Maximal regularity for the thermoelastic plate equations with free boundary conditions

Universität Konstanz

Maximal regularity for the thermoelastic plate equations with free boundary conditions

Robert Denk Yoshihiro Shibata

Konstanzer Schriften in Mathematik Nr. 352, März 2016

ISSN 1430-3558

© Fachbereich Mathematik und Statistik Universität Konstanz

Fach D 197, 78457 Konstanz, Germany

Konstanzer Online-Publikations-System (KOPS) URL: http://nbn-resolving.de/urn:nbn:de:bsz:352-0-325845

Maximal regularity for the thermoelastic plate equations with free boundary conditions

Robert Denk

and Yoshihiro Shibata

February 25, 2016

Abstract

We consider the linear thermoelastic plate equations with free boundary conditions in theLpin time andLq in space setting. We obtain unique solvability with optimal regularity for the inhomo- geneous problem in a uniformC⁴-domain, which includes the cases of a bounded domain and of an exterior domain with C⁴-boundary. Moreover, we prove uniform a priori-estimates for the solution.

The proof is based on the existence ofR-bounded solution operators of the corresponding generalized resolvent problem which is shown with the help of an operator-valued Fourier multiplier theorem due to Weis.

1 I ntroduction

Let Ω be a domain in theN-dimensional Euclidean spaceR^N with boundary Γ. In the present paper we consider the linearized thermoelastic plate equations given by

utt+ ∆²u+ ∆θ=f1 in (0,∞)×Ω,

θ_t−∆θ−∆u_t=f₂ in (0,∞)×Ω (1-1)

with initial conditions

u|t=0 =u0 in Ω, ut|t=0 =u1 in Ω, θ|t=0 =θ₀ in Ω.

(1-2) In (1-1)–(1-2), we omit all physical constants for simplicity of presentation. These equations model the behaviour of a thin plate with the elastic properties being influenced by the temperature (see, e.g., [14]).

In (1-1),u=u(t, x) stands for the vertical displacement at timetand at the pointx= (x₁, . . . , x_N)∈Ω while θ = θ(t, x) describes the temperature relative to a constant reference temperature. For (1-1), several boundary conditions are of interest, see, e.g. [15] for a survey on physically relevant boundary conditions. In the present paper, we consider free boundary conditions which are given by

∆u−(1−β)∆⁰u+θ=g1 on (0,∞)×Γ,

∂ν ∆u+ (1−β)∆⁰u+θ

=g2 on (0,∞)×Γ,

∂νθ=g3 on (0,∞)×Γ.

(1-3)

∗Universit¨at Konstanz, Fachbereich f¨ur Mathematik und Statistik, 78457 Konstanz, Germany e-mail address: robert.denk@uni-konstanz.de

†Department of Mathematics and Research Institute of Science and Engineering Waseda University, Ohkubo 3-4-1, Shinjuku-ku, Tokyo 169-8555, Japan.

e-mail address: yshibata@waseda.jp

Partially supported by JSPS Grant-in-aid for Scientific Research (S) # 24224004 and Top Global University Project Subjectclass(2010): 35K35; 35J40; 42B15

Keywords: Thermoelastic plate equations; generation of analytic semigroups; maximalLp-Lq regularity; R-bounded solution operator, operator-valued Fourier multipliers

In (1-3),β ∈[0,1) is a parameter which is fixed throughout this paper, ∆ and ∆⁰ stand for the Laplace operator in Ω and the Laplace-Beltrami operator on Γ, respectively, and ∂ν denotes the derivative in outer normal direction. Note that the termθcan be omitted in the second line of (1-3) if we replaceg2

byg₂−g₃.

There is a rich literature on the thermoelastic plate equations under various kinds of boundary conditions. Exponential stability of the associated semigroup in L₂ (in the case of a bounded domain) has been proved by Kim [12], Munos Rivera and Racke [22], Liu and Zheng [21], Lasiecka and Triggiani [15] –[16], and Shibata [26]. For a survey on general von Karman evolution equations, we refer to Chuesov and Lasiecka [2]. It turns out that the generated semigroup is even analytic, see also Liu and Renardy [20], Liu and Liu [18], and Liu and Yong [19] in the L2-setting. This means that the effect from the heat equation inθis strong enough to obtain analytic behaviour of the whole system although the first equation in (1-1) is a simply dispersive equation (the product of two Schr¨odinger equations) with respect tou.

Most of the results mentioned above are obtained in an L2-setting, where energy methods are avail- able. However, as the original equations modelling thermoelastic plates are non-linear, anLp-approach is also relevant in order to handle equations with low regularity of the data. Therefore, several results on (1-1) inLp-spaces were obtained. In the whole-space case, analyticity of the generated semigroup in L_p was shown by Denk and Racke [4]. In the case of the half-space and of bounded domains, equations (1-1) with Dirichlet boundary conditions

u=∂_νu=θ= 0 on (0,∞)×Γ

were studied by Naito and Shibata [24] and by Naito [23]. It was shown that in Lp an analytic C⁰- semigroup is generated and that even maximal Lp-Lq-regularity holds which is the key property for the analysis of the non-linear equations. By Denk, Racke and Shibata [5], [6] energy estimates for the generated semigroup inLq were shown.

