Local energy decay estimate of solutions to the thermoelastic plate equations in two- and three-dimensional exterior domains

Local energy decay estimate of solutions to the thermoelastic plate equations in two- and three-dimensional exterior domains

Robert Denk^∗, Reinhard Racke^∗ and Yoshihiro Shibata^†

Abstract

In this paper we prove frequency expansions of the resolvent and local energy decay estimates for the linear thermoelastic plate equations:

utt+ ∆²u+ ∆θ= 0 and θt−∆θ−∆ut= 0 in Ω×(0,∞),

subject to Dirichlet boundary conditions: u|Γ = Dνu|Γ = θ|Γ = 0 and initial conditions (u, ut, θ)|t=0 = (u0, v0, θ0). Here Ω is an exterior domain (domain with bounded comple- ment) in Rⁿ with n = 2 or n = 3, the boundary Γ of which is assumed to be a C⁴- hypersurface.

1 Introduction and main results

Let Ω be an exterior domain (domain with bounded complement) in Rⁿ with n= 2 or n= 3, the boundary Γ of which is assumed to be a C⁴-hypersurface. In this paper, we consider the linear thermoelastic plate equations

u_tt+ ∆²u+ ∆θ= 0 and θ_t−∆θ−∆u_t= 0 in Ω×R+ (1.1) subject to the initial conditions

u(x,0) =u₀(x), u_t(x,0) =v₀(x), θ(x,0) =θ₀(x) (x∈Ω) (1.2) and Dirichlet boundary conditions

u|_Γ=Dνu|_Γ =θ|_Γ = 0. (1.3)

Here D_ν =Pn

j=1ν_jD_j (D_j =∂/∂x_j), and ν= (ν₁, . . . , ν_n) denotes the unit outer normal to Γ.

In (1.1),u stands for a mechanical variable denoting the vertical displacement of the plate, whileθstands for a thermal variable describing the temperature relative to a constant reference temperature ¯θ. The thermal effect introduces a damping. In fact, when Ω is a bounded reference configuration, the exponential stability of the associated semigroup under several different kind of boundary conditions have been proved by Kim [5], Mun˜oz Rivera and Racke [18], Liu and Zheng [14], Avalos and Lasiecka [1], Lasiecka and Triggiani [7, 8, 9, 10] and Shibata [22]. Also, the analyticity of the semigroup has been shown, cf. Liu and Renardy [12] and then it has been studied by Russell [20], Liu and Liu [11], Liu and Yong [13], Mun˜oz Rivera and Racke [19] in

∗Department of Mathematics and Statistics, University of Konstanz, 78457 Konstanz, Germany

†Department of Mathematical Sciences, School of Science and Engineering, Waseda University, Ohkubo 3-4-1, Shinjuku-ku, Tokyo 169-8555, Japan

the L₂ or Hilbert space setting (see also the book of Liu and Zheng [15] for a survey). In the L_p-setting this was investigated in our paper [4], where sufficiently strong a priori estimates for the resolvent in Lp-spaces have been proved. Before [4], Denk and Racke [3] studied the Cauchy problem for (1.1) in the whole spaceRⁿ, also giving decay rates of solutions, and Naito and Shibata [16] studied the initial boundary value problem for (1.1) with Dirichlet boundary condition in the half-spaceRⁿ+.

There were not yet any decay estimates for exterior domains. The purpose of this paper is to study the local energy decay of solutions to problem (1.1) – (1.3). To formulate the problem (1.1) – (1.3) in the semigroup setting, introducing the unknown functionv =ut, we rewrite it in matrix form:

U_t=AU in Ω×R+, U|_t=0=U₀, BU|_Γ = 0, (1.4) where we have set

U =



 u v θ



, U₀ =



 u₀ v₀ θ0



, A=





0 1 0

−∆² 0 −∆

0 ∆ ∆



, BU =



 u D_νu

θ



. (1.5)

To study the initial boundary value problem (1.4), we consider the corresponding resolvent problem:

(λI−A)U =F in Ω, BU|_Γ= 0, (1.6)

whereIdenotes the 3×3 unit matrix. We shall give an expansion of the resolvent with respect to the frequency parameter λ(Theorem 1.3). Then, representing the semigroup via the resolvents (essentially: Laplace transform) will give the local energy decay result (Theorem 1.4).

To state our main results precisely, we introduce several spaces and some symbols at this point. Throughout this paper, let n ∈ {2,3}. For a general domain O ⊂ Rⁿ, p ∈ (1,∞) and any integerm,Lp(O) and W_p^m(O) stand for the usual Lebesgue space and Sobolev space, respectively. Letk · k_Lp(O) andk · k_Wm

p (O)denote their norms. For a general domainOwithC¹ boundary ∂O, we introduce the spacesW_p,0² (O) and W_p,D^m (O) (m= 2,4) as follows:

W_p,0² (O) ={u∈W_p²(O)|u|_∂O = 0},

W_p,D^m (O) ={u∈W_p^m(O)|u|_∂O =D_νu|_∂O= 0} (m= 2,4), (1.7) where ν = (ν₁, . . . , ν_n) denotes the unit outer normal to ∂O. Let H_p(O) and D_p(O) be the spaces defined by the following formulas:

H_p(O) ={F =^T(f, g, h)|f ∈W_p,D² (O), g ∈L_p(O), h∈L_p(O)},

D_p(O) ={U =^T(u, v, θ)|u∈W_p,D⁴ (O), v ∈W_p,D² (O), θ∈W_p,0² (O)}. (1.8) Here and hereafter, ^TM denotes the transposed of M. We define the norms k · k_Hp(O) and k · k_Dp(O) by the following formulas:

kFk_Hp(O)=kfk_W2

p(O)+k(g, h)k_Lp(O) (F =^T(f, g, h)∈ H_p(O)), kUk_Dp(O)=kuk_W4

p(O)+k(v, θ)k

W2

p(O) (U =^T(u, v, θ)∈ D_p(O)). (1.9) LetA_O be the operator whose domain isD_p(O) and whose operation is defined by the formula:

A_OU =AU forU ∈ D_p(O). (1.10)

In [4] we proved the following theorem.

