i s n o n - n e g a t i v e ) (14) and (15) t o g e t h e r e x p r e s s t h a t
N
e
L
N
u
u^ > 0
in
ft
and t h a t
>
0
s o
u
u^. = 0 ( a . e )
outside D
t
.
L e t us m o t i v a t e t h e c o n c e p t o f weak s o l u t i o n b y s k e t c h i n g how
arises from condition
of
bution language
39D
3
"at
XQ
that equation
t?
*(
l e n g t h measure o f 3 D )
a r c
t
/ tpds
on
F
2
Ag
D
= - 2TT 6g — ^ -
.
• ( a r c l e n g t h measure o f 3D )
t
i t to zero outside
D^, a n d w h e r e
,
g^
is extended to
t
when p a s s i n g f r o m t h e o u t s i d e o f
normal d e r i v a t i v e
f r o m t h e i n s i d e o f D.
into
itself).
all
hence
a l s o e q u a l s t h e jump o f t h e o u t w a r d normal d e r i v a t i v e
9 n
3D
that
is understood that the Green's f u n c t i o n
by p u t t i n g
2
distribution
t
where i t
]R
.
i n t h e r i g h t member i s t h e
Combine t h i s w i t h t h e f a c t
(18)
distri-
t
= — 5 ^ —
3D
( 1 1 ) i n ( B ) c a n be e x p r e s s e d i n
as
Here the second f a c t o r
(17)
interpretation
it.
Observe f i r s t
(16)
( 8 ) a n d a l s o be g i v i n g a p h y s i c a l
it
o f g^
across
(besides being j u s t
T h e n we g e t
the
-25-
(19)
9
^
Xx n
D
= AA g
\
=
t
n
+
+
Now ( 1 3 )
(20)
is
with
u
t
= Jg
0
t
2 T T6 6
0
2 7 1
is essentially
(19)
integrated with respect to time.
That
i s ( 1 3 ) . (14) and (15) a r e c l e a r l y a l s o s a t i s f i e d
for
dx
D
T
integration of
(19)
gives
(21)
*D • D
X
t
=
0
A
t
u
+
which e s s e n t i a l l y
(20) s i n c e
g
* 0
2 7 T t
6
> 0 , and
Q
g^
with x ) .
between (21) and (13)
i s smoothed o u t t o - [ j j - j X p
t h e f a c t t h a t 6g € H
(22)
(C)
t
u. - / g
0
where
D
for x £ [ 0 , t ]
t
(D
T
increases
T
The d i f f e r e n c e
u^ i n
= 0 outside
-\
r
in (13).
This replacement
r
(Q)
i s t h a t t h e D i r a c measure §Q
(equivalently
i s n o t i n t e n d e d t o be e x a c t l y
1
g ^€H (ft)
D
Q
(20) but
is necessitated
) . Thus, the
by
function
rather
dx
n
T
(23)
\
(
Z
)
(Here g ^ z , ? )
fjGD.)
dition
fact,
I
~~ W
9
D
T
(
Z
'
C
)
^
i s t h e G r e e n ' s f u n c t i o n w i t h r e s p e c t t o an a r b i t r a r y
T h i s m o d i f i c a t i o n o f gp d o e s n o t a f f e c t
since
one has
it
i s e a s i l y s e e n t h a t g^ =
point
the moving boundary
i n a n e i g h b o u r h o o d o f 3D.
conIn
-26-
(24)
g (z)
= g (z)
D
+
D
g (z)
log|z[
+ l o g [ zJ
D
--yyj
- -jjj-y J
/
log|z-c|da
l o g |z-c | d a
f o r z £ ID.
?
9 (z)
z€D
0
zefls D
D
ND.
N e x t l e t us g i v e a d i r e c t p h y s i c a l m e a n i n g t o ( 1 3 ) , o r r a t h e r t o
i n t h e c o n t e x t o f h e a t c o n d u c t i o n p r o b l e m s and S t e f a n p r o b l e m s .
(25)
sider the ordinary
heat
c(T)|T
+ q
=
V
(k(T)VT)
> 0
and q = q ( x , y , t )
Thus,
con-
equation
,
d e s c r i b i n g t h e h e a t f l o w i n some m a t t e r .
ture, c = c(T)
(19)
the s p e c i f i c
Here T = T ( x , y , t )
heat, k = k(T) > 0
is the
the thermal
temperaconductivity
is a source term (the heat production per u n i t volume).
d e n s i t y o f t h e m a t t e r i s a s s u m e d t o be c o n s t a n t a n d i s t a k e n a s u n i t y .
i s a l s o assumed t h a t
If c(T)
an o r d i n a r y
If c(T)
(26)
c(T)
c
and
k do n o t d e p e n d e x p l i c i t l y
and k ( T ) a r e s m o o t h , s t r i c t l y
heat conduction
is of the
= L • S(T-Tg)
on
positive functions
(25)
describes
problem.
form
+ a smooth, s t r i c t l y
Then Tg i s t h e m e l t i n g t e m p e r a t u r e , L > 0
and
equation
T>Tg
T(x,y,t)
positive
function
the latent heat, the
d e f i n e t h e s o l i d and l i q u i d phases r e s p e c t i v e l y and t h e
= Tg d e f i n e s t h e moving boundary s e p a r a t i n g t h e s e
Now I c l a i m t h a t o u r e q u a t i o n
c(T)
= 6(T)
.
solid.
inequalities
phases.
(27)
It
x,y,t.
( 2 5 ) c a n be s e e n t o d e s c r i b e a t w o - p h a s e S t e f a n p r o b l e m o f a m e l t i n g
T<Tg
The
(19) is of the type
(25)
with
two
-27-
Thus our moving boundary problem w i l l
type
(of
be a S t e f a n p r o b l e m o f
" g e n e r a l i z e d type" because c l a s s i c a l l y
h e a t t o be e v e r y w h e r e
strictly
positive).
generalized
one r e q u i r e s t h e
Our problem w i l l
also in
be a o n e - p h a s e p r o b l e m s i n c e o u r b o u n d a r y c o n d i t i o n s a r e s u c h
T = constant
specific
practice
that
(= 0) i n one phase ( t h e s o l i d p h a s e ; t h e d e s c r i p t i o n o f
s o l i d p h a s e as t h a t d e f i n e d b y T < T g c a n n o t be t a k e n l i t e r a l l y
in
the
this
case).
T o p r o v e t h e c l a i m we i n t r o d u c e t h e e n t h a l p y
the temperature function
T
= / c ( T ' ) dT
(28)
H(T)
(29)
T
(J>(T) = / k ( T ' ) d T '
(30)
(25)
defined
H
and
by
and
1
respectively.
(The lower l i m i t s of
these
<J>
(heat function)
the integrals
are immaterial.)
In terms
of
becomes
|£ = A * + q .
Now t h e
(31)
c(T)
= 6(T)
(32)
k(T)
= 1
choices
yield
(33)
<j>(T) = T
(34)
H(T)
= the Heaviside f u n c t i o n of
0
1
for
T
T < 0
T > 0
Thus by o b s e r v i n g
that x
i s e s s e n t i a l l y t h e H e a v i s i d e f u n c t i o n o f gp
t
t
( t h e r e a r i s e s a t i n y p r o b l e m o f how t o d e f i n e t h e H e a v i s i d e f u n c t i o n a t t h e
D
-28-
jump p o i n t ,
but t h i s problem is
g
(19) and
, Xn
n
i
n
in any case u n i m p o r t a n t
<(>, H i n ( 3 Q ) c a n be c h a n g e d b y a d d i t i v e
w i t h o u t a n y t h i n g h a p p e n i n g ) we s e e t h a t
(35)
(j,
(36)
H =
(37)
q = 2TT6
=
= g
T
X
(19)
with
t
q
proves the
claim.
is c l e a r from the formulation of
t
D
Dj e f i
(C) that
( g i v e n Dg) a weak
c a n be u n i q u e a t m o s t up t o s e t s o f t w o - d i m e n s i o n a l
t
measure z e r o ;
set
constants
and
It
tion
(30) goes o v e r i n t o
functions
D
n
u
This
since the
for
(13) - (15)
such t h a t
D
almost everywhere.
is not affected
Xn>
t
=
Xn
if
Lebesgue
i s r e p l a c e d by
in the sense of d i s t r i b u t i o n s ,
i.e.
will
on.
I f we h a d w a n t e d t o we c o u l d h a v e r e m e d i e d t h i s s i t u a t i o n b y
representatives
another
t
T h u s p h r a s e s s u c h a s " u n i q u e up t o n u l l - s e t s "
a p p e a r i n some t h e o r e m l a t e r
solu-
i n each e q u i v a l e n c e c l a s s o f D:s
(calling
choosing
eR
D a n d D'
D
K,r
equivalent
if x
n
= Xn' a . e . ) .
f o r such r e p r e s e n t a t i v e s :
T h e r e i s a t l e a s t one n a t u r a l
i n each e q u i v a l e n c e c l a s s t h e r e
choice
is a unique
maximal e l e m e n t , t h e u n i o n o f a l l i t s members, o r , what amounts t o
same,
let DeJL
be r e p r e s e n t e d b y
the
K,r
(38)
D
1
= ]R ^
2
supp(1 - x )
D
,
w h e r e s u p p means s u p p o r t
the r e p r e s e n t a t i v e
boorhood
U
D' o f
in the sense o f d i s t r i b u t i o n s .
D consists of those points
with the property
|U n D| = I U I
choices of
representative
D , namely
words,
which have a n e i g h -
.
One r e a s o n w h y we h a v e c h o s e n n o t t o f o r m u l a t e
canonical
z
In o t h e r
d o m a i n s i s t h a t we l a t e r
(C)
shall
in terms of
work w i t h
these
other
-29-
(39)
D. = D
U { z e ft : u . ( z ) > 0 }
n
and i t
i s not q u i t e c l e a r w h e t h e r t h e s e two c h o i c e s always are the
A nice feature of
parameter
in i t :
one f o r each
t e [0,T]
.
( c f . p . 45 f ) ,
t
In f a c t ,
if
D. ( i n R
x
D
K
it
, r
(42)
g r a d u.
+
= 0
t
since
t
and
and
(13)
u^
satisfy
(C)
then
i m p l i e s t h a t Au £ L
solve the f o l l o w i n g
i s assumed t h a t 3 D
) and a c o n t i n u o u s l y
in
u
u
D
t
free
= 3(ft^D ).
t
differentiate
function
that
(40)
(41)
if
differentiate
boundary problem (at least
u . o n D. s u c h
t
t
.
and i t f o l l o w s t h a t
Find a region
problems,
E a c h o f t h e s e p r o b l e m s c a n be v i e w e d as a f r e e
3D
m u s t be c o n t i n u o u s l y
t
i s t h a t t h e t i m e v a r i a b l e a p p e a r s o n l y as a
( C ) j u s t c o n s i s t s o f a number o f u n c o u p l e d
boundary problem f o r
u
(C)
same.
D
and
= 0
on
3D.
S a k a i , i n [ 1 9 ] , works w i t h a n o t h e r r e p r e s e n t a t i v e , namely t h a t
d e f i n e d b y ( i ) - ( i i i ) on p . 14 ( w h i c h he s h o w s t o be u n i q u e ) .
domain
-30-
e . Two l i n e a r
complementarity
problems
These two problems c o n s t i t u t e
formulation
step
b e t w e e n t h e weak
o f t h e moving boundary p r o b l e m and v a r i a t i o n a l
The e x p r e s s i o n
([6])
an i n t e r m e d i a t e
"linear complementarity
and has been a d o p t e d
D
(i.e.
D
G HQ(Q)
function
p r o b l e m " has been u s e d by
Elliott
here.
put Q -
Let r , R , T > 0 ,
inequalities.
Dg e f i p
defined
and l e t
r
DQCp
be a g i v e n d o m a i n
) . For each t e [ 0 , T ]
introduce
with
the
by
(43)
(Since the r i g h t
function
member a b o v e b e l o n g s
in Hg(ft).
Au
(45)
u
+ A^
t
( 4 3 ) defines a unique
condition
(C)
becomes
t
> 0
t
/u
(46)
(fi)
)
Expressed with the aid of
(44)
,-1
to H
t
• ( A u + Aijjj.) = 0
.
t
Now, t h e f i r s t
but w i t h
of our l i n e a r complementarity
( 4 4 ) weakened t o the i n e q u a l i t y
Au
problems
t
+ A^
o n e i s t h e same t h i n g e x p r e s s e d i n t h e f u n c t i o n
say
v
t
t
is j u s t
(44)-( 4 6 ) ,
< 0 , and t h e
= u
t
+
second
T h u s we
that
t h e map [ 0 , T ] 3 t - » u e H J ( G ) s a t i s f i e s
(D1)
t
(47)
u
> 0
(48)
Au
t
(49)
/u
t
t
+ Ai|; < 0
t
•(Au + A^ )
t
a n d we s a y
t
that
= 0
,
(D1) if
for all
te[0,T]
-31-
t h e map [ 0 , T ] 3 t v
(D2)
£ Hg(ft)
t
satisfies
(D2)
if
for all
t£[0,T]
(50)
(51)
Av^ < 0
(52)
J ( v - * )
t
-Av
t
= 0
t
Comments: A d i f f e r e n c e
is that
the regions
the i n i t i a l
tained
T
d o m a i n DQ w h i c h i s now g i v e n b e f o r e h a n d a n d i s
(43))
given functions iJ^CH
a n d we w i l l
for iJJ £Hj(ft)
t
u £Hg(ft)
t
.
satisfy
the problems
t
That
(53)
I.
= {z£ft
(C)
(54)
Av
+
= 0
Consider
: v. ( z )
ft
grad
v
We
will
involving
i s , we w i l l
for
implicitly
con-
arby
t o r e f e r t o ( D 1 ) and ( D 2 )
also refer to
all
the
use f o r m u l a t i o n s
(given tg)
for
of the form given
.
( D 1 ) and ( D 2 )
correspondences
such as
Similar
"let
remarks hold
=
( D 1 ) a n d ( D 2 ) c a n be c o n s i d e r e d as f r e e
( D 2 ) f o r example.
for
(D2)
\
g r a d \p.
Introducing
boundary
the coincidence
set
iMz)}
implies
I
and
(55)
(56)
without
problem
in
(C)
and ( E 1 ) and ( E 2 ) .
one can see t h a t
determined
(not necessarily
(D1) f o r t = tg"
For f i x e d t £ [ 0 , T ]
conditions.
(ft)
not of the form ( 4 3 ) .
-» U ^ J V J .
(B),
(D1) and ( D 2 ) a r e m e a n i n g f u l
sometimes take the l i b e r t y
isolated values of
[0,T] 3t
(A),
.
is c l e a r that the conditions
bitrary
versions
have d i s a p p e a r e d f r o m the f o r m u l a t i o n , e x c e p t
in the function ^
It
for
D
as c o m p a r e d w i t h t h e e a r l i e r
on
91
that outside
1^.,
v^. s a t i s f i e s
the
over-
-32-
T h u s we h a v e a f r e e b o u n d a r y c o n d i t i o n f o r
91^. .