The proof of maximal Lp-regularity for the linearized system and a rather complete analysis of the non-linear thermoelastic plate equations can be found in a recent paper by Lasiecka and Wilke [17]. In that paper, the boundary conditions

u= ∆u=θ= 0 on (0,∞)×Γ

are studied. From a mathematical point of view, these boundary conditions are easier to handle. This is due to the fact that the operator ∆² appearing in the first line of (1-1) can then be interpreted as the square of the Dirichlet-Laplace operator, and solvability of (1-1) can be shown by abstract operator- theoretic methods. For the boundary conditions (1-3) studied in the present paper, such an abstract approach seems to be not available, and one needs a thorough analysis of the (localized) solution opera- tors.

The purpose of this paper is to prove maximalLp-Lq-regularity of the initial boundary value problem (1-1)-(1-3). In our approach, setting v = ut we rewrite (1-1) as a first-order system acting on U = (u, v, θ)^>, whereM^> denotes the transposed ofM, and being of the form

U_t−A(D)U =F in (0, T)×Ω, B(D)U =G on (0, T)×Γ, U|t=0=U₀ in Ω (1-4) with operator-matricesA(D) andB(D) being defined by

A(D) :=





0 1 0

−∆² 0 −∆

0 ∆ ∆



, B(D) :=





∆−(1−β)∆⁰ 0 1

∂ν(∆ + (1−β)∆⁰) 0 0

0 0 ∂ν



.

SettingF = (0, f₁, f₂)^>,U₀= (u₀, u₁, θ₀)^>,G= (g₁, g₂, g₃)^>, andU = (u, u_t, θ)^>in (1-4) represents the equations (1-1)-(1-3). To prove maximalL_p-L_q-regularity of problem (1-4), we prove the existence of an R-bounded solution operator of the problem:

λU−A(D)U =F in Ω, B(D)U =G on Γ (1-5)

with F = (0, f1, f2)^> ∈ Lq(Ω)² and G = (g1, g2, g3)^> ∈ H_q²(Ω)×H_q¹(Ω)², which is the generalized resolvent problem corresponding to problem (1-4).

To state the main results pricesely, at this point we introduce several symbols used throughout the paper. N, R, andC denote the sets of all natural numbers, real numbers, and complex numbers, respectively. SetN0=N∪ {0}. For any multi-indexκ= (κ1, . . . , κ_N)∈N^N0, we write|κ|=κ1+· · ·+κ_N and ∂_x^κ = ∂₁^κ1· · ·∂_N^κN with ∂_i = ∂/∂x_i. For any scalar function f and vector-valued function g = (g₁, . . . , g_k), let

∇f = (∂1f, . . . , ∂_Nf), ∇<sup>`</sup>f = (∂<sub>x</sub><sup>α</sup>f | |α|=`),

∇g= (∂igj|i= 1, . . . , N, j= 1, . . . , k), ∇^{`</sup>g= (∇<sup>`}g1, . . . ,∇^`gk).

For any domain D, let Lq(D), H_q^m(D) (m ∈ N) and B_q,p^s (D) (s∈ (0,∞)\N) be the Lebesgue space, Sobolev space and Besov space, whilek·kL_q(D),k·kH_q^m(D)andk·kB^s_q,p(D)denote their norms, respectively.

We write H_q⁰(D) =L_q(D) andB^s_qq(D) =W_q^s(D). Let X andY be Banach spaces, and letL(X, Y) be the space of all bounded linear operators fromX to Y. We use the abbreviationL(X) =L(X, X). For an intervalJ = (0, T) with T ∈(0,∞], Lp(J, X) denotes the X-valued Lebesgue space andH_p^m(J, X) (m∈N) theX-valued Sobolev space, whilek · k_Lp(J,X)andk · k_Hm

p(J,X)denote their norms, respectively.

For any domain V in C, C(V, X) denotes the set of all X-valued functions f = f(λ) defined for λ = γ+iτ ∈V which are continuously differentiable with respect toτ whenλ∈V. Let Σϑand Σϑ,λ₀ be the sets inCdefined by

Σ_ϑ:={λ∈C\ {0} | |argλ|< ϑ}, Σ_ϑ,λ0 :={λ∈Σ_ϑ| |λ| ≥λ₀} (1-6) for any 0< ϑ≤πandλ₀>0. LetX^d={f = (f₁, . . . , f_d)^> |f_i ∈X (i= 1, . . . , d)}, while the norm of X^d is written byk · kX instead ofk · k_Xd for short. In particular, we write

kGk_Hm

q(D)×H_q^`(D)² =kg1k_Hm

q(D)+k(g2, g3)k_H`

q(D) forG= (g1, g2, g3)^> ∈H_q^m(D)×H_q^`(D)². Let

Gq(D) :={(F, G)|F = (0, f₁, f₂)^>, (f₁, f₂)∈L_q(D)², G= (g₁, g₂, g₃)^>∈H_q²(D)×H_q¹(D)²}, k(F, G)k_Gq(D):=k(f1, f₂)kL_q(D)+kGkH_q²(D)×H_q¹(D)²,