Theorem 1.1. Let 1< p <∞. Let ρ(A_Ω) be the resolvent set of A_Ω. Let C+={λ∈C|Reλ≥0}

where C denotes the set of all complex numbers. Then, ρ(A_Ω)⊃C+\ {0}.

Moreover, for any λ₀ >0 there exists a constant C depending onλ₀, p and Ωsuch that for anyλ∈C+ with |λ| ≥λ0 andF ∈ H_p(Ω) there holds the estimate:

|λ|k(λI− A_p)⁻¹Fk_Hp(Ω)+k(λI− A_p)⁻¹Fk_Dp(Ω) ≤CkFk_Hp(Ω).

In view of Theorem 1.1, by standard arguments in the theory of analytic semigroups (cf.

Vrabie [24]) we know that for anyσ >0 there exists aθσ ∈(0, π/2) such that

ρ(A_Ω)⊃ {λ∈Σθσ | |λ|> σ}, (1.11) where we have set

Σ={λ∈C\ {0} | |argλ|< π−}. (1.12) Moreover, there exists a constantCσ depending onσ such that

|λ|k(λI− A_Ω)⁻¹Fk_Hp(Ω)+k(λI− A_Ω)⁻¹Fk_Dp(Ω)≤CσkFk_Hp(Ω) (1.13) for any λ∈Σ_θσ with|λ|> σ and F ∈ H_p(Ω). Let us define a setU by the formula

U = [

σ>0

{λ∈Σ_θσ | |λ|> σ}. (1.14) From (1.11) we see that

ρ(A_Ω)⊃ U. (1.15)

By (1.13), we have the following theorem.

Theorem 1.2. Let 1 < p < ∞. Then, A_Ω generates an analytic semigroup {T_Ω(t)}t≥0 in H_p(Ω).

Let bbe a number such thatB_b ⊃Rⁿ\Ω, whereB_b ={x∈Rⁿ| |x|< b}. Set Ω_b =B_b∩Ω.

We introduce the following spaces:

Lp,b(Ω) ={f ∈Lp(Ω)|f(x) = 0 for |x|> b}, H_p,b(Ω) =H_p(Ω)∩(L_p,b(Ω))³

={F =^T(f, g, h)|f ∈W_p,D² (Ω)∩L_p,b(Ω), g, h∈L_p,b(Ω)}.

(1.16)

Replacing Ω byRⁿ, we defineL_p,b(Rⁿ) andH_p,b(Rⁿ). For functionsU =^T(u, v, θ) we will write kUk_Dp,loc(Ωb):=kU|_Ωbk_Dp(Ωb).

For Banach spaces X and Y, L(X, Y) denotes the set of all bounded linear operators from X into Y and L(X) = L(X, X). For any domain ω in C, Anal (ω, X) denotes the set of all holomorphic functions defined onω with their values inX. We set

ω_τ :={λ∈C| |λ|< τ}, ω˙_τ :=ω_τ\(−∞,0].

The following two theorems are our main results.

Theorem 1.3. Let n∈ {2,3},1< p <∞ and letb be a number such that Bb−3 ⊃Rⁿ\Ω. Let U be the same set as in (1.14). Set L_p,b(Ω) =L(H_p,b(Ω),D_p,loc(Ω_b)).

(a) In the case n = 2 there exist a constant τ > 0 and an operator-valued function G ∈ Anal( ˙ω_τ,L_p,b(Ω)) such that for any F ∈ H_p,b(Ω)and λ∈ω˙_τ∩ U there holds the equality:

(λI− A_Ω)⁻¹F =G(λ)F in Ω_b.

Moreover, there exist operators G1, G2∈ L_p,b(Ω) and an operator-valued function G₃∈Anal( ˙ω_τ,L_p,b(Ω))

such that

G(λ) =G1+ (logλ)⁻¹G2+G3(λ) for any λ∈ω˙τ, kG₃(λ)Fk_D

p,loc(Ωb) ≤C|logλ|⁻²kFk_H

p(Ω) for any λ∈ω˙_τ and F ∈ H_p,b(Ω). (1.17) (b) In the case n = 3 there exist a constant τ > 0 and operator-valued functions G_j ∈ Anal(ωτ,L_p,b(Ω)) (j = 1,2) such that for any F ∈ H_p,b(Ω) and λ ∈ ωτ ∩ U there holds the equality:

(λI− A_Ω)⁻¹F =λ¹²G₁(λ)F+G₂(λ)F in Ω_b. (1.18) For wave equations, elasticity or Maxwell equations, a collection of references for results on low frequency asymptotics is given in the work of Pauly [17].

With the expansion of the resolvent in terms of the frequency parameter above, we shall obtain the following local energy decay result.

Theorem 1.4. Let1< p <∞and letbbe the same constant as in Theorem 1.3. Let{T_Ω(t)}t≥0

be the semigroup associated with problem (1.1) – (1.3) which is given in Theorem 1.2. Then, we have

kT_Ω(t)Fk_Dp,loc(Ωb)≤

(Cp,bt⁻¹(logt)⁻²kFk_Hp(Ω) if n= 2,

C_p,bt⁻³²kFk_Hp(Ω) if n= 3 (1.19) for anyt≥1 and F ∈ H_p,b(Ω).

