Advances in Pure Mathematics, 2011, 1, 42-48
doi:10.4236/apm.2011.13010 Published Online May 2011 (http://www.scirp.org/journal/apm)
Analyticity of Semigroups generated by Degenerate
Mixed Differential Operators
Adel Saddi
Department of Mat hematics, College of Education for Girls, King Khalid University, Abha, Saudi Arabia
Received January 18, 2011; revised March 14, 2011; accepted March 20, 2011
Abstract
In this paper we are interested in studying the dissipativity of degenerate mixed differential operators in-
volving an interface point. We show that, under particular interface conditions, such operators generate ana-
lytic semigroups on an appropriate Hilbert space H. To illustrate the results an example is discussed.
Keywords: Adjoint, Interface, Dissipative Operators, Analytic Semigroups
1. Introduction
The evolution of a physical system in time is usually
described in a Banach space by an initial value problem
for a differential equation on the form:
 

0
d0, 0
d0
Ut LU tt
tUU





(1)
Such problems are well posed in Banach space
X
if
and only if the operator L generates a 0 C-semigroup

0
tt
T on
X
[1]. Here the so lution

Ut is given by
 
00
for
t
Ut TUUDL.
Problems involving interface arise naturally in many
applied situation such as acoustic wave in ocean [2] and
also as heat conduction in non homogeneous bodies. A
systematic study of interface problems involving ordi-
nary differential operator was done in [3].
Several authors have been interested to differential
operators with matrix coefficients. Such operators arise
in diverse range of applications (e.g. in Quantum phys-
ics), some examples in harmonic analysis have been
treated in [4-6] and for an example in semigroups theory
we refer to [7-8].
In this paper, inspired in the works of A. Saddi and O.
A. Mahmoud Sid Ahmed [9] and also that of T. G.
Bhaskar and R. Kumar [10], we establish with suitable
assumptions the analyticity of semigroups generated by a
class of differential operators involving matching inter-
face conditions in the setting of complex Hilbert space.
As it is well known, in order that an operator L gen-
erates an analytic semigroup it suffices that it satisfies
the m-dissipativity and we must have (see [11])
,,0,0LU UmLU U 

 e (2)
The paper is organized as follows: In section 2 we in-
troduce the different notions and notations which we
shall need in the sequel. In section 3 we study the mixed
operator L and its adjoint L and we investigate some
of its properties. In section 4 we study the dissipativity of
the operator


L


and its adjoint for some suitable
real number

. We show that, under particular interface
conditions, such operators generate strongly continuous
semigroups. Using the previous results we conclude in
section 5 with the aim of the paper about generation of
analytic semigroups of operators with respect some reg-
ular interface conditions. Finally we discuss an example
as an application to our results.
2. Notations and Preliminaries
Let


n
MC be the space of all square n order matrix
with complex coefficients, and

n
GL C the subset of


n
MC consisting of invertible matrices. The adjoint of
a matrix


n
MCA is denoted by *
A
.
Let




12
,0 ,0,,
I
aI bwhere 0ab 

 , and


*\0
kk
II. For 1, 2k and an interval
,
kk
X
I denote by


2,
k
LXC the complex Hilbert
space defined by
 


2
2, :,measurabled
k
kk X
LXuXut t





CC
endowed with the canonical inner product

A. SADDI 43
 
,d
k
kX
uvutvt t (3)
We set also,
  
2
22
,,ex is t a n d a b s o l u te l y
con tinuous on ,and
k
kkk
uLX uu
HX XuLX
 



 


C
C
Consider now the product Hilbert space


21,LXC

22,LXC equipped with the inner product
112 2
12
,, ,UVu vuv (4)
for all
 
22
12 1212
,, ,,,Uuu VvvLXLXCC
Fix now

22
12
,,LI LIHC C and denote its
subspace
 
22
12
H
HI HI Let k L be the dif-
ferential operator defined on k
I
by
, 1,2
kkkkkk
Luaubu k





(5)
where k
a and k b are two real measurable functions on
k
I
. We make the following assumptions: For k = 1,2

1:k
ha
is continuous and 0
k
a on k
I
, ,
kk
ab

are
absolutely continuous on k
I
.


