# Maximal Regularity for Flexible Structural Systems in Lebesgue Spaces

Authors: Claudio Fernández; Carlos Lizama; Verónica Poblete
Journal: Mathematical Problems in Engineering
 (2010)

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## Abstract

We study abstract equations of the formλu′′′(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 Lp-spaces based on the technique of Fourier multipliers. Finally, an application with the Dirichlet-Laplacian in a bounded smooth domain is
given.


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## Body

## 1. 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 characterizingLp-maximal regularity (or well-posedness) 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 ofvibrations 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(1.1)αu′′′(t)+u′′(t)=βAu(t)+γAu'(t)+f(t),α,β,γ∈ℝ,
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 Hölder 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. Section2, 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 Lp(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. Preliminaries

Letα,β,γ>0 be given. In what follows we denote(2.1)k(t)=1α∫0t(t-s)e-s/αds=-α+t+αe-t/α,t∈ℝ+,a(t)=βk(t)+γα∫0te-s/αds=-(αβ-γ)+βt+(αβ-γ)e-t/α,t∈ℝ+.In order to give an operator theoretical approach to (1.1) we introduce the following definition.Definition 2.1.
LetA 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)⊂D(A) and AR(t)x=R(t)Ax for all x∈D(A),t⩾0.(R3)
The following equation holds:(2.2)R(t)x=k(t)x+∫0ta(t-s)R(s)Axds
for all x∈D(A),t⩾0. In this case, R(t) is called the (α,β,γ)-regularized family generated by A.Remark 2.2.
It is proved in [10], in the more general context of (a,k)-regularized families, that an operator A is the generator of an (α,β,γ)-regularized family if and only if there exists ω⩾0 and a strongly continuous function R:ℝ+→ℬ(X) such that   {(λ2+αλ3)/(β+γλ):Reλ>ω}⊂ρ(A) and
(2.3)H(λ)x:=1β+γλ(λ2+αλ3β+γλ-A)-1x=∫0∞e-λtR(t)xdt,Reλ>ω,x∈X.Because of the uniqueness of the Laplace transform, we note that an(α,β,γ)-regularized family corresponds to an (a,k)-regularized family studied in [10]. In fact, we have(2.4)â(λ)=β+γλλ2+αλ3,k̂(λ)=1λ2+αλ3,∀Reλ>ω.As in the situation ofC0-semigroups, we have diverse relations of an (α,β,γ)-regularized family and its generator. The following result is a direct consequence of [10, Proposition 3.1 and Lemma 2.2].Proposition 2.3.
LetR(t) be an (α,β,γ)-regularized family on X with generator A. Then the following hold.(a)
For allx∈D(A) one has R(·)x∈C3(ℝ+;X).(b)
Letx∈X and t⩾0. Then ∫0ta(t-s)R(s)xds∈D(A) and(2.5)R(t)x=k(t)x+A∫0ta(t-s)R(s)xds.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 thatq:ℂ+→ℂ is holomorphic and satisfies supReλ>0|λq(λ)|<∞ and let b>0. Then there exists f∈C(ℝ+) with supt>0|e-ωtt-bf(t)|<∞ such that q(λ)=λb∫0∞e-λtf(t)dt for all Reλ>0.Lemma 2.5.
Suppose thatq:ℂ+→ℂ is holomorphic and satisfies |λq(λ)|+|λ2q′(λ)|⩽M for all Reλ>0. Then there exists a bounded function f∈C(ℝ+) such that q(λ)=∫0∞e-λtf(t)dt for all Reλ>0.We introduce the means(2.6)∥(x1,…,xn)∥R:=12n∑ϵj∈{-1,1}n∥∑j=1nϵjxj∥
for x1,…,xn∈X.Definition 2.6.
LetX, Y be Banach spaces. A subset 𝒯 of ℬ(X,Y) is called R-bounded if there exists a constant c⩾0 such that
(2.7)∥(T1x1,…,Tnxn)∥R⩽c∥(x1,…,xn)∥R
for all T1,…,Tn∈𝒯,x1,…,xn∈X,n∈ℕ. The least c such that (2.7) is satisfied is called the R-bound of 𝒯 and is denoted as R(𝒯).The notion ofR-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].Remark 2.7.
(a)
Let𝒮,𝒯⊂ℬ(X,Y) be R-bounded sets, then 𝒮+𝒯:={S+T:S∈𝒮,T∈𝒯} is R-bounded.(b)
Let𝒯⊂ℬ(X,Y) and 𝒮⊂ℬ(Y,Z) be R-bounded sets, then 𝒮·𝒯:={S·T:S∈𝒮,T∈𝒯}⊂ℬ(X,Z) is R-bounded and
(2.8)R(𝒮·𝒯)⩽R(𝒮)·R(𝒯).(c)
Also, each subsetM⊂ℬ(X) of the form M={λI:λ∈Ω} is R-bounded whenever Ω⊂ℂ is bounded.We recall that those Banach spacesX for which the Hilbert transform is bounded on Lp(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 Wps(Ω),1<p<∞ (see [25]), Lebesgue spaces Lp(Ω,μ),1<p<∞, Lp(Ω,μ;X), 1<p<∞, when X is a UMD space, and the Schatten-von Neumann classes Cp(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 thatX is a UMD space and let 1<p<∞. Let M∈C1(ℝ∖{0};ℬ(X)) be such that the following conditions are satisfied. (i)
The set{M(ρ)}ρ∈ℝ∖{0} is R-bounded.(ii)
The set{ρM'(ρ)}ρ∈ℝ∖{0} is R-bounded.
Then the operatorT defined by
(2.9)Tf=(M(·)[f̂(·)])∨wheref∈𝒮(X)
extends to a bounded operator from Lp(ℝ,X) to Lp(ℝ,X).