T h i s f r e e boundary
lem i s known as "the o b s t a c l e p r o b l e m " ( t h e f u n c t i o n
obstacle).
f.
See [ 1 1 ] o r
Two v a r i a t i o n a l
We s h a l l
valent,
respectively,
Au. +
the
.
inequalities
i n e q u a l i t i e s w h i c h a r e t o be e q u i -
to the l i n e a r complementarity
n
a n d \p
be as on p . 3 0
f
t h e map [ 0 , T ] 9 t
(57)
f o r example
f o r m u l a t e two v a r i a t i o n a l
L e t r , R , T , ft , D
(E1)
[12]
describes
prob-
u e Hg(ft)
t
satisfies
.
problems
T h e n we s a y
(E1)
if,
(D1) and
that
for all
te[0,T],
< 0
and
(58)
•
/ V(u - u ) • Vu. > 0
t
ft
Z
i
u e HQ(Q)
We s a y
(E2)
Z
for
all
~
with
Au + A ^
< 0 .
t
that
t h e map [ 0 , T ] 3 t -> v , € H ' ( f t ) s a t i s f i e s
(59)
and
•i
(60)
/ V ( v - v . ) • Vv.
ft
1
v£ O f t )
z
z
with
> 0
for
"
v > ii.
.
all
(E2)
if,
for all
(D2).
t e [0,T],
-33-
g . T h e moment
Let
ft
(thus 31=
say
inequality
denote the class of a l l
with
U
ft
R,r
r,R>0
inequality,
/ cp -
(61)
D
R,r
D <= K
as on p . 23 ) , a n d l e t
T>0.
with
OeD
T h e n we
that
t h e map [ 0 , T ] 3t
(F)
IR
bounded open s e t s
t
if,
-* D^G %
satisfies
( F ) , or s a t i s f i e s
f o r e a c h t £ [0 , T ] ,
/ c p > 2-rrt • i p ( 0 )
D
0
for every function
Remarks:
2
2
cp £ H (]R )
By d e f i n i t i o n
which is subharmonic
(see e.g.
[20], Sect.
i s r e q u i r e d t o be u p p e r s e m i c o n t i n u o u s
perty with
r e s p e c t t o small
involving
(F)
( T h e o r e m 10, ( F )
n
n
n
2irt
(n = 0)
for
n >1
pro-
2
i n ( F ) , H (]R ) , i s p e r h a p s n o t t h e most
( C ) ) works f a i r l y
a n d ± Im z
n
we s e e t h a t ( F ) i m p l i e s t h e moment
ID. I = I D I +
function
and t o have t h e s u b - m e a n v a l u e
is a choice f o r which the proof of our only
By c h o o s i n g cp = ± Re z
Dg U DJ.
.
discs.
The choice of the t e s t class
one, but i t
in
30) a subharmonic
2
natural
t h e moment
and
(n>0)
theorem
well.
in a neighbourhood
property:
of
-34-
R E L A T I O N S BETWEEN T H E MOVING BOUNDARY C O N D I T I O N S , AND
E X I S T E N C E AND U N I C I T Y
We s h a l l
RESULTS
now s e t up t h e r e l a t i o n s
ditions
(A) - (F)
in Section
unicity
of solutions
of
I I and t h e r e b y a l s o p r o v e e x i s t e n c e
(C) - (F)
a l s o p r o v e some m o n o t o n i c i t y
Thm 2
Thm 1
=>
(B)
=^>
(for
( A ) and ( B ) o n l y u n i c i t y ) .
and c o m p a r i s o n
The diagramme below i l l u s t r a t e s
(A)
between the moving boundary
Thm 3
Thm 5
(D1)
Thm
T h m 10
(D2)
and
We w i l
theorems.
the contents of this
(C)
con-
sections.
(E1)
4
Thm 5
<^>
Minimal
properties
and
monotonicity
properties
(Thm 7
with cor )
(E2)
r
Have u n i q u e
solutions
with
regularity
properties
(Thm 6 )
s
Thus the c o n d i t i o n s
infer
(A) or
(C) follows
solutions.
(C) - (F) are a l l
(B) from them.
equivalent,
b u t we a r e n o t a b l e
The e x i s t e n c e and u n i c i t y o f s o l u t i o n s
from the elementary f a c t that
( E 1 ) and ( E 2 ) have
unique
to
to
-35-
a.
(A)
(B)
T h e o r e m 1: S u p p o s e t h e map ( a , b ) 3 t - »
also satisfies
Proof:
D e S
t
satisfies
(A).
Then
it
(B).
A l l we h a v e t o p r o v e i s t h a t t h e
formula
8g
J cp d x d y
(1)
= - / cp
3D,
oo
holds f o r a l l
cp e C ( F
function of t.
(2)
Let
ds
an
7
) a n d t h a t t h e r i g h t member o f i t
x , y be t h e c o o r d i n a t e v a r i a b l e s
is a continuous
2
i n TR a n d l e t
S = 5(s,t)
n = n(s,t)
denote the components o f e ( s , t ) £ | R
9g
95(s,t) _
D
u
at
ax
t
(p.21-22).
Then ( A ) ( i i i )
becomes
(C(s,t))
(3)
9n(s,t) _
u
at
a
t
(C(s,t))
y
T h u s t h e r i g h t member o f
99
3g
D
- / * ^ ~
3D,
't
(1)
d s
=
is
ag
D
- / vf-iir
a n d we o n l y h a v e t o p r o v e t h e
(4)
w / ^ x d y
7
TR .
9y
d y
t
To prove
= ^ ( f
formula
dy-|n x)
d
( 4 ) we c o n s i d e r t h e e x p r e s s i o n c p d x d y as a C°° 2 - f o r m on
7
S i n c e a 2 - f o r m on 1R i s a u t o m a t i c a l l y c l o s e d ( i . e . d ( c p d x d y ) = 0 )
-36-
and hence e x a c t
(IR
= a dx + b d y
CO
being simply connected), there
(a = a ( x , y ) , b = b ( x , y ) )
such t h a t
T
= ]R/Z
(the range of the v a r i a b l e s ) .
Stokes' formula at the f i r s t
/ cp dx d y = 4r
D,
j
[dx
+
b -
3
+
s t e p , we g e t
b ( C
(
2
3s3t
3s3t
3t 3S
ds + /
¥
n
3s3t
2
, t ) ) % ^
s
^3x * 3 t
f
3y 3 t 3s
9x
t
13?
(
ds
3n
3n 9 ^
"3? ' "3t L ? )
¥
D
r
S
=
9U
^
3s
3x 3t 3s.
3s
/95
J
;
M
3y 3 t 3s
J
3y * 3 t 3 s
3y 3t
¥
I ^ * "9T
*
v
J..
^
9t
3t
8s'
a
( t h e map <; = ( £ , n )
is a continuous
differentiate
)
This
proves Theorem
and c ( « , t )
function of
1.
ds
s
*
t
t.
assumptions
is a diffeomorphism
from the above computations
with respect to
;
t
is of class C
It also follows
3y ' 3 t 3 s .
J
3n . >
" W
x
3x * 3t
ds
9t
d
{
The above computation is v a l i d under the g i v e n r e g u l a r i t y
(1)
using
2
?
3s3t
/
T
Then,
t
2
9
/
T
i.e.
/ a dx + b d y
3D
i ( d s , t ) ) % ^
dt
s u c h t h a t da) = c p d x d y ,
= ip .
Now, l e t
4r
on ]R
i s a C°° 1 - f o r m
¥ ->- 3 D ) .
t h a t t h e r i g h t member
and t h a t hence
/ cpdxdy
^
t
of
is continuously
-37-
(B) •» (C)
b.
T - > E S s a t i s f i e s (B) a n d c h o o s e r , R , T > 0 s o t h a t
D c:c= D a n d D <=<=B , a n d l e t ft = B . T h e n
EFT
Theorem 2: Suppose ( a , b ) 3
[0,T] c
(a,b),
for all
TE[0,T]
N
T
Q
p
T
R
a n d [ 0 , T ] 3 t -» D. E R
„ satisfies
, r
D
K
T,
N
R
(C)
T
R
.
i
Further,
t
= / g
0
u
dt
n
T
(a v e c t o r - v a l u e d
defined
9D
( z )
=
u^EHGCFT) o c c u r r i n g i n (C) i s
the function
i n t e g r a l ) , where
g
i s t h e "smoothed o u t G r e e n ' s
n
function"
by
H ^ T
]
/ 9 D
(
Z
»
C
)
D
A
?
=
r
g (z)
D
+ loglzl
- -^p—y-
/ l o g I z - c I da^
for z £ B
g (z)
z
D
E
D
p
^ B
Z E ft v
Proof:
— — —
We f i r s t
D c=<= D c=cz B
r
t
Now
^
fort£[0,T].
R
to prove that
have t o prove t h a t
D
<=
< 0
9 n
on
D. E J L ^
t
K , r
Since
B <=<= D
r
for
X < t
3D
so t h a t f o r m u l a
t
for
Q
and
r
D
TE[0,T],
i.e.
Djccz B
it
R
that
suffices
.
( 1 1 . 1 1 ) shows
that
/ cp d x d y > 0
D
t
f o r a l l cpGC'TQR ) s u c h t h a t
function of
/ cp d x d y <
D
t,
X
t
Thus
/ cpdxdy is a non-decreasing
D
i.e.
/ cp d x d y
D.
cp>0.
for
X < t ,
t
-38-
2
oo
for all
cp£C (]R )
with
c
larity
requirements
D
r
E F I
T
R
for
ip>0.
It
i s e a s y t o see ( i n v i e w o f t h e
on 3 D ) t h a t t h i s
implies t h a t D c = D
t
t£[0,T]
is
T
for
t
regu-
x<t.
Thus
proved.
i
N e x t we h a v e
g
EHG(ft)
is continuously d i f f e r e n t i a b l e
n
due t o t h e r e g u l a r i t y
and so
D ^ c c ft ,
Now l e t
A
t
u
t
on
t
= / g
0
n
( D
t
T
Thus,
E S ) .
implies
1
g
£ H
n
dx
(ft),
g ^£H^(ft).
N
be d e f i n e d b y ( 1 1 . 1 3 ) ,
i.e.
( T E T O . T ] ) .
to complete the proof o f Theorem 2, t h e r e are t h r e e
First
n
i n a l l ft ,
i s bounded
n
3D
ft^D^
2
to prove.
t
= 0
u GHj(fi)
Then in o r d e r
u
assumption f o r
g^
g
is continuous
, and a t
3D.
Vg
t
+
-\- ^'l^j\
\
=
.
In f a c t
i n ft ^ 3 D
things
that
,
T
and t h e n t h a t
(11.14)
and ( 1 1 . 1 5 )
( 1 0 ) means b y d e f i n i t i o n
hold.
that
t
<u ,P)
T
=
/
<g
,P>
n
0
-1
for all
1
(ft) = H G ( f t ) .
P £ H
1
that the function
(see the l a s t
tainly
written
(u ,Acp) = ^<g
t
i.e.
using
[0,T]
It will
3 x -+ ( g
f o l l o w f r o m t h e c o m p u t a t i o n s t o come
, P ) E ]R
n
p i e c e o f t h e Remark on p.
1
-1
D
(9)
S i n c e A : H G ( f t ) -> H
,Acp) d x
for all
is continuous
40 f ) ,
T
exists.
t
dx
x
(ft)
f o r each P E H ^(ft)
so t h e i n t e g r a l
i s an i s o m o r p h i s m ,
tp£H (ft)
Q
in (11)
( 1 1 ) c a n be
cer-
-39-
1
(
D "
x
D "
X
t
2
0
T
r
(for
prove
D
d
s
-
=
^ - ^
t
3
-X
D
(
2
d
T
tp£C~(ft).
(
f
o
r
veH^n))
a 1 1
Green's
.
(second) formula
gives
d
s
" i ;
t
=
,
\ V
A
/
k
p
,
\
=
t
t
D
0
M
=
( 1 1 . 1 1 ) we g e t
t
J < P = /( D
0
h~
Do.
7
T
t
3
9
/cp-g
~3l
3D
P
n
1
=
s A t p )
%
Combining t h i s w i t h
t
=
( 1 3 )for
\
a u ; ' ^
t
u
t p >
tp£C"(n))
8
<X
'
x
We f i r s t
1
Z
'"nrTT D
t
D
T
? r
ds)dT
*
*TFT ]D
'
r
r
x
c p >
i
+
0
C
%
T
'
A
c
p
)
d
x
A
Thus
( 1 3 ) is proved f o r
tp£C~(ft)
.
To extend i t to a l l
tp£H (ft)
0
j(q~ ,Atp}dx o f ( 1 3 ) d e p e n d
0
T
c o n t i n u o u s l y w i t h r e s p e c t t o t h e H g - n o r m on tp ; f o r
C ( ^ ) i s dense in
we n e e d o n l y t o p r o v e t h a t t h e r i g h t member
1
c
1
H QJ (( Q )
a n d tT hH eI l e f t member o f
respect to
( 1 3 )obviously
1
is H -continuous
with
Q
tp
We h a v e
t
/<g
D
t
t
, A t p ) d x | •= | / ( g , , t p ) d T | < J
D
O x
N||
O
•
/ | | G
0
II
D
x
dx
x
O
| (g
x
Q
t
, t p ) [ d x < / || G
O
X
Q
||
H
• IM|-
1
Q
H
1
Q
dx
-40-
and t h u s i t
i s enough to prove
that
t
/ II 9
N
0
x
u
II dx < »
.
*/
T o p r o v e t h i s we e s t i m a t e ||g
n
t
u
II3D
" 3 D
t
1
,
^
2
>
x
V
(
"9D
D
§
ft
t
93
}
L
x
/
g"n
3D
•
\-%
{
'
}
t
A
(
'.
9 D
" 3 D
x
]
t
x
D
ds
T
=
-
/
t
g
3D
'—7r=^-
n
D
9g
D
t
ds
<
max
' 3D
3 n
g
• (-
n
D
t
G
Du
||
n
u
0
bounded f o r
—
D
^
ds)
3 n
X
X
t
< g
n
/
3D
X
• max
3D
x
- g
n
" /
9g
3 n
t "
u
=
33
D
^
°t
Since g
IIg
x<t
Q
D
X
2TT
|| .
for
T H (ft)
X
9g
=
n
ft
t
-
2
- g
for
n
te[0,T]
this
shows
that
T
< 2Tr«max g
"
3D
T
n
Q
te[0,T]
<
for all
0 0
te[0,T]
.