 
2*
2
10 122
0
:,, ,
llim i,m
kk k
xx
haaLI
baxbax



 
C
exist inR and

kk
ba

 is bounded on k
I
.
Let 1
A and 2 A two matrices in


2
GL C For

2
kk
H
Iu, denote
 

,,,12
k
k
k
k
ux
xxk
ux I





u
The interface condition at the singular point 0x

, is
given by
  
12312
000:: 0hAuuA.
Note that this work can be easily generalized to de-
generate matrix differential operators. Here the operator
may have non-regu lar co ef f icien ts an d may be sing u lar at
the extremities of intervals and especially at the interface
point. In particular with this meaning this study is a
proper extension of [9].
3. Mixed Operator


,
L
DL and
its Adjoint
In order to study the operator L, we introduce its Green
formula. We will be able to obtain some characteristic
properties. According to ([12], p. 189) the corresponding
formal Lagrange adjoint expression of ,1,2
k
Lk

are
given as

*, 1,2
kk kkkk
Lu aubu k
 
  (6)
We consider the operator
 
,LDL given by



 

12 1122
12
,,
for ,
LULu uLuLu
Uuu DL

 (7)
where








12
,00,0
ab
DLU uuH 


and






11 22
,
aabb
uubaua bu 
  (8)
a

and b

are here two fixed real numbers.
It is easy to show that



,LDL is a densely defined
closed unbounded linear operator in H and hence has a
unique adjoint (see for example Theorem 3.6 [5]).
For




12 12
,, ,UuuVvv DL, and 0a



b


, a simple calculation gives the Green’s Formula.




 

11 1
2
11 1111 111
2222222
1111 1
22
11
22 22
22
222
d
d
d
d
b
a
b
a
a
b
abxvxx
abxvxx
av uavubvu
av uavubvu
avbvu xx
avbvu
u
x
u
u
x
u












 




 




 



  




















Using the conditions 0
a

 and 0
b

 we get,
 



 

 

 

1
1112 22
12
b22 222
111111
12
2
22
22
1
122
00
0
1111 1
222
0
,,
a ba a
a ba a
,
lim li
d
d
m
xx
a
b
a
LU Vuvuv
vvu
vvu
avbv ux
LL
b
a
xx
x
xavbv ux









 




 


















vBuvBu
where ,1,2
kk

B are matrix functions defined on *
k
I
,
given by
0
kkk
kk
baa
a

 



B
The matching interface condition

00, and the
notation



**
1,1,2,
kk k
k

CAB imply

A. SADDI 44
 


 




 



 

 
1112 22
12
b2
*
1112 2
222222
1111111
122
11
2
00
0
1
22
11
0
**
1
22
12
,
lim lim
,,
a ba a
a ba
d
d
a
,
xx
a
b
ba
a
LU Vuvuv
u
u
x
x
abuxx
abuxx
uu
LL
vvb
vv a
vv
vv
baLu 
















 








 








Au ACvCvu


 




**
1122 2
12
*
11 121222
00
lim im
,
l
xx
vuv
x
x
L





Cvv ACAu u
(9)
where





  
*111
2
11 1
*22222
,
a
bb
aabaa a
a
v
ba avb
v
v


 

  


with these simplifications, we obtain the following result.
Proposition 3.1 Let

 
,LDL be the operator de-
fined as in (7) and (8 ). Then its adjoint


,LDL

is a
densely defined closed unbounded operator given by

 


**
1** 2
,00,0
ab
DLV vvH   
 



** **
121122
*
12
,,
for ,
LV LvvLvLv
Vvv DL

 (10)
where
 
12
*1
00
2
li0m lim
xx
x
x


 CCvv.
Proof. Let
 
,
M
DM be the operator given by
 


*
1** 2
,00,0
ab
DMV vvH   
 


***
121122
12
,,
for ,
MVL vvLvLv
Vvv DM


One has to prove that *
M
L

and



*
DM DL.
From Green’s formula, it follows that



*
DM DL.
To show the opposite inclusion, it remains to verify that

*
12 121212
,,,,, ,LuuvvuuLvv
for all


*
12 12
,,,uuDL vvDL. From (9) this
is true if one proves that
 


 



**
21
**
11112 22
00
2
lim lim0
ba
xx
ub ua
xx




 Cv uuACvA
If we choose

12
,uu DL verifying


1
ua


20ub

, then we get



 




*
11122
0122 0
lim lim0
xx
xx




CvAuvuCA
Now from Gree n’s formula, we obtain


**
21
0
ba
bauu 


An appropriate choice of

12
,uu DL, implies
**
0
ba


 and


*00

. This yields



*MDL D hence the proof is achieved.
4. m-Dissipativity of


,
L
DL
Recall first the definition due to Pazy [3].
Definition 4.1 A linear closed densely defined opera-
tor




,
M
DM on a complex Hilbert space is called
m-dissipative if



for all , ,0
and is surjective for some 0.
uDM Muu
M 
 


e
It is our aim to show, under certain assumptions on the
coefficients of , 1,2,
k
Lk

that the mixed operator is
m-dissipative. The next technical lemma ma y b e f oun d in
[9].
Lemma 4.1 Let ,
f
g two numerical functions of
class 1 C on


