We define and analyse graph spaces of first-order linear differential operators. In particular we consider the density of the set of smooth functions and the construction of a trace operator

Jensen, Max

French

Oxford University Research Archive (ORA)

Density and trace for graph spaces of first-order linear operators

Max Jensen

CiteSeerX

Density and Trace for Graph Spaces of First-Order Linear Operators

Mathematical Institute Eprints Archive

Report No. 02/05
Density and Trae for Graph Spaes of
First-Order Linear Operators
Max Jensen
4th Marh 2002
Abstrat
We dene and analyse graph spaes of rst-order linear dierential opera-
tors. In partiular we onsider the density of the set of smooth funtions and
the onstrution of a trae operator.
Oxford University Computing Laboratory
Wolfson Building
Parks Road
Oxford, England OX1 3QD
Introdution
Let us begin with an example. Suppose that 
 is the open square (0; 1)
2
 R
2
on whih
we dene the dierential operator
L : L
1
(
)! D
0
(
); v 7! 
1
v + 
2
v;
D
0
(
) denoting the spae of distributions. Then the harateristis of L pass at an angle of
45
Æ
or =4 through the domain, as indiated on the left gure below. We wish to onsider
the boundary value problem for data presribed on the set

 

 := 
 \ f(x
1
; x
2
) : x
1
= 0g \ f(x
1
; x
2
) : x
2
= 0g:
The method of harateristis allows one to solve this problem for smooth and also non-
smooth data on 
 

. Clearly, disontinuities in the boundary data lead to disontinuous
solutions whih do not lie in W
1;q
(
). Therefore we fous our attention on the graph spae
W
q
L
(
) of L, whih is dened as the vetor spae of all funtions v 2 L
q
(
) for whih the
value L(v) lies in L
q
(
) as well. In the ontext of this example it means that we require
the existene of a weak derivative in the harateristi diretion only, while admitting
disontinuities in other diretions.
1
10
1


10


0
x
2
x
2
x
1
x
1
a
0
a
b
0
b
In the main part of this report we address the issue of density of C
1
-funtions for general
rst-order linear dierential operators on Lipshitz domains. We are also interested in
dening a trae for members ofW
q
L
(
). However we notie that ertain restritions emerge
through the weaker requirements on dierentiability, ompared with W
1;q
(
); onsidering,
for instane, the harateristi boundary at the upper left orner of the domain depited
in the right plot; on the restrition to the slie a
0
b
0
in normal diretion, a solution of the
boundary value problem is idential to the boundary data on ab  
 

, negleting the
ompression fator os(=4). Therefore when the boundary data lie merely in L
q
(
) it
appears unreasonable to expet existene of a trae of the solution in the sense of W
1;q
-
funtions.
Our aim in this report is to formalise these observations and to introdue the graph spae
and its assoiated trae operator in a rigorous manner. We shall onlude the disussion
with a few omments about related publiations.
1
Denition and Density
Consider a bounded Lipshitz domain 
  R
n
with boundary 
 and unit outward normal
 = (
1
; : : : ; 
n
). Choose a onjugate pair q, q
0
, i.e., q; q
0
2 R suh that
1 < q <1; q
0
=
q
q   1
:
Given a tensor B 2 [W
1;1
(
)℄
mmn
and a matrix C 2 [W
1;1
(
)℄
mm
, we are interested
in the graph spae of the dierential operator
L : [L
q
(
)℄
m
! [D
0
(
)℄
m
;v 7! 
k
(B
ijk
v
j
) + C
ij
v
j
:
Here and throughout the text we employ the Einstein summation onvention. We also fol-
low the notional onvention to typeset entries of tensors, inluding vetors and matries, in
itali letters; tensors will be typeset in bold letters. Further, we set, for every manifoldM ,
L
q
(M) := [L
q
(M)℄
m
; W
r;q
(M) = [W
r;q
(M)℄
m
:
Then the graph spae of L is the set
W
q
L
(
) := fv 2 L
q
(
) : L(v) 2 L
q
(
)g;
whih is normed by the mapping
kvk
L;q
= kvk
L
q
(
)
+ kL(v)k
L
q
(
)
:
Our rst investigations onern the density of smooth funtions in W
q
L
(
). Consider the
approximate identity Æ 7!  
Æ
, Æ > 0, with non-negative funtions  
Æ
(x) = Æ
 n
 
