Lq-solutions to the Cosserat spectrum in bounded and exterior domains

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L

^q

-solutions to the Cosserat spectrum in bounded and exterior domains

by

Stephan Weyers,

November 2005

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Abstract

In the present paper we consider forλ∈R the existence of non-trivial solutions of the following problems:

1. classical Cosserat spectrum

u∈C²(G)ⁿ∩C⁰(G)ⁿ

∆u=λ∇divu u

∂G = 0 2. weak Cosserat spectrum

u∈Hb_•^1,q(G)ⁿ ∇u,∇φ

G = λ

divu,divφ

G ∀φ∈Hb_•^1,q⁰(G)ⁿ

whereG⊂Rⁿis a bounded or an exterior domain and Hb•^1,q(G) (cf. Definition 2.1) is the suitable space for weak solutions.

This problem was investigated firstly by Eugene and Francois Cosserat. It is a special case of the Lame equation and describes the displacement of a homogeneous isotropic linear static elastic body without exterior forces.

In this paper we characterize the weak Cosserat spectrum for bounded or exterior domainsG⊂Rⁿ(n≥2) and 1< q <∞(for the definition ofHb•^2,q(G) cf. Definition 3.1)

Theorem 14.1. Letn≥2, 1< q <∞,k∈N,k≥2,k > ⁿ_q and let G⊂Rⁿ be a bounded or an exterior domain with∂G∈C^k+2.

1. The set W :=

λ∈R : there is 06=u∈Hb_•^1,q(G)ⁿ, such that for all φ∈Hb_•^1,q⁰(G)ⁿ holds

∇u,∇φ

G = λ

divu,divφ

G

is finite or countably infinite.

2. Forλ∈R\{1,2} the space V_λ :=

u∈Hb_•^1,q(G)ⁿ :

∇u,∇φ

G = λ

divu,divφ

G ∀φ∈Hb_•^1,q⁰(G)ⁿ is finite dimensional.

3. For every sequence (λm)⊂W withλm 6=λl for m6=lholds λ_m →2 (m→ ∞)

4.

{∇s : s∈Hb_•^2,q(G)} ⊂ V₁

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Thereforeλ= 1 is an eigenvalue of infinite multiplicity andλ= 2 is an accumulation point of eigenvalues of finite multiplicity.

In this generality the result is new.

E. and F. Cosserat [Co1-Co9] studied the classical Cosserat spectrum for certain types of domains like a ball, a spherical shell or an ellipsoid. In chapter 16 we use their approach for explicit solutions.

General results are due to Mikhlin [Mi, 1973], who investigated the Cosserat spectrum for n = 3 and q = 2, and Kozhevnikov [Ko2, 1993], who treated bounded domains in the casen = 3 and q = 2. Kozhevnikov’s proof is based on the theory of pseudodifferential operators.

Faierman, Fries, Mennicken and M¨oller [FFMM, 2000] gave a direct proof for bounded domains,n≥2 and q= 2.

Michel Crouzeix gave 1997 a simple proof for bounded domains, in casen= 2,3 and q= 2.

In this paper we use the idea of Crouzeix to proof the results for bounded and exterior domains,n≥2 and 1< q <∞.

The following regularity theorem is new:

Theorem 15.5. Letn≥2, 1< q <∞,k∈N,k≥2,k > ⁿ_q and let G⊂Rⁿ be either a bounded or an exterior domain with∂G∈C^k+3. Assume thatu∈Hb•^1,q(G), λ∈R\{1,2} and

∇u,∇φ

G = λ

divu,divφ

G for allφ∈Hb_•^1,q⁰(G)ⁿ Then

1. u∈Hb•^1,˜^q(G)ⁿ and ∇u∈H^k,˜^q(G)ⁿ² for all 1<q <˜ ∞, 2. u∈C^k(G),

3. ∆u = λ∇divu

It is amazing, that the eigenspaces of eigenvalues λ /∈ {1,2} don’t depend on q!

Further we get important results for the classical Cosserat spectrum: λ = 2 is an accumulation point of eigenvalues, too. λ= 1 is also a classical eigenvalue, because fors∈C₀^∞(G) withu:=∇sholds ∆u=∇divu.

Now we like to describe, how we proved these results.

Starting point was the paper [Si] of Christian G. Simader. He proved, that in the case of the upper half space H ={(x⁰, xn) ∈Rⁿ:xn>0} there exists exactly two eigenvalues, namely λ = 1 and λ = 2. Simader used the paper [MueR] and the decomposition (cf. Theorem 4.2)

L^q(H) =A^q(H)⊕B^q(H) He was able to solve explicitly inHb•^1,q(H)ⁿ the equation

∇T_q(p),∇φ

H =

p,divφ

H ∀φ∈Hb_•^1,q⁰(H)ⁿ

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for a givenp∈L^q(H), and proved by direct calculations, that forp₀ ∈A^q(H) holds divT_q(p0) =p0

and forph∈B^q(H)

divT_q(p_h) = 1 2p_h

We tried to carry over these results to the slightly perturbated half space Hw = {(x⁰, xn) ∈ Rⁿ : xn > w(x⁰)} (for w ∈ C₀²(Rⁿ⁻¹)) and to domains with compact boundary. Therefore we considered for suitableµ >0 and

ρµ∈C₀^∞(R), 0≤ρµ≤1, ρµ(t) =

0 ,ift≥4µ 1 ,ift≤2µ the isomorphism

f : H_w →H, f(x) = x⁰, x_n−w(x⁰)ρ_µ(x_n) and the Piola transform (cf. [Cia, p.37PP])

P : Hb_•^1,q(H_w)ⁿ→Hb_•^1,q(H)ⁿ, (P v)(y) =

detf⁰(f⁻¹(y))−1

f⁰(f⁻¹(y))v(f⁻¹(y)) and converted the inner products

∇P⁻¹v,∇P⁻¹φ

Hw respectively

div (P⁻¹v),div (P⁻¹φ)

Hw

by means of the transformation rule in inner products ∇v,∇φ

H+B₁(v, φ) respectively

divv,divφ

H +B₂(v, φ)

By the properties of Piola’s transform B₂ defines a compact operator, if divv ∈ B^q(H), but we are not able to prove that forB₁ too.