Xq(Ω) :={H= (F⁰, G, G⁰, g⁰⁰₁)| F⁰ = (f1, f2)^>∈Lq(Ω)²,

G= (g1, g2, g3)^> ∈H_q²(Ω)×H_q¹(Ω)², G⁰= (g⁰₁, g₂⁰, g⁰₃)^>∈H_q¹(Ω)×Lq(Ω)², g⁰⁰₁ ∈Lq(Ω)}, kHk_Xq(D):=k(f1, f2)k_Lq(D)+kGk_H2

q(D)×H_q¹(D)+kG⁰k_H1

q(D)×Lq(D)²+kg₁⁰⁰k_Lq(D). (1-7) For any λ ∈ C and (F, G) ∈ Gq(D), let Hλ(F, G) := (f1, f2, G, λ^1/2G, λg1) with F = (0, f1, f2)^> and G= (g1, g2, g3)^>. In particular, G⁰ and g⁰⁰₁ are the corresponding variables toλ^1/2Gand λg1. For any exponent q∈(1,∞), let q⁰ =q/(q−1) be the dual exponent ofq. The lettersC and c denote generic positive constants and the constant C_a,b,... depends on a, b, . . .. The values of the constantsC, c and C_a,b,... may change from line to line.

Next, we introduce two definitions (see, e.g., [3], [13]).

Definition 1.1. A family T ⊂ L(X, Y) of operators is called R-bounded if for one (and then all) p∈[1,∞) there exists a constantC >0 such that for allm∈N, (Tk)k=1,...,m⊂ T, and (xk)k=1,...,m⊂X we have

m

X

k=1

rkTkxk

_L

p([0,1],Y)

≤C

m

X

k=1

rkxk

_L

p([0,1],X)

.

Here the Rademacher functions rk, k ∈N, are given by rk: [0,1]→ {−1,1}, t7→ sign(sin(2^kπt)). The smallest suchC is called theR-bound ofT onL(X, Y) which is written byR_L(X,Y)(T) in what follows.

Note that we omit the dependence onpin the notation of theR-bound.

Definition 1.2. A domain Ω is called a uniformC⁴-domain if there exist positive constantsα,β andK such that for anyx₀∈Γ there exist a coordinate numberj and aC⁴-functionh(x⁰) defined onB_α⁰(x⁰₀) such thatkhk_H4

∞(B_α⁰(x⁰₀))≤K and

Ω∩Bβ(x0) ={x∈R^N |xj> h(x⁰) (x⁰∈B⁰_α(x⁰₀))} ∩Bβ(x0), Γ∩B_β(x₀) ={x∈R^N |x_j=h(x⁰) (x⁰∈B⁰_α(x⁰₀))} ∩B_β(x₀).

Here,x⁰ has been defined byx⁰= (x1, . . . , xj−1, xj+1, . . . , x_N) forx= (x1, . . . , x_N), B_α⁰(x⁰₀) ={x⁰ ∈R^N−1| |x⁰−x⁰₀|< α}, Bβ(x0) ={x∈R^N | |x−x0|< β}.

In what follows, Ω is assumed to be a uniform C⁴-domain. Let ι : L_1,loc(Ω) → L_1,loc(R^N) be an extension operator possessing the following properties:

(e-1) For any 1 < q < ∞ and f ∈ H_qⁱ(Ω) we have ιf ∈ H_qⁱ(R^N), ιf = f in Ω and kιfk_Hi q(R^N) ≤ C_qkfk_Hi

q(Ω) fori= 0, , . . . ,4.

(e-2) For any 1< q <∞andf ∈H_q¹(Ω),k(1−∆)^−1/2ι(∇f)k_Lq(RN)≤C_qkfk_Lq(Ω).

Here, (1−∆)^−1/2 is the operator defined by (1−∆)^−1/2f =F⁻¹[(1 +|ξ|²)^−1/2Ff] with the help of the Fourier transformF and its inverse transformF⁻¹defined by

(Fϕ)(ξ) = Z

Rⁿ

ϕ(x)e^−ixξdx (ξ∈Rⁿ), (F⁻¹ϕ)(x) = 1 (2π)ⁿ

Z

Rⁿ

ϕ(ξ)e^ixξdξ (x∈Rⁿ).

For the existence of such an extension operator, we refer, e.g., to Schade and Shibata [25, Appendix A].

In what follows, suchιis fixed. LetW⁻¹_q (Ω) be the space defined by W⁻¹_q (Ω) =

f ∈L_1,loc(Ω)| kfk_W−1

q (Ω)=k(1−∆)^−1/2ιfk_Lq(RN)<∞ . Finally, we state the main results of this paper.

Theorem 1.3 (MaximalLp-Lq-regularity). Let T >0. Let 1< p, q <∞. Assume thatΩis a uniform C⁴-domain in R^N. Then, there exists a number λ1 such that for any initial data U0 = (u0, u1, θ0)^> ∈ Bq,p^4−2/p(Ω)×Bq,p^2−2/p(Ω)², right-hand sideF = (0, f1, f2)^> with(f1, f2)^>∈Lp((0, T), Lq(Ω)²)and bound- ary data G= (g₁, g₂, g₃)^> with

G∈H_p¹((0, T), L_q(Ω)×W⁻¹_q (Ω)²)∩L_p((0, T), H_q²(Ω)×H_q¹(Ω)²)

satisfying the compatibility condition: G|_t=0 =B(D)U₀ on Ω, problem (1-4) admits a unique solution U = (u, ut, θ)^> with

u∈

2

\

`=0

H_p^{`</sup>((0, T), H<sub>q</sub><sup>4−2`}(Ω)), θ∈

1

\

`=0

H_p^{`</sup>((0, T), H<sub>q</sub><sup>2−2`}(Ω)) possessing the estimate:

2

X

`=0

k∂_t^`uk_L

p((0,T),H^4−2`_q (Ω))+

1

X

`=0

k∂_t^`θk_L

p((0,T),H_q^2−2`(Ω))≤e^γTn

ku₀k_B4−2/p

q,p (Ω)+kθ₀k_B2−2/p q,p (Ω)

+k(f1, f2)k_{Lp((0,T),Lq(Ω)}2)+kGk_Lp((0,T),H2

q(Ω)×H_q¹(Ω)²)+k∂tGk_L

p((0,T),Lq(Ω)×W⁻¹q (Ω)²)

o

with some positive constantγ >0independent of T.