1
(Æ
 1
x),
satisfying the ondition that the support of  
1
lies within the unit ball entred at the
origin. We write, for v 2W
q
L
(
),
v
Æ
= (v
Æ
1
; : : : ; v
Æ
n
) = (v
1
  
Æ
; : : : ; v
n
  
Æ
):
Theorem 1: Let 

0
 
 and suppose that, for v 2 W
q
L
(
), 1 < q < 1, the support
supp(v) is a subset of 

0
. Then, for every " > 0 there exists a funtion v
"
2 [C
1
0
(

0
)℄
m
with
kv   v
"
k
L;q
< ":
Proof. Step I: Transformation of (Bv)   
Let u 2 L
q
0
(

0
) and assume Æ < dist(supp(v); 

0
) and Æ
0
< dist(

0
; 
). We denote
dierentiation with respet to

x by


k
, this implies


k
 
Æ
(x 

x) =  
k
 
Æ
(x 

x):
2
Hene, if we extend u to 
 by setting u(x) = 0 outside 

0
,
Z


u
i
(
k
(B
ijk
v
j
)   
Æ
) dx=
Z


u
i
(x)
Z




k
(B
ijk
(

x) v
j
(

x)) 
Æ
(x 

x) d

xdx
=
Z


lim
Æ
0
!0
u
Æ
0
i
(x)
Z




k
(B
ijk
(

x) v
j
(

x)) 
Æ
(x 

x) d

xdx
= lim
Æ
0
!0
Z




k
(B
ijk
(

x) v
j
(

x))
Z


u
Æ
0
i
(x) 
Æ
(x 

x) dx d

x
= lim
Æ
0
!0
Z


 B
ijk
(

x) v
j
(

x)


k
Z


u
Æ
0
i
(x) 
Æ
(x 

x) dx d

x
= lim
Æ
0
!0
Z


u
Æ
0
i
(x)
Z


 B
ijk
(

x) v
j
(

x)


k
 
Æ
(x 

x) d

x dx
=
Z


u
i
(x)
Z



k
(B
ijk
(

x) 
Æ
(x 

x)) v
j
(

x) d

xdx:
In the ourse of integration by parts we used that


x 7!
Z


u
Æ
0
i
(x) 
Æ
(x 

x) dx

2 D(
):
As L
q
0
(

0
) is the dual spae of L
q
(

0
) and supp( 
Æ
 
k
(B
ijk
v
j
))  

0
, we obtain the
identity of L
q
(

0
) funtions

x 7! (
k
(B
ijk
v
j
)   
Æ
)(x)

=

x 7!
Z



k
(B
ijk
(

x) 
Æ
(x 

x)) v
j
(

x) d

x

:
Step II: Transformation of (Bv
Æ
)
As above, we have
Z


u
i

k
(B
ijk
( 
Æ
 v
j
)) dx=
Z


u
i
(x) 
k
 
B
ijk
(x)
Z


v
j
(

x) 
Æ
(x 

x) d

x

dx
=
Z


u
i
(x)
Z



k
(B
ijk
(x) 
Æ
(x 

x)) v
j
(

x) d

x dx:
Therefore, in L
q
(

0
),

x 7! 
k
(B
ijk
( 
Æ
 v
j
))(x)

=

x 7!
Z



k
(B
ijk
(x) 
Æ
(x 

x)) v
j
(

x) d

x

:
Step III: Boundedness of B( 
Æ
)v
Suppose that Æ < dist(supp(v); 