## 3. Existence of Solutions

Letα,β,γ∈(0,∞). Consider the equation(3.1)u′′(t)+αu′′′(t)=βAu(t)+γAu'(t)+f(t),
with initial conditions u(0)=u'(0)=u′′(0)=0, where A is the generator of an (α,β,γ)-regularized family R(t). By a solution of (3.1) we understand a function u∈C(ℝ+;D(A))∩C3(ℝ+;X) such that u'∈C(ℝ+;D(A)) and verify (3.1).Proposition 3.1.
LetR(t) be an (α,β,γ)-regularized family on X with generator A. If f∈Lloc1(ℝ+,D(A2)), then u(t) given by
(3.2)u(t)=∫0tR(t-s)f(s)ds,t⩾0
is a solution of (3.1).Proof.
Given thatx∈D(A), we obtain from Proposition 2.3 that R(·)x, and hence u, is of class C3(ℝ+,X). For all x∈D(A), we have
(3.3)R'(t)x=(1-e-t/α)x+∫0t[β+(γ/α-β)e-(t-s)/α]R(s)Axds.
If x∈D(A2), then R'(t)x∈D(A). Moreover,
(3.4)R′′(t)x=1αe-t/αx+γαR(t)Ax+∫0t(βα-γα2)e-(t-s)/αAR(s)xds,R′′′(t)x=-1α2e-t/α+γαR'(t)Ax+βαR(t)Ax-γα2AR(t)x+∫0t(γα3-βα2)e-(t-s)/αAR(s)xds.
Since f∈Lloc1(ℝ+,D(A2)), from (3.2), we have that u(t),u'(t)∈D(A) and
(3.5)u'(t)=∫0tR′(t-s)f(s)ds,u′′(t)=∫0tR′′(t-s)f(s)ds,u′′′(t)=R′′(0)f(t)+∫0tR′′′(t-s)f(s)ds.
Hence,
(3.6)u′′(t)+αu′′′(t)-βAu(t)-γAu'(t)=∫0tR′′(t-s)f(s)ds+f(t)+α∫0tR′′′(t-s)f(s)ds-βA∫0tR(t-s)f(s)ds-γA∫0tR'(t-s)f(s)ds.
By the other side, for all x∈D(A2), we obtain
(3.7)R′′(t)x+αR′′′(t)x-βAR(t)x-γAR'(t)x=1αe-t/αx+γαAR(t)x+∫0t(βα-γα2)e-(t-s)/αAR(s)xds-1αe-t/α+γAR'(t)x+βAR(t)x-γαAR(t)x+∫0t(γα2-βα)e-(t-s)/αAR(s)xds-βAR(t)x-γAR'(t)x=0.
Since f(t)∈D(A2) and A is closed, from (3.6) we conclude that u(t) verify (3.1).The following remarkable result provides a wide class of generators of(α,β,γ)-regularized families. In what follows we denote(3.8)φ(λ):=1â(λ)=λ2(1+αλ)β+γλ.Theorem 3.2.
Let-B be a positive self-adjoint operator on a Hilbert space H such that
(3.9)αβ⩽γ.
Then B is the generator of a bounded (α,β,γ)-regularized family on H.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 Reλ>0. If Imφ(λ)≠0, then clearly φ(λ)∈ρ(B). If Imφ(λ)=0, then we claim that Reφ(λ)>0. In fact, for λ=a+bi∈ℂ,a>0, with a direct computation we obtain