In p a r t i c u l a r
||g || i s
t
n
U
and so (15)
is
true.
t
/ g dx d i d
0
x
e x i s t , t h e n i t w o u l d be t r i v i a l l y t r u e t h a t t h e r i g h t member o f ( 1 3 ) d e t
p e n d s c o n t i n u o u s l y on c p . A s u f f i c i e n t c o n d i t i o n f o r
/ g dx t o e x i s t
Remark:
I f we k n e w a p r i o r i
that the vector-valued
integral
n
n
i s t h a t t h e map [ 0 , T ] 3 t - > g
n
'We a r e t h e n u s i n g t h e f a c t ,
D cD,
for
x <t .
x
t
1
eHg(ft)
be c o n t i n u o u s
already proven
0
(see
( p . 37-38
)
T
[ 1 8 ] , p . 73 f ) .
that
-41-
It
seems v e r y
reasonable
that this
i s the case but
I h a v e no p r o o f f o r
I t m a y , h o w e v e r , be s e e n f r o m t h e c o m p u t a t i o n on p .
t + cjp
is continuous
from the l e f t ,
i.e.
that
t
Moreover,
it
follows
f r o m p . 39
that
40 a t
least
CJQ -> g^ a s
x
t
it.
that
i ^ t
.
t + i
is weakly continuous,
t
i s c o n t i n u o u s f o r each p e h H ( f t ) .
In f a c t , from
u
i.e.
that
t -* ( g
,p)
n
t
u
\
{
M
~1
=
D
we s e e t h a t
t
<g
n
u
( p = Atp, t p e c T ( f i ) ) ,
continuous
"
S
x
i D '
l
p
>
,p) i s c o n t i n u o u s f o r a dense set o f
p:s
t
s i n c e t h e f i r s t t e r m i n t h e r i g h t member o f
by a s s u m p t i o n
then f o l l o w s
N O
(
(p.
automatically
22 ) ,
and t h e c o n t i n u i t y
f r o m t h e b o u n d e d n e s s o f ||g
(End of Remark.)
It
remains
from (10)
for all
||
D
(16)
(t£[0,T]).
to prove
(11.14)
and ( 1 1 . 1 5 ) .
g^
>0
(11.14)
follows
immediately
.
T
/ u . ( 1 - x
t
(11.15)
first
) = <u ,1-x
D
t
But
{
p = 1- x
observe
t
> = / (9
t
0
D
t
by c h o o s i n g
in
D
(11)
that
.1-X
D
D
T
>DT
t
.
now
%
>
"
1
T
since
V
t
D
T
C
=
I
(ft)
Z
(now p r o v e n ) , s i n c e
To prove
p€ H
is
-1
§
D •
(
1
" D
X
FTT
( p . 37-38 )
+
)
=
/
n
t
%
-
0
f
°
r
T
e
C
0
»
t
]
ft^-D^.T
and
g
D
= 0
outside
.
Thus
•
-42-
/ (g
O
,i-x
D
>dr =
D
x
t
and ( 1 1 . 1 5 )
is
proven.
This finishes
-
c
the p r o o f o f Theorem 2.
( C ) => ( D )
Let
r,R,T,ft,D
Theorem 3:
t
t
t
=
t
+
^t
be t h e f u n c t i o n s d e f i n e d b y
Proof:
(D2)
u G HJ(A)
T
satisfies
( C ) and
(11.13), (11.43)
let
and
( D 1 ) a n d [ 0 , T ] 3 t -* v e H j ( f t )
t
T h i s was p r o v e d a l r e a d y on p . 30
- 1 < 0 , a n d ( D 1 ) «* ( D 2 ) i s
n
u
satisfies
r
.
sisted of replacing the equality
X
R
•
Then [ 0 , T ] 3 t satisfies
be a s on p. 3 0 .
t
t
u
Q
S u p p o s e t h e map [ 0 , T ] 3 t -* D e f L
u ,^ ,v eHj(fl)
v
o
( t h e s t e p ( C ) => ( D 1 ) j u s t
con-
i n ( 1 1 . 1 3 ) b y an i n e q u a l i t y b a s e d on
obvious).
t
A l t h o u g h o b v i o u s we f o r m u l a t e t h e e q u i v a l e n c e b e t w e e n ( D 1 ) a n d
as a s e p a r a t e
(D2)
theorem.
•i
Theorem 4: L e t
cessarily
u^v^^eH^f!)
d e f i n e d by ( 1 1 . 4 3 ) ) .
(D1) i f and o n l y
if
be r e l a t e d b y v
Then
t
= u
t
+ if>
t
[0,T] 3 t-» u e Hj(n)
[ 0 , T ] 3 t -* v . e H ^ ( f i )
t
satisfies
(D2).
(^
t
not
satisfies
ne-
-43-
d
-
(D) * (E)
Let r,R,T,ft,Dg
and
be as on p. 3 0 .
T h e o r e m 5:
(i)
t h e map [ 0 , T ] 3 t - » u € H J ( « ) s a t i s f i e s
t
satisfies
(ii)
if
it
( E 2 ) i f and o n l y
if
it
(D1) .
t h e map [ 0 , T ] 3 t - > v
satisfies
( E 1 ) i f and o n l y
t
e HJ(ft) s a t i s f i e s
(D2) .
Proof:
( i ) «- : S u p p o s e t -> u
/ u -(Au +
t
to\> )
t
t
satisfies
< 0 = J u -(Au
t
( D 1 ) . T h e n , b y (11.47) a n d (11.49)
,
+A^ )
t
t
•
1
for all ueH (ft)
with
Q
Au +
< 0.
Thus
< 0.
Since
(by subtracting
/ u «A^ )
t
t
/ u »A(u - u ) < 0,
t
t
or
/ Vu «V(u - u ) > 0 ,
t
t
for all ueHj(ft)
s h o w s t h a t t -* u
( i ) => : S u p p o s e
with
t
Au + A i p
t
for all ueHg(ft)
(11.58)).
t -* u^ s a t i s f i e s
t
with
t
(E1).
this
Then
+A^ )
t
Au + A i ^ < 0 .
Here the c h o i c e u =
is > 0 , while the choice
Hence
(11.57) i s (11.48)
satisfies (E1).
/ u «(Au + A^ ) < / u «(Au
t
t
((17) is just a reformulation
of
s h o w s t h a t t h e r i g h t member o f ( 1 7 )
u = 2 ( u ^ + 4^) - i j ^ s h o w s t h a t i t i s
< 0.
-44-
/ U «(Au
t
+ A^ )
t
= 0 .
t
Writing
p = Au + A ^
and u s i n g
t
(18),
(17)
becomes
/ u -p < 0
t
for all
p e H (ft)
T h i s shows
u
t
with p < 0 .
_ 1
that
> 0 .
Now ( 2 0 ) ,
(11.57) and ( 1 8 ) c o n s t i t u t e t h e c o n d i t i o n s
h a v e p r o v e n t h a t t -» u
( i i ) <= : S u p p o s e
/ ( v - i|» )«Av
t
t -> v
< 0 =
t
satisfies
(D1)
satisfies
t
/(v -^ )-Av
t
t
i n ( D 1 ) a n d s o we
.
(D2).
T h e n , by ( 1 1 . 5 1 ) and
(11.52)
t
1
for all
v£Hg(ft)
/ V(v - v ) - W
t
for all
that
> 0
t
veHg(ft)
with
S u p p o s e t -* v
/ (v-^ )«Av
t
for all
<
t
v > ip .
t
vGH^(ft) with
(E2)
/ (v -* ).Av
t
t
t
= 0
t
t
< 0 , or
satisfies
t
t
(11.50) t h i s
shows
(E2).
Then
t
v > ^
shows t h a t
is
.
. ((21)
s h o w s t h a t t h e r i g h t member o f
f
/(v-v )«Av
Since (11.59)
t
/(v -^ )»Av
v = 2 ( v . - X/J.) + ip
Thus
T
,
t -» v^ s a t i s f i e s
( i i ) => :
v > 4> .
with
it
is
(11.60)).
(21) is > 0 ,
is
< 0.
Thus
T h e c h o i c e v = 4>
while the
choice
T
-45-
T h u s , w i t h tp = v - ip
(23)
/ tp»Av
Av
i p £ H (ft)
with
cp > 0 .
T h i s shows
that
< 0 .
+
Since (11.59),
( 2 4 ) and (22) c o n s t i t u t e t h e c o n d i t i o n s
we h a v e p r o v e n t h a t
This finishes
e.
becomes
< 0
t
for all
(24)
(21)
t
t -» v
t
satisfies
(D2)
(D2)
.
t h e p r o o f o f Theorem 5.
E x i s t e n c e , u n i q u e n e s s and r e g u l a r i t y o f s o l u t i o n s t o
Let r,R,T,ft,Dg
t h e o r e m we s h a l l
in
and ^
recall
t
be as on p . 3 0 .
some g e n e r a l
(E)
Before formulating the
regularity
next
results.
i
Let 2<p<°°
are
(25)
cp£Hg(ft)
.
Then the f o l l o w i n g
implications
true:
A cp e L ( f t )
=> cp e H
p
Here
that
cp£C ' ( f t )
a
its first
with exponent
sup
and l e t
z,ceK
2
> (ft)
P
means t h a t
cp i s c o n t i n u o u s l y d i f f e r e n t i a t e
order partial
derivatives
a (0<a<1),
i.e.
z - ? a
f o r each compact
are l o c a l l y Holder
in
y-derivative
and
continuous
< + oo
K c ft , a n d s i m i l a r l y f o r t h e
ft
Sep
3y
-46-
The f i r s t
operator
A
implication
in (25) is a consequence of the f a c t t h a t
is uniformly e l l i p t i c .
( [ 1 2 ] , Theorem 4 . 1 0 , p.
34).
T h e s e c o n d i m p l i c a t i o n f o l l o w s f r o m a w e l l - k n o w n lemma b y
( [ 1 2 ] , Lemma A 9 ,
Aip £ L ( f t ) f o r a l l
and
i/j e C ' ( f t ) f o r a l l
p
t
(11.43) of
^
we s e e t h a t
p < ~ a n d s o , b y ( 2 5 ) , ye
Thus
1
Sobolev
p.56).
Now f r o m t h e d e f i n i t i o n
t
a
A^ eL°°(ft)
u n i q u e n e s s and r e g u l a r i t y
T h e o r e m e 1.1
in [ 2 ] .
concerns the f u n c t i o n
v^
H »P(ft)
for all
2
theorem f o r solutions of v a r i a t i o n a l
More p r e c i s e l y ,
the part
existence,
inequalities,
of Theorem 6 which
i s a d i r e c t c o n s e q u e n c e o f T h e o r e m e 1.1
D e s p i t e o f t h i s we w i l l
u^.
in
include a proof o f Theorem 6 h e r e .
We p r o v e
is very easy to
i s n i c e t o be s e l f - c o n t a i n e d t o s u c h a l o w p r i c e ;
prove
a n d we p r o v e
o f s o l u t i o n s b e c a u s e we h a v e a p r o o f o f t h a t p a r t w h i c h i s
l e r and more e l e m e n t a r y t h a n t h e s t a n d a r d p r o o f s and w h i c h perhaps i s
T h e o r e m 6:
(i)
The v a r i a t i o n a l
u ,
t
v eHj(fi)
t
inequalities
(te
( E 1 ) and ( E 2 ) have u n i q u e
u
t
(iii)u ,
t
a n d v^. a r e r e l a t e d b y
v £H ' (fl)
2
t
for all
(This
p
solutions
[0,T]).
( T h i s depends o n l y on t h e f a c t t h a t ^
(ii)
for all
[2]
f o l l o w by t h e n a p p l y i n g T h e o r e m s 4 and 5.
e x i s t e n c e and u n i q u e n e s s o f s o l u t i o n s because t h i s
regularity
p < °°
a < 1 .
while those parts concerning
and i t
.
t
Theorem 6 below is e s s e n t i a l l y a special case of a standard
e.g.
the
£Hg(ft).)
v^. = u^ + i j ^ .
p < <» . I n p a r t i c u l a r
a < 1 .
p a r t d e p e n d s on i>. e H ' ( f t ) , a l l
2
p
p < °° . )
u ,
t
v £C ' (ft)
t
1
a
simpnew.
-47-
Proof:
(i)
follows from standard Hilbert
f o r example.
Fix
te[0,T]
K = {u £ Hj(n)
: Au + A i p
t
and
< 0}
space t h e o r y .
Consider
(E1)
let
.
i
Then
K
is a closed convex set
and t h a t
c a n be
it
in Hg(ft). That i t
formulated
tp€Hg(ft)
(u,cp) > - ( ^ , c p )
for all
cp£Hg(ft)
t
(28) shows t h a t
K
i
with
cp > 0 ,
with
cp > 0 .
is the i n t e r s e c t i o n of a set of
c l o s e d h a l f s p a c e s and so i s c l o s e d ( w e a k l y and
The c o n d i t i o n
€ K
i n ( E 1 ) now b e c o m e s
t
>
0
for
all
u£K
(K b e i n g c l o s e d and c o n v e x )
•i
u £Hg(ft),
t
Find
namely t h a t
u. e K
such t h a t
u
( w e a k l y and
strongly)
strongly).
(for fixed
t£[0,T])
z
Thus
(i)
(ii)
follows
,
this condition
i n f ||u|| .
ueK
is proved ( w i t h a s i m i l a r argument f o r
Alternative
u
t
(ii)
it
is s u f f i c i e n t
a n d v^ b e l o n g s t o
1: F o r t h e f a c t t h a t
t o p r o v e t h a t one o f
H ' (ft)
2
.
p
We g i v e t w o
v £H ' (ft)
t
2 < p < ° ° ) we may r e f e r t o t h e e x i s t i n g
equalities
(E2)).
by c o m b i n i n g T h e o r e m 5 and T h e o r e m 4 .
In view of
functions
i s s a t i s f i e d by a u n i q u e
w h i c h s o l v e s t h e minimum norm p r o b l e m
t
||u.|| =
z
(iii):
or
and
(u-u ,Uj.)
and
obvious
t
for all
t
is
i s c l o s e d f o l l o w s f r o m t h e f a c t t h a t t h e c o n d i t i o n Au + AiJ> < 0
<Au + Aif^.cp) < 0
u
is convex
2
p
alternatives.
(whenever ^
literature
the
e H ' (ft),
2
on v a r i a t i o n a l
o f the kind ( E 2 ) , f o r example [ 2 ] , Theoreme
1.1.
p
in-
-48-
Alternative
2: We c a n p r o v e t h a t
u^£H
' (ft)
p
by a r a t h e r s i m p l e
and
e l e m e n t a r y a r g u m e n t w h i c h a l s o m i g h t be o f i n d e p e n d e n t i n t e r e s t .
g o e s as
follows:
We w a n t t o s h o w t h a t i f
then the s o l u t i o n
Au + Ai/; < 0
lfieHj(fi) n H ' ( f i )
2
uGHg(ft)
f o r some 2 < p < ° o
p
,
of
,
- /v(u'-u)«Vu > 0
for all
U'GHJM
A u ' + Aip < 0
with
also belongs to H ' ( f t ) .