0
). We onsider the operator
T : L
1
(
)! L
1
(
); (Tv)
i
(x) =
Z


(B
ijk
(

x)  B
ijk
(x)) 
k
 
Æ
(x 

x) v
j
(

x) d

x:
Sine B
ijk
lies in W
1;1
(
), it is a Lipshitz ontinuous funtion for whih
kBk := kBk
[W
1;1
(
)℄
mmn
3
is a Lipshitz onstant. Therefore we have ontinuity of T in the L
1
-vetor norm:
kTvk
L
1
(
)
=
m
X
i=1
Z





Z


(B
ijk
(

x)  B
ijk
(x)) 
k
 
Æ
(x 

x) v
j
(

x) d

x



dx

m
X
i=1
Z


Z





B
ijk
(

x) B
ijk
(x)
Æ



jÆ 
k
 
Æ
(x 

x)j jv
j
(

x)j d

x dx

m
X
j=1
kBk 
Z


Z


jÆ 
k
 
Æ
(x 

x)j dx jv
j
(

x)j d

x
= kBk  k
k
 
1
k
L
1
(
)
 kvk
L
1
(
)
;
where we used that Æ 
k
 
Æ
(x) = Æ
 n
(
k
 
1
)(Æ
 1
x) and the transformation of variables
x 7! Æx. In the ase of v 2 L
1
(
), the same bound holds for the L
1
-norm, sine
kTvk
L
1
(
)
=max ess-sup
i

x



Z


(B
ijk
(

x) B
ijk
(x)) 
k
 
Æ
(x 

x) v
j
(

x) d

x



max ess-sup
i

x
Z





B
ijk
(

x)  B
ijk
(x)
Æ



jÆ 
k
 
Æ
(x 

x)j jv
j
(

x)j d

x
kBk  kvk
L
1
(
)

Z


jÆ 
k
 
Æ
(x 

x)j d

x
= kBk  k
k
 
1
k
L
1
(
)
 kvk
L
1
(
)
:
Next we apply the Riesz-Thorin Interpolation Theorem to derive a bound for v 2 L
q
(
).
For the reader's onveniene we have stated the theorem in the Appendix; we apply it with
 = 1  1=q = 1=q
0
and p = q, so that, for v 2 L
q
(
),



Z


(B
ijk
(

x)  B
ijk
( )) 
k
 
Æ
(( ) 

x) v
j
(

x) d

x



L
q
(
)
= kTvk
L
q
(
)
 2  kBk  k
k
 
1
k
L
1
(
)
 kvk
L
q
(
)
:
Step IV: Boundedness of (Lv)
Æ
  Lv
Æ
Using that  
Æ
(x)  0 for all x 2 R
n
, we dedue that



Z



k
(B
ijk
(

x) B
ijk
( )) 
Æ
(( ) 

x) v
j
(

x) d

x



L
q
(
)
(1)


m
X
i=1
Z



Z


j
k
((B
ijk
(

x)  B
ijk
(x))j j 
Æ
(x 

x)) v
j
(

x)j d

x

q
dx

1=q
 2  kBk  k 
Æ
 jvj k
L
q
(
)
 2  kBk  k 
1
k
L
1
(
)
 kvk
L
q
(
)
:
Hene using the produt rule
k
k
(B
ijk
v
j
)   
Æ
  
k
(B
ijk
v
Æ
j
)k
L
q
(
)
=



Z



k
 
(B
ijk
(

x)  B
ijk
( )) 
Æ
(( ) 

x)

v
j
(

x) d

x



L
q
(
)
 2  kBk  k 
1
k
W
1;1
(
)
 kvk
L
q
(
)
:
4
Finally as in (1)
k(C
ij
v
j
)   
Æ
  C
ij
v
Æ
j
k
L
q
(
)
=



Z


((C
ij
(

x)  C
ij
( )) 
Æ
(( ) 

x)) v
j
(

x) d

x



L
q
(
)
 2  kCk
[L
1
(
)℄
mm
 k 
1
k
L
1
(
)
 kvk
L
q
(
)
:
Step V: Weakly onverging sequene
The last step shows that the sequene
s
1
(`) = L(v)   
1=`
  L(v   
1=`
)
is bounded in L
q
(
). Hene by the Banah-Alaoglu Theorem there exists a sequene
t
1
: N ! N suh that s
1
Æ t
1
is weakly onverging to an element

v 2 L
q
(
). But, for
w 2 L
q
0
(
) and

` = 1=t
1
(`),
Z


v
i
w
i
dx= lim
Æ!0
Z


v
i
w
Æ
0
i
dx
= lim
Æ!0
lim
`!1
Z


 
L(v)   

