7 P roof of Theorem 1.3







AR(ζ0, D)S0(λ)( ˜ζ0F⁰) 0

0 0







 +

∞

X

`=1









AR(ζ`, D)S`(λ)( ˜ζ`F<sup>0</sup>,ζ˜`G,ζ˜`G<sup>0</sup>,ζ˜`g₁⁰⁰) BR(ζ`, D)S`(λ)( ˜ζ`F<sup>0</sup>,ζ˜`G,ζ˜`G<sup>0</sup>,ζ˜`g₁⁰⁰) λ^1/2B_R(ζ_{`</sub>, D)S<sub>`}(λ)( ˜ζ_{`</sub>F<sup>0</sup>,ζ˜<sub>`}G,ζ˜_{`</sub>G<sup>0</sup>,ζ˜<sub>`}g⁰⁰₁) λBR1(ζ`, D)(S`1(λ)( ˜ζ`F<sup>0</sup>,ζ˜`G,ζ˜`G<sup>0</sup>,ζ˜`g₁⁰⁰)







 .

By (6-1), (6-7) and Proposition 2.2, we have

R_L(Xq(Ω))({(τ ∂τ)^sU(λ)|λ∈Σϑ,λ2})≤Cqκ1λ^−1/2₂ (s= 0,1) (6-14) for anyλ2≥λ1. We chooseλ2so large that

Cqκ1λ^−1/2₂ ≤1/2. (6-15)

By (6-13), (6-14) and (6-15), we have

kHλU(λ)(F, G)k_Xq(Ω)≤¹₂kHλ(F, G)k_Xq(Ω). (6-16) By the equivalence of the norms kHλ(·)k_Xq(Ω) and k · k_Gq(Ω) (cf. (5-19)), the inverse (I−U(λ))⁻¹ = I+P∞

n=1U(λ)ⁿ exists in L(Gq(Ω)). LetV =A(λ)Hλ(I−U(λ))⁻¹(F, G), and then by (6-12)V solves the equation:

Lλ(D)V = (F, G) on Ω×Γ. (6-17)

The uniqueness of solutions follows from the existence of solutions of the dual problem, so thatV is the unique solution of the equation (6-17).

On the other hand, by (6-14) and (6-15), I− U(λ) =I+P∞

n=1U(λ)ⁿ exists in C(Σϑ,λ₂,L(Xq(Ω))) and satisfies the estimate:

R_L(Xq(Ω))({(τ ∂τ)^s(I− U(λ))⁻¹|λ∈Σϑ,λ₂})≤2 (s= 0,1). (6-18) Moreover, by (6-13),

Hλ(I−U(λ))⁻¹=Hλ+

∞

X

n=1

HλU(λ)ⁿ= (I+

∞

X

n=1

U(λ)ⁿ)Hλ= (I− U(λ))⁻¹Hλ. (6-19) LetB_i(λ) =A_i(λ)(I− U(λ))⁻¹ (i= 1,2). Then, by (6-9), (6-18) and Proposition 2.2,

R_L(X

q(Ω),Hq^4−j(Ω))({(τ ∂τ)^s(λ^j/2B1(λ))|λ∈Σϑ,λ₁})≤Cqκ1 (s= 0,1, j = 0,1,2,3,4), R_L(X

q(Ω),Hq^2−j(Ω))({(τ ∂τ)^s(λ^j/2B2(λ))|λ∈Σϑ,λ₁})≤Cqκ1 (s= 0,1, j = 0,1,2).

Moreover, settingB(λ) = (B1(λ), λB1(λ),B2(λ))^>, by (6-19) we see that B(λ)Hλ(F, G) =A(λ)Hλ(I−U(λ))⁻¹(F, G),

and thereforeV =B(λ)Hλ(F, G) is the unique solution of the equation (1-5) (cf. (6-17)). This completes the proof of Theorem 1.4 replacingλ₀byλ₂.

7 P roof of Theorem 1.3

To prove Theorem 1.3, we start with

Lemma 7.1. Let 1< p, q <∞. For any θ0∈Bq,p^2−2/p(Ω),u0 ∈Bq,p^4−2/p(Ω), and u1 ∈B^2−2/pq,p (Ω), there exist ω andwsuch that ω|t=0=θ0,w|t=0=u0, and∂tw|t=0=u1 inΩ,

ω∈

1

\

`=0

H_p^{`</sup>((0,∞), H<sub>q,p</sub><sup>2−2`}(Ω)), w∈

2

\

`=0

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

and

1

X

`=0

k∂_t^`ωk_L

p((0,∞),Hq^2−2`(Ω)≤Ckθ0k_B2−2/p q,p (Ω),

2

X

`=0

k∂_t^`wk_L

p((0,∞),H_q,p^4−2`(Ω)≤C(ku0k_B4−2/p

q (Ω)+ku1k_B2−2/p q,p (Ω)).

with some positive constantC.