2
((31)
p
is
(E1)
end, consider the following a u x i l i a r y
t h e c o n d i t i o n s on
u a r e more
i
Find
r
It
ueHg(ft)
min(0,Aip)
variational
inequality,
To
in
this
which
restrictive:
satisfies
< Au + A^ < 0 ,
- /v(u'-u)«Vu > 0
with
which
for a fixed t e [ 0 , T ] . )
min(0,AiJj)
for all
u'eHj(fl)
< A u ' + Aip < 0 .
S i n c e , j u s t as f o r
(E1),
t h e c o n s t r a i n t s on
u (and u')
above
delimit
A
a c l o s e d c o n v e x s u b s e t o f H g ( f t ) , (32)
Now t h i s
u
solution a priori
are of the form
belongs to
f^ < Au < f 2
can show t h a t t h e s o l u t i o n o f
(31)
we a r e d o n e .
(D1),
has a u n i q u e s o l u t i o n
H ' (ft) since the
with
u of
(32)
2
conditions
p
f ,j ^
€ L (ft).
p
ueHg(ft)
Therefore
u
on
i f we
i s t h e same as t h e s o l u t i o n
A n d t o s h o w t h i s we n e e d o n l y show t h a t
.
of
satisfies
i.e.
'u > 0
- Au + tup < 0
/u«(Au + A^) = 0
since these conditions characterize
t h e s o l u t i o n o f (31)
(Theorem 5 ( i ) ) .
-49-
U£HQ(Q.)
Thus l e t
u
satisfies
(33) - (35) above.
/ u « ( A u ' + AI/J)
for
all
be t h e s o l u t i o n o f ( 3 2 ) a n d we s h a l l
< /
prove
that
F i r s t we g e t f r o m ( 3 2 ) t h a t
U»(AU+AIJJ)
u' G H Q ( ^ )
min(0,AI(J)
< A u ' + AI(J < 0 ,
with
min(0,AIP)
< p < 0 .
p = 0
in (37).
This
with
or
/ u«p < / u«(Au + AIP)
for all
peH~ (ft)
1
First
choose
gives
/ u«(Au + AIP) > 0 .
Then
P
choose
min(0,A\P)
on
N
Au +
o n ft ^ N
=
AIP
where
N = {zeft
: u ( z ) < 0}
(Observe that
since
u£H ' (ft),
2
u
p
i s a continuous f u n c t i o n , so t h a t
i s a w e l l - d e f i n e d o p e n s e t i n ft . )
/
u *min(0,AIJ;)
N
< / u*(Au + A I P ) ,
We g e t
or
N
/ U»(AU + A^ -
min(0,A^))
> 0
S i n c e A U + M> - m i n ( 0 , A I P )
> 0
N
A U + AIP - m i n ( 0 , A I P )
AU
<
0
on
N.
=0
on
and
N.
U < 0
This
shows
on
N
that
(42) forces
N
-50-
B u t now
u =0
on
3N U 3ft
i s c o n t i n u o u s and b e l o n g s t o
u > 0
that
on
N
I
(by the d e f i n i t i o n of
Hg(ft)).
Therefore
Au < 0
N . Comparing w i t h the d e f i n i t i o n o f
i s t h e empty s e t .
N, and s i n c e
N
on
we m u s t
u
N implies
conclude
Thus
u > 0 .
Thus
of
(33)
is proven.
( 3 2 ) ) , and c o m b i n i n g
( 3 4 ) we know f r o m t h e b e g i n n i n g
(33) and ( 3 4 ) w i t h ( 3 8 ) y i e l d s
(it
is
(35).
2
T h i s ends t h e p r o o f t h a t t h e s o l u t i o n o f
and c o m p l e t e s t h e A l t e r n a t i v e
This also finishes
Remark:
Observe t h a t ,
equivalent
(31) belongs to H ' ( f t ) ,
p
2 program.
although i t
is t r i v i a l
that
(D1) and (D2)
that
statement
( E 1 ) and ( E 2 )
it
(ii)
o f Theorem 6 i s not completely
Minimize
||v-^ ||
when
Minimize
||v||
when
t
(=v
Av < 0
t
=
u ^ + I ^ )
:
•i
(veH (ft))
Q
1
v > ty. ( v € H ( f t ) )
n
.
the
obvious.
variational
says f o r example t h a t the f o l l o w i n g
p r o b l e m s h a v e t h e same s o l u t i o n
it
( E 1 ) and (E2) a r e e q u i v a l e n t under
I n t e r m s o f t h e minimum norm p r o b l e m s a s s o c i a t e d w i t h t h e
equalities
are
v ^ = u^ + I J ^ ( a s s t a t e d i n T h e o r e m 4 ) ,
is not a complete t r i v i a l i t y
i.e.
D
t h e p r o o f o f Theorem 6.
via the r e l a t i o n
same r e l a t i o n ,
part
two
-51-
f
-
( D ) => ( C ) .
By t h e T h e o r e m s 4 , 5 a n d 6 we know t h a t
solutions
u^
ferentiate
and
v
t
(t£[0,T]),
(D1) and (D2) have
that these are continuously
f u n c t i o n s and a r e r e l a t e d by
v^ = u
+ ^
t
.
result
dif-
Our goal
is to prove that these solutions give r i s e to a solution of
preparation for this
unique
(C).
now
As a
( T h e o r e m 9 ) we n e e d t w o l e m m a s , Lemma 7
a n d Lemma 8 .
Lemma 7:
(i)
i
L e t u £ H g ( f t ) be t h e s o l u t i o n o f ( D 1 ) .
Then u < u
1
ueHg(ft) which s a t i s f y
Au + A i p < 0 a n d u > 0 .
t
t
for
all
t
(ii)
Let
v eH
t
i
vtHg(ft)
Remark:
first
(i)
Q
1
(ft)
be t h e s o l u t i o n o f ( D 2 ) .
which s a t i s f y
says that
conditions,
v > ^
and
(ii)
< v
that satisfy
is a consequence of
however,
(ii).
be some f u n c t i o n
include a proof here.
i n H '^(ft)
1
^eHg(ft).
P r o o f o f Lemma 7:
the functions
u
obviously
satisfying
the conditions
in ( i i ) ,
in
In t h i s proof the
S i n c e we know t h a t t h e f u n c t i o n s
the r e l a t i o n v = u + ^
(i i ) .
(11.49).
or at
( f o r some p < <=°), a l t h o u g h Lemma 7 i s
r e s p e c t i v e l y a r e r e l a t e d by
We p r o v e
function,
Due t o t h e i m p o r t a n c e o f Lemma 7
be a s s u m e d t o be t h a t f u n c t i o n d e f i n e d b y ( 1 1 . 4 3 ) ,
satisfying
two
( m o r e o r l e s s ) a s p e c i a l c a s e o f Lemma 1.1
\p. w i l l
tween
the
(ii).
we w i l l ,
(ii)
all
the t h i r d c o n d i t i o n ,
f o r us
for arbitrary
for
(11.47) and ( I I . 4 8 ) , i n (D1) t h e r e i s a s m a l l e s t
o f Lemma 7 i s
[ 1 3 ] , and ( i )
t
Av < 0.
in the class of functions
namely t h a t f u n c t i o n which a l s o s a t i s f i e s
-Similarly for
Then v
v ^ = u^ + y
u^ a n d v
t
function
least
true
in ( i )
( T h e o r e m 4 ) , and
since
s e t s up a o n e - t o - o n e c o r r e s p o n d e n c e
the conditions
it suffices
i n ( i ) and t h o s e
and
be-
functions
t o p r o v e one o f ( i ) and
(ii).
-52-
With v
t
a n d we s h a l l
I(v )
and
This
as i n t h e s t a t e m e n t o f t h e l e m m a , p u t
p r o v e t h a t w > 0.
= {zeft
t
v
: v (z)
.
t
is a w e l l - d e f i n e d closed set in
and so ( T h e o r e m 6)
and
ft
s i n c e , by a s s u m p t i o n , ^
v^ a r e c o n t i n u o u s
Denote by suppAv^ t h e s u p p o r t o f
Av
SUppAV^ c= I ( v )
(44) shows
ft
Everywhere
t
Since - ( v
3ft
t
in
-
(v
^ I(v )
t
ft
on
v
> -
t
I(v ),
3(ft x
(by
t
t
it
t
that
.
(v
t
- ip )
t
.
n
( i n some
Kv ))
c: 3ft u I ( v )
t
e
t
now be t h a t
m
i i
n
m
u
m
.
(45) t o g e t h e r
with
(47) imply w > 0 in
p r i n c i p l e f o r superharmonic f u n c t i o n s )
t h e minimum p r i n c i p l e
n o t n e e d be a n i c e f u n c t i o n b u t i s j u s t an e l e m e n t o f H ( f t ) .
Q
follows.
and
this
i n a l l ft .
H o w e v e r , some c a r e i s n e e d e d i n a p p l y i n g
p r o c e e d f o r e x a m p l e as
a n d on
sense)
then gives the desired conclusion, w > 0
w
i n ft v
- ip^.) i s a c o n t i n u o u s f u n c t i o n w h i c h v a n i s h e s on I ( v ^ )
The argument w i l l
ft ^ * ( t )
>0
T h e n , v^ - ip
we h a v e
- ^ )
t
(46) shows t h a t
w > 0
P
that
in
w = ( v - 4> )
H -' (ft)
.
t
Aw = A v < 0
G
a s a d i s t r i b u t i o n on ft ( s o ,
t
being a non-negative continuous f u n c t i o n which is
f r o m ( 1 1 . 5 1 ) and ( 1 1 . 5 2 )
t
functions.
b y d e f i n i t i o n , s u p p A v ^ i s a r e l a t i v e l y c l o s e d s e t i n ft ) .
follows
v^,
Let
= ^ (z)}
t
w = v -
since
We c a n
-53-
C h o o s e an a r b i t r a r y
I(v )
u 8ft
t
e - (v
IP )
>
T
ft
such
0
in
Then t h e r e
E
0
>
in
< 0
in
ft
\ I(v )
(by
( 4 6 ) ) we h a v e
By
(45)
,
t
i n a n e i g h b o u r h o o d o f t h e c o m p a c t s e t ft V N ( C ft \
( 4 9 ) a n d ( 5 9 ) now show
E
0
>
d i n a r y minimum p r i n c i p l e f o r C
family of regularizations
of
f o r e x a m p l e by a p p l y i n g t h e
superharmonic functions
w + E
where
Since
h (z)
E > 0
=
in
P
(51)
was a r b i t r a r y ,
a n d Lemma 7 i s
proven.
Corollary
Let IP, IJ/eHj(ft)
u,u'
7.1:
and v , v '
choices
be t h e s o l u t i o n s
= IP, Y
1
w h e r e h £ C™( F )
Z
n H ' (ft)
2
of
p
and
, h > 0
a n d / h = 1) .
some p < O O ) , a n d
.
If
AIP < AIP' then
u < u'
(ii)
If
IP > I P '
AIP < A I P ' )
(iii)
(= ( i )
(or
and ( i i )
if
.
then
v > v'
.
combined)
I f AIP < AIP' then
u' -
(IP -
IP')
< u < u
1
and
letting
(D1) and (D2) c o r r e s p o n d i n g t o
(i)
v
1
+
suitable
we c o n c l u d e t h a t w > 0 i n
(for
(IP -
IP')
or-
P
O
h ( / P ) ,
t o some
( s a y t o (w + E ) * h ,
O
N O ,
I(v
that
i n ( a n e i g h b o u r h o o d o f ft ^ N) U N = ft ,
2
P
of
N
and i n p a r t i c u l a r
w +
N
N .
( i n the sense of d i s t r i b u t i o n s ) .
A ( w + e)
is a neighbourhood
that
w + E > e - ( v ^ - IP^.) e v e r y w h e r e
Since
w +
-
t
in
E > 0.
> v > v'
.
let
the
ft,
-54-
Proof:
(i)
(ii)
We h a v e A u ' + Aip < A u
1
lemma g i v e s
= ip,
( w i t h ip
We h a v e v > ip > ip
( w i t h i|> = ip', v
t
t
+ Aip
and
1
and
u^ = u )
A v < 0.
= v )
v
1
t
< 0
1
u < u'
Thus
< v .
1
u ' > 0.
Thus ( i )
of
the
.
(ii)
o f t h e lemma
gives
(Aip < Aip' i m p l i e s ip > y'
as
1
ip, ip' e H g ( f i )
28.10
(iii)
(i)
for
b y a weak maximum p r i n c i p l e .
v < v ' + (ip - i p ) , w h i c h
(ii)
i s the second p a i r of i n e q u a l i t i e s
pair
i s t h e n o b t a i n e d by s u b t r a c t i n g
Let
r,
R, T ,
ft,
D
Q
and ip
in ( i i i ) .
t
be a s on p . 30,
t
£ Hg(ft)
be t h e s o l u t i o n o f (D1)
[0,T] 3 t - »
t
£ Hj(n)
the s o l u t i o n of
,J
A8 = -
(D2).
Further let
, Xin
r
Then, for
u
v
t
Proof:
A
^t
=
< u
T
+ 2ir(t-T)'9 > V
t
r
T < t,
- 27r(t-T)*6 < u
>
t
v
and
t
t
From
X
D,
0
1 + 2irt--n D 7 r D
x
1 -
r
=
2TTt'A0
0
we g e t
for
x < t
and
ip
T
- ip
t
= 2ir(t-x)9
Now C o r o l l a r y
7.2
.
follows from ( i i i )
first
let
and
1
f u n c t i o n d e f i n e d by
The
ip .
[ 0 , T ] 3 t -* u
v
together
1
with
7.2:
Corollary
example.)
g i v e s u + ip < u ' + ip, i . e .
Corollary
See [ 2 0 ] ,
of Corollary
7.1
.
e e H j ( f t ) be t h e
-55-
Corollary
D
c c D
r
7 . 3 : L e t r,
0
c
c c
D
R , T , ft be a s on p . 3 0 , l e t
a n d l e t i|» , 4>'
r e s p e c t t o Dg a n d Dg' r e s p e c t i v e l y .
corresponding
u
< uj.
t
solutions
and
v
of
t€ [0,T]
Proof:
S i n c e Aip^ < Ai|^
W i t h r,
3 t -* v
be t h e s o l u t i o n
I(v )
t
D
and ( i i )
v ^ , vj.
be
the
Then
ft,
te[0,T]
of Corollary
Dg a n d
^
the c o r o l l a r y
follows
7.1 .
a s on p . 3 0 ,
let
HJ(A)
of
( D 2 ) and
= {z eft : v ( z )
t
= Dg U (ft x
t
for all
R, T ,
e
t
( D 1 ) and ( D 2 ) .
u|. a n d
.
immediately from ( i )
[0,T]
u^,
Let
with
> vj.
t
for all
Lemma 8 :
DQ s a t i s f y
Q
be d e f i n e d b y ( I I . 4 3 )
t
R
D ,
put
= ip (z)}
,
t
l(v ))
.
t
Then
suppAv^ c
(ft ^ D g ) n I ( v ^ )
D^ c ft ^ s u p p A v ^
where
in both
and
,
inclusions
the d i f f e r e n c e - s e t s
have measure
zero.