(3.10)Reφ(λ)=(a2-b2)(1+αa)(β+γa)-2ab2α(β+γa)+αγb2(a2-b2)+2ab2(1+αa)(β+γa)2+γ2b2,Imφ(λ)=αb(a2-b2)(β+γa)+2ab(1+αa)(β+γa)-γb(a2-b2)(1+αa)+2ab3αγ(β+γa)2+γ2b2.
Note thatImφ(λ)=0 if and only if b=0 or α(a2-b2)(β+γa)+2a(1+αa)(β+γa)-γ(a2-b2)(1+αa)+2ab2αγ=0.
Sinceαβ⩽γ, we have that
(3.11)α(a2-b2)(β+γa)+2a(1+αa)(β+γa)-γ(a2-b2)(1+αa)+2ab2αγ=2αγab2+b2(γ-αβ)+γa2+3αβa2+2βa+2αγa2>0.
Hence, Imφ(λ)=0 if and only if b=0. Since a>0, a direct calculation gives
(3.12)Reφ(λ)=a2(1+αa)β+γa>0,
proving the claim. We conclude that φ(λ)∈ρ(B) for all Reλ>0. Hence (see Kato [26, Section V.3.5]),
(3.13)∥(φ(λ)-B)-1∥=1dist(φ(λ),σ(B))∀Reλ>0.
Note that
(3.14)supReλ>0(|λ|2+1dist(φ(λ),σ(B)))<M,
since dist(φ(λ),σ(B)) has order |λ|2. Define Q(λ)=(1/(β+γλ))(φ(λ)-B)-1. We have by (3.14) and (3.13) that for all Reλ>0(3.15)∥λQ(λ)∥=∥λ(β+γλ)(φ(λ)-B)-1∥⩽|λ||β+γλ|1dist(φ(λ),σ(B))<M.
On the other hand,
(3.16)λ2Q'(λ)=-γλβ+γλ[λQ(λ)]+[λQ(λ)][λ2(φ(λ)-B)-1][λâ(λ)′â(λ)]1λ2â(λ),
where
(3.17)1λ2â(λ)=1+αλβ+γλ,λâ(λ)'â(λ)=-2αγλ2+(γ+3αβ)λ+2β(1+αλ)(β+γλ)
and, by (3.14),
(3.18)∥λ2(φ(λ)-B)-1∥⩽|λ2|dist(φ(λ),σ(B))<M
for all Reλ>0. We conclude that supReλ>0∥λ2Q'(λ)∥<∞.
By Lemma2.5 there exists a strongly continuous family R(t) such that ∥R(t)∥⩽K and Q(λ)=R̂(λ) for Reλ>0. In consequence, for all Reλ>0 we have
(3.19)R̂(λ)=φ(λ)λ2(1+αλ)(φ(λ)-B)-1=1β+γλ(λ2+αλ3β+γλ-B)-1,
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 onL2(Ω) and σ(Δ)⊂(-∞,0), we obtain the following corollary.Corollary 3.3.
LetΩ be a bounded domain in ℝn with smooth boundary ∂Ω, and assume that αβ⩽γ. Then the Dirichlet-Laplacian operator Δ with domain H2(Ω)∩H01(Ω) is the generator of an (α,β,γ)-regularized family on X=L2(Ω).Remark 3.4.
In Theorem3.2 the condition αβ⩽γ is fundamental to have φ(λ)∈ρ(B) for all λ∈ℂ with Reλ>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 caseαβ>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)=1Figure 1Figure 2