As another approach we searched for a relationship of Green’s function G of the Laplace operator (cf. Definition 17.4) to the reproducing kernel R in B^q(G) (cf.

Definition 18.2), because with

Definition 11.1. Letn≥2 and let G⊂Rⁿbe either a bounded or an exterior domain with∂G∈C¹.

1. LetT_q:L^q(G)→Hb•^1,q(G)ⁿ be defined by (cf. Theorem 2.9) ∇T_q(p),∇φ

G =

p,divφ

G for all φ∈Hb_•^1,q⁰(G)ⁿ 2. LetZ_q:L^q(G)→L^q(G), Z_q(p) := div (T_qp)

holds

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Theorem 11.3. Letn≥2 and let G⊂Rⁿ be either a bounded or an exterior domain with∂G∈C¹. Assume λ∈R. Then there is u∈Hb•^1,q(G)ⁿ with

∇u,∇φ

G=λ

divu,divφ

G for all φ∈Hb_•^1,q⁰(G)ⁿ if and only if there isp∈L^q(G) with

λ Zq(p) =p In this case one can choose p= divu.

IfG⊂Rⁿ is a bounded domain, for p∈L^q(G) formally holds with u:=T_q(p) u(x) =

Z

G

G(x, y) (−∆u)(y)dy= Z

G

G(x, y) (−∇p)(y)dy

Z_q(p)(x) = divu(x) = −

n

X

i=1

Z

G

(∂_x_iG)(x, y) (∂_y_ip)(y)dy

= −

n

X

i=1

Z

∂G

(∂_x_iG)(x, y)

| {z }

=0

p(y)N_i(y)dω_y+ Z

G

p(y)

n

X

i=1

∂_y_i∂_x_iG(x, y)dy

= Z

G

p(y)

n

X

i=1

∂yi∂xiG(x, y)dy

If therefore

n

X

i=1

∂_y_i∂_x_iG(x, y)−1

2R(x, y)

was a compact operator, the assertion about the Cosserat spectrum would follow by the spectral theorem for compact Hermitian operators. We couldn’t prove that directly. There are results about the relationship of Green’s function of the bilaplace operator ∆² to the reproducing kernel in B²(G) (see [ELPP, Theorem 4.3, p.113]), but we couldn’t find the relationship above.

After solving the Cosserat spectrum in another way we can prove the relationship indirectly:

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Theorem 19.1 Let n≥2, 1< q < ∞, k∈ N,k > 1 +ⁿ_q and letG ⊂Rⁿ be a bounded domain with∂G∈C^2+k. Let

G(x, y) =S(x−y) +h(x, y)

be Green’s function of the Laplace operator inGand letRbe the reproducing kernel inB^q(G). Then

Z_q(p)(x) = p(x) +

n

X

i=1

Z

G

p(y)∂_y_i∂_x_ih(x, y)dy a.e. for p∈B^q(G) Therefore

n

X

i=1

∂_y_i∂_x_ih(x, y) + 1

2R(x, y) is a compact operator.

For the unit sphereB1 (chapter 20) and for the half space one can prove this result directly. In this cases reproducing kernel and Green’s function are known explictly.

It is an interesting question, whether it is possible to prove this directly in general, too.

Finally we found the paper [Cr] of Michel Crouzeix. His sketch of a proof forbounded domains andq= 2 is very short. He proved, that for p∈B²(G) holds¹

kZ₂(p)−1

2pk_1,2;G ≤ Ckpk_2;G (∗)

1. It suffices to prove (∗) for p∈H^k,2(G)∩B²(G), because H^k,2(G)∩B²(G) is dense in B²(G) with respect tok·k_2;G.

2. Obviously

kZ₂(p)− 1

2pk_2;G ≤ Ckpk_2;G 3. Choose a sufficiently smoothζ :Rⁿ→R with

ζ

∂G = 0, ∇ζ

∂G =N (outer unitary normal)

1Crouzeix considered only p∈L²(G) withR

Gp dx= 0. That means no loss of generality, because for arbitraryp∈L²(G) forφ∈H₀^1,2(G) holds:

hp− Z

G

p dx,divφi_G= D

p,divφ E

G

Furthermore only forp ∈L^q(G) withR

Gp dx= 0 there is a constantC >0, which doesn’t depend onp, such that

kpk_q;G ≤ C sup

06=φ∈H₀^1,q⁰(G)

D p,divφE k∇φk_q0

(see [St, Satz 8.2.1, p.256]). For our purpose the restriction toL^q-functions with mean value zero is not necessary.

