Maximal Regularity for Flexible Structural Systems in Lebesgue Spaces

Abstract

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^𝑝-spaces based on the technique of Fourier multipliers. Finally, an application with the Dirichlet-Laplacian in a bounded smooth domain is given.

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 characterizing 𝐿^𝑝-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 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, (1.1) 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 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. 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. Preliminaries

Let 𝛼,𝛽,𝛾>0 be given. In what follows we denote

1𝑘(𝑡)=𝛼^𝑡₀(𝑡−𝑠)𝑒^−𝑠/𝛼𝑑𝑠=−𝛼+𝑡+𝛼𝑒^−𝑡/𝛼,𝑡∈ℝ₊,𝛾𝑎(𝑡)=𝛽𝑘(𝑡)+𝛼^𝑡₀𝑒^−𝑠/𝛼𝑑𝑠=−(𝛼𝛽−𝛾)+𝛽𝑡+(𝛼𝛽−𝛾)𝑒^−𝑡/𝛼,𝑡∈ℝ₊. (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)⊂𝐷(𝐴) and 𝐴𝑅(𝑡)𝑥=𝑅(𝑡)𝐴𝑥 for all 𝑥∈𝐷(𝐴),𝑡⩾0.(R3) The following equation holds:𝑅(𝑡)𝑥=𝑘(𝑡)𝑥+^𝑡₀𝑎(𝑡−𝑠)𝑅(𝑠)𝐴𝑥𝑑𝑠 (2.2) 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 1𝐻(𝜆)𝑥∶=(𝜆𝛽+𝛾𝜆²+𝛼𝜆³𝛽+𝛾𝜆−𝐴)⁻¹𝑥=^∞₀𝑒^−𝜆𝑡𝑅(𝑡)𝑥𝑑𝑡,Re𝜆>𝜔,𝑥∈𝑋. (2.3)

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

𝑎(𝜆)=𝛽+𝛾𝜆𝜆²+𝛼𝜆³,1𝑘(𝜆)=𝜆²+𝛼𝜆³,∀Re𝜆>𝜔. (2.4)

As in the situation of 𝐶₀-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. Let 𝑅(𝑡) be an (𝛼,𝛽,𝛾)-regularized family on 𝑋 with generator 𝐴. Then the following hold.(a)For all 𝑥∈𝐷(𝐴) one has 𝑅(⋅)𝑥∈𝐶³(ℝ₊;𝑋).(b)Let 𝑥∈𝑋 and 𝑡⩾0. Then ∫^𝑡₀𝑎(𝑡−𝑠)𝑅(𝑠)𝑥𝑑𝑠∈𝐷(𝐴) and 𝑅(𝑡)𝑥=𝑘(𝑡)𝑥+𝐴^𝑡₀𝑎(𝑡−𝑠)𝑅(𝑠)𝑥𝑑𝑠. (2.5)

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 𝑞∶ℂ₊→ℂ is holomorphic and satisfies sup_Re𝜆>0|𝜆𝑞(𝜆)|<∞ and let 𝑏>0. Then there exists 𝑓∈𝐶(ℝ₊) with sup_𝑡>0|𝑒^−𝜔𝑡𝑡^−𝑏𝑓(𝑡)|<∞ such that 𝑞(𝜆)=𝜆^𝑏∫^∞₀𝑒^−𝜆𝑡𝑓(𝑡)𝑑𝑡 for all Re𝜆>0.

Lemma 2.5. Suppose that 𝑞∶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)=∫^∞₀𝑒^−𝜆𝑡𝑓(𝑡)𝑑𝑡 for all Re𝜆>0.

We introduce the means

‖(𝑥₁,…,𝑥_𝑛)‖_𝑅1∶=2^𝑛_{𝜖𝑗∈{−1,1}𝑛}‖^𝑛_𝑗=1𝜖_𝑗𝑥_𝑗‖ (2.6) for 𝑥₁,…,𝑥_𝑛∈𝑋.