Proof. To prove the lemma, we consider the shifted heat equation:

∂tv+v−∆v=f in (0,∞)×R^N, v|t=0=v0. (7-1) Employing the same argument as in the previous sections, it is easy to prove that for anyf ∈L_p((0,∞), Lq(R^N)) andv0∈B^2−2/pq,p (R^N), problem (7-1) admits a unique solutionv∈T1

`=0H<sub>p</sub><sup>`</sup>((0,∞), H_q^2−2`(R^N)) possessing the estimate:

1

X

`=0

k∂_t^`vk_L

p((0,∞),Hq^2−2`(R^N))≤C(kv0k_B2−2/p

q,p (R^N)+kfkL_p((0,∞),Lq(Ω))). (7-2) Assume thatv₀∈B^4−2/pq,p (R^N) andf ∈T1

`=0H<sub>p</sub><sup>`</sup>((0,∞), H_q^2−2`(R^N)). Then, for any multi-indexα∈N^N0

with|α| ≤2 we have

∂t∂_x^αv+∂_x^αv−∆∂_x^αv=∂_x^αf in (0,∞)×R^N, ∂_x^αv|t=0=∂_x^αv0. Thus, the unique existence of solutions of (7-1) yields that

v∈

1

\

`=0

H_p^{`</sup>((0,∞), H<sub>q</sub><sup>4−2`}(R^N)),

1

X

`=0

k∂_t^`vk_L

p((0,∞),H^4−2`q (R^N))≤C(kv₀k_B4−2/p

q,p (R^N)+kfk_Lp((0,∞),H2 q(R^N))).

(7-3)

Moreover, the relation: ∂_t²v=−∂_tv+ ∆∂_tv+∂_tf yields that∂_t²v∈L_p((0,∞), L_q(R^N)) and k∂²_tvk_{Lp((0,∞),Lq(R}N))≤C(kv0k_B4−2/p

q,p (R^N)+

1

X

`=0

k∂_t^`fk_L

p((0,∞),Hq^2−2`(R^N))). (7-4) Letι_hbe an extension map as given in Introduction satisfying property (e-1). Letω be a solution of the shifted heat equation:

∂tω+ω−∆ω= 0 in (0,∞)×R^N, ω|t=0=ιhθ0. Since ιhθ0 ∈Bq,p^2−2/p(R^N), by (7-2) with f = 0 there exists a uniqueω ∈T1

`=0H<sub>p</sub><sup>`</sup>((0,∞), H_q^2−2`(R^N)) satisfying the estimate

1

X

`=0

k∂^`_tωk_L

p((0,∞),Hq^2−2`(R^N))≤Ckιhθ0k_B2−2/p q,p (R^N). Sinceι_hθ₀=θ₀ on Ω andkιhθ₀k_B2−2/p

q,p (R^N)≤Ckθ0k_B2−2/p

q,p (Ω) with some constantC >0, the restriction ofω on Ω is the function satisfying the required properties.

Next, let f be a solution of the shifted heat equation:

∂tf+f−∆f = 0 in (0,∞)×R^N, f|t=0=v0

withv₀=ι_hu₁+ι_hu₀−ι_h∆u₀. Sincev₀∈Bq,p^2−2/p(R^N) and kv₀k_B2−2/p

q,p (R^N)≤C(ku₁k_B2−2/p

q,p (Ω)+ku₀k_B4−2/p q,p (Ω)), there exists a uniquef ∈T1

`=0H<sub>p</sub><sup>`</sup>((0,∞), H_q^2−2`(R^N)) satisfying the estimate:

1

X

`=0

k∂_t^`fk_L

p((0,∞),H_q^2−2`(R^N)) ≤C(ku1k_B2−2/p

q,p (Ω)+ku0k_B4−2/p

q,p (Ω)). (7-5)

Letwbe a solution of the shifted heat equation:

∂tw+w−∆w=f in (0,∞)×R^N, w|t=0=ιu0. (7-6) Then, by (7-3), (7-4) and (7-5), there exists a unique w ∈ T2

`=0H<sub>p</sub><sup>`</sup>((0,∞), H_q^4−2`(R^N)) satisfying the estimate:

2

X

`=0

k∂_t^`wk_L

p((0,∞),H_q^4−2`(R^N)) ≤C(ku1k_B2−2/p

q,p (Ω)+ku0k_B4−2/p q,p (Ω)).