The difficulty in proving Theorem 1.3 arises from the facts that the expansion formula of the resolvent operator (λ−∆)⁻¹ inR² has the singularity logλand that of (λ−∆²)⁻¹ inRⁿ has the singularities λ⁻¹logλ when n = 2 and λ⁻¹² when n = 3, respectively. Therefore, we can not use the usual compact perturbation method to obtain the expansion formula in the exterior domain. To prove Theorem 1.3, first of all employing the Seeley argument [21] about the invertibility ofI+Kλ,Kλ being a compact operator valued holomorphic function inλ, we shall show that (λI− A_Ω)⁻¹ has an expansion formula nearλ= 0 which starts fromλ^s(logλ)^β in two dimensional case and λ^s2 in three dimensional case for some integers s and β. Then, by a contradiction argument based on the uniqueness theorem we shall show that s = 0 and β = 0. Our strategy of the proof of Theorem 1.3 follows R. Kleinmann and B. Vainberg [6] and W. Dan and Y. Shibata [2], where the low frequency expansions of the Laplace operator and Stokes operator in the two dimensional case were obtained.

We will prove Theorems 1.3 and 1.4 in Sections 2–3 for the (somewhat simpler) case n= 3.

Modifications for the case n= 2 are indicated in Sections 4 and 5.

2 Expansion formulas in three dimensions

We start with the three-dimensional case by showing an expansion formula of the resolvent in the whole-space.

Theorem 2.1. Let 1 < p < ∞ and b > 0. Let L_p,b(R³) be the set of all bounded linear operators from H_p,b(R³) into D_p,loc(B_b) and ρ(A_R3) the resolvent set of A_R3. Then, there exist constants ∈(0, π/2) and operator-valued functions H_j(λ)∈Anal (C,L_p,b(R³)) (j = 1,2) such thatρ(A_R3)⊃Σ and

(λI− A_R3)⁻¹F =λ⁻¹²E₀F+E₁F +λ¹²H₁(λ)F+λH₂(λ)F in B_b (2.1) for anyλ∈Σ andF ∈ H_p,b(R³). Here,Σ is the set defined in (1.12),

E₀F =



 αR

R³g dx+βR

R³h dx 0

0



, E₁F =





E₃²∗(−∆f +g+h)

−f E₃¹∗(h−∆f)



, E₃¹(x) = 1

4π|x|, E₃²(x) =−|x|

8π,

(2.2)

∗ stands for the convolution operator, is given in (2.6), and α and β are non-zero constants given in (2.11) in the proof below.

Remark 2.2. E₃¹(x) and E₃²(x) are fundamental solutions to−∆ and ∆² in R³, respectively.

Proof. For F ∈ H_p(R³), we set U(λ) = (λI− A_R3)⁻¹F. Let ˆU(λ)(ξ) = ^T(ˆuλ(ξ),vˆλ(ξ),θˆλ(ξ)) be the Fourier transform of U(λ). Then, from Naito and Shibata [16], we have the following formulas:

ˆ u_λ(ξ) =

3

X

j=1

h A⁰_j +A¹_j +A²_j

(λ+γ_j|ξ|²)|ξ|²|ξ|²fˆ(ξ) + A⁰_j +A¹_j

(λ+γ_j|ξ|²)|ξ|²g(ξ) +ˆ A⁰_j

(λ+γ_j|ξ|²)|ξ|²ˆh(ξ) i

,

ˆ v_λ(ξ) =

3

X

j=1

h−(A⁰_j +A¹_j)|ξ|²

λ+γj|ξ|² fˆ(ξ) + A¹_j+A¹₂

λ+γj|ξ|²ˆg(ξ) + A¹_j

λ+γj|ξ|²ˆh(ξ)i , θˆλ(ξ) =

3

X

j=1

h A⁰_j|ξ|²

λ+γj|ξ|²f(ξ)ˆ − A¹_j

λ+γj|ξ|²g(ξ) +ˆ A⁰_j +A²_j λ+γj|ξ|²ˆh(ξ)

i .

(2.3)

Here,γ_j (j= 1,2,3) are numbers such that

3

Y

j=1

(t+γj) =t³+t²+ 2t+ 1 for any t∈C, (2.4) 0< γ₁ <1,γ₃ is the complex conjugate ofγ₂ and Reγ₂ = (1−γ₁)/2>0; andA⁰_j,A¹_j and A²_j (j = 1,2,3) are complex numbers such that

λ^k Q3

j=1(λ+γj|ξ|²) =

3

X

j=1

A^k_j

(λ+γj|ξ|²)|ξ|^4−2k (k= 1,2,3)

for any ξ∈R³ and λ∈Cwithλ+γ_j|ξ|²6= 0 (j= 1,2,3). We have the following formulas:

3

X

j=1

A⁰_j =

3

X

j=1

A¹_j = 0,

3

X

j=1

A²_j = 1,

3

X

j=1

A⁰_j γj

= 1,

3

X

j=1

A¹_j γj

=

3

X

j=1

A²_j γj

= 0. (2.5)

Since γ₂ and γ₃ are complex conjugate and Reγ₂ >0, we may assume that 0 <argγ₂ < π/2.