Concerning the compatibility conditions, we remark that G|_t=0 and B(D)U₀ both belong to the space Bqp^2−2/p(Ω)×(Bpq^1−2/p(Ω))² in the case p > 2. For simplicity, we assume that the compatibility conditions holds in Ω. In view of the nonlinear equation, we typically assume 2< p <∞, N < q <∞, and 2/p+N/q < 1. In this case, the traces of G|t=0 and B(D)U0 on the boundary exist, and the compatibility condition can be formulated with respect to these traces.

In this paper, to prove Theorem 1.3, we prove the existence ofR-bounded solution operators associ- ated with problem (1-5). Namely, we prove

Theorem 1.4 (Existence of R-bounded solution operators). Let 1 < q < ∞. Assume that Ω is a uniform C⁴-domain in R^N. Let Gq(Ω) and Xq(Ω) be defined as in (1-7). Then, there exist a number ϑ > π/2, a positive numberλ0, and an operator family Bi(λ) (i= 1,2) with

B₁(λ)∈ C(Σ_ϑ,λ0,L(X_q(Ω), H_q⁴(Ω))), B₂(λ)∈ C(Σ_ϑ,λ0,L(X_q(Ω), H_q²(Ω))),

such that problem(1-5)admits a unique solutionU =B(λ)Hλ(F, G)withB(λ) = (B1(λ), λB1(λ),B2(λ))^>

for any(F, G)∈Gq(Ω) andλ∈Σϑ,λ₀, where Hλ(F, G) = (f1, f2, G, λ^1/2G, λg1)forF= (0, f1, f2)^> and G= (g1, g2, g3)^>, and there hold the estimates:

R_L(X

q(Ω),H_q^4−j(Ω))({(τ ∂τ)^s(λ^j/2B1(λ))|λ∈Σϑ,λ₀})≤C (s= 0,1, j = 0,1,2,3,4), R_L(X

q(Ω),H_q^2−j(Ω))({(τ ∂τ)^s(λ^j/2B2(λ))|λ∈Σϑ,λ₀})≤C (s= 0,1, j = 0,1,2). (1-8)

2 A nalysis in the whole space

The purpose of this section is to prove the existence of anR-bounded solution operator associated with the resolvent problem:

λU−A(D)U =F in R^N (2-1)

withF = (0, f1, f2)^>. One main tool for the proof is the following lemma due to Denk and Schnaubelt [7, Lemma 2.1] and Enomoto and Shibata [9, Theorem 3.3].

Lemma 2.1. Let 1 < q < ∞ and let Λ be a set in C. Let m = m(λ, ξ) be a function defined on Λ×(R^N\ {0})which is infinitely differentiable with respect toξ∈R^N\ {0} for eachλ∈Λ. Assume that for any multi-indexα∈N^N0 there exists a constantCα depending onαandΛsuch that

|∂_ξ^αm(λ, ξ)| ≤C_α|ξ|^−|α| (2-2)

for any (λ, ξ)∈Λ×(R^N \ {0}). Let K_λ be an operator defined by K_λf =Fξ⁻¹[m(λ, ξ)Ff(ξ)]. Then, the family of operators {Kλ|λ∈Λ}isR-bounded on L(Lq(R^N))and

R_L(Lq(RN))({Kλ|λ∈Λ})≤Cq,N max

|α|≤N+1Cα (2-3)

with some constantCq,N depending only onq andN. The symbol of the operator matrixA(D) is given by

A(ξ) :=





0 1 0

−|ξ|⁴ 0 |ξ|² 0 −|ξ|² −|ξ|²



,

i.e., we haveA(D) =F⁻¹A(·)F. Thus, by the Fourier transform equation (2-1) is transformed to

(λI−A(ξ)) ˆU(ξ) = ˆF(ξ) (2-4)

with ˆU = (Fu,Fv,Fθ)^> and ˆF = (0,Ff1,Ff2)^>. The analysis of the inverse matrix (λI−A(ξ))⁻¹ was essentially done in [24]. As we need some variants of the results in [24], we summarize the main properties of (λI−A(ξ))⁻¹ and give a short indication of the proofs.

In the following, defineγ₁, γ₂, γ₃ by the equality

p(t) :=t³+t²+ 2t+ 1 = (t+γ1)(t+γ2)(t+γ3) (2-5) with γ1 ∈ R, γ2 = ¯γ3 and Imγ2 >0. Then γ1 ∈ (0,1), Reγ2 = Reγ3 ∈(0,¹₂), and det(λ−A(ξ)) = Q3

j=1(λ+γj|ξ|²) (see [24, Lemma 2.3]). Letϑ0> π/2 andϑ1> π/2 be chosen in such a way that λγ⁻¹_i ∈Σϑ₁ (i= 1,2,3) forλ∈Σϑ₀. (2-6)

Then the inequality

|λγ_i⁻¹+|ξ|²| ≥c(|λ|+|ξ|²) (2-7) holds for anyλ∈Σϑ₀ andξ∈R^N with some positive constantc.

