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Applied Mathematics, 2010, 1, 283-287 doi:10.4236/am.2010.14036 Published Online October 2010 (http://www.SciRP.org/journal/am) Copyright © 2010 SciRes. AM Analyticity of Semigroups Generated by Singular Differential Matrix Operators Ould Ahmed Mahmoud Sid Ahmed1, Adel Saddi2 1Department of Mat hem at ic s, C oll e ge of Scie nce, Al j ouf University, Aljouf, Sa u di Arabia 2Department of Mathematics, College of Educa tion for Girls in Sarat Ebeidah, King Khalid University, Abha, Saudi Arabia E-mail: sidahmed@ju.edu.sa, adel.saddi@fsg. rnu.tn Received March 9, 2010; August 3, 2010; August 6, 2010 Abstract In this paper we prove the analyticity of the semigroups generated by some singular differential matrix op- erators of the form 2 2 () =, dBxd x dx dx in the Banach space ([0,],( )), d CMC with suitable boundary con- ditions. To illustrate the work an example is discussed. Keywords: Dissipative Operators, Positive Operators, Spectrum, Analytic Semigroups, Evolution Equations 1. Introduction As we will see in the sequel the problem of characte- rizing operator matrices generating strongly continuous semigroups or analytic semigroups is quite difficult. Th e main problem consists in finding appropriate assumptions on the matrix entries allowing general results but still including the concrete examples we have in mind. The evolution of a physical system in time is usually described in a Banach space by an initial value problem of the form 0 () ()=0, 0 (0) =. dU tUt t dt UU (1.1) Problem of type (1.1) is well posed in a Banach space X if and only if the operator (,())D generates a 0 C-semigroup >0 () tt Ton X . Here the solution ()Ut is given by 0 ()= t Ut TU for the initial data 0()UD . For operator semigroups we refer to [1-5] and to [6] for the theory of op erator matrices. The harmonic analysis for a class of differential operators with matrix coefficients was treated in [7,8]. In this work we are interested in a generalization of the analyticity and the positivity of the semigroup generated by a matrix singular differential operator (,())D. A similar study was realized in [9] for a class of differential operators with matrix coefficients and interface. For the scalar case we refer to [10]. This paper is organized as follows. In the second section we introduce some notations and give prelimina- ries results. In the third section we investigate some properties of the operator (,())D in particular we prove that it is closed, densely defined, dissipative and satisfies the positive minimum principle. Some spectral properties of the operator (,())D are obtained in this section. In section fourth we established the analyticity of the semigroup generated by an operator in the form (,())D. In this case the problem (1.1) has a unique solution for all 0 UX. In section five we present a concrete example of application of the results obtained. 2. Notations and Preliminary Results Let () d M C be the space of dd matrices with complex coefficients and d I the identity matrix. The norm of a matrix 1, =() ij d ijd A aMC wil l b e defi ned by 1, || ||=|| max ij ijd A a (2.1) The reason of this special choice will be justified in Lemma 2.1 and Lemma 2.2. A matrix 1, =() ij d ijd A aMC is called nonnega- tive (resp positive) if all entries ij a are real numbers and nonnegative (resp positive). In this case we write 0A, (.