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4. and define forp∈H^k,2(G)∩B²(G) andu:=T_q(p)∈H₀^1,2(G)ⁿ w:= u∇ζ − 1

2p ζ 5. One can show thatw∈H₀^1,2(G) and

∆w= 2∇u· ∇∇ζ+u· ∇∆ζ−1

2p∆ζ ∈L²(G) Therefore w∈H^2,2(G) and

kwk_2,2;G ≤ Ckpk_2;G 6. Furthermore

∇w∇ζ−(divu−1

2p)∈H₀^1,2(G) 7. Then

k∇(divu−1

2p)k_2;G ≤ k∇(∇w∇ζ)k_2;G+k∇

∇w∇ζ−(divu−1 2p)

k_2;G

≤ C˜kpk_2;G and finally (∗).

8. Then for bounded domains by Rellich’s imbedding theoremZ₂−¹₂I is a compact operator, and by the spectral theorem for compact self-adjoint operators the assertion follows.

Now we describe for each part of the proof, which additional work was necessary to carry over the results to exterior domains and the case 1< q <∞.

1. is proved in chapter 9. We need elliptic regularity theorems (Theorem 7.7 and 7.8) and for exterior domains the asymptotic behavior of harmonic functions (Lemma 8.9).

2. follows immediately by Definition 11.1 and Theorem 2.8.

3. For ζ ∈ C₀^k(Rⁿ) to hold we need in Theorem 6.1 ∂G ∈ C^k+1. It is possible, that it suffices to assume that ∂G∈C^k. But our proof is very elementary.

4. The definition ofw was the ingenious idea of Crouzeix.

5. is proved in Lemma 12.2 respectively 13.2. We need p ∈ C⁰(G) and u ∈ C¹(G)ⁿ. This is shown in Lemma 12.1 respectively 13.1 by means of Sobolev’s imbedding theorems. Therefore we must assume ∂G ∈ C^k+2 with k > ⁿ_q in Theorem 14.1.

6. In Lemma 12.3 respectively 13.3 we need again the regularity ofp and u and a few theorems about differentiable functions (chapter 5).

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7. Then the estimate follows by Theorem 2.8.

8. Analogously to Rellich’s imbedding theorem for bounded domains the imbedding H^1,q(G) ∩B^q(G) in B^q(G) is compact in exterior domains (Theorem 10.1). The proof is based on the asymptotic behavior of harmonic functions (Theorem 8.7). For real Banach spaces the spectral theorem B.9 is applicable.

Finally we derive Theorem 14.1.

The regularity of the solutions (Theorem 15.5) can be proved as follows: If u ∈ Hb•^1,q(G)ⁿ,λ∈R\{1,2}and

∇u,∇φ

G = λ

divu,divφ

G for allφ∈Hb_•^1,q⁰(G)ⁿ then withp:= divuby Theorem 11.3 and 11.4 holds

p∈B^q(G), λ Zq(p) =p Withµ:= ₁₋^λλ

2

∈Rwe derive by (∗) respectively Theorem 12.4 and 13.4 p = µ

Z_q(p) − 1 2p

∈H^1,q(G) For 1< q < nwithq^∗ = _n−q^nq holdsp∈L^q^∗(G) and then

p = µ

Zq∗(p) − 1 2p

∈H^1,q^∗(G)

By induction we derive p ∈ H^1,s(G) for a certain n < s < ∞, whence p ∈ C⁰(G).

Because of the asymptotic behavior ofB^q-functions in exterior domains (Theorem 8.12) we derive further

p∈H^1,˜^q(G)∩C⁰(G) ∀1< q <∞ and

∇u∈H^1,˜^q(G)ⁿ²∩C¹(G) ∀1< q <∞

For the regularity of higher derivatives we use use the density ofH^k,q(G)∩B^q(G) inB^q(G) with respect to k·k_q;G (Theorem 9.1 and 9.2) and the inequality (Lemma 15.3)

kZ_q(π)− 1

2πk_k,q;G ≤ C_kkπk_k−1,q;G Then we can prove

p = µ^k

Zs − 1 2I

k

p ∈H^k,s(G) for alln < s <∞. Therefore Theorem 15.5 holds.

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Part I: Preliminaries

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1 Notations

Forx0 ∈Rⁿ, 0< r < R we denote

B_r(x₀) :={x∈Rⁿ:|x−x₀|< r} B_r:={x∈Rⁿ:|x|< r}

A_r,R(x₀) :={x∈Rⁿ:r <|x−x₀|< R} A_r,R:={x∈Rⁿ:r <|x|< R}

Further we define for an openG⊂Rⁿ and k∈N

C^k(G) := {f ∈C^k(G) : forα ∈Nⁿ0, |α| ≤k there isf^(α)∈C⁰(G) withf^(α)|_G=D^αf}

C₀^k(G) := {f ∈C^k(G) : supp(f)⊂G}

If 1< q <∞we always use the notation q⁰ := q

q−1 Forf ∈L^q(G) denote

kfk_q;G:=

Z

G

|f|^qdx ¹_q

Usually we don’t strictly distinguish between a function and the corresponding equiv- alence class inL^q(G). For example the notation

f ∈L^q(G)∩C⁰(G) means, that there is a (unique) continuous representative.

