Maximal Regularity for Flexible Structural Systems in Lebesgue Spaces

We study abstract equations of the form λu′′′ t u′′ t c2Au t c2μAu′ t f t , 0 < λ < μ which is motivated by the study of vibrations of flexible structures possessing internal material damping. We introduce the notion of α; β; γ -regularized families, which is a particular case of a; k regularized families, and characterize maximal regularity in L-spaces based on the technique of Fourier multipliers. Finally, an application with the Dirichlet-Laplacian in a bounded smooth domain is given.


Introduction
During the last few decades, the use of flexible structural systems had steadily increased importance.The study of a flexible aerospace structure involves problems of dynamical system theory governed by partial differential equations.
We consider here the problem of characterizing L p -maximal regularity or wellposedness for a mathematical model of a flexible space structure like a thin uniform rectangular panel, for example, a solar cell array or a spacecraft with flexible attachments.This problem is motivated by both engineering and mathematical considerations.
The study of vibrations of flexible structures possessing internal material damping was first derived by Bose and Gorain 1 .The consideration of external forces leads to more general equations of the form αu t u t βAu t γAu t f t , α, β, γ ∈ R, 1.
where A is a closed linear operator acting in a Banach space X and f is an X-valued function.
We emphasize that the abstract Cauchy problem associated with 1.1 is in general ill posed; see, for example, 2 .Also it is well known that in order to analyze well-posedness, a direct approach leads to better results than those obtained by a reduction to a first-order equation.
Maximal regularity in H ölder spaces for 1.1 has been recently characterized in 3 .In case α 0, there are more literatures.For example, stability of the solution was studied by Gorain in 4 .In 5 , Gorain and Bose studied exact controllability and boundary stabilization.More recently, Batkai and Piazzera 6, page 188 have obtained the exact decay rate.We note that well-posedness in Lebesgue spaces in the case of a damped wave equation has been only recently considered by Chill and Srivastava in 7 , and in Hölder spaces by Poblete 8 .We note that the class studied in 8 includes equations with delay.In particular, well-posedness of the homogeneous abstract Cauchy problem has been observed in 9 for α 0 under certain assumptions on A.
This paper is organized as follows.Section 2, collects results essentially contained in 10 and standard literature on R-boundedness and maximal regularity see 11 and 12 .
In Section 3 we study, by an operator theoretical method, sufficient conditions for existence of solutions for 1.1 .We obtain two results: a description of the solution by means of certain regularized families Proposition 3.1 and the existence of such families in the particular case of positive self-adjoint operators Theorem 3.2 .In Section 4, we succeed in characterizing well-posedness of 1.1 in terms of R-boundedness of a resolvent set which involves A Theorem 4.2 .This will be achieved in the Lebesgue spaces L p R, X , where X is a UMD space see below the definition .The methods to obtain this goal are those incorporated in 13 where a similar problem in case of the first-order abstract Cauchy problem has been studied.Our main result Theorem 4.2 is a combination of the well-known and deep result due to Weis 14 stated in Theorem 2.8 and a direct calculation involving the parameters α, β, and γ.

2.1
In order to give an operator theoretical approach to 1.1 we introduce the following definition.
Definition 2.1.Let A be a closed and linear operator with domain D A defined on a Banach space X.One calls A the generator of an α, β, γ -regularized family {R t } t 0 ⊂ B X if the following conditions are satisfied.R1 R t is strongly continuous on R and R 0 0.

R2 R t D A
Because of the uniqueness of the Laplace transform, we note that an α, β, γregularized family corresponds to an a, k -regularized family studied in 10 .In fact, we have As in the situation of C 0 -semigroups, we have diverse relations of an α, β, γregularized family and its generator.The following result is a direct consequence of 10, Proposition 3.1 and Lemma 2.2 .Proposition 2.3.Let R t be an α, β, γ -regularized family on X with generator A. Then the following hold.
Results on perturbation, approximation, asymptotic behavior, representation, as well as ergodic-type theorems for α, β, γ -regularized families can be also deduced from the more general context of a, k -regularized families see 10, 15-18 .
We will need the following results on Laplace transform see 19, Theorem 2.5.1 and Corollary 2.5.2 for a detailed proof .