Let us define by the formula:

= argγ₂. (2.6)

Since λ+γ_j|ξ|² 6= 0 for any λ ∈ Σ and ξ ∈ R³, by Fourier multiplier theorem we have U(λ) =^T(u_λ, v_λ, θ_λ)∈ D_p(R³). Moreover, for any ⁰ with < ⁰ < π/2 there exists a constant C depending on ⁰ such that

2

X

j=0

|λ|^2−j2 k∇^j(∇²uλ, vλ, θλ)k_Lp(R3) ≤CkFk

Hp(R3),

|λ|k∇u_λk_Lp(R3)+|λ|²ku_λk_Lp(R3) ≤Ck(|λ|f, g, h)k_Lp(R3)

(2.7)

for anyλ∈Σ⁰ (cf. Naito-Shibata [16]), where∇^jw= (D^αw| |α|=j). From these observations, we see thatρ(A_R3)⊃Σ.

Now, restricting ourselves to the case where F ∈ H_p,b(R³), we shall derive an expansion formula of (λI − A

R³)⁻¹F by using the formula (2.3). Let F_ξ⁻¹ denote the Fourier inverse transform, and then we have

F_ξ⁻¹[(λ+|ξ|²)⁻¹](x) = e⁻

√ λ|x|

4π|x| , F_ξ⁻¹[(λ+|ξ|²)⁻¹|ξ|⁻²](x) =−λ⁻¹e⁻

√λ|x|

4π|x| − 1 4π|x|

(2.8)

for any λ∈C\(−∞,0]. Since we havee⁻

√

λ|x|=P∞ j=0(−√

λ|x|)^j/(j!), we have F_ξ⁻¹[(λ+|ξ|²)⁻¹](x) = 1

4π|x|−λ¹²

4πH₁¹(λ|x|²) +λ|x|

8π H₂¹(λ|x|²), (2.9) F_ξ⁻¹[(λ+|ξ|²)⁻¹|ξ|⁻²](x) = λ⁻¹²

4π − |x|

8π + λ¹²|x|²

4π H₁²(λ|x|²)−λ|x|³

4π H₂²(λ|x|²), (2.10) where we have set

H₁²(z) =

∞

X

j=0

z^j

(2j+ 3)!, H₂²(z) =

∞

X

j=0

z^j (2j+ 4)!, H₁¹(z) = 1 +zH₁²(z), H₂¹(z) = 1 + 2zH₂²(z).

Now, we assume thatF ∈ H_p,b(R³). Sinceλ+γ_j|ξ|²=γ_j(λγ_j⁻¹+|ξ|²), using (2.10) and (2.5), from (2.3) we have

u_λ(x) =hX³

j=1

A⁰_j +A¹_j

√γj

1 4π

Z

R³

g dx+X³

j=1

A⁰_j

√γj

1 4π

Z

R³

h dxi

λ⁻¹² +E₃²∗(−∆f +g+h)

+λ¹²

hnX³

j=1

A⁰_j+A¹_j +A²_j

γ_j^3/2 H₁²(γ_j⁻¹λ|x|²) |x|²

4π o

∗(−∆f)

+ nX³

j=1

A⁰_j+A¹_j γ_j^3/2

H₁²(γ_j⁻¹λ|x|²) |x|²

4π o

∗g+ nX³

j=1

A⁰_j γ_j^3/2

H₁²(γ_j⁻¹λ|x|²) |x|²

4π o

∗h i

+λhnX³

j=1

A⁰_j +A¹_j +A²_j

γ_j² H₂²(γ_j⁻¹λ|x|²)|x|³ 4π

o∗(−∆f)

+ nX³

j=1

A⁰_j+A¹_j

γ_j² H₁²(γ_j⁻¹λ|x|²) |x|³

4π o

∗g+ nX³

j=1

A⁰_j

γ²_j H₁²(γ_j⁻¹λ|x|²) |x|³

4π o

∗h i

. Setting

α=

3

X

j=1

A⁰_j+A¹_j

√γ_j , β =

3

X

j=1

A⁰_j

√γ_j, (2.11)

we have the first line of the formula (2.1) with (2.2). Using the fact that E₃¹ ∗(−∆f) = f to obtain the formula forvλ(x), by (2.3), (2.5) and (2.9) we have

v_λ(x) =−f +λ¹²hX³

j=1

A⁰_j +A¹_j 4πγ_j^3/2

H₁¹(γ_j⁻¹λ|x|²)

∗(−∆f)

−X³

j=1

A¹_j +A²_j 4πγ_j^3/2

H₁¹(γ_j⁻¹λ|x|²)

∗g−X³

j=1

A¹_j 4πγ_j^3/2

H₁¹(γ_j⁻¹λ|x|²)

∗h i

−λhnX³

j=1

A⁰_j +A¹_j

γ_j² H₂¹(γ_j⁻¹λ|x|²)|x|

8π

o∗(−∆f)

−nX³

j=1

A¹_j+A²_j

γ_j² H₂¹(γ_j⁻¹λ|x|²)|x|

8π o

∗g−nX³

j=1

A¹_j

γ²_j H₂¹(γ_j⁻¹λ|x|²)|x|

8π o

∗hi ,

θ_λ(x) =E¹₃∗(h−∆f)−λ¹² hX³

j=1

A⁰_j 4πγ_j^3/2

H₁¹(γ⁻¹_j λ|x|²)

∗(−∆f)

−X³

j=1

A¹_j 4πγ_j^3/2

H₁¹(γ⁻¹_j λ|x|²)

∗g+X³

j=1

A⁰_j+A¹_j 4πγ_j^3/2

H₁¹(γ_j⁻¹λ|x|²)

∗hi

+λ

hnX³

j=1

A⁰_j

γ_j²H₂¹(γ_j⁻¹λ|x|²) |x|

8π o

∗(−∆f)

−nX³

j=1

A¹_j

γ_j²H₂¹(γ⁻¹_j λ|x|²) |x|

8π o

∗g+ nX³

j=1

A⁰_j+A¹_j

γ_j² H₂¹(γ_j⁻¹λ|x|²) |x|

8π o

∗h i

. This completes the proof of Theorem 2.1.