Moreover,
A
v
t
=
Proof:
all
" X(ft-D )nl(v )
0
Recall
p<°° .
=
t
( p . 45-46
Thus i p , v
defined closed set
t
in
x
D "
t
1
•
and Theorem 6) t h a t ^ ,
t
are continuous
t
ft
and
D
t
functions,
i s an o p e n
set.
v £H ' (ft)
2
t
I(v )
t
p
for
is a well-
-56-
Observe next that the d e f i n i t i o n
(D2) f o r
since the solution of
and so
I(v )
ip^, v
= Ai|>.
(58)
t
G H ' ^P ( f t )
v^=ty^
We c l a i m t h a t
Av.
t=
of
0
is consistent for t =
is
a.e.
i m p l i e s tthhaatt Aip.
on I ( v ^ )
and w h i c h w i l l
on
and
Av
t
are
n o t be p r o v e n
Suppose u G H ' P ( f t )
left.
Then a l l
2
partial
(1<p<°°)
derivates
in p a r t i c u l a r
lemma, w h i c h i s
Au=0
and t h a t u = 0
of order
<2
a.e.
I.
i n H ^ ' P e v e r y such d e r i v a t i v e o f order
on
<2
f o r m o f an L - f u n c t i o n , a n d t o s a y t h a t
non-trivial
on a c l o s e d
vanish almost
derivative.
set
every-
For a
function
have a r e p r e s e n t a t i v e
i t vanishes
p
a.e.
t o s a y t h a t a n y o n e , and t h e n a l l , o f i t s r e p r e s e n t a t i v e s
in
on a s e t means
vanishes
a.e.
set.)
F o r a p r o o f o f t h e a b o v e lemma we may r e f e r t o [ 1 2 ] ,
p . 53 .
Two a p p l i c a t i o n s
Applications
o f t h e lemma t o u = v
f o l l o w by combining
suppAv
t
<= l ( v )
(58)
t
- ^
Av. < 0
iiD~r D
x
> 0
on
D,
r
0
- 1
ft v
settles
(58).
w i t h the f o l l o w i n g
,
27Tt • - r
Lemma A . 4 ,
o f t h a t lemma y i e l d o u r l e m m a .
Lemma 8 w i l l
t
p
here.
( B y " d e r i v a t i v e " we a l w a y s mean d i s t r i b u t i o n
on t h a t
L -functions.
I (v.)
Lemma:
I,
Aipg<0),
implies
is a consequence of the f o l l o w i n g
w h e r e on
(in view of
V Q = ^ Q
0
= ft .
n
Now
(54)
D,
0
Now
facts:
-57-
( 5 9 ) was p r o v e n on p . 52 ,
D
(together with
Aip = A v
t
t
hence by
(63)
Aip
( 1 1 . 5 1 ) and ( 6 1 )
is
(11.43)
imply that almost everywhere
on I ( v t )
w
e
n
a
v
e
< 0 ,
(61)
= - x^^
t
is
cz D g ) .
r
(58) and (60)
(62)
(60)
D
( a . e . on
I(v ))
( a . e . on
I(v ))
t
hence
^
\
A
= - ^ D
Q
In view of
(
6
5
)
(66)
A
v
t
" ft-D
=
X
* I(v )
x
Q
Since
(ft^Dg)
suppAv
t
(59),
fl I ( v )
cz ( f t ^ D )
Q
if
E
=
" (ft-D )
X
n I(v )
t
]R
closed in
ft
, (65) shows
(56)
i n (66) has measure z e r o .
then suppx
n
E
<= E a n d E ^ s u p p x
i s o b t a i n e d f r o m (55) by t a k i n g
t
and (57)
is
D..
r,
R, T ,
(67)
[0,T]
3 t -* u
t
e H^ft)
(68)
[0,T]
3 t -
t
e
v
Hj(ft)
ft,
and
(GeneE
has
Dg a n d ip
t
complements
(65) above t o g e t h e r
T h i s p r o v e s Lemma 8 .
Theorem 9: With
that
zero.)
(and using the d e f i n i t i o n o f D ) ,
the d e f i n i t i o n of
*
t
that the d i f f e r e n c e - s e t
is proven.
ft
.
Lebesgue measure
(55)
n I(v )
Q
is a closed set in
n-dimensional
Thus
(64) shows t h a t i n a l l
is r e l a t i v e l y
t
It also follows
rally,
t
t
as on p . 3 0 ,
let
with
-58-
be t h e s o l u t i o n s o f
(D1) and (D2) r e s p e c t i v e l y
( s o t h a t v^ = u^ + i ^ ) .
Define
D
t
= D
U (n^I(v ))
Q
= D
t
for t £ ( 0 , T ] .
suffices
Then, if
that R
[ 0 , T ] 3 t -»
D
U { Z £ ft : u ( z )
Q
t
R
i s l a r g e enough o r
> 2T + p , i f
t
a
R
)
> 0}
D
Q
c
B
p
T
r
(C).
in (C)
is identical
u^. ( = v ^ - i f ^ ) a b o v e
Proof:
———
F i r s t we m u s t p r o v e t h a t
B cc D c c
f
B <=c= D^
with
B
t
the
Further,
D. £ I L ^
t
K , r
(t £ [0,T])
R
follows from
r
(it
) , t h e map
i s w e l l - d e f i n e d and s a t i s f i e s
that
small enough
the function
"u^." a p p e a r i n g
.
for all
t £ [0,T],
i.e.
.
D c c DQ
and
r
Dg C
D^ .
To prove
Dj.c=c:B
we s h a l l a p p l y C o r o l l a r y 7 . 3 .
Choose p such
D c=B <=<=B
and l e t
DA = B . T h e n D c c D c D A c c B
, and
( J p R
U p
r
U
U
R
R
that
n
D
ip'^, u | , v£
with
u^ < u |
I'(vj.) c
where
IB
,
or
= {z£
= B
> 0
p
ft:
vj.(z)
for
t £ (0,T]
t
I = 2irt + I E I
= ij^(z)}
that
is defined
,
7.3
v^. - ip^ < v.j. - if^ .
t
Now we c l a i m
(v!)
t
d e f i n e d as i n C o r o l l a r y
(t£[0,TJ)
I(v ) ,
where I ' ( v j J
ft^T
n
by
i.e.
,
.
D
we h a v e b y C o r o l l a r y
T h i s shows
that
7.3
-59-
Once ( 7 2 )
is proven
PY = 2 T + P
D
= D
t
t
(72)
: u!(z)
z
=
B
X
(for T E ( 0 , T ]
(71) and (72) t h e n
U (nM'(vj.))
Q
is equivalent
P
B ^
Q
P
= B
~
X
D A
~
2
U
T
* ~ I L T T
X
D
r
and ( 7 6 ) f o l l o w s
from (77).
((77)
is a solution
P
to
(A)
is
and
s u g g e s t e d by t h e f a c t
and hence t o
( B ) ,
IP'
Thus
t€[0,T]
It
GHJ(Q)
(72)
if
1
that u|EHG(ft)
B
= B
P
T
=
2T +
Av
t
which in view of
(11.49)
d e f i n e d by A ^
=
t
X
D
-
" 1 +
T
d e f i n e d by (77)
2IRT
'
i s p r o v e n , a n d s o we h a v e p r o v e n t h a t
remains
AIP
P
<
2
R
,
2
to prove
=
X
D t
"
(11.43)
combined w i t h
This completes
DG<= B
P
=
P
DQ
of
and
B
E &
R
r
solves
D
Q
is
(11.13).
(77)
.
=
B
P
.)
for
(C).*/
Lemma 8 ( 5 7 )
shows
1
(11.14)
above.
t h e p r o o f o f T h e o r e m 9.
u^ g i v e n b y ( 6 7 )
X
.
.
(11.13) - (11.15)
*/
With
with
0
(D1) w i t h
is
( C ) ,
t h a t t h e map
t
u | = / 9 R dx
0
P
domain and t h a t t h e r e f o r e by Theorem 3
is a l s o easy to check d i r e c t l y
+
.
R
that
x
as i n i t i a l
t
«=<=B
'
r
Au
<=D
T
u
It
that
give
= D U
to proving
and assuming
> 0} = B
P^
t - » B
follow
c a n be p r o v e d s i m p l y b y c o m p u t i n g t h e f u n c t i o n u { e x p l i c i t l y .
1
i t i s f o u n d t h a t u£
is the f u n c t i o n in HG(ft)
f o r which
In f a c t
u
c D
t
and t h i s
A
since
U (J)NL(v ))
Q
will
t
< R )
To prove
{z£ft
D <=c:]DP
is
( 1 1 . 4 7 ) , and
(11.15)
that
-60-
Corollary 9.1:
_ _ _ _ _ _
enough
(R
Let
r,
R, T ,
ft
and
D
be a s on
n
p. 3 0 w i t h
R
large
u
> 2T + p
suffices,
if
DgC]D )
p
.
Then t h e r e e x i s t s
a
solu-
tion
[0,T]
of
3 t -> D
(C).
t
This
e ^
R
j
r
solution
a distribution
i s u n i q u e up t o n u l l - s e t s ,
f o r each t £ [ o , T ] .
Moreover,
u^ € H g ( f t ) be t h e f u n c t i o n a p p e a r i n g
a s an e l e m e n t o f H ^ f t ) ,
D
t
= D
for
U { Z e ft : u ( z )
Q
t€[0,T].
(In
as
let
(C).
Then
(78) above
(79)
u^ r e f e r s
u^ i s
unique
c a n be c h o s e n t o be
to the continuous
representative
t
The e x i s t e n c e of the s o l u t i o n
Theorem 9 (combined w i t h the f a c t
Theorem 9 a l s o
For t = 0 (79)
s a y t -» D
(for
(D1)
is unique
u
t
= 0 in t h a t case
a solution of
of
(79) f o r
(this
u
q
solutions
from
(D1)).
te(0,T].
follows
from
s u p p o s e we h a v e t w o s o l u t i o n s o f
(11.13).
t h e same \|> ) b y T h e o r e m 3 .
5 and 6 ) .
Then
Thus
t
distributions,
of the
since
(Theorems
immediately
c a n be c h o s e n a s i n
t -*• D | ( w h e r e D Q = D ) , a n d l e t
the corresponding
(D1)
(78) f o l l o w s
that there exists
As t o t h e u n i c i t y ,
and
T
shows t h a t
is true
for example).
u
t
u =u|
t
T h u s , a l s o by
or almost everywhere.
This
and
t
and
(11.13)
(78),
u^. £ H ( f t ) be
Q
uj.
both
solve
since the s o l u t i o n
(11.13),
shows t h e u n i c i t y
Xp
=
Xpi
of
as
part
corollary.
Corollary
9.2:
3 t -
satisfy
(i.e.
unique
1 S
u .)
Proof:
[0,T]
Xn.
> 0}
t
all
of
and i n
in
i.e.
(C).
D ^D.
T
D
t
Let
e X
R
j
r
Then, for
x < t,
has measure z e r o , o r
D
T
<= D
Xn
T
except for a
5 Xn
x
i
t
n
null-set
the d i s t r i b u t i o n
sense),
-61-
Proof:
Let
u^
and
by t h e s o l u t i o n
v^
be t h e s o l u t i o n s
(80) of
(C)
D
t
= D
=
D
U
e 9. : u ( z )
{Z
T < t
is
7.2
u
< u
T < t,
for
t
showing t h a t
9.3:
„
K i
eft
be t w o s o l u t i o n s
(except
Proof:
and
9
[ 0 , T ] 3 t -* Dj.
R > r
of
( C ) , and suppose t h a t
for null-sets)
The p r o o f
for all
is similar
t £
D Q <= D Q .
Corollary
9.4:
3 t ->
(a,b)
(Unicity
D ,
T
to that of Corollary
9 . 2 , but with
Proof:
.
of
Then
(A) or
D
t
(A))
t
(C).
close to
Dj.
The u n i c i t y
£ft
t > t
t^ = 0.
we o b t a i n f o r
chosen a r b i t r a r i l y
3 t -> D ,
solutions):
implies the
Suppose
Q
(t £
(a,b)
(a,b)).
s u i t a b l e c h o i c e s o f r,
b)
some t p £
By a p p l y i n g T h e o r e m 2 ( a n d T h e o r e m 1
two
R, T > 0 ( T c a n be
solutions
R > r
part of Corollary
9.1 c o m b i n e d w i t h t h e
a s s u m p t i o n s on D^. a n d Dj. t h e n g i v e s t h a t
This
7.2.
( B ) , and suppose t h a t f o r
= Dj. f o r a l l
We may a s s u m e t h a t
in case of
of
of classical
Corollary
Dj. € S
a r e two s o l u t i o n s
= Dj.
T h e n D^ c Dj.
[0,T].
7.3 used i n t h e l a s t a r g u m e n t i n s t e a d o f C o r o l l a r y
[0,T]
for
Let
[ 0 , T ] 3 t -» D ,
L
t
D^ c= D^
as w a s t o be p r o v e d .
Corollary
D
zero
> 0 } .
t
Now, b y C o r o l l a r y
uniqueness
=
t
Q
Then the
produced
b y c h a n g i n g D^. w i t h a s e t o f m e a s u r e
we c a n a s s u m e t h a t
U (n^I(v ))
Q
(D1) and (D2)
(as in Theorem 3 ) .
p a r t o f T h e o r e m 9 shows t h a t ,
i f necessary,
of
corollary.
= Dj.
regularity
for t £ [0,T]
.
-62-
( C ) «*
9-
(H
T h e o r e m 10:
A map
[ 0 , T ] 3 t -> D
satisfies
(C)
ER
t
R > R
i f and o n l y
if
it
satisfies
(F).
Proof:
=* : S u p p o s e ( 8 1 ) s a t i s f i e s
(C).
Since
D. c c ft = D
1
it
i s enough t o p r o v e
subharmonic
Let
and u
u
in
= 0
t
continuous
?
t£[0,T]
?
cp £ H g ( f i ) fl H ( F
)
which
are
D^.
£ WQ{Q)
t
(11.61) f o r a l l
for all
D
a.e.
be t h e f u n c t i o n d e f i n e d b y ( 1 1 . 1 3 ) .
an ft ^ D
(since
(by
t
(11.14)
and ( 1 1 . 1 5 ) ) .
Au^ £ L°°(ft)), i n p a r t i c u l a r
2
1
Now, l e t
Then u
Moreover,
> 0,
t
u
t
is
b o u n d e d ( a s u^ = 0 on 3ft),
2
cp £ H ' ( f t ) n H ( F ) be s u b h a r m o n i c i n D . .
t h e n Acp > 0
o
2
~
in D in the sense of d i s t r i b u t i o n s .