>0)respA and we call 1, ||=| | ij ijd Aa O. A. M. S. AHMED ET AL. Copyright © 2010 SciRes. AM 284 the absolute value of A . For 1, =() ij d ijd A aMC and 1, =ij ijd Aa () d M C, we will write 0.ABifAB and >>0.ABifAB A matrix-valued functions is said to be continuous, differentiable or integrable if all its elements are continuous, differentiable or integ r able fu n ctions. If th e matrix () A x is integrable over [,]ab, then ()(). bb aa A xdxAx dx For two matrix functions 1, ()=() ij ijd Axa x and 1, ()=() ij ijd Bxb x , we shall write the negligibility 0 ()=(()), A xoBx xx if, 0 ()= () ij ij axobxxx for all 1, .ij d Similarly we define the notions of domination ()=(()) A xOBx and the equivalence ()~ () A xBx (see for more details [11]). Let X be the space ([0,],( )) d CMC of continuous, matrix-valued functions on [0, ] . On the space X we define the norm ||.|| , X by 0 ||||= sup ||()||, Xx f fx for all . f X Note that the normed space (,||.||) X X is a Banach space. We denote by (]0,],( )),=1,2, kd CMCk the space of all ktimes continuously differentiable matrix- valued functio ns U defined on ]0, [ such that () lim( ) p xUx exists and finite forall0pk. Consider a singular second order differential operator (,())D with matrix coefficients defined on X by 2 2 () =, dBxd UU U x dx dx where Uand Bare matrix-valued functions, with the domain 2 0 ()= {([0,],())(]0,], ()),()= 0}. lim d dx DUCMC MC Ux We assume that B is a real valued continuous and bounded matrix-valued function on [0,[ and if (0) =d BI we add the assumption |() (0)| D RxBx B (2.2) in a neighborhood of 0, for nonnegative constants diagonal matrices R and D. Here D x is the diagonal matrix with diagonal entries =,=1,2, di i x id , where , =1,2,, i di d are the diagonal entries of D. We will now recall some results needed in the sequel. Lemma 2.1 Let () d A MC . The following properties hold 1) 0A if and only if 0AB for all 0.B 2) ||||||, A BAB hence | ||||| A BAB if 0A. Lemma 2.2 Let ,() d A BMC . The following proper- ties hold 1) |||| A B implies . A B 2) ||= . A A 3) |||||| ||, A EdA where 1, =,=1,1,. ij ij ijd Ee eijd Proposition 2.1 Let A be a nonnegative matrix with spectral radius ()rA. 1) The resolvent (,)RA is positive whenever >().rA 2) If ||>()rA , then |(,)|(| |,).RAR A Proof. 1) We use the Neumann series representation for the resolvent 11 11 1 0 (,)=() =() =||>(). d n n n RAAI A AforrA If 0A then 0, n A for all n, hence for >()rA , we have 1 =0 (,)= 0 lim k n k nk A RA since the finite sums are positive and convergence holds in every entry. 2) For ||>(),rA we have 1 =0 1 =0 |(,)|= lim =(||,). lim || k n k nk k n k nk A RA ARA Theorem 2.1 ([12], Perrons Theorem) If A is a nonnegative matrix, then ()rA is an eigenvalue of A with positive eigenvector. Definitio n 2 .1 1) An operator (, ())SDS on a Banach Lattice (,)X is called positive if 0()={(),0}.SuforalluD SvD Sv 2) A semigroup >0 () tt T on X is positive if and only if t T is positive for all >0t. Remark 2.1 An operator (, ())SDS defined on a ()K space( K compact) satisfies the positive minimum principle if for every (),uDSu positive and x K ,()=0ux , then ()()0.Sux Theorem 2.2 ([1]) O. A. M. S. AHMED ET AL. Copyright © 2010 SciRes. AM 285 Let (, ())SDS be the generator of a semigroup >0 () tt T on ()CK , then the semigroup is positive if and only if (, ())SDS satisfies the positive minimum principle. 3. The Diagonal Case In this section we assume that the matrix function B is diagonal. 3.1. Characterization of the Operator (,())D The proofs of Proposition 3.1 and Proposition 3.2 can be deduced from the scalar case given in Proposition 2.2 and Prop osi tion 2.3 from [10]. Proposition 3.1 Let >(0) d IB and 2(]0,],( )) d UXC MC . Then ()UD if and only if 2([0, ],()) d UMC for some >0 and (0) =(0)= 0UU . Proposition 3.2 Let (0) =d BI and 2(]0,],( )) d UXC MC . Then ()UD if and only if 1([0,]),( )) d UC MC for some >0 , (0) = 0U and 0 0 1(() ())=0. lim x xUxUtdt x Moreover, if ()UD then ()= (log)Ux oxE and 1()=Uxo logxE x as 0,x where E is the constant matrix introduced in Lemma 2.2. Proposition 3.3 The