Definition 2.6. Let 𝑋, 𝑌 be Banach spaces. A subset 𝒯 of ℬ(𝑋,𝑌) is called 𝑅-bounded if there exists a constant 𝑐⩾0 such that ‖(𝑇₁𝑥₁,…,𝑇_𝑛𝑥_𝑛)‖_𝑅⩽𝑐‖(𝑥₁,…,𝑥_𝑛)‖_𝑅 (2.7) for all 𝑇₁,…,𝑇_𝑛∈𝒯,𝑥₁,…,𝑥_𝑛∈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 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].

Remark 2.7. (a)Let 𝒮,𝒯⊂ℬ(𝑋,𝑌) be 𝑅-bounded sets, then 𝒮+𝒯∶={𝑆+𝑇∶𝑆∈𝒮,𝑇∈𝒯} is 𝑅-bounded.(b)Let 𝒯⊂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 𝑅(𝒮⋅𝒯)⩽𝑅(𝒮)⋅𝑅(𝒯). (2.8)(c)Also, each subset 𝑀⊂ℬ(𝑋) of the form 𝑀={𝜆𝐼∶𝜆∈Ω} is 𝑅-bounded whenever Ω⊂ℂ is bounded.

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 𝑊^𝑠_𝑝(Ω),1<𝑝<∞ (see [25]), Lebesgue spaces 𝐿^𝑝(Ω,𝜇),1<𝑝<∞, 𝐿^𝑝(Ω,𝜇;𝑋), 1<𝑝<∞, 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∈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 operator 𝑇 defined by
𝑇𝑓=(𝑀(⋅)[𝑓(⋅)])^∨where𝑓∈𝒮(𝑋) (2.9) extends to a bounded operator from 𝐿^𝑝(ℝ,𝑋) to 𝐿^𝑝(ℝ,𝑋).

3. Existence of Solutions

Let 𝛼,𝛽,𝛾∈(0,∞). Consider the equation

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,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. Let 𝑅(𝑡) be an (𝛼,𝛽,𝛾)-regularized family on 𝑋 with generator 𝐴. If 𝑓∈𝐿¹_loc(ℝ₊,𝐷(𝐴²)), then 𝑢(𝑡) given by 𝑢(𝑡)=^𝑡₀𝑅(𝑡−𝑠)𝑓(𝑠)𝑑𝑠,𝑡⩾0 (3.2) is a solution of (3.1).