The next step in the proof of our main results consists in an expansion formula for the resolvent operator in Ω near λ= 0. We will show the following theorem.

Theorem 2.3. Let 1 < p < ∞ and b be a positive number such that Bb−3 ⊃ R³\Ω. Let U and L_p,b(Ω) be the same sets as in (1.14) and Theorem 1.3, respectively. Then, there exist a constant τ >0, an integer sand operators G_j(λ)∈Anal (ωτ,L_p,b(Ω)) (j= 1,2)such that

(λI− A_Ω)⁻¹F =λ^s2G₁(λ)F +λ^s+12 G₂(λ)F in Ω_b for anyλ∈ω_τ ∩ U and F ∈ H_p,b(Ω).

In what follows, we shall prove Theorem 2.3. For a given functionf defined on Ω,ιf denotes the zero extension off to the whole space R³ andrf denotes the restriction of f to the domain Ωb = Ω∩Bb. From Denk, Racke and Shibata [4] (also Simader [23]), we know the unique existence of a solution U₀ =^T(u₀, v₀, θ₀)∈ D_p(Ω_b) of the equation:

−AU₀=F in Ω_b, BU₀|_∂Ωb = 0 (2.12) for any F ∈ H_p(Ω_b), Here, ∂Ω_b = Γ∪S_b,S_b={x∈R³ | |x|=b} and BU0|_∂Ωb = 0 means that

u₀ =D_νu₀=θ₀ = 0 on Γ andS_b,

where Dν = (x/|x|)· ∇ on S_b. Let us define the operatorSΩ_b by the formula: SΩ_bF =U0 and write S_ΩbF = (u_Ωb, v_Ωb, θ_Ωb) as long as no confusion occurs. Let E₀, E₁, H₁(λ) and H₂(λ) be the same operator as in Theorem 2.1 and set

H(λ) =λ⁻¹²E₀+E₁+λ¹²H₁(λ) +λH₂(λ). (2.13) In what follows, we write H(λ)F = (u_λ,R³, v_λ,R³, θ_λ,R³). Let ϕ be a function in C₀^∞(R³) such thatϕ(x) = 1 for|x|< b−2 andϕ(x) = 0 for|x|> b−1. With these preparations, we introduce the operator Φ as follows:

Φ(λ)F = (1−ϕ)H(λ)ιF +ϕS_ΩbrF. (2.14) By Theorem 2.1, we have

Φ(λ)F = (1−ϕ)(λI− A_R2)⁻¹ιF +ϕS_ΩbrF (2.15) when λ∈Σ. And therefore, applying λI−A to Φ(λ)F, we have

(λI−A)Φ(λ)F =F+T(λ)F in Ω, BΦ(λ)F|_Γ= 0 (2.16) for any λ∈Σ, where T(λ)F is defined by the formula:

T(λ)F =





0

−L³_ϕ(u_λ,R² −uΩb)−L¹_ϕ(θ_λ,R² −θΩb) L¹_ϕ(θ_λ,R²−θ_Ωb) +L¹_ϕ(v_λ,R² −v_Ωb)



, (2.17)

L³_ϕ(w) = ∆²(ϕw)−ϕ∆²w, andL¹_ϕ(w) = ∆(ϕw)−ϕ∆w. If we consider (2.16) only on Ω_b, the operators in both sides of (2.16) are analytic with respect toλ∈C\(−∞,0], and therefore by analytic continuation we have

(λI−A)Φ(λ)F =F +T(λ)F in Ω_b, BΦ(λ)F|_Γ= 0 (2.18) for anyλ∈C\(−∞,0]. If (I+T(λ))⁻¹ exists, then Φ(λ)(I+T(λ))⁻¹F solves equations (2.16) and (2.18).

Lemma 2.4. Let U and Σ be the same sets as in (1.14)and Theorem 2.1, respectively. Then, (I+T(λ))⁻¹ exists as a bounded linear operator on H_p,b(Ω) for anyλ∈ U ∩Σ.

Proof. Letλ∈Σ∩ U. Since the second and third components ofT(λ)F belong toW_p¹(Ω) and suppT(λ)F ⊂ Db−2,b−1 = Bb−1 \Bb−2, by Rellich’s compactness theorem T(λ) is a compact operator onH_p,b(Ω). Therefore, to prove the lemma it suffices to show thatI+T(λ) is injective.

LetF be an element ofH_p,b(Ω) such that (I+T(λ))F = 0. SetU = Φ(λ)F, and then by (2.18) we have

(λI−A)U = 0 in Ω, BU|_Γ= 0.

Since SΩ_brF ∈ D_p(Ω_b) and (λI− A_R3)⁻¹ιF ∈ D_p(R³) forλ∈Σ (cf. (2.7)), by (2.15) we have U ∈ D_p(Ω). SinceU ⊂ρ(A_Ω) as follows from (1.15), we have U = 0, which implies that

(1−ϕ)(λI− A_R3)⁻¹ιF +ϕS_ΩbrF = 0 in Ω. (2.19) Recalling that ϕ(x) = 1 for |x|< b−2 and ϕ(x) = 0 for |x|> b−1, by (2.19) we have

(λI− A_R3)⁻¹ιF = 0 for|x|> b−1, SΩbrF = 0 for|x|< b−2.

If we setV(x) = (S_ΩbrF)(x) forx∈Ω_b and V(x) = 0 forx6∈Ω, thenV(x) belongs to D_p(B_b) and satisfies the equation:

(λI−A)V =ιF inB_b, BV|_Sb = 0.