`
  L
 
v   

`

w
Æ
i
dx
= lim
Æ!0

Z


L(v)w
Æ
i
dx  lim
`!1
Z


L
 
v   

`

w
Æ
i
dx

= lim
Æ!0
lim
`!1
Z


 
v
j
  v
j
  

`
 
C
ij
w
Æ
i
  B
ijk

k
w
Æ
i

dx
= lim
Æ!0
Z


lim
`!1
 
v
j
  v
j
  

`
 
C
ij
w
Æ
i
  B
ijk

k
w
Æ
i

dx = 0;
and so

v = 0.
Step VI: Strongly onverging sequene
Given " > 0, we an selet a sequene t
2
: N! N, suh that for t
3
= t
1
Æ t
2
and

` = 1=t
3
(`)
kv  v   

`
k
L
q
(
)
<
"
3  2
`
; kL(v)  L(v)   

`
k
L
q
(
)
<
"
3  2
`
:
Using Mazur's Theorem, f. [4, Theorem 3.13℄, there exists a nite onvex ombination
v
"
=
s
X
`=1

`
v   

`
;
s
X
`=1

`
= 1; 
`
2 [0; 1℄; s 2 N;
suh that




s
X
`=1

`
L(v)   

`

  L(v
"
)



L
q
(
)
=



s
X
`=1

`
(L(v)   

`
  L(v   

`
))



L
q
(
)
<
"
3
:
Hene
kv   v
"
k
L
q
(
)

s
X
`=1

`
kv   v   

`
k
L
q
(
)
<
1
X
`=1
"
3  2
`
<
"
3
:
5
Similarly, but by using the triangle inequality
kL(v)  L(v
"
)k
L
q
(
)
<
"
3
+
"
3
:
But then kv   v
"
k
L;q
< " and v
"
2 [C
1
0
(

0
)℄
m
. ////
The transformations in Steps I, II and the rst bound in Step III are based on ideas in [1℄.
We an extend the theorem to all funtions in W
q
L
(
) using the tehniques introdued by
Meyers and Serrin in [2℄, see also [7, p. 54℄. However, due to the existene of funtions
v 2 W
q
L
(
) whose support does not lie ompatly in 
, it beomes neessary to admit
C
1
(
) as lass of approximating funtions instead of C
1
0
(

0
), 

0
 
.
Theorem 2: The spae [C
1
(
)℄
m
\W
q
L
(
) is dense in W
q
L
(
), 1 < q <1.
Proof. Let 

i
be open subsets in 
 suh that 

i
 

i+1
and
1
[
i=1


i
= 
:
Let F be a partition of unity of 
 subordinate to the overing (

i+1
n

i 1
)
i2N
, where 

 1
is taken as the empty set. This means that F is a family of funtions f
j
, j 2 N, suh that:
1. for eah f
j
there exists an i suh that supp(f
j
)  

i+1
n 

i 1
;
2. eah ompat set K  
 intersets the support of nitely many f
j
only;
3. for all x 2 
 holds
P
j
f
j
(x) = 1.
Let

f
i
be the sum of all f
j
2 F for whih i is the smallest index suh that supp(f
j
) 


i+1
n 

i 1
. Then the

f
i
sum to one, too. Choose " > 0. For i 2 N and v 2W
q
L
(
) there
exists, aording to the last theorem, a funtion v
";i
in [C
1
0
(

i+1
n 

i 1
)℄
m
suh that
kf
i
 v   v
";i
k
L;q
<
"
2
i
:
Sine 

j
is ompat, the supports of only nitely many v
";i
interset 

j
. Hene the sum
v
"
=
1
X
i=1
v
";i
is dened and is a member of [C
1
(
)℄
m
. Notie that beause of the layout of the supports
of the v
";i
, the sequene
j 7!