Let

H^k,q(G) :={u:G→R|u measurable, D^αu ∈L^q(G) for all |α| ≤k}

With

kuk_k,q;G:=



 X

|α|≤k

kD^αuk^q_q;G





1 q

foru∈H^k,q(G) H^k,q(G) is a Banach space. Let

H₀^k,q(G) :=C₀^∞(G)^k·k^k,q;G Underlined terms always denote vectors

u:= (u₁, . . . , u_n) Often we use the notation

u∈H^1,q(G) instead of u∈H^1,q(G)ⁿ or ∇u∈L^q(G) instead of ∇u∈L^q(G)ⁿ

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if no confusion could arise. Further we use the notations hf, gi_G :=

Z

G

f g dx

h∇f,∇gi_G :=

n

X

i=1

Z

G

(∂if) (∂ig)dx f , g

G :=

n

X

i=1

Z

G

f_i g_idx

∇f ,∇g

G :=

n

X

i,j=1

Z

G

(∂ifj) (∂igj)dx ∇²f,∇²g

G :=

n

X

i,j=1

Z

G

(∂_i∂_jf) (∂_i∂_jg)dx and

kfk_q;G :=

n

X

i=1

kf_ik^q_q;G

!¹_q

k∇fk_q;G :=

n

X

i=1

k∂_ifk^q_q;G

!¹_q

k∇²fk_q;G :=





n

X

i,j=1

k∂_i∂_jfk^q_q;G





1 q

k∇fk_q;G :=





n

X

i,j=1

k∂_jfik^q_q;G





1 q

if the expressions are well defined.

The inner product inRⁿ we denote most of the time by x y :=

n

X

i=1

xiyi ifx, y∈Rⁿ but sometimes we write

hx, yi :=

n

X

i=1

x_iy_i ifx, y∈Rⁿ

Anexterior domain is a domainG⊂Rⁿ withRⁿ\G compact and 0∈Rⁿ\G.

A⊂⊂B means: A, B⊂Rⁿ open,A bounded andA⊂B Let

kfk_∞;G:= sup

x∈G

|f(x)|

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and

Sn−1:={x∈Rⁿ:|x|= 1}=∂B₁ ω_n:=|S_n−1|_n−1 IfX is a real normed vector space, we denote by

X^∗ :=

(

F^∗ :X→R sup

kxk6=0

F^∗(x) kxk <∞

)

its dual space.

The property (GA) denotes forG⊂Rⁿ:

(GA) There is an open∅ 6=K ⊂Rⁿ withG=Rⁿ\K

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2 The space H b

_•^1,q

(G)

Definition 2.1. Letn≥2, 1≤q <∞ and let G⊂Rⁿ satisfy (GA). Then Hb_•^1,q(G) := {u:G→R|umeasurable, u∈L^q(G∩B_R)∀R >0,

∇u∈L^q(G) and for each η∈C₀^∞(Rⁿ) holds ηu∈H₀^1,q(G)}

Definition 2.2. Letn≥2, 1≤q <∞ and let G⊂Rⁿ satisfy (GA). Then Hb₀^1,q(G) := {u:G→Rmeasurable|u∈L^q(G∩B_R)∀R >0, ∇u∈L^q(G)

and there exists a sequence (u_i)⊂C₀^∞(G) so that ku−u_ik_q,G∩B_R →0 ∀R >0 and k∇u− ∇u_ik_q,G→0}

Theorem 2.3. Letn≥2, 1≤q <∞and let G⊂Rⁿ satisfy (GA). Then (a) H₀^1,q(G)⊂Hb₀^1,q(G)⊂Hb•^1,q(G)

(b) Foru∈Hb•^1,q(G) by k∇uk_q,G a norm is defined onHb•^1,q(G).

(c) Equipped with k∇ · k_q,G-norm Hb•^1,q(G) is a Banach space being reflexive for 1 < q < ∞. If q = 2 then Hb•^1,2(G) is a Hilbert space with inner product h∇u,∇vi foru, v∈Hb•^1,2(G).

(d) Hb₀^1,q(G) is a closed subspace ofHb•^1,q(G) andHb₀^1,q(G) =C₀^∞(G)^k∇·k^q,G Proof. see [Si/So, Theorem I.2.2, p.27]

Theorem 2.4. Letn≥2, 1≤q <∞and let G⊂Rⁿ satisfy (GA). Then Hb_•^1,q(G) = {u:G→R measurable|u∈L^q(G∩B_R)∀R >0, ∇u∈L^q(G)

and there exists a sequence (ui)⊂C₀^∞(G) so that ku−uik_q,G∩B_R +k∇u− ∇u_ik_q,G∩B_R →0 ∀R >0}

Proof. see [Si/So, Theorem I.2.4, p.29]

Theorem 2.5. Let n≥ 2, 1 ≤q <∞ and let G⊂ Rⁿ be open and bounded.

Then

Hb_•^1,q(G) =H₀^1,q(G) Proof. Easy consequence of Definition 2.1

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Theorem 2.6. Let 2≤n≤q <∞and let G⊂Rⁿ satisfy (GA). Then Hb_•^1,q(G) =Hb₀^1,q(G)

Theorem 2.7. Let n ≥ 2 and let G ⊂ Rⁿ be an exterior domain. Suppose 1≤q < n. Choose r >0 withRⁿ\G⊂Br and let

ϕ_r∈C^∞(Rⁿ), 0≤ϕ_r≤1, ϕ_r(x) =

0 ,if|x| ≤r 1 ,if|x| ≥2r Then

1. Hb₀^1,q(G)⊂Hb•^1,q(G) and Hb₀^1,q(G)6=Hb•^1,q(G)

2. Hb•^1,q(G) =Hb₀^1,q(G)⊕ {αϕ_r : α∈R} in the sense of a direct decomposition.

Theorem 2.8 (Variational inequality in Hb_•^1,q(G)). LetG⊂Rⁿ (n≥2) be either a bounded or an exterior domain and let∂G∈C¹. Let 1< q <∞. Then there exists a constantCq=C(q, G)>0 so that

k∇uk_q,G ≤ Cq sup

06=φ∈Hb•^1,q⁰(G)

h∇u,∇φi

k∇φk_q⁰_,G ∀u∈Hb_•^1,q(G)

Proof. see [Si/So, Theorem II.1.1, p.45]

Theorem 2.9 (Functional representation in Hb_•^1,q(G)). Let G⊂Rⁿ (n≥2) be either a bounded or an exterior domain and let∂G∈C¹. Let 1< q <∞.