Since (λI−A_R³)⁻¹ιF also satisfies the above equation, by the uniqueness of solutions we have V = (λI−A_R3)⁻¹ιF inB_b, and therefore S_ΩbF = (λI−A_R3)⁻¹ιF in Ω_b, which inserted into (2.19) implies that

0 = (λI− A_R3)⁻¹ιF +ϕ(SΩbF −(λI− A_R3)⁻¹ιF) = (λI− A_R3)⁻¹ιF in Ω.

Therefore,F = (λI−A)(λI− A_R3)⁻¹ιF = 0 in Ω, which completes the proof of the lemma.

By Lemma 2.4 we have

(λI− A_Ω)⁻¹= Φ(λ)(I+T(λ))⁻¹ (2.20) forλ∈Σ∩ U.

Now, we shall discuss the invertibility of (I+T(λ)) for λ∈ω˙σ with someσ > 0, where we have set

˙

ωσ ={λ∈C\ {0} | |λ|< σ and |argλ|< π}.

For this purpose, we introduce an auxiliary operator:

Φ₀F = (1−ϕ)E₁ιF +ϕS_ΩbrF

forF ∈ H_p,b(Ω), where E₁ is the same operator as in Theorem 2.1. Note that

−AE₁ιF =ιF in R³.

We writeE₁ιF =^T(u_0,R³, v_0,R³, θ_0,R³) unless any confusion may occur. ApplyingA to Φ0F, we have

−AΦ₀F =F+T₀F in Ω, BΦ₀F|_Γ= 0, (2.21)

where

T₀F =





0

−L³_ϕ(u_0,R³ −u_Ωb)−L¹_ϕ(θ_0,R³ −θ_Ωb) L¹_ϕ(θ_0,R³−θΩ_b) +L¹_ϕ(v_0,R³ −vΩ_b)



.

Since the second and third members of T₀F belong to W_p¹(Ω) and suppT₀F ⊂ Db−2,b−1, by Rellich’s compactness theorem T0 is a compact operator on H_p,b(Ω). According to Theorem 2.1, we set

u_λ,R3 =u_0,R3 +λ⁻¹²T(αg+βh) +U_λ,R3, v_λ,R³ =v_0,R³ +V_λ,R³,

θ_λ,R³ =θ_0,R³+ Θ_λ,R³, whereT a=R

R³a dxand

T(U_λ,R3, V_λ,R3,Θ_λ,R3) =λ¹²H₁(λ)ιF +λH₂(λ)ιF. (2.22) Then, we have

(I+T(λ))F = (I+T₀)F +λ⁻¹²(∆²ϕ)^T(0, T(αg+βh),0) +R(λ)F (2.23) where

R(λ)F =





0

−L³_ϕ(U_λ,R³)−L¹_ϕ(Θ_λ,R³) L¹_ϕ(Θ_λ,R3) +L¹_ϕ(V_λ,R3)



. (2.24)

In view of (2.22) and (2.24), there exist operatorsRj(λ)∈Anal (C,L(H_p,b(Ω))) (j= 1,2) such that

R(λ)F =λ¹²R₁(λ)F +λR₂(λ)F (2.25) for any λ∈C\(−∞,0]. In particular, we have

λ→0limkR(λ)k_L(H

p,b(Ω))= 0. (2.26)

Here,k · k_L(H

p,b(Ω)) denotes the operator norm ofL(H_p,b(Ω)). SinceT0 is a compact operator on H_p,b(Ω), by Seeley’s lemma [21] there exists a finite range operatorBsuch thatI+T0−Bhas an inverse operator (I+T0−B)⁻¹ ∈ L(H_p,b(Ω)). SetG_λ =I+T0−B+R(λ) andG0 =I+T0−B, and then

(I+T(λ))F =G_λF+BF+λ⁻¹²(∆²ϕ)^T(0, T(αg+βh),0) (2.27)

G_λ= (I+R(λ)G⁻¹₀ )G₀. (2.28)

By (2.26) there exists aτ0>0 such thatkR(λ)G⁻¹₀ k_L(H

p,b(Ω) ≤1/2 for anyλ∈ω˙τ0, and therefore by Neumann series expansion we have

G⁻¹_λ =G⁻¹₀ (I+R(λ)G⁻¹₀ )⁻¹ =G⁻¹₀

∞

X

j=0

(−R(λ)G⁻¹₀ )^j (λ∈ω˙τ0). (2.29) In view of (2.25), we see that there exist aτ₁ >0 and operators G_j(λ)∈Anal (ω_τ1,L(H_p,b(Ω)) (j = 1,2) such that

G⁻¹_λ =λ¹²G1(λ) +G2(λ) for any λ∈ω˙τ1. (2.30)

We define the operator ˜B by the formula ˜BF = (∆²ϕ)^T(0,R

R³(αg+βh)dx,0). As both operators B and ˜B are finite range operators, we can choose h₁, . . . ,h_m ∈ H_p,b(Ω) which are linearly independent overC in such a way that

BF =

m

X

j=1

β_j(F)h_j, BF˜ =

m

X

j=1

β˜_j(F)h_j

with βj(F), ˜βj(F)∈ C. To represent βj(F), ˜βj(F) ∈C in more convenient way, we introduce h^∗₁, . . ., h^∗_m ∈ H_p,b(Ω)^∗ such that < h_j,h^∗_k >=δ_jk, where < ·,·> is the dual paring between H_p,b(Ω) and its dual space H_p,b(Ω)^∗ and δjk denote the Kronecker delta symbols. By using these symbols, we write