It was shown in [24, Section 2] that for allλ∈Σϑ₀ we have (λI−A(ξ))⁻¹= 1

det(λI−A(ξ))





λ(λ+|ξ|²) +|ξ|⁴ λ+|ξ|² |ξ|²

−(λ+|ξ|²)|ξ|⁴ λ(λ+|ξ|²) λ|ξ|²

|ξ|⁶ −λ|ξ|² λ²+|ξ|⁴



. Sinceγ1γ2γ3= 1 as follows from Vieta’s formula, we have det(λI−A(ξ)) =Q3

i=1(λγ_i⁻¹+|ξ|²), and then a solutionU = (u, λu, θ)^> of problem (2-1) is given byU =F⁻¹ (λI−A(ξ))⁻¹Ff(ξ)

, i.e., u(x) =F⁻¹h λ+|ξ|²

Q3

i=1(λγ⁻¹_i +|ξ|²)Ff₁(ξ)i

+F⁻¹h |ξ|² Q3

i=1(λγ_i⁻¹+|ξ|²)Ff₂(ξ)i , θ(x) =−F⁻¹h λ|ξ|²

Q3

i=1(λγ_i⁻¹+|ξ|²)Ff1(ξ)i

+F⁻¹h λ²+|ξ|⁴ Q3

i=1(λγ⁻¹_i +|ξ|²)Ff2(ξ)i .

(2-8)

Let the operatorS₀(λ) acting onf be defined by S0(λ)f :=F⁻¹hY³

i=1

(λγi+|ξ|²)−1

Ff(ξ)i . By Lemma 2.1 and (2-7),

R_L(Lq(RN))

n

(τ ∂τ)^s(λ^j/2∂_x^αS0(λ))|λ∈Σϑ₀

o≤Cj,α (s= 0,1) (2-9) forj∈N0andα∈N^N0 withj+|α|= 6. Moreover, by (2-8)u=S1(λ)Fandθ=S2(λ)F forF = (f₁, f₂)^>

with

S1(λ)F := (λ−∆)S0(λ)f1−∆S0(λ)f2, S₂(λ)F :=λ∆S₀(λ)f₁+ (λ²+ ∆²)S₀(λ)f₂.

At this point, we introduce some fundamental properties ofR-bounded operators and Bourgain’s results concerning Fourier multiplier theorems with scalar multiplier.

Proposition 2.2. a) LetX andY be Banach spaces, and letT andSbeR-bounded families inL(X, Y).

Then, T +S={T+S |T ∈ T, S∈ S}is also anR-bounded family inL(X, Y)and R_L(X,Y)(T +S)≤ R_L(X,Y)(T) +R_L(X,Y)(S).

b) LetX,Y andZ be Banach spaces, and letT andS beR-bounded families inL(X, Y)andL(Y, Z), respectively. Then, ST ={ST |T∈ T, S∈ S}also anR-bounded family inL(X, Z) and

R_L(X,Z)(ST)≤ R_L(X,Y)(T)R_L(Y,Z)(S).

c) Let 1 < p, q < ∞ and let D be a domain in R^N. Let m =m(λ) be a bounded function defined on a subset Λ in C and let Mm(λ) be a map defined by Mm(λ)f =m(λ)f for any f ∈Lq(D). Then, R_L(Lq(D))({Mm(λ)|λ∈Λ})≤CN,q,Dkmk_L∞(Λ).

d) Let n = n(τ) be a C¹-function defined on R\ {0} that satisfies the conditions |n(τ)| ≤ γ and

|τ n⁰(τ)| ≤ γ with some constant c > 0 for any τ ∈ R\ {0}. Let Tn be the operator-valued Fourier multiplier defined by Tnf =F⁻¹(nF[f])for any f with F[f]∈D(R, Lq(D)). Then,Tn is extended to a bounded linear operator from Lp(R, Lq(D)) into itself. Moreover, denoting this extension also by Tn, we have

kTnk_L(Lp(R,L_q(D)))≤C_p,q,Dγ.

Here,D(R, Lq(D))denotes the set of allLq(D)-valued C^∞-functions on Rwith compact support.

Proof. The assertions a) and b) follow from [3, p.28, Proposition 3.4], and the assertions c) and d) follow from [3, p.27, Remarks 3.2] (see also Bourgain [1]).

Since

∂_ξ^α(τ ∂_τ)^s λ^j/2(iξ)^β Q3

i=1(λγ_i⁻¹+|ξ|²)

≤C_α|ξ|^−|α|

for anys∈ {0,1},j ∈N0 andβ ∈N^N0 with j+|β|= 6 and (λ, ξ)∈Σ_ϑ0×(R^N \ {0}) as follows from (2-7), by Lemma 2.1 and Proposition 2.2 a),

R_L(Lq(RN))({(τ ∂_τ)^s(λ^j/2∂_x^α(λ−∆)S₀(λ))|λ∈Σ_ϑ0})≤γ₀ (s= 0,1, j= 0,1,2,3,4 j+|α|= 4), R_L(Lq(RN))({(τ ∂_τ)^s(λ^j/2∂_x^α∆S₀(λ))|λ∈Σ_ϑ0})≤γ₀ (s= 0,1, j= 0,1,2,3,4, j+|α|= 4), R_L(Lq(RN))({(τ ∂τ)^s(λ^j/2∂_x^α(λ∆S0(λ)))|λ∈Σϑ₀})≤γ0 (s= 0,1, j = 0,1,2, j+|α|= 2), R_L(Lq(RN))({(τ ∂τ)^s(λ^j/2∂_x^α(λ²+ ∆²)S0(λ))|λ∈Σϑ₀})≤γ0 (s= 0,1, j= 0,1,2, j+|α|= 2) for some constantγ0>0. Combined with Proposition 2.2 c), this yields the following result.