## 4.Lp-Well-Posedness

Having presented preliminary material onR-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 Lp-maximal regularity.The definition ofLp-maximal regularity which we investigate in this section is given as follows.Definition 4.1.
One says that (3.1) has Lp-maximal regularity (or is Lp-well posed) on ℝ+ if for each f∈Lp(ℝ+,X) there is a unique function u∈W3,p(ℝ+,X)∩W1,p(ℝ+,[D(A)])∩Wp(ℝ+,[D(A)]) such that (3.1) holds a.e.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.
LetX be a UMD space, 1<p<∞, and let A be the generator of a bounded (α,β,γ)-regularized family R(t). The following statements are equivalent. (i)
Equation (3.1) has Lp-maximal regularity on ℝ+.(ii)
b(ρ):=-ρ2((1+iαρ)/(β+iγρ))∈ρ(A) for all ρ∈ℝ∖{0} and the set(4.1){ρ3β+iγρR(b(ρ),A)}ρ∈ℝ∖{0}isR-bounded.Proof.
(i)⇒(ii). By (3.1) and Definition 4.1 together with Proposition 3.1, the convolution operator with kernel
(4.2)K4(t):=R′′′(t)χ(0,∞)(t),t∈ℝ,
is a bounded operator from Lp(ℝ,X) to Lp(ℝ,X). Note that the Fourier transform R̃(ρ) exists for ρ≠0 because R(t) is bounded and R̂(λ)(Reλ>0) can be analytically extended from Reλ>0 to the imaginary axis. Then the symbol of this convolution operator is given by
(4.3)M(ρ)=ρ3β+iγρR(b(ρ),A),ρ∈ℝ∖{0},
and the conclusion follows from [11, Proposition 3.17].
(ii)⇒(i). Define N(ρ):=(1/(β+iγρ))R(b(ρ),A) and
(4.4)N1(ρ):=AN(ρ).
We check that the set {N1(ρ)}ρ∈ℝ∖{0} is R-bounded.
Since(b(ρ)-A)R(b(ρ),A)=I, we have that AR(b(ρ),A)=b(ρ)R(b(ρ),A)-I. Replacing in (4.4)
(4.5)N1(ρ)=b(ρ)β+iγρR(b(ρ),A)-1β+iγρI=-1+iαρβ+iγρρ2N(ρ)-1β+iγρI.
Note that
(4.6)|1+iαρβ+iγρ|2=1+α2ρ2β2+γ2ρ2<1β2+α2γ2,|1β+iγρ|2=1β2+γ2ρ2<1β2.
Since the sum ofR-bounded sets is R-bounded, see [11], we obtain that {N1(ρ)} is R-bounded.
We now check that the set{ρN1′(ρ)}ρ∈ℝ∖{0} is R-bounded. With a direct computation, we obtain
(4.7)N1′(ρ)=b'(ρ)N(ρ)+b(ρ)N'(ρ)+iγ(β+iγρ)2I=2αγ(β+iγρ)2ρ3N(ρ)-γ+3αβ(β+iγρ)2iρ2N(ρ)-2β(β+iγρ)2ρN(ρ)-iγβ+iγρb(ρ)N(ρ)-2αγρ2-(γ+3αβ)iρ-2ββ+iγρρb(ρ)N(ρ)N(ρ)+iγ(β+iγρ)2I.
Hence
(4.8)ρN1′(ρ)=2αγρ(β+iγρ)2ρ3N(ρ)-γ+3αβ(β+iγρ)2iρ3N(ρ)-2β(β+iγρ)2ρ2N(ρ)+iγ-αγρ(β+iγρ)2ρ3N(ρ)+(2αγρ2-(γ+3αβ)iρ-2β)1+iαρ(β+iγρ)2ρ4N(ρ)N(ρ)+iγρ(β+iγρ)2I=2αγρ(β+iγρ)2ρ3N(ρ)-γ+3αβ(β+iγρ)2iρ3N(ρ)-2β(β+iγρ)2ρ2N(ρ)+iγ-αγρ(β+iγρ)2ρ3N(ρ)+2αγ1+iαρ(β+iγρ)2ρ3N(ρ)ρ3N(ρ)-(γ+3αβ)i-αρ(β+iγρ)2ρ3N(ρ)ρ2N(ρ)-2β1+iαρβ+iγρρ3N(ρ)ρN(ρ)+iγρ(β+iγρ)2I=αγρ-3αβi(β+iγρ)2ρ3N(ρ)-2β(β+iγρ)2ρ2N(ρ)+2αγ1+iαρ(β+iγρ)2ρ3N(ρ)ρ3N(ρ)-(γ+3αβ)i-αρ(β+iγρ)2ρ3N(ρ)ρ2N(ρ)-2β1+iαρβ+iγρρ3N(ρ)ρN(ρ)+iγρ(β+iγρ)2I.
Since the set {ρ3N(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 T1 defined by
(4.9)T1f=(N1(·)[f̂(·)])∨wheref∈𝒮(X)
extends to a bounded operator from Lp(ℝ,X) to Lp(ℝ,X).
Define
(4.10)N2(ρ):=ρβ+iγρAR(b(ρ),A).
We will prove that the sets {N2(ρ)}ρ∈ℝ∖{0} and {ρN2′(ρ)}ρ∈ℝ∖{0} are R-bounded.