operator (,())D is a densely defined, closed, dissipative and satisfies the positive minimum princi ple. Proof. Put 1 2 () 00 0() ()= 0 00() d bx bx Bx bx and for =1,2, ,id let () =i i bx uu u x with 20 ()= {([0, ],)(]0, ],),()=0}. lim i xi D uC CCux Then 1, =() ij ijd UuD if and only if () ij i uD for all ,=1,2,.ij d Hence from ([10]. Lemma 2.4), the operator (,())D is a densely defined and closed. Let us show that (,())D is dissipative: ||||||||,for>0and() XX UUU UD Let 1, =() ij ijd UuD and >0 . According to [10], for all 1,ijd we have |||| |||| ijiji ij uuu then ||||||||, XX UUU and hence the dissipativity holds. In order to prove that (, ())D satisfies the positive min i mu m p r in ci p le , a ss u me t h at 1, =() ij ijd UuD and positive such that 0 ()=0Ux for some 0[0, [.x If 0>0xthen 0 ij uand 0 ()=0, ij ux for all i, j = 1, 2, ,d. Hence 0 '() =0 ij ux and 0 '( )0 ij ux for all i, j = 1, 2,,d. That means 0 ()0Ux. If 0=0x then 0 ()=0Ux. 3.2. Spectral Analysis of the Operator (,())D The purpose of next th eorem is to deduce under re asonable hypothesis on the coefficients of B a precise description of the spectrum of the operator (,())D. Theorem 3.1 If (0) d I B , then the spectrum of (,( ))D is contained in ],0]. Proof. Let 1, ,= () ij ijd CU uD and 1, =( ). iji j d Vv X We have()=()= for all,=1,2,, iij ij UVu v ij d 1 = =1 =(), =1,2,, (). ijiij id i i uvfor allijd and This is sufficient to note that the spectrum of (,( ))D verifies = =1 ()( ) id i i and then the result hold by ([10], Lemma 2.6). Theorem 3.2 The operator (,())D with (0) d I B , generates an analytic semigroup of angle . 2 Moreover, the semigroup is positive and contractive. Proof. For 0< < , put * ={;|()|}zC argz . It is clear from theorem 3.1 that () and then for and =1,2, ,id, the operator () i is invertible and verifies 1 || ()|||| i ii C for some positive cons- tant i C (see [10]). So () is invertible and veri- O. A. M. S. AHMED ET AL. Copyright © 2010 SciRes. AM 286 fies 11 max || ()|||| i id X C . Hence the operator (,())D with (0) d I B, generates an analytic semigroup of angle . 2 For the positivity using the rela- tion between the operators , =1,2,, iid and and from the fact that each operator (,( )), ii D =1,2, ,id, generates an analytic positive and contrac- tive semigroup, we deduce the result. 4. The Non Diagonal Case In this section we consider the operator () ='' ' Bx UU U x Assume that 1 ()=() ()()BxPxDxPx , where D is diagonal matrix function and P is nonsingular matrix function. If P is constant, put 1 =VPU , then =,= = ' ''' '' UPVU PVandUPV so we obtain 1() ='' ' Dx PPVV V x Hence, from Theorem 3.2 we can easily verify the following Proposition 4.1 The operator (,())D with (0) d ID , generates an an alytic semi g rou p of a ngle . 2 Remark that the condition (0) d ID is equivalent to the fact that the spectrum ((0))B of the constant matrix (0)B verifies ((0))],1].B We turn now to the general case in which we proceed with a perturbation argument. For this we recall the following definition. Definiti on 4.1 ([2].Definition 2.1). Let :() A DA XX be a linear operator on the Banach space X . Any operator :()BDB X X is called A -bounded if )()DAD B and if there exist constants ,ab in such that ||||||||||||( ).BU aAU bUforallUDA (5.1) The A -bound of B is 0=inf{ 0:0(5.1)}aathere existsbsuch thatholds Proposition 4.2 ([2].Theorem 2.10). Let the operator (, ()) A DA generates an analytic semigroup on a Banach space X . Then there exists a constant >0 such that (,()) A BDA generates an analytic semigroup for every A -bounded operator B having A -bound 0<.a Introduce now the operators 00 (,())D and (,( ))D given by 2 02 (0)( )(0) == dBd BxBd and x dxx dx dx with, 0 20 0 ()=( )={([0,], ())(]0, ],()),()=0}. lim dd x DD UC MC CMCUx Then we have 0 = and if we choose 0 ()=()DD, we obtain the principal result of the paper. Theorem 4.1 Assume that (0)B is diagonalizable and ((0))],1]B or (0) =d BI and (2.2) holds. Then the operator (,())D, generates an analy- tic semigroup of angle . 