,

 such that
f
is real then



   
22 2
2d
d
fggx x
f
gf fxgxxg











e(11)
In what follows, consider the following function ma-
trices


*
2on ,1,2
kk
MIkCT given by
0
kkk
kk
baa
a





T
Theorem 4.1 Assume that the matrices , 1,2
kkA
satisfy the condition





**
11
1
11
1122 2
00
lim lim
xx
xx





AT
A
AAT (12)
Then there exists a real 0

 such that the operator






,LDL   is m-dissipative.
Proof. Let


12
,Uuu DL , we have
 


   
 






1112 22
12
2
22 2
2
2
11122 2
00
02
11111
2
22
2
111
1
222
02
,, ,
1
2
1
2
lim ()lim
d
d
b
xx
b
a
a
LUULuuLuu
ab bbub
ab u
auu xa
ab
aaaaa
uu x
bauua xx
bauua x
u
ux
























 



A. SADDI 45
Then, by using Lemm a 4.1, we get,
123
,LU USSS e
where
  
    




2
1222 2
2
11
0
1
22
11 11
0
1
1
2
1
2
dd
b
a
b
a
Sabbbabub
aa bau
mux xm
a
x
a
x
a
u


















with
  

 

11111
222 22
2,
2
2,
2
a
a
x
maabxaxxI
a
x
mabxaxxI
x
b
b
x



 


For 1,2k and 0

, we have,

 

  
   
2
2
2
2
22
2
2
2
kkkkkk k
kk kkk
k
kk kk
muxm xu xm xuxu x
mxux mxuxux
ux
mxux mxux 








 


e
So,

 
  
22
2
2
111
2
2
2
d
d
kk
kkkk
II
kk
k
kk
Smuxxmxux
ux
mxux x





















  










211111
0
2
0
1
11
22222
12212
00
1lim 2
2
lim 2
1lim lim
2
x
x
xx
Sbauaux
bau
u
aux
x
u
uu
x







 



e
e
uT Tuuu







022
311111
22
222
022
12d
2
2d
a
b
Sbauauxx
bauau xx
 

 
 


Thus we obtain,
 

  
 







11 2
2
1
2
2
2
222
2
1
122
00
1
,2
d
d
1lim lim
2
k
k
kk
I
k
k
kk
kkk
I
k
xx
LU Ubax
mx
mxux x
axmx ux x
x
x









 





 









e
uuTTuu
For sufficiently small

, such that

2
20
kk
am  ,
we obtain,
 

  








 

  

1
11 111
1
2222 222
2
2
1
2
2
2
1
11
0
1
0
2
1
2
2
2
22
11
1
,2
d
1lim
2
lim
d
k
k
kk
I
k
k
kk
x
x
kk
I
k
k
kk
LU Ubax
mx
mxux x
x
x
bax
mx
mxux x
uu u 
















 





 

















 
















e
Au AAu
Au
TA
TAAAu

22
2
,uU  
where,
 
2
2
1,2
max sup
k
k
kk k
kxI
mx
bax mx 
 




















Thus, we have shown that

L

 is dissipative. For
showing that


L


is m-dissipative, we have to
show that


L



is also dissipative. The interface
term vanishes, since


12
,vv verifies the condition,








11 1
12222
1
111
00
lim lim0,
xx
xx






CT vCCTCv
which itself is a consequence of (12). So, using same
techniques as above, for all



*
12
,Vvv DL, we
get,



 









 


** *
1112 22
12
22
221 11
00
0
11
22
1111122 2222
22
11 1222222
0
1
,, ,
lim lim
dd
ba
xx
a
b
LVVLvvL vv
avaavv xavvx
avv
bavbvabva
bvaxxb vaxxavv



 
 
 


 



A. SADDI 46
Then one has,
 
2
*
1
1
,2kkk
I
k
LV Vbax


 



e
 


22
2d
kk k
mx mxvxx 


 



2
2
22 2
112
,vv vVv  
This implies that both

L

 and

L



are
dissipative, thus

L

 is m-dissipative and hence
the theorem is proved.
5. Analyticity of the Semigroup Generated
by
 
,
L
DL
The purpose of this section is to prove the analyticity of
the semigroup generated by



,LDL . For This goal
we impose some additional conditions on the matrices
,1,2
k
Ak. In the following we recall a theorem due to
Fattorini [11].
Theorem 5.1 Let
 
,
A
DA be a densely defined
operator in a Hilbert space such that for any


uDA:
,,forso0me0,eAuu mAuu    Then

,
A


DA generates an analytic semigroup of contrac-
tions.
With the help of Theorem 5.1, we will establish our
main result.
Theorem 5.2 Assume that the matrices k
B
and
,1,2
kkT defined respectively in sections 3 and 4, sat-
isfy the conditions