In fact, note thatN2(ρ)=ρN1(ρ)=-((1+iαρ)/(β+iγρ))ρ3N(ρ)-(ρ/(β+iγρ))I. Hence the set {N2(ρ)} is R-bounded. Moreover, we have
(4.11)ρN2′(ρ)=ρ2N1′(ρ)+ρN1(ρ)=αγρ2-3αβρi(β+iγρ)2ρ3N(ρ)-2β(β+iγρ)2ρ3N(ρ)+2αγρ+iαρ2(β+iγρ)2ρ3N(ρ)ρ3N(ρ)-(γ+3αβ)i-αρ(β+iγρ)2ρ3N(ρ)ρ3N(ρ)-2β1+iαρβ+iγρρ3N(ρ)ρ2N(ρ)+iγρ2(β+iγρ)2I+N2(ρ),
obtaining that the set {ρN2′(ρ)}ρ∈ℝ∖{0} is R-bounded. By Theorem 2.8 we conclude that the operator T2 defined by
(4.12)T2f=(N2(·)[f̂(·)])∨wheref∈𝒮(X)
extends to a bounded operator from Lp(ℝ,X) to Lp(ℝ,X).
Finally, define
(4.13)N3(ρ):=ρ2β+iγρR(b(ρ),A)=ρ2N(ρ).
The set {N3(ρ)}ρ∈ℝ∖{0} is R-bounded from hypothesis and also note that the set {ρN3′(ρ)}ρ∈ℝ∖{0} is R-bounded, since
(4.14)ρN3′(ρ)=2ρ2N(ρ)-iγβ+iγρρ3N(ρ)-2αγβ+iγρρ3N(ρ)ρ3N(ρ)+γ+3αββ+iγρiρ3N(ρ)ρ2N(ρ)+2ββ+iγρρN(ρ)ρ3N(ρ).
Again by Theorem 2.8 we conclude that the operator T3 defined by
(4.15)T3f=(N3(·)[f̂(·)])∨wheref∈𝒮(X)
extends to a bounded operator from Lp(ℝ,X) to Lp(ℝ,X). From (4.9), (4.12), and (4.15) and since it is clear that (3.1) has Lp-maximal regularity if the convolution operator with each one of the kernels
(4.16)K1(t)≔AR(t)χ(0,∞)(t),K2(t)≔AR′(t)χ(0,∞)(t),K3(t)≔R′′(t)χ(0,∞)(t),t∈ℝ,
is a bounded operator from Lp(ℝ,X) to Lp(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.Corollary 4.3.
The solutionu of (3.1), under the conditions given by Theorem 4.2, satisfies the following maximal regularity property: u,u'∈Lp(ℝ+;[D(A)]) and Au,Au',u′′,u′′′∈Lp(ℝ+;X). Moreover, there exists a constant C>0 independent of f∈Lp(ℝ+;X) such that
(4.17)∥u∥p+∥u'∥p+∥u′′∥p+∥u′′′∥p+∥Au∥p+∥Au′∥p+⩽C∥f∥p.The proof follows by the closed-graph theorem.As an example, we consider forA=Δ the vibration equation subject to the action of an external force. Explicitly, we consider(4.18)vtt(t,x)+λvttt(t,x)=c2(Δv(t,x)+μΔvt(t,x))+f(t,x)in]0,T]×Ω,v(t,x)=0on]0,T]×Ω,v(0,x)=0inΩ,vt(0,x)=0inΩ,vtt(0,x)=0inΩ
in a smooth bounded region Ω⊂ℝn. Also, we assume that f∈L2(ℝ;L2(ℝn)). We have the following application in the Hilbert space setting.Theorem 4.4.
LetΩ be a bounded domain with smooth boundary ∂Ω in ℝn. Suppose that 0<λ<μ. Then the initial value problem (4.18) defined on L2(Ω) with Dirichlet boundary conditions has L2-maximal regularity on ℝ+.Proof.
Letα=λ, β=c2, and γ=c2μ and note that αβ<γ if and only if λ<μ. By Corollary 3.3 we conclude that Δ generates a bounded (α,β,γ)-regularized family on L2(Ω).
Note that we haveb(ρ)=-ρ2((1+iαρ)/(β+iγρ))∈ρ(Δ) and there exists a constant C>0 such that
(4.19)∥ρ3β+iγρR(b(ρ),Δ)∥=∥ρb(ρ)1+iαρ(b(ρ)-Δ)-1∥=|ρ||1+iαρ||b(ρ)|dist(b(ρ),λ1(Ω))⩽C,
for all ρ∈ℝ. Here λ1(Ω) is the first eigenvalue of the Dirichlet-Laplacian. Hence, by Theorem 4.2 the assertion follows.Remark 4.5.
In Figure3, we show b(ρ) in case λ=3, μ=4, and c2=1.Figure 3



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*Source: 196956-2010-04-14.xml*