Analyticity of Semigroups generated by Degenerate Mixed Differential Operators

Such problems are well posed in Banach space X if and only if the operator L generates a 0 C -semigroup   0 t t T  on X [1]. Here the solution   U t is given by     0 0 for t U t TU U D L   . 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 ordinary 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 physics), 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 interface conditions in the setting of complex Hilbert space. As it is well known, in order that an operator L generates an analytic semigroup it suffices that it satisfies the m -dissipativity and we must have (see [11])


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: Such problems are well posed in Banach space X if and only if the operator L generates a 0 C -semigroup   0 t t T  on X [1].Here the solution   U t is given by . 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 ordinary 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 physics), some examples in harmonic analysis have been treated in [4][5][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 interface 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]) The paper is organized as follows: In section 2 we introduce 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 regular interface conditions.Finally we discuss an example as an application to our results.

Notations and Preliminaries
Let   n M C be the space of all square n order matrix with complex coefficients, and endowed with the canonical inner product We set also, , , exist and absolutely con tinuous on , and for all where k a and k b are two real measurable functions on k I .We make the following assumptions: The interface condition at the singular point 0 x  , is given by         . Note that this work can be easily generalized to degenerate matrix differential operators.Here the operator may have non-regular coefficients and may be singular at the extremities of intervals and especially at the interface point.In particular with this meaning this study is a proper extension of [9].

L D L 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 We consider the operator , , for , where a  and b  are here two fixed real numbers.
It is easy to show that     , L D L is a densely defined closed unbounded linear operator in H and hence has a unique adjoint (see for example Theorem 3.6 [5]). For , and 0 , a simple calculation gives the Green's Formula.
The matching interface condition   0 0   , and the notation where with these simplifications, we obtain the following result.
, L D L be the operator defined as in (7) and (8).Then its adjoint is a densely defined closed unbounded operator given by for , where be the operator given by , , for , To show the opposite inclusion, it remains to verify that hence the proof is achieved.
and is surjective for some 0.
It is our aim to show, under certain assumptions on the coefficients of , 1, 2, k L k  that the mixed operator is m-dissipative.The next technical lemma may be found in [9].
In what follows, consider the following function matrices Then there exists a real 0   such that the operator Then, by using Lemma 4.1, we get, and 0   , we have, For sufficiently small  , such that   we obtain, which itself is a consequence of (12).So, using same techniques as above, for all  is m -dissipative and hence the theorem is proved.

Analyticity of the Semigroup Generated by
The purpose of this section is to prove the analyticity of the semigroup generated by For This goal we impose some additional conditions on the matrices , 1,2 k A k  .In the following we recall a theorem due to Fattorini [11].
Theorem 5.1 Let

   
, A D A be a densely defined operator in a Hilbert space such that for any  .Using the identity Using the relation Under the assumption ( 13) and the interface condition, we get We have also, for sufficiently small  , Thus the proof is achieved and the result of the Theorem is obtained.
where k c is a piecewise continuous function on k I .For more detail in perturbation theory of linear operators we refer to [7] and [13].
In the following an example is given to demonstrate the effectiveness of our results.The end points conditions are taken to be 1, 1, 0, 1, 1, 0 for some real constants Then for all 0 u H  , the above evolution partial differential system has a unique solution which is analytic in time for 0 t  .The following functions are a concrete example for the above system.
x a x t t b x x t Recall first the definition due to Pazy[3].Definition 4.1 A linear closed densely defined operator

)
is densely defined, then from Theorem 5.1, to show that it generates an analytic semigroup, it suffices to verify that Re , an analytic semigroup for all L -Bounded operatorsB .In particular the result remains true if we choose easily to verify that the conditions of Theorem 5.2 are fulfilled for the operator