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Advances in Pure Mathematics, 2011, 1, 42-48
doi:10.4236/apm.2011.13010 Published Online May 2011 (http://www.scirp.org/journal/apm)
Copyright © 2011 SciRes. 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
E-mail: [email protected]
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
MCA is denoted by *
A
.
Let
12
,0 ,0,,
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
Copyright © 2011 SciRes. APM
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,LXC 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
Iu, 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
Copyright © 2011 SciRes. APM
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
kkA
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
Copyright © 2011 SciRes. APM
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
Copyright © 2011 SciRes. APM
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
kkT 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
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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
.
6. References
[1] K. Ito and F. Kappel, “Evolution Equations and Ap-
proximations,” Series on Advances in Mathematics for
Applied Sciences, World Scientific Publishing Company,
River Edge, Vol. 61, 2002.
[2] C. A. Boyes, “Acoustic Waveguides,” Application to
Oceanic Sciences, Wiley, New York, 1984.
[3] A. Pazy, “Semigroups of Linear Operators an Applica-
tions to Partial Differential Equations,” Applied Math
Sciences, Springer, New York, Vol. 44, 1983.
[4] N. H. Mahmoud, “Partial Differential Equations with
Matricial Coefficients and Generalized Translation Op-
erators,” Transactions of the American Mathematical So-
ciety, Vol. 352, No. 8, 2000, pp. 3687-3706.
[5] N. H. Mahmoud, “Heat Equations Associated with Ma-
trix Singular Differential Operators and Spectral Theory,”
Integral Transforms and Special Functions, Vol. 15, No.
3, 2004. pp. 251-266.
doi:10.1080/10652460310001600591
[6] J. Weidmann, “Spectral Theory of Ordinary Differential
Operators,” Lecture Notes in Mathematics, Springer,
Berlin, Vol. 1258, 1987.
[7] K. J. Engel and R. Nagel, “One-Parameter Semigroups
for Linear Evolution Equations,” Springer-Verlag, New
York, 2000.
[8] R. Nagel, “One-Parameter Semigroups of Positive Op-
A. SADDI
Copyright © 2011 SciRes. APM
48
erators,” Lecture Notes in Mathematics, Springer- Verlag,
Berlin, Vol. 1184, 1986.
[9] A. Saddi and O. A. Mahmoud Sid Ahmed, “Analyticity
of Semigroups Generated by a Class of Differential Op-
erators with Matrix Coefficients and Interface,” Semi-
group Forum, Vol. 71, No. 1, 2005, pp. 1-17.
doi:10.1007/s00233-004-0173-6
[10] T. G. Bhaskar and R. Kumar, “Analyticity of Semigroup
Generated by a Class of Differential Operators with In-
terface,” Nonlinear Analysis, Vol. 39, No. 6, 2000, pp.
779-791. doi:10.1016/S0362-546X(98)00237-5
[11] H. O. Fattorini, “The Cauchy Problem,” Addison Wesley,
Massachusetts, Vol. 18, 1983.
[12] H. Chebli, “Analyse Hilbertienne,” Centre de Publication
Universitaire, Tunis, 2001.
[13] T. Kato, “Perturbation Theory for Linear Operators,”
Springer-Verlag, Berlin, 1966.