Theorem 2.3. Let 1 < q < ∞ and λ0 >0. Let ϑ0 be the number given in (2-6). Then, there exist operator families Si(λ) (i= 1,2) with

S1∈ C(Σϑ0,λ0,L(Lq(R^N)², H_q⁴(R^N))), S2∈ C(Σϑ0,λ0,L(Lq(R^N)², H_q²(R^N)))

such that problem (2-1)admits a unique solutionU = (S1(λ)F⁰, λS1(λ)F⁰,S2(λ)F⁰)^> for anyλ∈Σ_ϑ0,λ0 andF = (0, f₁, f₂)^> with F⁰= (f₁, f₂)^>∈L_q(R^N)², and there hold the estimates:

R_L(L

q(R^N)²,H_q^4−j(R^N))

(τ ∂τ)^s(λ^j/2S1(λ))|λ∈Σϑ₀,λ₀ ≤Cλ₀γ0 (s= 0,1, j= 0,1,2,3,4), R_L(L

q(R^N)²,H_q^2−j(R^N))

(τ ∂τ)^s(λ^j/2S2(λ))|λ∈Σϑ₀,λ₀ ≤Cλ₀γ0 (s= 0,1, j= 0,1,2), with some constantCλ₀ >0.

3 S olution operators in the half-space

Let R^N+ ={x = (x1, . . . , x_N) ∈ R^N | xN > 0} and R^N0 = {x= (x1, . . . , xN) ∈ R^N | x_N = 0}. The purpose of this section is to prove the existence of R-bounded solution operators of the generalized resolvent problem:

λU−A(D)U =F inR^N+, B(D)U =G onR^N0 (3-1) in the half-space R^N+ with U = (u, λu, θ)^>, F = (0, f1, f2)^> and G = (g1, g2, g3)^>. The boundary condition in (3-1) is represented componentwise by

∆u−(1−β)∆⁰u+θ=g1 onR^N0 ,

∂_N(∆u+ (1−β)∆⁰u) =g2 onR^N0 ,

∂_Nθ=g₃ onR^N0 .

(3-2)

Here, ∆⁰=PN−1

j=1 ∂_j². Then, the main result of this section is

Theorem 3.1. Let 1< q <∞ andλ0>0. Then, there exist a numberϑ > π/2 and operator families Ti(λ) (i= 1,2)with

T1∈ C(Σϑ,L(Xq(R^N+), H_q⁴(R^N+))), T2∈ C(Σϑ,L(Xq(R^N+), H_q²(R^N+))) such that problem (3-1)admits a unique solution

U = T1(λ)Hλ(F, G), λT1(λ)Hλ(F, G),T2(λ)Hλ(F, G)^>

for any (F, G)∈Gq(R^N+)andλ∈Σϑ, and there hold the estimates:

R_L(X

q(R^N₊),H_q^4−j(R^N₊))

(τ ∂τ)^s(λ^j/2T1(λ))|λ∈Σϑ,λ₀ ≤CN,q,λ₀ (s= 0,1, j= 0,1,2,3,4), R_L(X

q(R^N+),Hq^2−j(R^N+))

(τ ∂τ)^s(λ^j/2T2(λ))|λ∈Σϑ,λ0 ≤CN,q,λ0 (s= 0,1, j= 0,1,2).

In what follows, we prove Theorem 3.1. Let ιh be the Lions extension operator of the form:

[ι_hf](x) :=

(f(x⁰, x_N) (x_N >0), P6

j=1ajf(x⁰,−jx_N) (x_N <0), (3-3) for any givenf onR^N+, wherex⁰= (x1, . . . , x_N−1), andaj are real numbers satisfying the relations:

6

X

j=1

(−j)^ka_j = 1 fork=−1,0,1, . . . ,4.

LetS(λ) = (S1(λ),S2(λ))^> be the solution operator given in Theorem 2.3, and let the operatorS+(λ) = (S+1(λ),S+2(λ))^> acting onF⁰= (f1, f2)∈Lq(R^N+)² be defined by

S+i(λ)F⁰=Si(λ)(ιhF⁰) (i= 1,2). (3-4) Obviously,

V = S+1(λ)(ιhF⁰), λS+1(λ)(ιhF⁰),S+2(λ)(ιhF⁰)>

satisfies the equation: λV −A(D)V =F inR^N+ withF = (0, f1, f2)^> and the estimate:

R_L(L

q(R^N₊)²,H_q^4−j(R^N₊))({(τ ∂τ)^s(λ^j/2S+1(λ))|λ∈Σ_ϑ0,λ0})≤C_λ0 (s= 0,1, j= 0,1,2,3,4), R_L(L

q(R^N+)²,H_q^2−j(R^N+))({(τ ∂τ)^s(λ^j/2S+2(λ))|λ∈Σϑ₀,λ₀})≤Cλ₀ (s= 0,1, j= 0,1,2). (3-5) SettingU =V+W yields thatW should solve the equations (3-1) replacingFandGby 0 andG−B(D)V. Since the second component ofW coincides withλtimes the first component, in what follows, it suffices to consider the equations:

λ²u+ ∆²u+ ∆θ= 0, λθ−∆θ−λ∆u= 0 inR^N+ (3-6) with non-homogeneous boundary condition (3-2).