βj(F) =< BF,h^∗_j >=< F, B^∗h^∗_j >, β˜j(F) =<BF,˜ h^∗_j >=< F,B˜^∗h^∗_j > . Setting`<sup>∗</sup><sub>aj</sub> =B<sup>∗</sup>h<sup>∗</sup><sub>j</sub> and`^∗_bj = ˜B^∗h^∗_j, we have

BF +λ⁻¹²(∆²ϕ)^T(0, T(αg+βh),0) =

m

X

j=1

< F, `<sup>∗</sup><sub>aj</sub>+λ<sup>−12</sup>`^∗_bj>h_j,

and therefore we have

(I+T(λ))F =GλF +

m

X

j=1

< F, `<sup>∗</sup><sub>aj</sub>+λ<sup>−12</sup>`^∗_bj>hj. (2.31)

Applying G⁻¹_λ to the both side of (2.31), we have G⁻¹_λ (I+T(λ))F =F +

m

X

j=1

< F, `<sup>∗</sup><sub>aj</sub>+λ<sup>−12</sup>`^∗_bj > G⁻¹_λ h_j = (I+N_λ)F (2.32)

where we have defined the operator N_λ by the formula:

N_λF =

m

X

j=1

< F, `<sup>∗</sup><sub>aj</sub>+λ<sup>−12</sup>`^∗_bj > G⁻¹_λ h_j. (2.33) Now, we shall show the existence of the inverse operator ofI+N_λ. For the notational simplicity, we set G⁻¹_λ hj = vλ,j and `<sup>∗</sup><sub>aj</sub> +λ<sup>−12</sup>`^∗_bj = Aλ,j. Since {h_j}^m_j=1 is linearly independent, so is {v_λ,j}^m_j=1. Let us consider the m×m matrix: M(λ) = (δ_jk+ <v_λ,k, A_λ,j >). By (2.30) the (j, k) component δ_jk+<v_λ,k, A_λ,j >is of the form: λ⁻¹²m_1jk(λ) +m_2jk(λ), wherem_1jk(λ) and m2jk(λ) are complex valued holomorphic functions defined onωτ1. LetD(λ) be the determinant of M(λ). In particular, we can say that D(λ) ≡ 0 on ω_τ1 or there exist an integer q₁, and functions D_j(λ) (j= 1,2) such that

D(λ) =λ^q21D₁(λ) +λ^q1+12 D₂(λ) forλ∈ω˙_τ1, (2.34) D1(0)6= 0, andDj(λ) (j= 1,2) are both holomorphic inωτ1. We shall show that

D(λ)6≡0 inωτ1. (2.35)

In fact, let λ ∈ U ∩Σ ∩ω_τ1 and assume that D(λ) = 0. Then there exists a vector x_λ =

T(x_λ1, . . . , x_λm)∈R^m\ {0} such that 0 =

m

X

k=1

(δ_jk+<v_λ,k, A_λ,j >)x_λ,k=x_λ,j+

m

X

k=1

<v_λ,k, A_λ,j > x_λ,k (2.36) for j = 1, . . . , m. Set F_λ = P_m

k=1x_λ,kv_λ,k ∈ H_p,b(Ω), and then F_λ 6= 0, because {v_λ,k}^m_k=1 is linearly independent. On the other hand, by (2.33) and (2.36)

N_λF_λ =

m

X

j=1

< F_λ, A_λ,j >v_λ,j =

m

X

j,k=1

x_λ,k<v_λ,k, A_λ,j >v_λ,j =−

m

X

j=1

x_λ,jv_λ,j =−F_λ, which implies that (I+Nλ)Fλ = 0. And therefore, by (2.32) and (2.31) (I+T(λ))Fλ = 0. On the other hand, by Lemma 2.4 I+T(λ) is invertible when λ∈ U ∩Σ, and therefore we have F_λ= 0. This leads to a contradiction. Therefore, we have (2.35), and then (2.34) holds.

From (2.34), there exist a constant τ2 (0 < τ2 ≤ τ1) and holomorphic functions Ej(λ) (j = 1,2) defined onω_τ2 such that

D⁻¹(λ) =λ^−q21E₁(λ) +λ^−q21+12E₂(λ) forλ∈ω˙_τ2. (2.37) By using this fact, we shall show the existence of (I+N_λ)⁻¹. We may assume thatD⁻¹(λ)6= 0 whenλ∈ωτ2\ {0}. Let us denote the (j, k) cofactor ofM(λ) byMjk(λ), which has the similar formula toD⁻¹(λ) in (2.37). We observe that

(I+N_λ)[G−D(λ)⁻¹

m

X

j=1 m

X

k=1

< G, A_λ,k> M_jk(λ)v_λ,j]

=G−D(λ)⁻¹

m

X

j,k=1

< G, A_λ,k> M_jk(λ)v_λ,j

+N_λG−D(λ)⁻¹

m

X

j,k=1

< G, A_λ,k > M_j,k(λ)N_λv_λ,j = (∗).

Since N_λv_λ,j =Pm

`=1 <v<sub>λ,j</sub>, A<sub>λ,`</sub> >v<sub>λ,`</sub> as follows from (2.33) and our short notation: `^∗_aj + λ⁻¹²`^∗_bj =A_λ,j, we can proceed as follows:

(∗) =G−D(λ)⁻¹

m

X

j,k=1

< G, Aλ,k > Mjk(λ)vλ,j +

m

X

k=1

< G, Aλ,k>vλ,k

−D(λ)⁻¹

m

X

j,k,`=1

< G, A_λ,k> M_jk(λ)<v_λ,j, A_{λ,`</sub> >v<sub>λ,`}

=G+

m

X

k=1

< G, A_λ,k>v_λ,k−D(λ)⁻¹

m

X

j,k,`=1

(δ_{`j</sub>+<v<sub>λ,j</sub>, A<sub>λ,`} >)M_jk(λ)< G, A_λ,k>

v_λ,`

=G+

m

X

k=1

< G, A_λ,k>v_λ,k−

m

X

k,`=1

δ_{`k</sub> < G, A<sub>λ,k</sub>>v<sub>λ,`}

=G.