Proof. Given that 𝑥∈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 𝑅^′(𝑡)𝑥=(1−𝑒^−𝑡/𝛼)𝑥+^𝑡₀[𝛽+(𝛾/𝛼−𝛽)𝑒^{−(𝑡−𝑠)/𝛼}]𝑅(𝑠)𝐴𝑥𝑑𝑠. (3.3) If 𝑥∈𝐷(𝐴²), then 𝑅^′(𝑡)𝑥∈𝐷(𝐴). Moreover, 𝑅^′′1(𝑡)𝑥=𝛼𝑒^−𝑡/𝛼𝛾𝑥+𝛼𝑅(𝑡)𝐴𝑥+^𝑡₀(𝛽𝛼−𝛾𝛼²)𝑒^{−(𝑡−𝑠)/𝛼}𝑅𝐴𝑅(𝑠)𝑥𝑑𝑠,_′′′1(𝑡)𝑥=−𝛼²𝑒^−𝑡/𝛼+𝛾𝛼𝑅^′𝛽(𝑡)𝐴𝑥+𝛼𝛾𝑅(𝑡)𝐴𝑥−𝛼²𝐴𝑅(𝑡)𝑥+^𝑡₀(𝛾𝛼³−𝛽𝛼²)𝑒^{−(𝑡−𝑠)/𝛼}𝐴𝑅(𝑠)𝑥𝑑𝑠. (3.4) Since 𝑓∈𝐿¹_loc(ℝ₊,𝐷(𝐴²)), from (3.2), we have that 𝑢(𝑡),𝑢^′(𝑡)∈𝐷(𝐴) and 𝑢^′(𝑡)=^𝑡₀𝑅^′𝑢(𝑡−𝑠)𝑓(𝑠)𝑑𝑠,_′′(𝑡)=^𝑡₀𝑅^′′𝑢(𝑡−𝑠)𝑓(𝑠)𝑑𝑠,_′′′(𝑡)=𝑅^′′(0)𝑓(𝑡)+^𝑡₀𝑅^′′′(𝑡−𝑠)𝑓(𝑠)𝑑𝑠. (3.5) Hence, 𝑢^′′(𝑡)+𝛼𝑢^′′′(𝑡)−𝛽𝐴𝑢(𝑡)−𝛾𝐴𝑢^′=(𝑡)_𝑡0𝑅^′′(𝑡−𝑠)𝑓(𝑠)𝑑𝑠+𝑓(𝑡)+𝛼^𝑡₀𝑅^′′′(𝑡−𝑠)𝑓(𝑠)𝑑𝑠−𝛽𝐴^𝑡₀𝑅(𝑡−𝑠)𝑓(𝑠)𝑑𝑠−𝛾𝐴^𝑡₀𝑅^′(𝑡−𝑠)𝑓(𝑠)𝑑𝑠. (3.6) By the other side, for all 𝑥∈𝐷(𝐴²), we obtain 𝑅^′′(𝑡)𝑥+𝛼𝑅^′′′(𝑡)𝑥−𝛽𝐴𝑅(𝑡)𝑥−𝛾𝐴𝑅^′=1(𝑡)𝑥𝛼𝑒^−𝑡/𝛼𝛾𝑥+𝛼𝐴𝑅(𝑡)𝑥+^𝑡₀(𝛽𝛼−𝛾𝛼²)𝑒^{−(𝑡−𝑠)/𝛼}1𝐴𝑅(𝑠)𝑥𝑑𝑠−𝛼𝑒^−𝑡/𝛼+𝛾𝐴𝑅^′𝛾(𝑡)𝑥+𝛽𝐴𝑅(𝑡)𝑥−𝛼𝐴𝑅(𝑡)𝑥+^𝑡₀(𝛾𝛼²−𝛽𝛼)𝑒^{−(𝑡−𝑠)/𝛼}𝐴𝑅(𝑠)𝑥𝑑𝑠−𝛽𝐴𝑅(𝑡)𝑥−𝛾𝐴𝑅^′(𝑡)𝑥=0. (3.7) 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

1𝜑(𝜆)∶==𝜆𝑎(𝜆)²(1+𝛼𝜆).𝛽+𝛾𝜆 (3.8)