2 Proof. Let ()UD and observe that 2 2 2 2 0 () = (0)( )(0) = =, dBxd UU U xdx dx dBdBxBd UU U x dxxdx dxUU Let >0, there exists >0 such that for all 0< <x we have ||()(0) ||<.Bx B The formula 1 0 () =() Ux Uxtdt x implies that ||()||||||,0<<. X RU xUx On the other hand from the Taylor expansion to order 2 at >x and for all ()UD there exists a constant >0C such that 0 ||()|||||||||| (0) |||||||||||| . XX XXX RU xUCU B UUCU x Since (0)B is diagonalizable and ((0))],1[B or B satisfies the condition (2.2) for (0) =, d BI the map (0) , Bd UU x dx from ()D into ([0,],( )) d CMC is continuous (see [10]. Remark 2.5), so we deduce that the operator is -bounded with -bound is equal to zero. Hence, the desired result follows by applying Theorem 3.2 and Proposition 4.2. O. A. M. S. AHMED ET AL. Copyright © 2010 SciRes. AM 287 5. Application and Example Assume now that the operator (,( ))D satisfies the assumptions of Theorem 4.1, it generates so an analytic semigroup, and consider the evolution equation problem 0 ()()=(), 0 (0) = dU tUtftt dt UU (5.2) Corollary 5.1 If =0f then the problem (5.2) has a unique solution for all 0 UX . This solution is of infinitely continuously differentiable on ]0, [. For general case we have by Pazy [3] and R.autry [13]. Corollary 5.2 If 1, =( ) iji jd ff , and for all 1 ,,(]0,[, ) ij ijfLTC and for every ]0, [tT there is a ,ij t and a continuous function ,:[0,[ [0,[ ij t such that , , 0 () ()()(||)< . ij ij t ij t ij ijt d ftfsts and Then the problem (5.2) has a classical solution. EXAMP LE Let the Banach space =([0,],( )) d X CMC and 0. Put ()= x x and define the linear transforma- tion P on X in itself by =.PUUo It is clear that P is invertible and 11 ()= .PP Consider now the operator (, ())D defined on X by 22 12 =()Ux Ux BxU with 2 0 ()= {(]0,],()), ,( )()=0}, lim d x DUXC MC UX oPUx and BX is a diagonal matrix real valued function satisfying (0)<(12). d BI A simple calculus gives 21 =()PP (5.3) where 1 11() =. Bx UU U x Put 2 0 ()={([0, ], ())(]0, ], ()), ()=0}. lim d dx DUC MCC MC Ux From Theorem 3.2 the operator (,())D generates an analytic semigrou p of angle 2 moreover, the semi- group is positive and contractiv e. Hence the relation (5.3) implies that the operator (, ())D generates an analytic semigroup of angle 2 and the semigroup is contractive if 1. 6. Acknowledgements The Authors wish to thank Professor A. Rhandi for many helpful discussions and comments on the manuscript. 7. References [1] R. Nagel, “One-Parameter Semigroups of Positive Op- erators,” Lecture Notes in Math, Springer-Verlag, 1986. [2] K. J. Engel and R. Nagel, “One-Parameter Semigroups for Linear Evolution Equations,” Springer-Verlag, 2000. [3] A. Pazy, “Semigroups of Linear Operators and Applica- tions to Partial Differential Equations,” Applied Math Sciences 44, Springer, 1983. [4] K. Ito and F. Kappel, “Evolution Equations and Ap- proximations,” Series on Advances in Mathematics for Applied Sciences, Vol. 61, 2002. [5] E. M. Ouhabaz, “Analysis of Heat Equations on Do- mains,” Princet on University Press, New Jersey, 2005. [6] K. J. Engel, “Operator Matrices and Systems of Evolution Equations.” (Preprint). [7] N. H. 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Raghavan, “Nonegative Matrices and Applicat ion s,” Ca mbri dg e Univer sity Pre ss, Cambr idg e, 1997. [13] R. Dautray and J.-L. Lions, “Analyse Mathématique et Calcul Numérique pour les Sciences et les Techniques,” Tome 3, Série Scientifique, Masson, 1985. |