Lemma 2.4. Suppose that
Lemma 2.5.Suppose that q : C → C is holomorphic and satisfies |λq λ | |λ 2 q λ | M for all Re λ > 0. Then there exists a bounded function f ∈ C R such that q λ ∞ 0 e −λt f t dt for all Re λ > 0.
We introduce the means The notion of R-boundedness was implicitly introduced and used by Bourgain 20 and later on also by Zimmermann 21 .Explicitly it is due to Berkson and Gillespie 22 and to Clément et al. 23 .
R-boundedness clearly implies boundedness.If X Y , the notion of R-boundedness is strictly stronger than boundedness unless the underlying space is isomorphic to a Hilbert space 24, Proposition 1.17 .Some useful criteria for R-boundedness are provided in 11, 24 .
a Let S, T ⊂ B X, Y be R-bounded sets, then S T : We recall that those Banach spaces X for which the Hilbert transform is bounded on L p R, X , for some p ∈ 1, ∞ , are called UMD spaces.For more information and details on the Hilbert transform and the UMD Banach spaces we refer to 12 .Examples of UMD spaces include Hilbert spaces, Sobolev spaces when X is a UMD space, and the Schatten-von Neumann classes C p H , 1 < p < ∞ of operators on Hilbert spaces.
After these preliminaries, we state the following operator-valued Fourier multiplier theorem.It is fundamental in our treatment.A proof can be founded in 11 .
Theorem 2.8.Suppose that X is a UMD space and let 1 < p < ∞.Let M ∈ C 1 R \ {0}; B X be such that the following conditions are satisfied.
Then the operator T defined by extends to a bounded operator from L p R, X to L p R, X .

Existence of Solutions
Let α, β, γ ∈ 0, ∞ .Consider the equation Proof.Given that x ∈ D A , we obtain from Proposition 2.3 that R • x, and hence u, is of class C 3 R , X .For all x ∈ D A , we have

3.4
Since f ∈ L 1 loc R , D A 2 , from 3.2 , we have that u t , u t ∈ D A and

3.6
By the other side, for all x ∈ D A

3.9
Then B is the generator of a bounded α, β, γ -regularized family on H.

Mathematical Problems in Engineering 7
Proof.Since −B is a positive self-adjoint operator in H, the spectrum σ B is a subset of the negative real axis and the resolvent operator μ − B −1 is defined at least for all negative non real μ.Let λ ∈ C such that Re λ > 0. If Im ϕ λ / 0, then clearly ϕ λ ∈ ρ B .If Im ϕ λ 0, then we claim that Re ϕ λ > 0. In fact, for λ a bi ∈ C, a > 0, with a direct computation we obtain Re ϕ λ
Since αβ γ, we have that

3.15
On the other hand, where 1 αλ β γλ 3.17 and, by 3.14 , for all Re λ > 0. We conclude that sup Re λ>0 λ 2 Q λ < ∞.By Lemma 2.5 there exists a strongly continuous family R t such that R t K and Q λ R λ for Re λ > 0. In consequence, for all Re λ > 0 we have and, by Remark 2.2, it shows that R t is a bounded α, β, γ -regularized family generated by B.
Since it is a known fact that the Dirichlet-Laplacian operator is a self-adjoint operator on L 2 Ω and σ Δ ⊂ −∞, 0 , we obtain the following corollary.

Corollary 3.3.
Let Ω be a bounded domain in R n with smooth boundary ∂Ω, and assume that αβ γ.Then the Dirichlet-Laplacian operator Δ with domain H 2 Ω ∩ H 1 0 Ω is the generator of an α, β, γregularized family on X L 2 Ω .Remark 3.4.In Theorem 3.2 the condition αβ γ is fundamental to have ϕ λ ∈ ρ B for all λ ∈ C with Re λ > 0, which is the key in the proof.Figure 1 is the typical situation, where we have mapped by ϕ the lines Re λ 1, 2, and 3 with α 3, β 1, and γ 4. Note that in case αβ > γ it can happen that ϕ λ ∈ σ B .For example, taking α 1, β 5, and γ 1, we obtain Figure 2 of ϕ λ for Im λ ∈ R and Re λ 1