v  
j
X
i=1
v
";i

+





j
= (v   v
"
)
+
j


j
of L
q
(
) funtions exhibits monotoni and pointwise onvergene to (v  v
"
)
+
as j !1.
Hene by the Monotoni Convergene Theorem (v   v
"
)
+
is L
q
-integrable, as well. The
6
same argument employed on (v   v
"
)
 
, (
k
B
ijk
(v
j
  (v
"
)
j
))
+
and (
k
B
ijk
(v
j
  (v
"
)
j
))
 
asserts that v
"
2W
q
L
(
). We onlude
kv  v
"
k
L;q

1
X
i=1


f
i
 v   v
";i


L;q
< ":
This proves the density of smooth funtions in W
q
L
(
). ////
Trae
To dene the trae of funtions in W
q
L
(
) we follow the onstrution for W
q
(div;
); om-
pare, for example, with [5℄. Neas [3℄ shows existene of a bounded extension operator
 
e
:W
1 1=q
0
;q
0
(
)!W
1;q
0
(
); g = ( 
e
g)j


; 1 < q <1:
Sine, for v 2W
1;q
(
) and w 2W
1;q
0
(
),
Z


w
i
B
ijk

k
v
j
dS =
Z


(
k
w
i
)B
ijk
v
j
dV +
Z


w
i

k
(B
ijk
v
j
) dV; (2)
we obtain, for all w in the image of  
e
with w =  
e
g,
jhg;B()  vi


j  k
k
w
i
k
[L
q
0
(
)℄
mm
kB
ijk
v
j
k
[L
q
(
)℄
mm
+kwk
L
q
0
(
)
k
k
(B
ijk
v
j
)k
L
q
(
)
C  kgk
W
1 1=q
0
;q
0
(
)
 (kB
ijk
v
j
k
[L
q
(
)℄
mm
+ k
k
(B
ijk
v
j
)k
L
q
(
)
);
setting C = k 
e
k and B() = (B
ijk

k
)
ij
. This shows that, for all v 2 W
1;q
(
), the
funtional
h;B()  vi :W
1 1=q
0
;q
0
(
)! R; g 7! hg;B()  vi


is a ontinuous mapping. Therefore, h;B()  vi belongs to the dual of W
1 1=q
0
;q
0
(
),
whih is the spae
(W
1 1=q
0
;q
0
(
))
0
= [(W
1 1=q
0
;q
0
(
))
0
℄
m
=W
 1=q;q
(
):
Also
kh;B()  vik
W
 1=q;q
(
)
 C  (kB
ijk
v
j
k
[L
q
(
)℄
mm
+ k
k
(B
ijk
v
j
)k
L
q
(
)
): (3)
Beause
kB
ijk
v
j
k
[L
q
(
)℄
mm
 kBk
[W
1
(
)℄
mmn
kvk
L
q
(
)
;
k
k
(B
ijk
v
j
)k
L
q
(
)
 kL(v)k
L
q
(
)
+ kCk
[L
1
(
)℄
mm
kvk
L
q
(
)
;
we obtain from (3) the ontinuity of the linear operator
 
L
:W
1;q
(
)!W
 1=q;q
(
);v 7! h;B()  vi


:
7
in k  k
L;q
. As W
1;q
(
) is dense in W
q
L
(
),  
B
extends to a ontinuous linear operator on
the graph spae.
Denition: The operator
 
L
:W
q
L
(
)!W
 1=q;q
(
);v 7! lim
"!0
h;B()  v
"
i


=: h;B()  vi


is alled trae operator for W
q
L
(
); here fv
"
g
">0
is smooth and onverging to v in W
q
L
(
).
Observe that the denition of the trae is independent of the extension operator  
e
as the
hoie of  
e
only inuenes the onstant C in (3). As 
k
(B
ijk
v
j
) 2 L
q
(
) , the identity (2)
is meaningful for all elements of W
q
L
(
): Choose v
"
as above and let w 2W
1;q
0
(
), then
hw;B()  vi