Applying partial Fourier transform

F⁰[u](ξ⁰, x_N) :=

Z

R^N−1

u(x⁰, x_N)e^−ix0ξ0dx⁰,

whereξ⁰ = (ξ1, . . . , ξ_N−1), to (3-6) and (3-2) yields an ordinary differential equation system inx_N >0:

λ²w+ (∂_N² − |ξ⁰|²)²w+ (∂_N² − |ξ⁰|²)τ= 0 (x_N >0), λτ−(∂²

N − |ξ⁰|²)τ−λ(∂²

N − |ξ⁰|²)w= 0 (x_N >0) (3-7) with initial conditions

(∂_N² − |ξ⁰|²)w(0) + (1−β)|ξ⁰|²w(0) +τ(0) =F⁰[g1](ξ⁰,0),

∂_N (∂_N² − |ξ⁰|²)w(0)−(1−β)|ξ⁰|²w(0)

=F⁰[g2](ξ⁰,0),

∂_Nτ(0) =F⁰[g₃](ξ⁰,0).

(3-8)

Here we have setw(ξ⁰, x_N) = (F⁰u)(ξ⁰, x_N) andτ(ξ⁰, x_N) = (F⁰θ)(ξ⁰, x_N).

We find solutions w and τ of (3-7)–(3-8). For this, we use the representation formula of w and τ which was derived in [24, Eq. (3.15)]. There it was shown that every stable solution of (3-7) has the form

w(ξ⁰, x_N, λ) =

3

X

i=1

Piexp(−Ai(ξ⁰, λ)x_N),

τ(ξ⁰, x_N, λ) =−λ

3

X

i=1

(γ_i²+ 2)Piexp(−Ai(ξ⁰, λ)x_N).

(3-9)

Hereγ1, γ2, γ3 are given by (2-5). The numbersAi appearing in (3-9) are defined by Ak(ξ⁰, λ) =

q

λγ_i⁻¹+|ξ⁰|² (k= 1,2,3), (3-10) andP1, P2, P3are constants which will determined later by the boundary conditions. Inserting (3-9) into the boundary conditions (3-8), we get a linear equation system for the coefficientsPi:

3

X

i=1

(A²_i −β|ξ⁰|²−λ(γ_i²+ 2))P_i=F⁰[g₁](ξ⁰,0),

3

X

i=1

(−A³_i +A_i(2−β)|ξ⁰|²)P_i=F⁰[g₂](ξ⁰,0),

3

X

i=1

λA_i(γ_i²+ 2)P_i=F⁰[g₃](ξ⁰,0).

NotingA²_i =λγ_i⁻¹+|ξ⁰|²andγ_i⁻¹−γ_i²−2 = _γ¹

i(1−γ_i³−2γi) =−γi byp(−γi) = 0, the linear equations above are re-written of the form:

∆(ξ⁰, λ)(P1, P2, P3)^>= (F⁰[g1](ξ⁰,0),F⁰[g2](ξ⁰,0),F⁰[g3](ξ⁰,0))^> (3-11) with

∆(ξ⁰, λ) :=







−γ1λ+ζ −γ2λ+ζ −γ3λ+ζ A1(−_γ^λ

1 +ζ) A2(−_γ^λ

2 +ζ) A3(−_γ^λ

3 +ζ) λA₁(γ²₁+ 2) λA₂(γ₂²+ 2) λA₃(γ₃²+ 2)





. (3-12)

Hereζ:= (1−β)|ξ⁰|². The matrix ∆(ξ⁰, λ) is called the Lopatinski˘ı matrix of (3-6), (3-2).

It is the most important of this paper to analyze the inverse matrix of the Lopatinski˘ı matrix. For this purpose, we introduce some classes of multipliers.

Definition 3.2. LetV be a domain inC, let Ξ⊂(R^N−1\{0})×V, and letm: Ξ→C,(ξ⁰, λ)7→m(ξ⁰, λ) beC¹ with respect toτ (whereλ=γ+iτ) andC^∞ with respect toξ⁰.

(1) m(ξ⁰, λ) is called a multiplier of orderswith type 1 on Ξ if there hold the estimates:

|∂_ξ^κ0⁰m(ξ⁰, λ)| ≤Cκ⁰(|λ|^1/2+|ξ⁰|)^s−|κ0|, |∂_ξ^κ0⁰(τ ∂τm(ξ⁰, λ))| ≤Cκ⁰(|λ|^1/2+|ξ⁰|)^s−|κ0| (3-13) for any multi-indexκ⁰∈N^N0⁻¹ and (ξ⁰, λ)∈Ξ with some constantCκ⁰ depending solely onκ⁰ and Ξ.