M o r e o v e r , s i n c e Atp £ L ( f t ) , t h e
2
above p r o p e r t i e s o f
u^ s h o w t h a t u « A c p £ L ( f t ) , u^« Acp = 0 a . e .
on
ft v. D.
and hence
/ u . • Acp = 0 . U s i n g t h e s e f a c t s a n d a l s o ( 1 1 . 1 3 )
ftvD
t
t
Z
z
t
and t h e submeanvalue p r o p e r t y o f
D
/cp - /cp = ( x
- X
»
D
t
0
t
Q
= < u
A
t
+ 2irt
n
n
D
D
<P>
.-j^jx,,^.
cp i n D
=
«P> =
1
=
( U j . , Acp)
+
27Tt - -jjj—p
(x
= / u . • Acp + 2lTt • - — - y
ft
'V
Z
»V>
D
/
D
=
cp =
r
1
=
/
D
. • Acp + 2fTt •
U
t
> 2rt
Thus
| J
Z
.-JPLJ
-i-r-P
V
/
D
CP
r
/ cp > 2irt • cp(0)
(81) s a t i s f i e s
(F)
.
>
.
r
we g e t
-63-
*= :
Suppose (81)
satisfies
(F).
Thus
/tp > 2-nt ' t p ( O )
/tp t
D
D
0
2
for every
see t h i s
/tp
D
-
tp e H ( F
implies
/tp
D
t
2
)
which is subharmonic
I D
As we s h a l l
presently
Q
/tp
r'
D
r
1
2
f o r a l l tp G H ( f t ) n H ( f t )
L e t us a s s u m e
Then
.
that
2 Tit '-r^-j
>
in
( 8 3 ) c a n be
which are subharmonic
( 8 3 ) f o r a moment a n d d e f i n e
in
D
t
.
U j . G H g ( f t ) by
(11.13).
written
( A u , tp) > 0
t
1
2
tp G H ( f t ) n H ( f t ) w i t h
for all
Atp > 0
7 ~
n
1
9
Hg(ft) n H (ft)
(bijectively)
in
onto L (ft),
(84)
D..
Since A
maps
i s t h e same a s ( w i t h p = Atp)
(u ,p) > 0
t
for
all
p G L (ft)
Since a l l
u
> 0
t
(in
The choice
K
*
(
x
t
ft)
1
)
=
t
•(x
D t
in
D
t
p G L (ft)
.
are allowed in
(85)
.
p = XQ
D "
p > 0
non-negative
(
and s o , s i n c e t h e
Ju
with
2
u
-
t'X
1
D
is also allowed in
t
- >
1
integrand
-D = 0•
(85).
This
" ° '
in
(87)
is
non-positive,
gives
we h a v e
-64-
This
proves that the function
u £Hg(ft)
d e f i n e d by ( 1 1 . 1 3 )
t
fies
( 1 1 . 1 4 ) and ( 1 1 . 1 5 )
( = (86) and ( 8 8 ) ) .
T h u s t h e map ( 8 1 )
fies
( C ) , a n d t h e p r o o f i s c o m p l e t e as s o o n as we h a v e d e d u c e d
from
(82).
satissatis-
(83)
2
2
F o r t h a t p u r p o s e o b s e r v e f i r s t t h a t t h e t e s t c l a s s H (]R ) f o r ( 8 2 )
2
2
2
2
c a n be r e p l a c e d b y H ( f t ) s i n c e t h e r e s t r i c t i o n m a p p i n g H (TR ) -* H ( f t )
i s o n t o ( [ 2 0 ] , T h e o r e m 2 6 . 7 ) . Now l e t
Atp > 0
i n D. a n d we s h a l l
tp £ H ( f t ) )
outside
in
ID
D^
but u n f o r t u n a t e l y
Since
in
tp i s c o n t i n u o u s
tp i n
ft
(as
w h i c h c o i n c i d e s w i t h tp
|D . T h i s f u n c t i o n
is also
p
subharmonic
i t d o e s n o t b e l o n g t o H ( f t ) s i n c e Atp w i l l
be a meas
2
p a r t on 3|D^ , h e n c e A t p € L ( f t ) ) .
Suppose
2
(82) were a p p l i c a b l e t o
property
(83).
and w h i c h i s harmonic
r
with
2
there is a continuous function
ure having a singular
that
prove
cp £ H ^ ( f t ) n H ( f t ) be g i v e n
tp.
nevertheless
T h e n we w o u l d o b t a i n , u s i n g t h e
f o r h a r m o n i c f u n c t i o n s a n d t h e f a c t t h a t tp > tp i n
|D
mean-value
(and
r
every-
where),
/tp
D
-
D
t
Thus
= /$ -
/IP
D
Q
/ t p > 2 7 T t - t p ( 0 ) = 27Tt
D
t
"
Q
' V l )
prove
(83)
it
applicable.
there will
equality
It
For with
be v a l i d
Thus
1
in place of
1
D
r D
l
r
,
£ HJ(J2)
Acp,. = p
2
(e\> 0)
to which (82)
cp i n ( 8 9 ) t h e f i r s t
cp) t h e r e w i l l
cp
.
Consider
Acp .
where
by
zero
I t c a n be d e c o m p o s e d
Acp , s u p p o r t e d
What we h a v e t o do i s t o s m o o t h o u t
be d e f i n e d
inequality
be e r r o r s w h i c h t e n d t o
where y is the s i n g u l a r p a r t of
,
is
follow.
be a f a m i l y o f p o s i t i v e m o l l i f i e r s ,
= e" h(z/e)
£
( 8 2 ) ) and i n t h e r e m a i n i n g e q u a l i t i e s and i n -
(83) w i l l
p = Acp - y .
£
$
(by
£
remains to c o n s t r u c t
as Acp = p + y
{h } ^
cp
(which are v a l i d for
as e -» 0 .
£
1
i s e n o u g h t o p r o v e t h a t t h e f u n c t i o n cp a b o v e
c a n be a p p r o x i m a t e d u n i f o r m l y b y f u n c t i o n s cp
h (z)
"
r
/tp .
J - , -
( 8 3 ) w o u l d be p r o v e n .
To a c t u a l l y
and
/tp > 277t - -
'-J-j-
h£C~(]R )
2
h>0
say of the
a n d /h = 1 ,
y .
kind
let
Thus
by
let
3D ,
r
-65-
( h e n c e tp^ G H ( f t ) , a s
tp
2
= cp - $
pel.
( f t ) ) , and
let
.
1
2
E x t e n d b o t h $^ a n d cp t o a l l ]R
b y p u t t i n g t h e m z e r o o u t s i d e ft .
2
$1 w i l l
be c o n t i n u o u s
(on ] R ) , hence a l s o $
2
2
will
Then
be s o .
Now d e f i n e
cp
= cp, + cp * h
1
^2
e
0
S i n c e cp
cp -* cp
£
cp * h
0
c.
since
£
is continuous
2
uniformly.
co 2
£ C (R ) .
p > 0
Thus
cp
in
£/
2
F u r t h e r cp
£
-»• cp u n i f o r m l y as e -» 0 ,
£ H (ft)
s i n c e cp. € H ( f t )
2
2
F i n a l l y we h a v e
and
Acp
£
= p + y * h
and
£
>0
in
they
approximate
-
D.
L
\i > 0 .
£ > 0, have the r e q u i r e d p r o p e r t i e s :
(83) is
and so
2
e
D^.
u n i f o r m l y as £ -»• 0
Thus
cp * h
and (82) a p p l i e s
to them.
p r o v e d a n d t h e p r o o f o f T h e o r e m 10 i s
complete.
-66-
SUMMARIZING R E S U L T S AND A P P L I C A T I O N S
a . Main
theorems
We now w a n t t o s u m m a r i z e t h e e s s e n t i a l
result of Section
a w a y w h i c h i s f r e e f r o m some o f t h e t e c h n i c a l i t i e s
occurrence of the parameters
T h e o r e m 12 i s a p r e l i m i n a r y
Lemma 1 1 :
e \
t
1
be t w o s o l u t i o n s
spectively),
for t e [0,
(
t
> t
Proof:
only
> r
of
(e.g.
O u r m a i n t h e o r e m i s T h e o r e m 13.
We b e g i n w i t h a l e m m a .
,
(C)
(with parameter values r,R,T
and r ' , R ' , T '
and suppose t h a t D = D Q . T h e n , modulo n u l l - s e t s ,
.
Moreover, u
t
= uj. i n
\
M
R
^ ^ \
a
x
{
re-
D = Dj. f o r
Q
t e [0, min(T.T')]
u
of Section I I I
and
r
[ O . T ] £ t -* Dj. e \ ,
u
theorem.
in
Let
3 t -* D
[0,T]
r,R,T).
III
t
r
^ )
min(TJ')].
are the functions
d e f i n e d by (11.13)
We may s u p p o s e t h a t t h e t r i p l e t s
i n one c o m p o n e n t ,
in
(r,R,T)
(C).)
and ( r ' , R ' , T ' )
differ
s i n c e t h e g e n e r a l c a s e t h e n i s o b t a i n e d as a
combination of these pure
cases.
*/
C a s e 1:
ft = B
R
Then i t
r*r'
and l e t
, R = R',T =T ' .
w£Hg(ft)
be t h e s o l u t i o n
i s n o t h a r d t o see t h a t
T o mean " n o t n e c e s s a r i l y
We may a s s u m e t h a t r < r ' .
r=r'
w>0
".
in
Put
of
ft
and w = 0 o u t s i d e
E) , .
-67-
DEFINE
UV
= U!
FOR T E
+ T • W
WE WANT
TO
SHOW THAT
THE
PARAMETER
THE
UNICITY
SET)
THE
[0,T]
PART
AND UJ
LEMMA
TRIPLE
OF
= U^
IN
T
-» DJ
(R,R,T).
IN
WITH
WHENEVER
COROLLARY
WHICH,
CASE
(TOGETHER
9.1
VIEW
THAT
THIS
DJ
OF THE
UJ)
SATISFIES
IS
= D
T
DONE
'
IT
OF
FOR
FOLLOWS FROM
(EXCEPT
PROPERTIES
(C)
FOR A NULL-
W,
PROVES
1.
i
USING
UV
THE
> U| > 0
(SINCE
W
PROPERTIES
OF
W WE GET
(FOR
TE[0,T])
= 0 OUTSIDE
(7)
AND
AS WE WANTED
CASE
2:
DEFINE,
(8)
TO
R = R',
FOR T E
DJ 3 ID
OBSERVE
IN
THAT
T -*
DJ,
UJ
SATISFIES
•
Xj, •
(C)
R
R
FOR
R,R,T
PROVE.
R * R',
[0,T]
T =T
,
U
*/
-JJY-j
, ) AND
SHOW THAT
U" =
T
0
£ HQ(R^)
,
A U j + 2lTt
(6),
UJ
B
N
^I)
D« =
N
D,
'0
L
1
.
WE MAY
SUPPOSE
THAT
R' < R .
,
-68-
Then U J £ H Q ( B )
(since U J . £ H Q ( B
t h a t t ~» D J ,
uj
satisfies
Corollary
shows t h a t
r
9.1
required conclusion
Case 3 : r = r',
i
(C) f o r
DJ = D
in this
R = R',
r
and i t
))
r,R,T.
(a.e.)
t
is
immediately
The u n i c i t y
and
u j = u^
verified
statement
which is
of
the
case.
T*T'.
The r e q u i r e d c o n c l u s i o n
follows
trivially
f r o m C o r o l l a r y 9.1
in
this
case.
T h e p r o o f o f Lemma 11 i s
complete.
p
T h e o r e m 1 2 : G i v e n an o p e n b o u n d e d s e t
a DQ £ %
(p.33))
[0,oo) 3 t -» D
t
€
there
with 0 € D
DQCIR
31
T > 0, f o r each r > 0 w i t h
R > 0
large
(R >
p
[0,T]
satisfies
(C)
sufficiently
of
T h e map ( 9 )
each D
t
(i)
(9) to
It
(ii)
( 9 ) has t h e f o l l o w i n g
satisfies
if
and f o r
DQcB )
p
the
each
re-
the parameter values
f o r each
that
properties:
(F).
of
(Strictly
(9) to [ 0 , T ]
speaking,
satisfies
T>0.)
D
c= D.
t
[0,OO) 3 t -» Dj. 6 31
another
r,R,T.
zero.
is formulated, the r e s t r i c t i o n
Modulo n u l l - s e t s
If
for
t h e moment i n e q u a l i t y ,
i
(iii)
r
i s u n i q u e w i t h t h e s e p r o p e r t i e s , w h e r e " u n i q u e " means
as ( F )
(F)
B <=<= DQ
suffices,
i s u n i q u e up t o a s e t o f m e a s u r e
Moreover,
given
i s a map
such t h a t f o r each
striction
(i.e.
Q
initial
modulo n u l l - s e t s
for
T < t
.
has t h e p r o p e r t i e s
d o m a i n DQ £ 51,
for all
and
t £ [0,«>)
D
.
Q
of
<= DQ
(9) with respect
then
D^ c Dj.
to
-69-
Proof:
Corollary
9.1
and c o r r e s p o n d i n g
gives a l o t of solutions of
the asserted p r o p e r t y .
The u n i c i t y of
into a global
9 . 1 , and t h e t h r e e a d d i t i o n a l
proves the
solution
properties
(i),
(ii)
9.3
D
Q
and
(9)
with
(9) f o l l o w s also immediately
f o l l o w f r o m T h e o r e m 10, C o r o l l a r y 9 . 2 a n d C o r o l l a r y
This
( f o r the given
t o t h e v a r i o u s a l l o w a b l e c h o i c e s o f r , R and T )
Lemma 11 s h o w s t h a t t h e s e m e l t t o g e t h e r
Corollary
(C)
and
from
(iii)
respectively.
theorem.
L e t us f i n a l l y
g i v e a f o r m u l a t i o n o f T h e o r e m 12 w h i c h i s
self-
i
contained
(i.e.
does n o t r e f e r
to (C) e t c . ) ,
in which the function
i s r e p l a c e d by what i t a p p r o x i m a t e s , t h e D i r a c measure
6 at the
o r i g i n , and w h i c h i s f o r m u l a t e d i n such a way t h a t t h e
domains
really
become
T h e o r e m 13:
a Dp £ E )
G i v e n an o p e n b o u n d e d s e t D
Q
<= R
w i t h 0 6 Dq ( i . e .
t
e
&
property:
F o r e a c h t e [0,«>) t h e r e i s a d i s t r i b u t i o n
(11)
X
in
" X
n
U
(12)
u
t
> 0
(13)
D
t
= D
where
]R
2
such
with
that
and
Q
U {z £ F
: u (z)
2
t
shows t h a t
> 0}
T h e map (10)
,
u^ h a s a r e p r e s e n t a t i o n
c o n t i n u o u s o u t s i d e 0 , a n d (13)
refers
in form of a f u n c t i o n ,
to any such
has t h e f o l l o w i n g a d d i t i o n a l
representative.
properties:
/cp - /cp > 2lTt'Cp(0)
D
compact
I
Q
(11)
(i)
u.