L p -Well-Posedness
Having presented preliminary material on R-boundedness and Fourier multipliers, we will now show how these tools can be used to handle 3.1 .Our main result give concrete conditions on the operator A under which 3.1 has L p -maximal regularity.The definition of L p -maximal regularity which we investigate in this section is given as follows.
The following is the main abstract result of this section.It completely characterizes the maximal regularity of solutions for 3.1 in Lebesgue spaces.Theorem 4.2.Let X be a UMD space, 1 < p < ∞, and let A be the generator of a bounded α, β, γregularized family R t .The following statements are equivalent.
Proof.i ⇒ ii .By 3.1 and Definition 4.1 together with Proposition 3.1, the convolution operator with kernel is a bounded operator from L p R, X to L p R, X .Note that the Fourier transform R ρ exists for ρ / 0 because R t is bounded and R λ Re λ > 0 can be analytically extended from Re λ > 0 to the imaginary axis.Then the symbol of this convolution operator is given by and the conclusion follows from 11, Proposition 3.17 .
ii ⇒ i .Define N ρ : 1/ β iγρ R b ρ , A and We check that the set Note that

4.6
Since the sum of R-bounded sets is R-bounded, see 11 , we obtain that {N 1 ρ } is R-bounded.
We now check that the set {ρN 1 ρ } ρ∈R\{0} is R-bounded.With a direct computation, we obtain

4.8
Since the set {ρ 3 N ρ } is R-bounded and the complex functions appearing in the above equality are bounded, we obtain the claim from the fact that the sum of R-bounded sets is again R-bounded.We employ now Theorem 2.8 to conclude that the operator T 1 defined by extends to a bounded operator from L p R, X to L p R, X .Define We will prove that the sets {N 2 ρ } ρ∈R\{0} and {ρN 2 ρ } ρ∈R\{0} are R-bounded.
In fact, note that obtaining that the set {ρN 2 ρ } ρ∈R\{0} is R-bounded.By Theorem 2.8 we conclude that the operator T 2 defined by 12 extends to a bounded operator from L p R, X to L p R, X .Finally, define The set {N 3 ρ } ρ∈R\{0} is R-bounded from hypothesis and also note that the set

4.14
Again by Theorem 2.8 we conclude that the operator T 3 defined by The proof follows by the closed-graph theorem.
As an example, we consider for A Δ the vibration equation subject to the action of an external force.Explicitly, we consider v tt t, x λv ttt t, x c 2 Δv t, x μΔv t t, x f t, x in 0, T × Ω, v t, x 0 on 0, T × Ω, v 0, x 0 in Ω, v t 0, x 0 in Ω, v tt 0, x 0 in Ω 4.18 in a smooth bounded region Ω ⊂ R n .Also, we assume that f ∈ L 2 R; L 2 R n .We have the following application in the Hilbert space setting.

1 . 3 . 1 .
u t αu t βAu t γAu t f t , 3.1 with initial conditions u 0 u 0 u 0 0, where A is the generator of an α, β, γregularized family R t .By a solution of 3.1 we understand a function u ∈ C R ; D A ∩ C 3 R ; X such that u ∈ C R ; D A and verify 3.Proposition Let R t be an α, β, γ -regularized family on X with generatorA.If f ∈ L 1 loc R , D A2 , then u t given by u t t 0 R t − s f s ds, t 0 3.2 is a solution of 3.1 .

Theorem 4 . 4 .Figure 3 Remark 4 . 5 .
Figure 3 15Mathematical Problems in Engineering 13 extends to a bounded operator from L p R, X to L p R, X .From 4.9 , 4.12 , and 4.15 and since it is clear that 3.1 has L p -maximal regularity if the convolution operator with each one of the kernelsK 1 t : AR t χ 0,∞ t , K 2 t : AR t χ 0,∞ t , K 3 t : R t χ 0,∞ t , t ∈ R, 4.16is a bounded operator from L p R, X to L p R, X see 11 , we conclude i and the proof is complete.Of course, R-boundedness in 4.1 can be replaced by boundedness in case X H is a Hilbert space.The solution u of 3.1 , under the conditions given by Theorem 4.2, satisfies the following maximal regularity property: u, u ∈ L p R ; D A and Au, Au , u , u ∈ L p R ; X .Moreover, there exists a constant C > 0 independent of f ∈ L p R ; X such that