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 Lp-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 Lp-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,
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 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.
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.
R(t) is strongly continuous on ℝ+ and R(0)=0.
R(t)D(A)⊂D(A) and AR(t)x=R(t)Ax for all x∈D(A),t⩾0.
The following equation holds:
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
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
â(λ)=β+γλλ2+αλ3,k̂(λ)=1λ2+αλ3,∀Reλ>ω.
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.
For all x∈D(A) one has R(·)x∈C3(ℝ+;X).
Let x∈X and t⩾0. Then ∫0ta(t-s)R(s)xds∈D(A) and
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 that q: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 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)=∫0∞e-λtf(t)dt for all Reλ>0.
We introduce the means
∥(x1,…,xn)∥R:=12n∑ϵj∈{-1,1}n∥∑j=1nϵjxj∥
for x1,…,xn∈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,…,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 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.
Let 𝒮,𝒯⊂ℬ(X,Y) be R-bounded sets, then 𝒮+𝒯:={S+T:S∈𝒮,T∈𝒯} is R-bounded.
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).
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 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 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.
The set {M(ρ)}ρ∈ℝ∖{0} is R-bounded.
The set {ρM'(ρ)}ρ∈ℝ∖{0} is R-bounded.
Then the operator T defined by
Tf=(M(·)[f̂(·)])∨wheref∈𝒮(X)
extends to a bounded operator from Lp(ℝ,X) to Lp(ℝ,X).
3. Existence of Solutions
Let α,β,γ∈(0,∞). Consider the equation
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.
Let R(t) be an (α,β,γ)-regularized family on X with generator A. If f∈Lloc1(ℝ+,D(A2)), then u(t) given by
u(t)=∫0tR(t-s)f(s)ds,t⩾0
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+∫0t[β+(γ/α-β)e-(t-s)/α]R(s)Axds.
If x∈D(A2), then R'(t)x∈D(A). Moreover,
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
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,
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
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
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
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
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 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.
Hence, Imφ(λ)=0 if and only if b=0. Since a>0, a direct calculation gives
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]),
∥(φ(λ)-B)-1∥=1dist(φ(λ),σ(B))∀Reλ>0.
Note that
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∥λQ(λ)∥=∥λ(β+γλ)(φ(λ)-B)-1∥⩽|λ||β+γλ|1dist(φ(λ),σ(B))<M.
On the other hand,
λ2Q'(λ)=-γλβ+γλ[λQ(λ)]+[λQ(λ)][λ2(φ(λ)-B)-1][λâ(λ)′â(λ)]1λ2â(λ),
where
1λ2â(λ)=1+αλβ+γλ,λâ(λ)'â(λ)=-2αγλ2+(γ+3αβ)λ+2β(1+αλ)(β+γλ)
and, by (3.14),
∥λ2(φ(λ)-B)-1∥⩽|λ2|dist(φ(λ),σ(B))<M
for all Reλ>0. We conclude that supReλ>0∥λ2Q'(λ)∥<∞.
By Lemma 2.5 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
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 on L2(Ω) and σ(Δ)⊂(-∞,0), we obtain the following corollary.
Corollary 3.3.
Let Ω 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 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
4. Lp-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 Lp-maximal regularity.
The definition of Lp-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.
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.
Equation (3.1) has Lp-maximal regularity on ℝ+.
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.
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,
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
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
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)
N1(ρ)=b(ρ)β+iγρR(b(ρ),A)-1β+iγρI=-1+iαρβ+iγρρ2N(ρ)-1β+iγρI.
Note that
|1+iαρβ+iγρ|2=1+α2ρ2β2+γ2ρ2<1β2+α2γ2,|1β+iγρ|2=1β2+γ2ρ2<1β2.
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 {ρN1′(ρ)}ρ∈ℝ∖{0} is R-bounded. With a direct computation, we obtain
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
T1f=(N1(·)[f̂(·)])∨wheref∈𝒮(X)
extends to a bounded operator from Lp(ℝ,X) to Lp(ℝ,X).
Define
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 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
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
T2f=(N2(·)[f̂(·)])∨wheref∈𝒮(X)
extends to a bounded operator from Lp(ℝ,X) to Lp(ℝ,X).
Finally, define
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
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
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
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 solution u 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
∥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 for A=Δ the vibration equation subject to the action of an external force. Explicitly, we consider
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.
Let Ω 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.
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,
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.
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.
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