= A u . + 2Trt.fi ,
n
L>t
given
t h e r e e x i s t s a u n i q u e map
with the f o l l o w i n g
support
r
unique.
[ 0 , ~ ) 3 t -* D
(10)
r-jpjX
t
for every
D
0
"
2
cp £ H ( R
2
)
which is subharmonic
in
D.
.
-70-
(ii)
D <= D.
T
t
for
(iii)
If
[0,oo)
3 t -» Dj. £
of
(10)
for another D ^ e H
T
< t
DQ C= DQ
then
D
for all
Proof:
c Dj.
t
Essentially,
.
K
i s a n o t h e r map w i t h t h e
te
and
[0,»)
properties
if
.
t h e main p a r t o f t h e theorem f o l l o w s by
combining
C o r o l l a r y 9 . 1 w i t h Lemma 11 j u s t a s i n t h e p r o o f o f T h e o r e m 1 2 .
need o n l y modify the f u n c t i o n s
in the s i t u a t i o n of C o r o l l a r y
w (z)= -t-(loglz|
t
t
T h i s means
Aw
1
- -pBs—I /
r
D
1
t
> 0
everywhere
w
t
= 0
outside
fit
9 . 1 , add t o
l o g Iz -
is e a s i l y
together
D
r
(in
p
)
]R )
by
and
.
seen t h a t the s o l u t i o n s
into a solution
(10)
(cf.
of
satisfying
Theorem 1 2 ) .
( C ) p r o v i d e d by C o r o l l a r y 9 . 1
( 1 1 ) - ( 1 3 )
properties of
f o r t h e new u^ .
( 1 0 ) also
follow
1 3 . 1 : L e t M Q J M ^ . M ^ , . . . a n d c be g i v e n c o m p l e x n u m b e r s
MQ a n d c r e a l
n
defined
o
easily
= M
with
and p o s i t i v e and suppose t h a t t h e r e i s a bounded domain
containing
/ z d x dy
D
w^
,
and t h e t h r e e a d d i t i o n a l
D e l
u^. t h e f u n c t i o n
Thus,
c
The u n i c i t y
Corollary
9.1 a l i t t l e .
da ) .
p
E
r
w
Then i t
in C o r o l l a r y
that
= - 2Trt.(6 -
t
1
u^
We
N
n
the o r i g i n such
for all
n > 0.
that
-71-
T h e n t h e r e e x i s t s a d o m a i n D' ( b o u n d e d a n d c o n t a i n i n g
/ dx dy
D'
/ z
D'
= M +c
and
n
Proof:
for all
n > 1 .
Apply the theorem with D
Q
= D.
T h e n D' = D
h a s t h e r e q u i r e d p r o p e r t y as i s s e e n b y c h o o s i n g
± Im z
in D
n
(and d e f i n i n g
1
cp e H (1R ) )
(i)
g i v e a few f u r t h e r
for
t =
cp = ± Re z
i n such a way
and
n
that
.
o f t h e map ( 1 0 )
now be t h a t
in
Theorem
i n T h e o r e m 13:
i s a g i v e n domain and
3 t -* D
[0,-)
1
properties
Thus the background s i t u a t i o n w i l l
Dg e &
D
t
properties
We s h a l l
(14)
cp o u t s i d e
i n t h e moment i n e q u a l i t y
b. A d d i t i o n a l
13.
that
U
dx dy = M
n
D) s u c h
e ft
t
i s t h e map u n i q u e l y d e t e r m i n e d b y t h e f o l l o w i n g
property:
there
exist
*/
distributions
(15)
„
X
T
" X
(16)
u
t
> 0
(17)
D
t
= D
D
Q
u^ ( t £ [ 0 , « > ) )
= Au
q
{z
U
e
t
+
I
(18)
u (z)
(19)
u (z)
(20)
u (0)
t
: u (z)
t
is
t
t
=
u^
(also called u.)
= 0
for
in the
IE ' s u c h
that
Zirt.fi
The d i s t r i b u t i o n
function
w i t h compact s u p p o r t
t = 0
> 0}
will
.
a l w a y s be r e p r e s e n t e d b y t h e
completely determined
pointwise
by
and
continuous outside
z = 0
+ oo
*/ We a r e g o i n g t o u s e some c o m p l e x v a r i a b l e t h e o r y
s o we now r e p l a c e |R b y (C .
2
in t h i s
section,
-72-
for
t > 0 .
possible
u^(z)
(18)
since
AUj.£
^-t'loglzl
L°° o u t s i d e
outside
0
0
for
In p a r t i c u l a r
and i s a s u p e r h a r m o n i c f u n c t i o n
(i)
It
satisfies
t h e moment i n e q u a l i t y
(ii)
D
t
i s an i n c r e a s i n g
(iii)
D
t
is
function of
Suppose
in
o f t h e map ( 1 4 ) :
,
t ,
t .
follows:
( t o have i n f i n i t e c o n n e c t i v i t y e . g . ) .
(a)
DQC=CZ
(b)
Dj. h a s f i n i t e c o n n e c t i v i t y
(c)
Dj. i s s i m p l y c o n n e c t e d f o r a l l
(d)
Asymptotically
t
a circular
(e)
DQ .
t .
i s c o n n e c t e d b u t o t h e r w i s e a l l o w e d t o be v e r y
D
for
all
as
t
>
0
t -» «>
t h e n one e x p e c t s
of analytic
way:
irregular
that
.
for all
t > 0 .
sufficiently
t h e shape o f
large
curves for all
t .
Dj. a p p r o a c h e s t h a t
disc .
3D. c o n s i s t s
to
function
c o u l d be made p r e c i s e f o r e x a m p l e i n t h e f o l l o w i n g
DQ
since
( 1 4 ) w h i c h w o u l d be n i c e t o p r o v e c a n be
Dj. b e c o m e s n i c e r w i t h i n c r e a s i n g
(iv)
(17) r e f e r s
i n c r e a s i n g a s a f u n c t i o n o f DQ f o r f i x e d
v a g u e l y s t a t e d as
is natural
Uj. i s a s u b h a r m o n i c
pointwise definition
One e x p e c t e d p r o p e r t y o f
is
Uj. .
Now T h e o r e m 13 g i v e s us t h r e e p r o p e r t i e s
(iv)
t = 0 , (19)
by ( 1 5 ) , and (20)
near the o r i g i n .
this representative of
With t h i s
Uj. = 0
is possible since
t > 0 .
of
-73-
We h a v e n o t b e e n a b l e t o p r o v e a n y c o m p l e t e f o r m o f
we s h a l l
holds
prove
( T h e o r e m 15 b e l o w )
things
t
' .
shows t o what e x t e n t
Lemma 14:
U
is o n l y t h a t assuming
(iv).
( a ) and I b l a r e t r u e ,
What
then
We b e g i n w i t h a lemma w h i c h among
the term DQ in (17) r e a l l y
is
e)
other
necessary.
Let
= { z € (E : u ( z )
> 0}
t
Then, for t > 0 ,
(i)
is
(ii)
If
connected.
N
( i i i ) D^
i s a component o f DQ then e i t h e r
is the union of
meet
(iv)
If
Proof:
DQ
(i)
and t h o s e components o f
is connected then
It
V
3V.
be a c o m p o n e n t o f
Since
u
t
> 0
in
in
V.
V
as we w a n t e d t o
I n D Q , and i n p a r t i c u l a r
N
N
i t m u s t be c o n s t a n t l y
in
N
or
U
Since
connected).
•k
u^ = 0
.
t
Then
D^ i s
connected.
in
contains
9V <= 3u" <= (E ^ U^., s o
u^
0.
u^
t
shows t h a t
0
c a n n o t be a s u b 0.
show.
in
N, u
equal
to
N , proving
is a superharmonic
t
u
t
0
> 0, i f
in
N.
u
t
function.
attains
Thus e i t h e r
the value 0
u^ > 0
(ii).
i s an i m m e d i a t e c o n s e q u e n c e o f
= DQ
(iv)
and
i s c o n n e c t e d and
in
T
D Q w h i c h do n o t
is a subharmonic f u n c t i o n outside
(ii)
since
U
But
0 £ V
Therefore,
D^ =
this
Thus
D
= 0 .
i s enough t o p r o v e t h a t each component o f
harmonic f u n c t i o n
(iii)
or N n
t
U^..
Thus l e t
on
N <= U
(ii)
and t h e d e f i n t i o n o f
D ,
T
U .
t
0 e D
Q
n U
t
This gives
(i)
and ( i i )
immediately
show t h a t
D
Q
<= U
t
(DQ being
(iv).
I
1
Sakai h a r p r o v e d t h a t a ) I s t r u e , u n d e r t h e h y p o t h e s i s t h a t DQ i s a
"domain w i t h q u a s i - s m o o t h b o u n d a r y " , meaning e s s e n t i a l l y t h a t
3Dg
i s o f c l a s s C1 b u t w i t h c e r t a i n t y p e s o f c o r n e r s a l l o w e d ( f i n i t e l y
m a n y ) . See [ 1 9 ] , T h e o r e m 3 . 7 .
S a k a i a l s o h a s r e s u l t s on b ) t o e )
f o r s o l u t i o n s ( i n h i s s e n s e ) o f t h e H e l e Shaw f l o w p r o b l e m .
C f . p . 15
in the introduction (of the present paper).
-74-
T h e o r e m 1 5 : S u p p o s e t h a t , f o r some f i x e d
t > 0, D. i s f i n i t e l y
connected
*/
and
.
DQCCZDJ.
and i s o l a t e d
Proof:
that
Then
L e t U" = { z € C
t
3DQ <= U
= U
.
T
This
: u (z)
> 0} .
t
Then
DQCC
D
T
= D
u
Q
shows
U"
T
i m p l i e s t h a t e a c h c o m p o n e n t o f DQ i n t e r s e c t s U^.,
o f Lemma 1 4 ,
DQ C=
. Thus
D^ i s c o n n e c t e d .
Now l e t y
be a component o f
9D
is an a n a l y t i c
curve o r a point.
sphere. Since
u" i s c o n n e c t e d t h e r e
t
which contains
V
Since
curves
.
T
In p a r t i c u l a r
Let
d i s j o i n t union o f a n a l y t i c
points.
and s o , b y ( i i )
D,
3Dj. i s a f i n i t e
U
T
= 3U
a n d we s h a l l
t
L e t I = (C u {«>}
show t h a t y
denote the
Riemann
i s e x a c t l y one component o f
(namely t h a t component which c o n t a i n s
denote t h a t component.
y
t
0).
Then i t i s easy t o see t h a t
i s c o n n e c t e d t h i s a l s o shows t h a t
V
i s simply
(t^y
3V = y .
connected.
Put
(21)
W = D
and
(22)
t
v
DQ
= U
= z - 4
for
z £ W Uy.
t
-
D
0
CZ
V
define
S(z)
D
T
^
Due t o t h e a s s u m p t i o n t h a t D
i s a neighbourhood o f y
cluster at y ) .
tion)
Since
in
t
is finitely
V ( t h e o t h e r components o f
DQ i s a c o m p a c t s u b s e t o f
i t follows that also W i s a f u l l
connected
9D
t
cannot
D ( a l s o b y assumpt
neighbourhood o f y
in V.
By " a n a l y t i c c u r v e " we mean t h e f o l l o w i n g : a s u b s e t o f (C i s a n a n a l y t i c
c u r v e i f i t i s t h e i m a g e o f ( s a y ) 3 D u n d e r some n o n - c o n s t a n t f u n c t i o n
holomorphic i n a neighbourhood o f 3 D . Thus an a n a l y t i c c u r v e i s allowed
t o i n t e r s e c t i t s e l f and t o have c u s p s and o t h e r s i n g u l a r i t i e s .
We s a y
t h a t t h e a n a l y t i c c u r v e i s n o n - s i n g u l a r i f t h e f u n c t i o n above can be
chosen t o be u n i v a l e n t i n a n e i g h b o u r h o o d o f 3 D .
-75-
As
is continuously d i f f e r e n t i a t e
f u n c t i o n on W U y.
a continuous
nimum ( u . = 0 ) .
3u
On
outside the o r i g i n
y c t v
u^ a t t a i n s
its
is
mi-
Therefore
t
=0
S(z)
In
on y
= z" o n
W
S(z)
>
so
y
•
is holomorphic
— = 1 - Au.
3z
= 1 - x
z
u
+ X
n
t
u
s i n c e , by
+
n
2 7 T t < s
0
=
(15),
in
0
W .
Now i t i s known t h a t t h e e x i s t e n c e o f a f u n c t i o n w i t h t h e
o f S ( z ) above g i v e s the d e s i r e d conclusion f o r
j u s t c o n s i s t s o f o n e p o i n t we a r e d o n e .
c o n n e c t e d and
f
S(z)
: D ->- V
3V = y)
- TJxJ
-» 0
as
I t c a n be s e e n t h a t
3D
by
(25)
y
simply
Let
f
(W) o f
3lD
f(c)
extends
analytically
defining
z, i n a n e i g h b o u r h o o d o f
Moreover i t
in the neighbourhood
implies that
for
i s seen
3D
in
I \ D .
that
•
T h i s shows t h a t y
theorem is
D .
is
€ D )
= S(f(1/?))
=y
V
c- - > 3 E
f(?)
f(3D)
(since
if
that
U
across
Otherwise
V c a n be mapped c o n f o r m a l l y o n t o
is holomorphic
and ( 2 3 ) shows
S(f(?))
. T o be p r e c i s e ,
be t h e i n v e r s e map .
Then S ( f ( ^ ) )
D
y
properties
proven.
i s an a n a l y t i c
c u r v e ( a s d e f i n e d on p . 7 4 )
and t h e
in
-76-
Remarks:
(T)
I t m i g h t seem t h a t T h e o r e m 15 i s n o t o f much v a l u e s i n c e i t
supposes knowledge a p r i o r i
conclusions
about i t .
h y p o t h e s e s o n l y on
a b o u t t h e unknown domain
DQ
a n d w i t h t h e c o n c l u s i o n t h a t 3Dj. i s
s h o u l d be e n o u g h t o a s s u m e a b o u t
that D
= U
on D
cal
t
f o r t > 0;
t
Lemma
very
is automatically
Firstly,
fulfilled
DQ <= DQ, w h e r e DQ
1
for all
(If
t h e n Dj. = B ^
p
that
whenever
t , and c l e a r l y
p
hypothesis
with p
2
for
= p
o f DQ, h o l o m o r p h i c
Then,
and (22) o f
on
3D
in
2
DQCC
t
t
DQCC
i s l a r g e enough
is
to
t
l a r g e enough
and
the f o l l o w i n g :
DQ ^ K
S(z)
(29)
W = Dj. ^ ( K U { 0 } ) c V
(30)
S(z)
3u.