Then for everyF^∗ ∈Hb•^1,q⁰(G)^∗ there exists a unique u∈Hb•^1,q(G) so that F^∗(φ) =h∇u,∇φi ∀φ∈Hb_•^1,q⁰(G)

Furthermore withC_q by Theorem 2.8 holds C_q⁻¹k∇uk_q,G≤sup

n

F^∗(φ) : φ∈Hb_•^1,q⁰(G) and k∇φk_q⁰_,G ≤1 o

≤ k∇uk_q,G

Proof. see [Si/So, Theorem II.1.2, p.45]

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3 The space H b

_•^2,q

(G)

Definition 3.1. Letn≥2, 1≤q <∞ and let G⊂Rⁿ satisfy (GA). Then Hb_•^2,q(G) := {u:G→R measurable|u, ∇u∈L^q(G∩B_R)∀R >0,

∇²u∈L^q(G) and for eachη∈C₀^∞(Rⁿ) holds ηu∈H₀^2,q(G)}

Definition 3.2. Letn≥2, 1≤q <∞ and let G⊂Rⁿ satisfy (GA). Then Hb₀^2,q(G) := {u:G→R|u, ∇u∈L^q(G∩BR)∀R >0, ∇²u∈L^q(G)

and there exists a sequence (u_i)⊂C₀^∞(G) so that ku−u_ik_q,G∩B_R+k∇u− ∇u_ik_q,G∩B_R →0 ∀R >0 and k∇²u− ∇²u_ik_q,G→0}

Theorem 3.3. Letn≥2, 1≤q <∞and let G⊂Rⁿ satisfy (GA). Then (a) H₀^2,q(G)⊂Hb₀^2,q(G)⊂Hb•^2,q(G)

(b) Foru∈Hb•^2,q(G) by k∇²uk_q,G a norm is defined onHb•^2,q(G).

(c) Equipped with k∇² · k_q,G-norm Hb•^2,q(G) is a Banach space being reflexive for 1 < q < ∞. If q = 2 then Hb•^2,2(G) is a Hilbert space with inner product h∇²u,∇²vi foru, v∈Hb•^2,2(G).

(d) Hb₀^2,q(G) is a closed subspace ofHb•^2,q(G) andHb₀^2,q(G) =C₀^∞(G)^k∇

2·k_q,G

Proof. see [MueR, Satz II.1, p.126]

Theorem 3.4. Letn≥2, 1≤q <∞and let G⊂Rⁿ satisfy (GA). Then Hb_•^2,q(G) = {u:G→R|u, ∇u∈L^q(G∩BR)∀R >0, ∇²u∈L^q(G)

and there exists a sequence (ui)⊂C₀^∞(G) so that

ku−u_ik_q,G∩B_R+k∇u− ∇u_ik_q,G∩B_R+k∇²u− ∇²u_ik_q,G∩B_R →0 for all R >0}

Proof. see [MueR, Lemma II.3, p.129]

Theorem 3.5. Let n≥ 2, 1 ≤q <∞ and let G⊂ Rⁿ be open and bounded.

Then

Hb_•^2,q(G) =H₀^2,q(G)

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Proof. Easy consequence of Definition 3.1

Theorem 3.6. Let 2≤n≤q <∞and let G⊂Rⁿ satisfy (GA). Then Hb_•^2,q(G) =Hb₀^2,q(G)

Proof. see [MueR, Satz II.3, p.133]

Theorem 3.7. Let n ≥ 2 and let G ⊂ Rⁿ be an exterior domain. Suppose 1≤q < n. Choose r >0 withRⁿ\G⊂Br and let

ϕ_r∈C^∞(Rⁿ), 0≤ϕ_r≤1, ϕ_r(x) =

0 ,if|x| ≤r 1 ,if|x| ≥2r Furthermore defineψri(x) :=ϕr(x)xi for alli= 1, .., n. Then

1. ϕ_r, ψ_ri∈Hb•^2,s(G) for all 1< s <∞,

2. Hb₀^2,q(G)⊂Hb•^2,q(G) and Hb₀^2,q(G)6=Hb•^2,q(G), 3. Hb•^2,q(G) =Hb₀^2,q(G)⊕ {αϕ_r : α∈R} ⊕ {Pn

i=1βiψri : βi∈R}for 1< q < ⁿ₂, 4. Hb•^2,q(G) =Hb₀^2,q(G)⊕ {Pn

i=1β_iψ_ri : β_i ∈R} for ⁿ₂ ≤q < n.

Proof. see [MueR, Lemma II.8, p.140] and [MueR, Satz II.4, p.144]

Theorem 3.8 (Variational inequality in Hb_•^2,q(G)). LetG⊂Rⁿ (n≥2) be either a bounded or an exterior domain and let∂G∈C². Let 1< q <∞. Then there exists a constantC_q=C(q, n, G)>0 so that

k∆uk_q,G ≤ Cq sup

06=φ∈Hb•^2,q⁰(G)

h∆u,∆φi

k∆φk_q⁰_,G ∀u∈Hb_•^2,q(G)

Proof. see [MueR, Hauptsatz, p.191]

Theorem 3.9 (Functional representation in Hb_•^2,q(G)). Let G⊂Rⁿ (n≥2) be either a bounded or an exterior domain and let∂G∈C². Let 1< q <∞.