From this observation and our short notations: G⁻¹_λ hj =v_λ,j and`<sup>∗</sup><sub>aj</sub>+λ<sup>−12</sup>`^∗_bj =A_λ,j, we have (I+N(λ))⁻¹G=G−D(λ)⁻¹

m

X

j,k=1

< G, `<sup>∗</sup><sub>ak</sub>+λ<sup>−12</sup>`^∗_bk > M_jk(λ)G⁻¹_λ h_k forλ∈ωτ2 \ {0}. By (2.35), we see that

(I+T(λ))⁻¹ = (I +N_λ)⁻¹G⁻¹_λ

which combined with (2.30) and (2.37) implies that there exist an integer q2 and operators T_j(λ)∈Anal (ω_τ2,L(H_p,b(Ω))) (j = 1,2) such that

(I+T(λ))⁻¹=λ^q22T1(λ) +λ^q2+12 T2(λ)

for anyλ∈ω_τ2\ {0}. Combining this fact with (2.20), (2.14) and Theorem 2.1 implies Theorem 2.3.

3 The proofs of Theorems 1.3 and 1.4 in the three-dimensional case

In what follows,b denotes a large number such thatBb−3 ⊃R³\Ω. To prove Theorem 1.3, we start with the following lemmas.

Lemma 3.1. Let `be a positive integer and n∈ {2,3}. If u∈ S⁰(Rⁿ)∩L_1,loc(Rⁿ) satisfies the homogeneous equation:

∆^`u= 0 in Rⁿ (3.1)

and the radiation condition:

u(x) =O(|x|^m) as |x| → ∞, (3.2)

for some non-negative integer m, then u is a polynomial of orderm.

Proof. Since u ∈ S⁰(Rⁿ), applying the Fourier transform to (3.1) we have |ξ|^2`u(ξ) = 0, whichˆ implies that supp ˆu(ξ) ⊂ {0}. By the structure theorem of distributions, ˆu(ξ) is represented as follows: ˆu(ξ) =P

|α|≤kcαδ^(α)(ξ) for some non-negative integer k, whereδ denotes the Dirac delta function and c_α are complex numbers. By the Fourier inverse transform, we have

u(x) = X

|α|≤k

c_α(−ix)^α,

which combined with (3.2) implies that u = u(x) should be a polynomial of order m. This completes the proof of the lemma.

Lemma 3.2. Let E₁ be the same operator as in Theorem 2.1. Given F = ^T(f, g, h), we set U =E₁F =^T(u, v, θ). If F ∈ H_p,b(R³) and

Z

R³

(g(x) +h(x))dx= 0, (3.3)

then

u(x) =O(1), ∇u(x) =O(|x|⁻¹), (3.4)

θ(x) =O(|x|⁻¹) (3.5)

as |x| → ∞.

Proof. Since R

R³(g(y) +h(y)−∆f(y))dy= 0 as follows from (3.3), by (2.2) we have u(x) = −1

8π Z

R³

(|x−y| − |x|)(g(y) +h(y)−∆f(y))dy.

By Taylor’s formula we have

|x−y| − |x|= Z 1

0

d

dθ|x−θy|dθ=−

3

X

i=1

Z 1 0

(x_i−θy_i)y_i|x−θy|⁻¹dθ,

and therefore

u(x) =

3

X

i=1

Z 1 0

nZ

R³

(x_i−θy_i)y_i

|x−θy| (g(y) +h(y)−∆f(y))dyo dθ,

which combined with the fact that g(y) +h(y)−∆f(y) = 0 vanishes for|y| ≥b implies (3.4).

Since

θ=E₃¹∗(h−∆f) = 1

4π|x|∗(h−∆f)

and since h(y)−∆f(y) vanishes for |y| ≥ b, we have (3.5), which completes the proof of the lemma.

Lemma 3.3. Let 1< p <∞. (1)If θ∈W_p,loc² (Ω)satisfies the homogeneous equation:

∆θ= 0 in Ω, θ|_Γ= 0 (3.6)

and the radiation condition:

θ(x) =O(|x|⁻¹) (3.7)

as |x| → ∞, then θ= 0.

(2) If u∈W_p,loc⁴ (Ω) satisfies the homogeneous equation:

∆²u= 0 in Ω, u|_Γ =D_νu|_Γ= 0 (3.8) and the radiation condition:

u(x) =O(1) (3.9)

as |x| → ∞, then u= 0.

Proof. (1) By L_p (1< p <∞) solvability in any C² bounded domain for the Dirichlet problem of the Laplace operator (cf. Simader [23]) and Sobolev’s imbedding theorem, we see that θ∈W_2,loc² (Ω). Let ρ be a function inC₀^∞(R³) such that ρ(x) = 1 for |x| ≤1 and ρ(x) = 0 for

|x| ≥2. Setρ_L(x) =ρ(x/L) for L > b. Then, we have

0 = (∆θ, ρ_Lθ)_Ω=−(∇θ, ρ_L∇θ)_Ω+ (1/2)(θ,(∆ρ_L)θ)_Ω (3.10) where (a, b)Ω=R

Ωa(x)b(x)dx. Since

|(θ,(∆ρL)θ)Ω| ≤ k∆ρk

L∞(R3)L⁻² Z

L≤|x|≤2L

|θ(x)|²dx,