= z - 4 g-i +
the
+ 2t.)
DJ.)
that
DQ
i s c o n n e c t e d and
curves.
Q
implies
there is a function S (z)
Q
where
K
i s a compact
de-
subset
and w i t h
.
n
to
and
_
X
D
Q
( Z )
DJ.
DQ<=<= D J . •
i n t h e p r o o f o f T h e o r e m 15 we c h a n g e t h e d e f i n i t i o n s
W
DJ.
DQCZC
Dj. = Uj. a n d t h e s e c o n d o n e
f i n e d a n d c o n t i n u o u s on (DQ ^ K) U 3 D
= 7
that 3D
union of non-singular a n a l y t i c
impUes that
( a n d c a n be r e p l a c e d b y )
S (z)
pathologi-
i s some d i s c c e n t e r e d a t t h e o r i g i n , we
3DQ i s a f i n i t e d i s j o i n t
The f i r s t
(so
hypotheses
(except perhaps in very
in which the hypothesis
S e c o n d l y , suppose ( i n place of
n
However, since the
s i n c e b y a p p l y i n g T h e o r e m 13 ( i i i )
h a v e Dj. <= Dj.
D^ = D
analytic.
probable.
c a n be d i s p e n s e d w i t h .
situation
drawing
is connected
s i t u a t i o n s ) , T h e o r e m 15 a t l e a s t makes t h e a s s e r t i o n
L e t us m e n t i o n t w o s i t u a t i o n s
(28)
DQ t h a t i t
14(iv)).
i n T h e o r e m 15 seem v e r y p l a u s i b l e
analytic
before
I t w o u l d be d e s i r a b l e t o h a v e a t h e o r e m w i t h
Then i t
t
Dj.
pre-
•( S ( z ) - z
0
)
( z e W U y ) .
(21)
-77-
In (30)
it
s a y by
S (z)
on
i s assumed t h a t S ( z )
is extended to
Q
Q
W U Y
= z"
for
z e(W U Y ) ^Dg
» holomorphic
in
W (since
• Then
(15)
W U Y i n some w a y ,
S(z)
shows t h a t
—
W ^ 9DQ
a n d 9DQ i s a n i c e c u r v e )
and
is
S(z)
= z on
—
continuous
= 0
in
9Z
y .
T h e r e s t o f t h e p r o o f o f T h e o r e m 15 w o r k s as b e f o r e , a n d s o
c o n c l u s i o n o f the theorem holds w i t h the changed
(T)
If
Y
"is a n y n o n - s i n g u l a r a n a l y t i c
arc
hypotheses.
there
t h e Schwarz f u n c t i o n f o r Y , d e f i n e d and h o l o m o r p h i c
of
=1
on
that the anticonformal
map
Y
is
such t h a t S ( z )
It
singular.
* =
is the r e f l e c t i o n
Y .
in
z near to 9
n
t
u
t
9D^ w h e r e i t
near the
is
non-
boundary.
, the r e f l e c t i o n of
z
in
is
9u.
Z
z
in a neighbourhood
S(z)
9D^ a t t h e p i e c e s o f
z € D^,
S(z),
i n t h e p r o o f o f T h e o r e m 15 i s
T h i s g i v e s an i n t e r p r e t a t i o n o f
Namely, by ( 2 2 ) , i f
is a function
The i n t e r p r e t a t i o n o f
z -> z * = S ( z )
is c l e a r that the f u n c t i o n S ( z )
the Schwarz f u n c t i o n f o r
3D.
Y . See [ 4 ] ,
the
z
- 4-—Mz) .
9z
Thus,
- g r a d u. ( z )
t
t
= - 2 - ^ - (z)
3 u
9 z
reflected point
(D
1
1
= - ^ ( z * - z ) =•*• .
_
-
z
to
its
z*).
T h e o r e m 15 i s s i m i l a r
dence s e t f o r
(the vector from
variational
t o t h e o r e m s on t h e r e g u l a r i t y
inequalities
of the kind ( E 2 ) .
of the
coinci-
There are
t h e o r e m s h a v i n g as c o n c l u s i o n t h a t t h e b o u n d a r y o f t h e c o i n c i d e n c e
i s an a n a l y t i c
curve, e.g.
Theorem 4.3 in [13]
These theorems cannot, however,
be i m m e d i a t e l y a p p l i e d t o o u r
since they always r e q u i r e the obstacle f u n c t i o n
analytic,
( s e e a l s o §4 i n
( o u r ty^) t o be
satisfied).
set
[11]).
problem
real
a h y p o t h e s i s w h i c h i s n o t s a t i s f i e d f o r us ( a l o n g w i t h
h y p o t h e s e s on t h e o b s t a c l e f u n c t i o n w h i c h a r e n o t
such
other
-78-
T h e l a s t l i n e s a b o v e s h e d some l i g h t on t h e h y p o t h e s i s D g c c D^
i n T h e o r e m 15.
it
Namely, although
is so o u t s i d e
ip
is not real
t
and t h e h y p o t h e s i s
DQ
to guarantee a p r i o r i
that 3D
t
DQ<=<= D^
analytic
Q,
in a l l
c a n be v i e w e d a s a w a y
avoids the set where
ip
is not
t
real
analytic.
c. Applications
to quadrature
domains
The n e x t theorem c o n c e r n s a c l a s s o f domains c a l l e d q u a d r a t u r e
mains.
We s a y t h a t D<=([
z . , . . . ,z
i
n
in
r . > 1, s u c h
D
fda
D
i s a q u a d r a t u r e domain i f
and complex numbers
a . . ,
J , K
J
f
v
,
u
;
J
f
which is a n a l y t i c
Q u a d r a t u r e domains and q u a d r a t u r e
c i t e some r e s u l t s .
= z
([1],
(iii)
identities
Let
D
If
in
for
D and c o n t i n u o u s l y e x t e n d i b l e
(D.
shall
Then
is a function
to
of
D
such
S(z)
that
z £ 3D
i s simply connected then
if
some ( a n d t h e n e v e r y )
onto
D
is a rational
curve
identities
Lemma 2 . 3 ) .
D
D
D .
f r o m w h i c h we
be a b o u n d e d d o m a i n i n
only
If
in
(i.e.
D i s a q u a d r a t u r e domain i f and o n l y i f t h e r e
meromorphic
(ii)
and i n t e g r a b l e
(31) a b o v e ) a r e t r e a t e d i n [ 1 ] and [ 9 ] ,
S(z)
0 < k < r . - 1,
- - j
( z . )
K
for every function
(i)
where 1 < j < n ,
- -
points
that
r.-1
= I
I
a.
j = l k=0
the kind
there exist
do-
D
i s a q u a d r a t u r e domain i f
c o n f o r m a l map o f t h e u n i t d i s c
function
( [ 1 ] , Theorem
1).
i s a q u a d r a t u r e d o m a i n t h e n 3D i s a c o m p l e t e
(i.e.
there exists a real
polynomial
P(x,y)
algebraic
such
that
and
ID
-79-
3D = { z = x + i y
: P(x,y)
except for a f i n i t e
set
= 0}
(the set
i n t h e r i g h t member o f
contain
i s o l a t e d p o i n t s not in 3D))
Theorem
3.4).
It follows
subclass
This
[9],
of all
bounded s i m p l y
connected
i s a l s o t r u e f o r domains o f h i g h e r c o n n e c t i v i t y
( [ 9 ] , Theorem
(to
some
3.3).
Now r e t u r n t o t h e s i t u a t i o n on p . 71
is a quadrature domain, say s a t i s f i e s
inequality
( [ 1 ] , Theorem 3 and
t h a t t h e q u a d r a t u r e d o m a i n s make up a d e n s e
( i n any reasonable t o p o l o g y )
domains.
extent)
from ( i i )
( 3 3 ) may
( T h e o r e m 13)
shows
and suppose t h a t DQ
(31).
Then, for t > 0 ,
there
t h e moment
that
/ f d a = J f d a + 2-rrt • f ( 0 )
D
t
D
=
0
"J"
(k)
I
X a. . f
( z . ) +2irt • f ( 0 )
j = 1 k=0
'
n
1
[K}
J
2
for all
K
2
f £ H (K
J
*/ '
)
' which are a n a l y t i c
i n D . ( A p p l y t h e moment
T
in-
e q u a l i t y t o ± Re f
and ± I m f . )
T h i s n e a r l y s h o w s t h a t D^ i s a q u a d r a t u r e domain.
We m u s t o n l y e x t e n d t h e v a l i d i t y o f ( 3 4 ) t o a l l f G L ( D ^ )
which are a n a l y t i c
i n D^.
I n s t e a d o f d o i n g s o , h o w e v e r , we s h a l l
o u r p r o o f t h a t D ^ i s a q u a d r a t u r e d o m a i n on t h e c h a r a c t e r i z a t i o n
base
(i)
(p.78).
T h e o r e m 16:
D
T
Suppose
DQ
is a quadrature domain.
*'We
I
is a quadrature domain.
In p a r t i c u l a r
3D
t
t e m p o r a r i l y admit complex-valued f u n c t i o n s
T h e n , f o r each
i s an a l g e b r a i c
2 2
in H ( F )
t>0,
curve.
-80-
L e t S Q ( Z ) be t h e f u n c t i o n g i v e n by ( i )
Proof:
on p . 7 8
f o r D = DQ .
T h e t h e o r e m i s p r o v e n as s o o n as we h a v e e s t a b l i s h e d t h e
of a s i m i l a r f u n c t i o n Sj.(z) f o r
Define, for z e D
3u
S
t
(
z
)
=
"
Z
1 ^
4
_
n
(
> - (
Z
=z
for all
Then Sj.(z)
On
3D
S (z)
t
0
(
z
)
-
z
'
)
z £ (C \ D
on S D
T
\
Since Sj.(z)
c u r v e ) and
meromorphic
(15)).
Thus, Sj.(z)
T
in a l l
=> D
U D Q U 3DQ
( D ^ D Q )
Dj. .
Thus
t
S j . ( z ) has a l l
I
j=1
I
k=0
then that of
follows.
f
[K)
a. .
'
J
K
(z.)
J
is
r.-1
J f da
D
=
where
b^ = 2-RRT
t
I
j=1
algebraic
actually
is
the required properties
identities
I
a,
k=0
'
f
[K>
v
J
and
K
(z.) +b f(zl)
1
J
z| = 0 .
1
1
,
for
SQ(Z),
in ( D ^ D Q )U D Q .
it follows that S (z)
t
DQ is
r.-1
=
is meromorphic
and
proven.
T h e o r e m 16 a r e r e l a t e d as
If that of
so
i n a n e i g h b o u r h o o d o f 3 D Q ( w h i c h i s an
is clear that the quadrature
f da
SQ(Z)
.
t
is continuous
the theorem is
DQ
by
and i n D Q , e x c e p t a t t h e o r i g i n and a t t h e p o l e s o f
DQ
3S.
— — = 0 (by using
3z
It
(C
i s c o n t i n u o u s , e x c e p t a t t h e o r i g i n and a t t h e p o l e s o f
= z
D
to all
.
Q
the l a s t t h r e e terms in (35) v a n i s h ,
t
In
J
S
S Q ( Z ) (continuously)
w h e r e we h a v e e x t e n d e d
Sg(z)
.
t
,
t
t
X D
+
D=D
existence
DQ
and
in
-81-
L e t u s now c o m b i n e T h e o r e m 16 w i t h t h e e x i s t e n c e p a r t o f T h e o r e m 13.
This yields
that
DQ
if
i s a bounded q u a d r a t u r e domain c o n t a i n i n g
o r i g i n with the quadrature
identity
exists
a q u a d r a t u r e domain
where
z.
= 0.
x
D^
0 € DQ and
theses
ZQ £ D Q .
zJj £ D Q
if
ZQ = 0
this
it
c a n be r e p l a c e d b y t h e s i n g l e
h y p o t h e s i s c a n be r e l a x e d t o z j
we may a p p l y t h e p r e v i o u s
i s t h e n a q u a d r a t u r e domain w i t h z| e
=
j ~
1 a. , f
r
j=1
£
D nB(zj;e)
0
k=0
'
J
(
k
)
(z.)+Tre
K
2
.f(zi)
J
z'.,...,z^
for all
s t e p some z '
with
z j £ 3DQ
those
lies
zj
rature
/ f da
D
£ 3DQ
since
DQ r e p l a c e d
= 0
and T r e < b
DQ U
D(z|;e)
2
by
1
;
and
.
many t i m e s f o r
finitely
i f we j u s t t a k e c a r e t o make t h e
f o r which
z j £ DQ
(to avoid that at
on t h e b o u n d a r y o f t h e c u r r e n t d o m a i n a t t h a t
T h u s we h a v e p r o v e d t h e f o l l o w i n g
Corollary
hypothesis
DQ i n d i c a t e d i n t h e l a s t l i n e s above a t once and s i m u l -
modification of
taneously
hypo-
1
T h e a b o v e r e a s o n i n g may be r e p e a t e d f i n i t e l y
many p o i n t s
point
1
1
DQUB(ZJ; )
(38),
identity
n
f da
that
with
DQU_D(Z^;E)
/
so s m a l l
reasoning
for e > 0
quadrature
identity
there
i s c l e a r t h a t the two
DQ U D ( z J j ; e )
the
b^ > 0
i s n o t i n any way a d i s t i n g u i s h e d
identities
1
Actually
1
with the quadrature
Since the o r i g i n
in the context of quadrature
( 3 7 ) , then g i v e n any
the
16.1:
Let
D
corollary
some
step).
o f Theorem 16.*/
be b o u n d e d d o m a i n i n
(L
admitting the quad-
identity
n
= 1
j=1
r
j "
I
a. . f
k=0
'
1
Then given f i n i t e l y
positive
n u m b e r s b^
J
(
k
)
K
(z.)
(f €
J
many p o i n t s z J j , . . .
b
m
there exists
L^D))** .
7
d
e C ,
z j £ 3D a n d
a bounded domain
D
1
M o r e g e n e r a l r e s u l t s , a l o n g t h e same l i n e s a s t h i s c o r o l l a r y ,
obtained in [19] .
L (D) denotes t h e space o f i n t e g r a b l e a n a l y t i c
functions
in
arbitrary
with
are
D.
-82-
for all
f € L'(D')
.
a
In p a r t i c u l a r
there exists
( c h o o s i n g D = 0 a b o v e ) , g i v e n z j , . . . , z ^ e (C, b ^ , . . . , b >
m
a bounded domain
m
/ f da = I
b.f(z'.)
D'
j=1
J
for
all
fe
L](D')
a
J
.
D
1
with
0
-83-
V.
[1]
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