Then for everyF^∗ ∈Hb•^2,q⁰(G)^∗ there exists a unique u∈Hb•^2,q(G) so that F^∗(φ) =h∇²u,∇²φi ∀φ∈Hb_•^2,q⁰(G)

Furthermore there exists a constantD_q=D(q, G) with D⁻¹_q k∇²uk_q,G≤supn

F^∗(φ) : φ∈Hb_•^2,q⁰(G) and k∇²φk_q⁰_,G≤1o

≤ k∇²uk_q,G

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Proof. see [MueR, Lemma III.15, p.164] and Theorem 3.8

Lemma 3.10. Let G⊂Rⁿ (n≥2) be either a bounded or an exterior domain and let 1< q <∞. Then

∇²u,∇²φ

=h∆u,∆φi for all u∈Hb_•^2,q(G), φ∈Hb_•^2,q⁰(G)

Proof. (a) Considerϕr, ψri from Theorem 3.7. It holds

∂_jϕ_r(x) =∂_k∂_jϕ_r(x) = 0 for|x| ≤r and for |x| ≥2r Furthermore

∂_jψ_ri(x) =∂_jϕ_r(x)x_i+ϕ_r(x)δ_ij

∂_k∂_jψ_ri(x) =∂_k∂_jϕ_r(x)x_i+∂_jϕ_r(x)δ_ik+∂_kϕ_r(x)δ_ij Therefore

∂j∂_kψri, ∂j∂_kϕr∈C₀^∞(A^r

2,3r) (b) Suppose G⊂Rⁿ is an exterior domain, Rⁿ\G⊂ B^r

2. By Theorem 3.6 and 3.7 foru∈Hb•^2,q(G) there arev∈Hb₀^2,q(G) and f ∈C^∞(Rⁿ) such that

∂_i∂_jf ∈C₀^∞(A^r

2,3r)⊂C₀^∞(G) and u=v+f

There is by Definition 3.2 a sequence (vk)⊂C₀^∞(G) withk∇²(vk−v)k_q;G →0.

Letφ∈Hb•^2,q⁰(G). Then

∇²u,∇²φ

G =

n

X

i,j=1

Z

G

∂i∂jv ∂i∂jφ dx +

n

X

i,j=1

Z

G

∂i∂jf

| {z }

∈C₀^∞(G)

∂i∂jφ dx

= lim

k→∞

n

X

i,j=1

Z

G

∂i∂jvk ∂i∂jφ dx −

n

X

i,j=1

Z

G

∂j(∂i∂if

| {z }

∈C₀^∞(G)

)∂jφ dx

= lim

k→∞

n

X

i,j=1

Z

G

∂_i∂_iv_k∂_j∂_jφ dx +

n

X

i,j=1

Z

G

∂_i∂_if ∂_j∂_jφ dx

= h∆v,∆φi_G+h∆f,∆φi_G=h∆u,∆φi_G

(c) Suppose G ⊂ Rⁿ is a bounded domain. Then Hb•^2,q(G) = H₀^2,q(G) and the assertion follows as in (b) withf = 0.

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Theorem 3.11 (Functional representation in Hb_•^2,q(G)). Let G⊂Rⁿ (n≥2) be either a bounded or an exterior domain and let∂G∈C². Let 1< q <∞.

Then for everyF^∗ ∈Hb•^2,q⁰(G)^∗ there exists a unique u∈Hb•^2,q(G) so that F^∗(φ) =h∆u,∆φi ∀φ∈Hb_•^2,q⁰(G)

Furthermore there exists a constantKq=K(n, q, G) with K_q⁻¹k∆uk_q,G≤supn

F^∗(φ) : φ∈Hb_•^2,q⁰(G) and k∆φk_q⁰_,G ≤1o

≤ k∆uk_q,G

Proof. Let F^∗ ∈ Hb•^2,q⁰(G)^∗ be given. By Theorem 3.9 there exists a unique u ∈ Hb•^2,q(G) and a constant D_q =D(q, G)>0 with

F^∗(φ) =h∇²u,∇²φi ∀φ∈Hb_•^2,q⁰(G) D⁻¹_q k∇²uk_q,G≤sup

n

F^∗(φ) : φ∈Hb_•^2,q⁰(G) andk∇²φk_q⁰_,G ≤1 o

≤ k∇²uk_q,G By Lemma 3.10 it holds

F^∗(φ) =h∆u,∆φi ∀φ∈Hb_•^2,q⁰(G) and

D_q⁻¹k∆uk_q,G ≤ K_nD⁻¹_q k∇²uk_q,G

≤ K_nsupn

F^∗(φ) : φ∈Hb_•^2,q⁰(G) and k∇²φk_q⁰_,G≤1o

= K_n sup

06=φ∈Hb•^2,q⁰(G)

F^∗(φ) k∇²φk_q⁰_,G

≤ K_n² sup

06=φ∈Hb•^2,q⁰(G)

F^∗(φ) k∆φk_q⁰_,G

= supn

F^∗(φ) : φ∈Hb_•^2,q⁰(G) and k∆φk_q⁰_,G≤1o

Remark By Theorem 3.9 and Lemma 3.10 we derive, that k∇²·k_q;G and k∆·k_q;G are equivalent norms onHb•^2,q(G).