Moreover, by (7-6),∂tw|t=0 =v0−ιhu0+ ∆ιhu0=ιhu1, so that the restriciton ofwon Ω satisfies the required properties, which completes the proof of Lemma 7.1.

In view of Lemma 7.1, to prove Theorem 1.3, it suffices to consider the equations (1-4) withU0= 0.

LetF = (0, f1, f2)^> andG= (g1, g2, g3) satisfy the regularity condition:

(f1, f2)∈Lp((0, T), Lq(Ω)²), G∈H_p¹((0, T), Lq(Ω)×W⁻¹_q (Ω)²)∩Lp((0, T), H_q²(Ω)×H_q¹(Ω)²) and the compatibility condition: G|t=0 = 0. In the following, given f(t,·) defined fort ∈[0, T], f0(t,·) denotes the zero extension off tot <0, that is,f₀(t,·) =f(t,·) for 0< t < T, andf₀(t,·) = 0 fort <0, andE_Tf denotes the extention of f toRdefined by

[ETf](t,·) =

(f0(t,·) fort < T ,

f₀(2T−t,·) fort≥T . (7-7)

Note thatE_Tf vanishes fort6∈[0,2T] and moreover, iff|_t=0= 0, then

∂_t[E_Tf](t,·) =







∂_tf(t,·) fort≤T ,

−(∂tf)(2T−t,·) fort≥T ,

0 fort6∈[0,2T].

(7-8)

SinceETf =f for 0≤t≤T, instead of the equations (1-4) withU0= 0, we consider the equations:

Ut−A(D)U =ETF inR×Ω, B(D)U =ETG onR×Γ. (7-9) Let L be the Laplace transform with respect to time variable t and let L⁻¹ be its inverse transform, which are defined by

L[f](λ,·) = Z ∞

−∞

e^−λtf(t,·)dt= Z

R

e^{−iτ t}e^−γtf(t,·)dt, L⁻¹[f](t,·) = 1

2πi

Z γ+i∞

γ−i∞

e^λtf(λ,·)dλ=e^γt 2π

Z ∞

−∞

f(γ+iτ,·)dτ,

whereλ=γ+iτ ∈C. Let L[ETF](λ,·) =Jλ andL[ETG](λ,·) =Kλ. SinceETF and ETGvanish for t 6∈[0,2T], J_λ andK_λ are entire functions with respect to λ∈C. Applying the Laplace transform to (7-9), we have

λV_λ−A(D)V_λ=J_λ in Ω, B(D)V_λ=K_λ on Γ, (7-10)

where we have setVλ=L[U](λ,·). Now,Vλ=B(λ)Hλ(Jλ, Kλ) is the unique solution of (7-10), so that the uniqueness of the solution yields that Vλ is holomorphic forλ∈Σϑ,λ0. LetBi(λ) (i= 1,2) be the operators given in Theorem 1.4 and setB(λ) = (B1(λ), λB1(λ),B2(λ))^>. Then, the unique solutionU of (7-9) is given by

U(t,·) =L⁻¹[B(λ)Hλ(Jλ, Kλ)] =e^γtF⁻¹[B(λ)Hλ(F[e^−γ·ETF],F[e^−γ·ETG])](t,·).

Let Λ^1/2 be the operator defined by

[Λ^1/2f](t,·) =L⁻¹[λ^1/2L[f](λ,·)](t), and let

ke^−γtfkL_p(R,X)=nZ ∞

−∞

e^−γtkf(t,·)kX)^pdto^1/p .

Since|λ|^1/2≤ |λ|(1 +|ξ|²)^−1/2 when|λ| ≥1 +|ξ|² and|λ|^1/2≤(1 +|ξ|²)^1/2when|λ| ≤1 +|ξ|², by the Bourgain theorem (cf. Proposition 2.2) we have

ke^γtΛ^1/2fk_Lp(R,Lq(RN))≤C(ke^γt∂t(1−∆)^−1/2fk_Lp(R,Lq(RN))+ke^γt(1−∆)^1/2fk_Lp(R,Lq(RN))), so that by using property (e-2) of the extension mapιhgiven in the introduction we have

ke^γtΛ^1/2fk_{Lp(R,Lq(Ω))}≤C(ke^γt∂tfk_L

p(R,W⁻¹q (Ω))+ke^γtfk_Lp(R,H1

q(Ω))). (7-11) And also, using the extension mapιh and Proposition 2.2, we have

ke^−γt∂_tfk_Lp(R,Hm

q(Ω))+ke^−γtfk_L

p(R,H_q^m+2(Ω))