**
111 1
11 1222
00
lim lim
xx
xx

 

AB AABA (13)






**
111 1
11 1222
00
lim lim
xx
xx




A
TA ATA (14)
Then the operator


,LDL generates an analytic
semigroup of contractions.
Proof. Since the operator



,LDL is densely de-
fined, then from Theorem 5.1, to show that it generates
an analytic semigroup, it suffices to verify that
Re ,
A
uuIm ,0,Au u  
for
A
LI

 and for
some 0 and 0

 .
This is equivalent to show that,eLuu
,mLuu   2 u  .
Holds for all

uDL. Using the identity
 



  









2
1111
11 222
2
11
2
22 2
2
11
2
2222
0
1
00
0
11
22
1,
liml m
d
2
12
i
d
b
xx
a
a
b
bbab
aaaa
uu uu
uuau
uu a
LU Uabub
ab ua
axax
ba xx
ba xxu







 


















then
 


     

112 21
00
0
2
11 21
12
0
1222
,limlim
dd
xx
a
b
LuuUUa xax
ba
uu
uu uu
x
xba xx
JJ


 

 





mm
m
where
 


    

11
00
2
0
1
11 2
1222
112 2
02
lim lim
and
dd
x
b
x
a
Jaxax
J
baxxb
uu uu
uaxuxuu


 


 



m
m
Using the relation


2

 mm for all
,



we deduce the expression
















111
0
0
*
11
1111 1
11 11
2
11
0
*
1
22 222
22 121 22
02
2lim
lim
lim
lim
x
x
x
x
uu uuJaax
aax
x
x
uu uu





















m
mAuABA Au
AuABA Au
Under the assumption (13) and the interface condition,
we get













*
111
0
**
111 1
111
00
2
222
2
0
2lim
lim lim
lim 0
x
xx
x
x
x
J
x
x




 









mAu
u
ABA ABA
A
We have also, for sufficiently small

,









 














0
211
22
0
11
22
0
1111
2222
022
11
22
0
12
22
0
0
0
22
11 11
22
22
22 22
d
d
d
d
d
d
d
d
b
b
b
b
a
a
a
a
Jbauu
uu
xx
ba xx
ba xx
ba xx
bauu xx
ba
u
uux x
ba
u
u
uux
u
x
bauuxx





 











 















m
m
m
It follows that, for 0

, we have

A. SADDI 47
 






2
22
2
1
22
21
22
2
1
2
,,
d
d
2
d
2
k
k
k
k
kkk
I
k
kkk
I
k
kk kkk
I
k
Lu umLuu
mx
baxmxuxx
bau xx
ba mauxx
u












 

















e∣∣
where



*
2
22
1,2supmax
2
k
kkk k
xI
kkk
ba m
mbax 







 









Thus the proof is achieved and the result of the Theo-
rem is obtained.
Corollary 5.1 The operator


,LBDL gener-
ates an analytic semigroup for all L-Bounded opera-
tors B. In particular the result remains true if we choose

12
,BRR defined on H by ,1,2,
kk kk
Rucu k
where k c is a piecewise conti nuous function on k
I
.
For more detail in perturbation theory of linear opera-
tors we refer to [7] and [13].
In the following an example is given to demonstrate
the effectiveness of our results.
Example 5.1 Let


11, 0I and


20, 1I, and
consider the following differential system
 
 

2*
111
11
2
2*
222
22 2
2
1
12 0
,
,
,0
uuu
axbxxI
tx
x
uuu
ax bx xI
tx
x
uu u
 


 



 










where ,, 1,2
kk
abk are real functions verifying the
previous assumptions 12 andhh.
The interface condition at 0x

is such that
 
12
12
0,0,,0,0, .
uu
atbtutut
xx



The end points conditions are taken to be
 

1
111
2
222
1,1, 0,
1,1, 0
utut
x
utut
x






 

for some real constants 1 ,,,abc

and2

.
The operator




,SDS is as follows


12
,,, 1,2
kkkk kk
SSSSuaubuk
 

and



12 1
12
122
(),0 0,
0
DSUu uH

 

uuAA
where 1
0
0
a
c



A and 2
0
0
b
c




A.
Then it is easily to verify that the conditions of Theo-
rem 5.2 are fulfilled for the operator



,SDS if


00
k
a for1, 2k

,

11
0
lim
xbax






22
0
lim
xba






x
and




12
000ba aa


Then for all 0 uH

, the above evolution partial dif-
ferential system has a unique solution which is analytic
in time for 0t.
The following functions are a concrete example for the
above system.
 
2
1111
0
11
sin d,,3
x
axttbx x
t







 
2
2
2222
1
1log,,
2
axxxba





with 10ba


.
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