Theorem 3.2. Let −𝐵 be a positive self-adjoint operator on a Hilbert space 𝐻 such that 𝛼𝛽⩽𝛾. (3.9) 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 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. (3.10)
Note that 𝐼𝑚𝜑(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.
Since 𝛼𝛽⩽𝛾, we have that
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. (3.11) Hence, 𝐼𝑚𝜑(𝜆)=0 if and only if 𝑏=0. Since 𝑎>0, a direct calculation gives 𝑎Re𝜑(𝜆)=²(1+𝛼𝑎)𝛽+𝛾𝑎>0, (3.12) proving the claim. We conclude that 𝜑(𝜆)∈𝜌(𝐵) for all Re𝜆>0. Hence (see Kato [26, Section V.3.5]), ‖(𝜑(𝜆)−𝐵)⁻¹1‖=dist(𝜑(𝜆),𝜎(𝐵))∀Re𝜆>0. (3.13) Note that sup_Re𝜆>0(|𝜆|²+1dist(𝜑(𝜆),𝜎(𝐵)))<𝑀, (3.14) 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>0𝜆‖𝜆𝑄(𝜆)‖=‖(𝛽+𝛾𝜆)(𝜑(𝜆)−𝐵)⁻¹‖⩽|𝜆|1|𝛽+𝛾𝜆|dist(𝜑(𝜆),𝜎(𝐵))<𝑀. (3.15) On the other hand, 𝜆²𝑄^′(𝜆)=−𝛾𝜆𝛽+𝛾𝜆[𝜆𝑄(𝜆)]+[𝜆𝑄(𝜆)][𝜆²(𝜑(𝜆)−𝐵)⁻¹][𝜆𝑎(𝜆)^′]1𝑎(𝜆)𝜆²,𝑎(𝜆) (3.16) where 1𝜆²=𝑎(𝜆)1+𝛼𝜆𝛽+𝛾𝜆,𝜆𝑎(𝜆)^′𝑎(𝜆)=−2𝛼𝛾𝜆²+(𝛾+3𝛼𝛽)𝜆+2𝛽(1+𝛼𝜆)(𝛽+𝛾𝜆) (3.17) and, by (3.14), ‖𝜆²(𝜑(𝜆)−𝐵)⁻¹‖⩽|𝜆²|dist(𝜑(𝜆),𝜎(𝐵))<𝑀 (3.18) for all Re𝜆>0. We conclude that sup_Re𝜆>0‖𝜆²𝑄^′(𝜆)‖<∞.
By Lemma 2.5 there exists a strongly continuous family 𝑅(𝑡) such that ‖𝑅(𝑡)‖⩽𝐾 and 𝑄(𝜆)=𝑅(𝜆) for Re𝜆>0. In consequence, for all Re𝜆>0 we have
𝑅(𝜆)=𝜑(𝜆)𝜆²(1+𝛼𝜆)(𝜑(𝜆)−𝐵)⁻¹=1(𝜆𝛽+𝛾𝜆²+𝛼𝜆³𝛽+𝛾𝜆−𝐵)⁻¹, (3.19) 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 𝐿²(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 𝐻²(Ω)∩𝐻¹₀(Ω) is the generator of an (𝛼,𝛽,𝛾)-regularized family on 𝑋=𝐿²(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 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 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

Figure 1 

Figure 2 

4. 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.

Definition 4.1. One says that (3.1) has 𝐿^𝑝-maximal regularity (or is 𝐿^𝑝-well posed) on ℝ₊ if for each 𝑓∈𝐿^𝑝(ℝ₊,𝑋) there is a unique function 𝑢∈𝑊^3,𝑝(ℝ₊,𝑋)∩𝑊^1,𝑝(ℝ₊,[𝐷(𝐴)])∩𝑊^𝑝(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. Let 𝑋 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 𝐿^𝑝-maximal regularity on ℝ₊.(ii)𝑏(𝜌)∶=−𝜌²((1+𝑖𝛼𝜌)/(𝛽+𝑖𝛾𝜌))∈𝜌(𝐴) for all 𝜌∈ℝ⧵{0} and the set{𝜌³𝛽+𝑖𝛾𝜌𝑅(𝑏(𝜌),𝐴)}_{𝜌∈ℝ⧵{0}}is𝑅-bounded. (4.1)