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4 The spaces A

^q

(G) and B

^q

(G)

Definition 4.1. Let G⊂Rⁿ be either a bounded or an exterior domain with

∂G∈C². Let 1< q <∞. Then

A^q(G) := {∆u : u∈Hb_•^2,q(G)}

B^q(G) := {h∈L^q(G) : hh,∆φi_G= 0 ∀φ∈Hb_•^2,q⁰(G)}

Theorem 4.2. Let G ⊂ Rⁿ be either a bounded or an exterior domain with

∂G∈C². Let 1< q <∞. Then

L^q(G) =A^q(G)⊕B^q(G) in the sense of a direct decomposition

Proof. see [MueR, Satz IV.2.1, p.201]

Remark By Weyl’s Lemma for h ∈ B^q(G) holds (for a representative) ∆h = 0.

For bounded domains even holds

B^q(G) ={h∈L^q(G) : ∆h= 0}

For exterior domains on the other hand there are harmonicL^q-functions, which are notinB^q(G) (see [MueR] or Lemma 4.4).

Lemma 4.3. Let G ⊂ Rⁿ be either a bounded or an exterior domain with

∂G∈C². Let 1< q <∞ and h∈B^q(G)∩H^1,q(G). Then h∇h,∇φi_G= 0

for eachφ∈Hb•^1,q⁰(G)

Proof. (a) For φ∈Hb₀^1,q⁰(G) there is a sequence (φk)⊂C₀^∞(G) such that k∇φ_k− ∇φk_q⁰_;G→0

Then

h∇h,∇φi_G = lim

k→∞h∇h,∇φ_ki_G=− lim

k→∞hh,∆φki_G= 0

(b) Forφ=ϕr by Theorem 2.7 holds∇ϕ_r∈C₀^∞(G)ⁿ and ϕr ∈Hb•^2,q⁰(G). Then h∇h,∇ϕ_ri_G=−hh,∆ϕ_ri_G= 0

(c) By Theorem 2.5 - 2.7 the assertion follows.

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Lemma 4.4. LetG⊂Rⁿ be an exterior domain with ∂G ∈C². Let ϕ_r,ψ_ri as in Theorem 3.7. Let

S(z) :=











1

(n−2)ωn|z|²⁻ⁿ , z6= 0, n≥3

−_2π¹ ln|z| , z6= 0, n= 2 0 , z= 0, n≥2 Then for alli, j= 1, . . . , nholds

h∂_jS,∆ψ_rii_G = −δ_ij

Proof. Forz6= 0 holds

∂_jS(z) = − 1 ωn

z_j

|z|ⁿ Then

h∂_jS,∆ψ_rii_G = h∂_jS,∆(x_iϕ_r)i_G

= −

S,∆

δijϕr+xi(∂jϕr)

G

==

∂mϕr∈C₀^∞(G) δ_ijh∇S,∇ϕ_ri_G− h∆S, x_i(∂_jϕ_r)i_G

∆S=0== δijh∇S,∇ϕ_ri_G∩B

2r

= −δ_ijh∆S, ϕ_ri_G∩B

2r

+δ_ij Z

∂B2r

n

X

l=1

(∂_lS)(z) ϕ_r(z)

| {z }

=1

z_l

|z|dω_z

= −δ_ij 1

ωn

Z

∂B2r

n

X

l=1

zl

|z|ⁿ zl

|z|dωz = −δ_ij

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5 Theorems about differentiable functions

Lemma 5.1. Letn≥2, R >0, h >0 and

Z_R,h⁺ := {x= (x⁰, xn)∈Rⁿ : |x⁰|< R, 0< xn< h}

Z_R,h⁻ := {x= (x⁰, xn)∈Rⁿ : |x⁰|< R, −h < x_n<0}

ZR,h := {x= (x⁰, xn)∈Rⁿ : |x⁰|< R, |x_n|< h}

E_R := {x= (x⁰, x_n)∈Rⁿ : |x⁰|< R, x_n= 0}

Suppose

f ∈C⁰ Z_R,h⁺

∩C¹ Z_R,h⁺ Let

F(x) :=

f(x⁰, x_n) , if 0≤x_n< h

−3f(x⁰,−x_n) + 4f(x⁰,−^x₂ⁿ) , if −h < xn<0 Then

F ∈C¹(ZR,h), F|_Z+ R,h =f Proof. (a) There aref_i∈C⁰

Z_R,h⁺

, i= 1. . . nwithf_i|_Z+

R,h =∂_if

(b) Fori= 1. . . n−1 and |x⁰|< Rlet (h_k)⊂R with 0<|h_k|< R− |x⁰|,h_k→0.

Let 0< x_n< h. Then (x⁰+h_ke_i, x_n)∈Z_R,h⁺ and f(x⁰+hkei, xn)−f(x⁰, xn) =

Z hk

0

(∂if)(x⁰+tei, xn)dt Forxn→0 we get

f(x⁰+hkei,0)−f(x⁰,0) =

Z hk

0

fi(x⁰+tei,0)dt

mean value theorem= fi(x⁰+ζ_kei,0)h_k withζk between 0 and hk (this implies |ζ_k| →0 )

Therefore

h→0lim

f(x⁰+he_i,0)−f(x⁰,0)

h =f_i(x⁰,0)

(c) For i = n and |x⁰| < R, 0 < xn < ^h₂ let (h_k) ⊂R with 0 < h_k < ^h₂, h_k → 0.