(2) m(ξ⁰, λ) is called a multiplier of orderswith type 2 on Ξ if there hold the estimates:

|∂_ξ^κ0⁰m(ξ⁰, λ)| ≤Cκ⁰(|λ|^1/2+|ξ⁰|)^s|ξ⁰|^−|κ0|, |∂_ξ^κ0⁰(τ ∂τm(ξ⁰, λ))| ≤Cκ⁰(|λ|^1/2+|ξ⁰|)^s|ξ⁰|^−|κ0|. (3-14) for any multi-indexκ⁰∈N^N0⁻¹ and (ξ⁰, λ)∈Ξ with some constantC_κ⁰ depending solely onκ⁰ and Ξ.

Let Ms,i(Ξ) be the set of all multipliers of order s with type i on Ξ (i = 1,2). In the standard case Ξ = (R^N−1\ {0})×V, we writeMs,i(V) instead of Ms,i(Ξ).

Obviously, Ms,i(Ξ) are complex vector spaces. Moreover, the following lemma follows from the inequality (|λ|^1/2+|ξ⁰|)^−|α0|≤ |ξ⁰|^−|α0|and the Leibniz rule immediately.

Lemma 3.3. Let s1,s2 be two real numbers. Then, the following three assertions hold.

a) Given mi∈Ms_i,1(Ξ) (i= 1,2), we have m1m2∈Ms₁+s₂,1(Ξ).

b) Given`i∈Ms<sub>i</sub>,i(Ξ) (i= 1,2), we have `1`2∈Ms₁+s₂,2(Ξ).

c) Givenni∈Ms_i,2(Ξ) (i= 1,2), we haven1n2∈Ms₁+s₂,2(Ξ).

Due to ∂_ξ^α0⁰ξ` = 0 for |α<sup>0</sup>| ≥1, we have (ξ<sup>0</sup>, λ)7→ξ` ∈M^1,1(Σϑ₀). Similarly, due to∂_ξ^α0⁰|ξ⁰|² = 0 for

|α⁰| ≥3 we obtain forζ:= (1−β)|ξ⁰|² (see (3-12))

|∂_ξ^α0⁰ζ| ≤C(|λ|^1/2+|ξ⁰|)^2−|α0|, (3-15) which yields thatζ∈M2,1(Σϑ₀). Here and in the following,ϑ0 is the number given in (2-6). By (2-7),

c(|λ|^1/2+|ξ⁰|)≤ReAi(ξ⁰, λ)≤ |Ai(ξ⁰, λ)| ≤C(|λ|^1/2+|ξ⁰|) (3-16) with some positive constantscandC, which furnishes that

A_i(ξ⁰, λ)^s∈Ms,1(Σ_ϑ0), (A_i(ξ⁰, λ) +|ξ⁰|)⁻¹∈M−1,2(Σ_ϑ0), (3-17) wheresis any real number. The property of the Lopatinski˘ı matrix ∆ is given in

Theorem 3.4. There exists a number ^π₂ < ϑ≤πsuch that the inverse matrix∆(ξ⁰, λ)⁻¹ exists for any λ∈Σϑ andξ⁰ ∈R^N−1. Let

∆(ξ⁰, λ)⁻¹= (gij(ξ⁰, λ))i,j=1,2,3. Then,

λgi1∈M0,1(Σϑ) (i= 1,2,3), λgij∈M−1,1(Σϑ) (i= 1,2,3, j= 2,3). (3-18) Moreover, there exists a positive constantσ0>0 such that

λg_i1∈M0,2(Ξ_ϑ,σ0) (i= 1,2,3),

3

X

i=1

g_i1∈M−2,2(Ξ_ϑ,σ0) (i= 1,2,3),

λg_ij ∈M−1,2(Ξ_ϑ,σ0) (i= 1,2,3, j= 2,3),

3

X

i=1

g_ij ∈M−3,2(Ξ_ϑ,σ0) (i= 1,2,3, j= 2,3)

(3-19)

with

Ξϑ,σ₀ ={(ξ⁰, λ)∈R^N−1×Σϑ |σ0|ξ⁰| ≥ |λ|^1/2, λ∈Σϑ}. (3-20) The proof of Theorem 3.4 is the highlight of this paper, But, it is postponed to Section 4 and using Theorem 3.4, we are going to investigate the solution operator of the parameter-dependent system (3-6) and (3-2). By (3-9), (3-11) and Theorem 3.4,

w(ξ⁰, x_N, λ) =

3

X

i,j=1

e^{−Ai(ξ0,λ)xN}g_ij(ξ⁰, λ)F⁰[g_j](ξ⁰,0),

τ(ξ⁰, x_N, λ) =−λ

3

X

i,j=1

e^{−Ai(ξ0,λ)xN}(γ_i²+ 2)gij(ξ⁰, λ)F⁰[gj](ξ⁰,0).

Let ψ be a C^∞ function on R such that ψ(t) = 1 for t < 1 and ψ(t) = 0 for t > 2, and set ϕ0(ξ⁰, λ) =ψ(c0|ξ⁰|/|λ|^1/2) andϕ∞(ξ⁰, λ) = 1−ϕ0(ξ⁰, λ). Note that

ϕ₀(ξ⁰, λ) =

(1 ifc₀|ξ⁰| ≤ |λ|^1/2,

0 ifc0|ξ⁰| ≥2|λ|^1/2, ϕ_∞(ξ⁰, λ) =

(0 ifc₀|ξ⁰| ≤ |λ|^1/2,

1 ifc0|ξ⁰| ≥2|λ|^1/2. (3-21)