= lim
"!0
hw;B()  v
"
i


= lim
"!0
Z


(
k
w
i
)B
ijk
v
";j
dV + lim
"!0
Z


w
i

k
(B
ijk
v
";j
) dV
=
Z


(
k
w
i
)B
ijk
lim
"!0
v
";j
dV +
Z


w
i
lim
"!0

k
(B
ijk
v
";j
) dV
=
Z


(
k
w
i
)B
ijk
v
j
dV +
Z


w
i

k
(B
ijk
v
j
) dV:
Comments
In [1℄ Friedrihs analyses the existene and uniqueness of linear symmetri hyperboli
systems with assoiated dierential operator L. Initially he onsiders the weak and strong
extension of these operators, but then shows that these extensions oinide. This allows
him to apply a duality argument by whih he proves the existene of a strong solution by
uniqueness of weak solutions. To ensure oinidene of the weak and strong extension he
introdues the integral operators
K
"
u(x) =
Z


k
"
(x;

x)u(

x) d

x;
whih generalize the idea of molliers. Indeed k
"
is used in the alulation as a ombination
of an approximate identity with terms of L and its dual L
0
. Friedrihs presents three
properties of k
"
whih are suÆient to prove onvergene of ertain smooth funtions v
"
to v. We only used one of the properties, namely boundedness. The other two properties
onern a suitable hoie of the support of k
"
and the mean of k
"
. The analysis in [1℄ is
given for lens-shaped domains sine then symmetries in boundary terms an be exploited.
In 1964, ten years after the publiation of [1℄, Meyers and Serrin proved that for W
r;q
(
)
the weak and strong extension are equivalent for general open sets 
. The proof of Theorem
2 in this report uses very similar arguments to those in [2℄. The only dierene is that
Meyers and Serrin were able to use mollied funtions diretly while for our problem the
strongly onvergent sequene obtained by Theorem 1 is required.
8
Appendix
We ite the Riesz-Thorin Interpolation Theorem in abbreviated form from [6℄.
Theorem 3: Let 1  p
0
; p
1
; q
0
; q
1
 1. Further let 0 <  < 1, and let p and q be dened
by
1
p
=
1  
p
0
+

p
1
;
1
q
=
1  
q
0
+

q
1
:
Let  and  be -nite measures. If T is a linear mapping with
T :L
p
0
()! L
q
0
() ontinuous with norm M
0
;
T :L
p
1
()! L
q
1
() ontinuous with norm M
1
;
then
kTfk
L
q
 2M
1 
0
M

1
kfk
L
p
8 f 2 L
p
0
() \ L
p
1
():
Hene the operator is extendable to a ontinuous linear mapping
T : L
p
()! L
q
()
with norm 2M
1 
0
M

1
.
Aknowledgement
I would like to thank Endre Suli for pointing out to me the work of K. O. Friedrihs and
for many helpful omments.
Referenes
[1℄ Kurt Otto Friedrihs. Symmetri hyperboli linear dierential equations. Comm. Pure
Appl. Math., 7:345{392, 1954.
[2℄ Norman G. Meyers and James Serrin. H = W . Pro. Nat. Aad. Si. U.S.A., 51:1055{
1056, 1964.
[3℄ Jindrih Neas. Les methodes diretes en theorie des equations elliptiques. Masson et
Cie,

Editeurs, Paris, 1967.
[4℄ Walter Rudin. Funtional analysis. MGraw-Hill In., New York, seond edition, 1991.
[5℄ Hermann Sohr. The Navier-Stokes equations. Birkhauser, Basel, rst edition, 2001.
[6℄ Dirk Werner. Funktionalanalysis. Springer-Verlag, Berlin, extended edition, 2000.
[7℄ William P. Ziemer. Weakly dierentiable funtions. Springer-Verlag, New York, 1989.
9


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Density and trace for graph spaces of first-order linear operators

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