≤Cm,q(ke^−γt∂tfk_Lp(R,Hm

q (Ω))+ke^−γtΛ^1/2∇fk_Lp(R,Hm

q(Ω))+ke^−γt∇²fk_Lp(R,Hm

q (Ω))) (7-12) for anym∈N∪ {0}. Therefore, using (7-11) and (7-12) and applying the Weis operator valued Fourier multiplier theorem with the help of Theorem 1.4, we have

2

X

`=0

ke^−γt∂_t^`uk_L

p(R,H_q^4−2`(Ω))+

1

X

`=0

ke^−γt∂_t^`θk_L

p(R,H_q^2−2`(Ω))

≤C(ke^−γtET(f1, f2)kL_p(R,L_q(Ω))+ke^−γtETGkL_p(R,H²_q(Ω)×H_q¹(Ω)²)

+ke^−γtΛ^1/2ETGkL_p(R,H_q¹(Ω)×Lq(Ω)²)+ke^−γt∂tETg1kL_p(R,L_q(Ω)))

≤C(ke^−γtE_T(f₁, f₂)kL_p(R,L_q(Ω))+ke^−γtE_TGkL_p(R,H²_q(Ω)×H_q¹(Ω)²)

+ke^−γt∂_tE_TGk_L

p(R,L_q(Ω)×W⁻¹_q (Ω)²))

(7-13)

for anyγ ≥λ0 with constant C independent of γ, where we have setU = (u, ∂tu, θ). Using (7-7) and (7-8) and noting thatG|t=0= 0, we have

ke^−γtET(f1, f2)k_{Lp(R,Lq(Ω))}≤Ck(f1, f2)k_{Lp((0,T),Lq(Ω))}, ke^−γtETGk_Lp(R,H2

q(Ω)×H_q¹(Ω)²)≤CkGk_Lp((0,T),H2

q(Ω)×H_q¹(Ω)²), ke^−γt∂tETGk_L

p(R,Lq(Ω)×W⁻¹q (Ω)²)≤Ck∂tGk_L

p((0,T),Lq(Ω)×W⁻¹q (Ω)²), which, combined with (7-13), furnishes that

2

X

`=0

ke^−γt∂_t^`uk_L

p(R,Hq^4−2`(Ω))+

1

X

`=0

ke^−γtθk_L

p(R,Hq^2−2`(Ω))≤CIT (7-14) with

I_T =ke^−γt(f₁, f₂)kL_p((0,T),L_q(Ω))+kGkL_p(R,H²_q(Ω)×H_q¹(Ω)²)+k∂tGk_L

p(R,L_q(Ω)×W⁻¹q (Ω)²).

Especially, for anyt <0, we have e^γ|t|

2

X

`=0

k∂_t^`uk_L

p((−∞,t),Hq^4−2`(Ω))+

1

X

`=0

kθk_L

p((−∞,t),Hq^2−2`(Ω))

≤

2

X

`=0

ke^−γt∂_t^`uk_L

p(R,H_q^4−2`(Ω))+

1

X

`=0

ke^−γtθk_L

p(R,H_q^2−2`(Ω))≤CIT

for anyγ≥λ0, so that lettingγ→ ∞, we have e^γ|t|

2

X

`=0

k∂_t^`uk_L

p((−∞,t),H_q^4−2`(Ω))+

1

X

`=0

kθk_L

p((−∞,t),H_q^2−2`(Ω))= 0 for anyt <0, which implies that (u, θ) = 0 fort <0.

Summing up, we have proved thatU = (u, ∂tu, θ)^> satisifes the equations:

U_t−A(D)U =F in Ω×(0, T), B(D)U =G on Γ×(0, T), U|t=0= 0 in Ω and the estimate:

2

X

`=0

ke^−γt∂_t^`uk_L

p((0,T),H_q^4−2`(Ω))+

1

X

`=0

ke^−γtθk_L

p((0,T),H^2−2`_q (Ω))≤CIT.

The uniqueness of the solutions follwos from the existence of solutions of the dual problem, which completes the proof of Theorem 1.3.

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Im Dokument Maximal regularity for the thermoelastic plate equations with free boundary conditions (Seite 32-37)