Proof. (i)⇒(ii). By (3.1) and Definition 4.1 together with Proposition 3.1, the convolution operator with kernel 𝐾₄(𝑡)∶=𝑅^′′′(𝑡)𝜒_(0,∞)(𝑡),𝑡∈ℝ, (4.2) 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)(Re𝜆>0) can be analytically extended from Re𝜆>0 to the imaginary axis. Then the symbol of this convolution operator is given by 𝜌𝑀(𝜌)=³𝛽+𝑖𝛾𝜌𝑅(𝑏(𝜌),𝐴),𝜌∈ℝ⧵{0}, (4.3) and the conclusion follows from [11, Proposition 3.17].
(ii)⇒(i). Define 𝑁(𝜌)∶=(1/(𝛽+𝑖𝛾𝜌))𝑅(𝑏(𝜌),𝐴) and
𝑁₁(𝜌)∶=𝐴𝑁(𝜌). (4.4) 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)
𝑁₁(𝜌)=𝑏(𝜌)1𝛽+𝑖𝛾𝜌𝑅(𝑏(𝜌),𝐴)−𝛽+𝑖𝛾𝜌𝐼=−1+𝑖𝛼𝜌𝜌𝛽+𝑖𝛾𝜌²1𝑁(𝜌)−𝛽+𝑖𝛾𝜌𝐼. (4.5) Note that |1+𝑖𝛼𝜌|𝛽+𝑖𝛾𝜌²=1+𝛼²𝜌²𝛽²+𝛾²𝜌²<1𝛽²+𝛼²𝛾²,|1|𝛽+𝑖𝛾𝜌²=1𝛽²+𝛾²𝜌²<1𝛽². (4.6)
Since the sum of 𝑅-bounded sets is 𝑅-bounded, see [11], we obtain that {𝑁₁(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
𝑁^′₁(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)²−𝜌𝑁(𝜌)𝑖𝛾𝛽+𝑖𝛾𝜌𝑏(𝜌)𝑁(𝜌)−2𝛼𝛾𝜌²−(𝛾+3𝛼𝛽)𝑖𝜌−2𝛽𝛽+𝑖𝛾𝜌𝜌𝑏(𝜌)𝑁(𝜌)𝑁(𝜌)+𝑖𝛾(𝛽+𝑖𝛾𝜌)²𝐼. (4.7)
Hence
𝜌𝑁^′₁(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+𝑖𝛼𝜌𝜌𝛽+𝑖𝛾𝜌³𝑁(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+𝑖𝛼𝜌𝜌𝛽+𝑖𝛾𝜌³𝑁(𝜌)𝜌𝑁(𝜌)+𝑖𝛾𝜌(𝛽+𝑖𝛾𝜌)²𝐼. (4.8) 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 𝑇₁𝑓=(𝑁₁(⋅)[𝑓(⋅)])^∨where𝑓∈𝒮(𝑋) (4.9) extends to a bounded operator from 𝐿^𝑝(ℝ,𝑋) to 𝐿^𝑝(ℝ,𝑋).
Define
𝑁₂𝜌(𝜌)∶=𝛽+𝑖𝛾𝜌𝐴𝑅(𝑏(𝜌),𝐴). (4.10) We will prove that the sets {𝑁₂(𝜌)}_{𝜌∈ℝ⧵{0}} and {𝜌𝑁^′₂(𝜌)}_{𝜌∈ℝ⧵{0}} are 𝑅-bounded.
In fact, note that 𝑁₂(𝜌)=𝜌𝑁₁(𝜌)=−((1+𝑖𝛼𝜌)/(𝛽+𝑖𝛾𝜌))𝜌³𝑁(𝜌)−(𝜌/(𝛽+𝑖𝛾𝜌))𝐼. Hence the set {𝑁₂(𝜌)} is 𝑅-bounded. Moreover, we have
𝜌𝑁^′₂(𝜌)=𝜌²𝑁^′₁(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+𝑖𝛼𝜌𝜌𝛽+𝑖𝛾𝜌³𝑁(𝜌)𝜌²+𝑁(𝜌)𝑖𝛾𝜌²(𝛽+𝑖𝛾𝜌)²𝐼+𝑁₂(𝜌), (4.11) obtaining that the set {𝜌𝑁^′₂(𝜌)}_{𝜌∈ℝ⧵{0}} is 𝑅-bounded. By Theorem 2.8 we conclude that the operator 𝑇₂ defined by 𝑇₂𝑓=(𝑁₂(⋅)[𝑓(⋅)])^∨where𝑓∈𝒮(𝑋) (4.12) extends to a bounded operator from 𝐿^𝑝(ℝ,𝑋) to 𝐿^𝑝(ℝ,𝑋).
Finally, define
𝑁₃𝜌(𝜌)∶=²𝛽+𝑖𝛾𝜌𝑅(𝑏(𝜌),𝐴)=𝜌²𝑁(𝜌). (4.13) The set {𝑁₃(𝜌)}_{𝜌∈ℝ⧵{0}} is 𝑅-bounded from hypothesis and also note that the set {𝜌𝑁^′₃(𝜌)}_{𝜌∈ℝ⧵{0}} is 𝑅-bounded, since 𝜌𝑁^′₃(𝜌)=2𝜌²𝑁(𝜌)−𝑖𝛾𝜌𝛽+𝑖𝛾𝜌³𝑁(𝜌)−2𝛼𝛾𝜌𝛽+𝑖𝛾𝜌³𝑁(𝜌)𝜌³+𝑁(𝜌)𝛾+3𝛼𝛽𝛽+𝑖𝛾𝜌𝑖𝜌³𝑁(𝜌)𝜌²𝑁(𝜌)+2𝛽𝛽+𝑖𝛾𝜌𝜌𝑁(𝜌)𝜌³𝑁(𝜌). (4.14) Again by Theorem 2.8 we conclude that the operator 𝑇₃ defined by 𝑇₃𝑓=(𝑁₃(⋅)[𝑓(⋅)])^∨where𝑓∈𝒮(𝑋) (4.15) 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 kernels 𝐾₁(𝑡)∶=𝐴𝑅(𝑡)𝜒_(0,∞)(𝑡),𝐾₂(𝑡)∶=𝐴𝑅^′(𝑡)𝜒_(0,∞)(𝑡),𝐾₃(𝑡)∶=𝑅^′′(𝑡)𝜒_(0,∞)(𝑡),𝑡∈ℝ, (4.16) is 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.

Corollary 4.3. 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 ‖𝑢‖_𝑝+‖𝑢^′‖_𝑝+‖𝑢^′′‖_𝑝+‖𝑢^′′′‖_𝑝+‖𝐴𝑢‖_𝑝+‖𝐴𝑢^′‖_𝑝+⩽𝐶‖𝑓‖_𝑝. (4.17)

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)=𝑐²(Δ𝑣(𝑡,𝑥)+𝜇Δ𝑣_𝑡𝑣(𝑡,𝑥))+𝑓(𝑡,𝑥)in]0,𝑇]×Ω,𝑣(𝑡,𝑥)=0on]0,𝑇]×Ω,𝑣(0,𝑥)=0inΩ,_𝑡𝑣(0,𝑥)=0inΩ,_𝑡𝑡(0,𝑥)=0inΩ (4.18) in a smooth bounded region Ω⊂ℝ^𝑛. Also, we assume that 𝑓∈𝐿²(ℝ;𝐿²(ℝ^𝑛)). We have the following application in the Hilbert space setting.

Theorem 4.4. Let Ω be a bounded domain with smooth boundary 𝜕Ω in ℝ^𝑛. 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. Let 𝛼=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 have 𝑏(𝜌)=−𝜌²((1+𝑖𝛼𝜌)/(𝛽+𝑖𝛾𝜌))∈𝜌(Δ) and there exists a constant 𝐶>0 such that
‖𝜌³𝛽+𝑖𝛾𝜌𝑅(𝑏(𝜌),Δ)‖=‖𝜌𝑏(𝜌)1+𝑖𝛼𝜌(𝑏(𝜌)−Δ)⁻¹‖=|𝜌||1+𝑖𝛼𝜌||𝑏(𝜌)|dist(𝑏(𝜌),𝜆₁(Ω))⩽𝐶, (4.19) 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 Figure 3, we show 𝑏(𝜌) in case 𝜆=3, 𝜇=4, and 𝑐²=1.

Figure 3 

Acknowledgments

The authors are supported by Laboratorio de Analisis Estocástico, Proyecto Anillo ACT-13. The third author is also partially financed by Proyecto Fondecyt de Iniciación 11075046.