Then

f(x⁰, xn+hk)−f(x⁰, xn) = Z hk

0

(∂nf)(x⁰, xn+t)dt Forxn→0 we get

f(x⁰, hk)−f(x⁰,0) = Z hk

0

fn(x⁰, t)dt=fn(x⁰, ζk)hk

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with 0< ζ_k< h_k. Therefore lim

h→0 h>0

f(x⁰, h)−f(x⁰,0)

h =f_n(x⁰,0) (d) Obviously

F|_Z+ R,h

∈C¹(Z_R,h⁺ ) F|_Z− R,h

∈C¹(Z_R,h⁻ ) (e) Leti= 1. . . n−1 and |x⁰|< R. Then

h→0lim

F(x⁰+hei,0)−F(x⁰,0)

h = lim

h→0

f(x⁰+hei,0)−f(x⁰,0)

h =

(b) f_i(x⁰,0) (f) Let|x⁰|< R. Then

lim

h→0 h>0

F(x⁰, h)−F(x⁰,0)

h = lim

h→0 h>0

f(x⁰, h)−f(x⁰,0)

h =

(c) fn(x⁰,0) Furthermore

h→0lim

h<0

F(x⁰, h)−F(x⁰,0)

h = lim

h→0 h<0

−3f(x⁰,−h) + 4f(x⁰,−^h₂)−f(x⁰,0) h

= lim

h→0 h<0

−3f(x⁰,−h) + 3f(x⁰,0)

h +4f(x⁰,−^h₂)−4f(x⁰,0) h

= lim

h⁰→0 h⁰>0

3f(x⁰, h⁰)−3f(x⁰,0)

h⁰ −2f(x⁰, h⁰)−2f(x⁰,0) h⁰

=

(c) f_n(x⁰,0) (g) So fori= 1, . . . , n−1 holds

∂_iF(x) =







f_i(x⁰, x_n) , x_n≥0

−3f_i(x⁰, xn) + 4fi(x⁰,−^x₂ⁿ) , xn<0 and

∂nF(x) =







f_n(x⁰, x_n) , x_n≥0 3fn(x⁰, xn)−2fn(x⁰,−^x₂ⁿ) , xn<0 That is

∇F ∈C⁰(Z_R,h)ⁿ

Theorem 5.2. LetG⊂Rⁿ be open and bounded with∂G∈C¹. Thenf ∈C¹(G) if and only if there exists ˜f ∈C₀¹(Rⁿ) such that ˜f|_G=f

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Proof. Letf ∈C¹(G) (the inversion is trivial). We denote again byf ∈C⁰(G) the continuation off. There are f_i ∈C⁰(G) withf_i|_G=∂_if

(a) For x₀ ∈ ∂G there exists an open V_x₀ ⊂ Rⁿ with x₀ ∈ V_x₀. Further there is R >0 and a diffeomorphismφ_x₀ :V_x₀ →Z_R,Rsuch that holds: φ_x₀(V_x₀∩∂G) =E_R andφx0(Vx0 ∩G) =Z_R,R⁺ .

LetVex0 :=φ⁻¹_x₀(Z^R

2,^R₄)

(b) WithVe_x by (a) (x∈∂G) holds:

∂G⊂ [

x∈∂G

Ve_x

Because of the compactness of∂Gthere arex1, . . . , xN ∈∂Gsuch that withVei :=Vexi

holds

∂G⊂

N

[

i=1

Vei

Let ˜V₀ :=G. Then

G⊂

N

[

i=0

Vei

Chooseϕ_i ∈C₀^∞(Ve_i) fori= 0,1, . . . , N such that

N

X

i=0

ϕi(x) = 1 for allx∈G Define

g_i:=ϕ_if Then

g0 ∈C₀¹(Rⁿ) and fori, j = 1, . . . , N holds

∂jgi = (∂jϕi)f +ϕi(∂jf) = g_j⁽ⁱ⁾|_G withg⁽ⁱ⁾_j := (∂_jϕ_i)f +ϕ_if_j ∈C⁰(G)

Define fori= 1, . . . , N

hi(x) := gi φ⁻¹_i (x)

for allx∈Z_R⁺

i,Ri∪ERi =φi(Vi∩G) Then

hi∈C⁰

Z⁺_Ri

2 ,^Ri₄

and

∂jhi

Z⁺_Ri

2 ,Ri 4

=

n

X

k=1

h

g⁽ⁱ⁾_k φ⁻¹_i ih

∂j φ⁻¹_i

k

i Z⁺_Ri

2 ,Ri 4

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By Lemma 5.1 there is a

˜h_i ∈C¹(Z_Ri

2 ,^Ri₄ ) such that ˜h_i _Z+

Ri2 ,Ri 4

=h_i By the Definition of the continuation in Lemma 5.1 we see

˜hi ∈C₀¹(ZRi,Ri) Let

f˜i(y) := ˜hi(φi(y)) for all y∈Vi =φ⁻¹_i (ZRi,Ri) Then

f˜_i∈C₀¹(V_i) and f˜_i _G∩V

i

=g_i At least

f˜:=g0+

N

X

i=1

f˜i ∈C₀¹(Rⁿ), f˜ G=f

Lemma 5.3. Letf ∈C¹(R) withf(0) = 0 and|f⁰(t)| ≤Lfor allt∈R. Further letG⊂Rⁿ be open and 1< q <∞. Then for u∈H^1,q(G) holds

f(u)∈H^1,q(G) ∇f(u) =f⁰(u)∇u

Proof. see [SiDGL, Satz 6.14]

Lemma 5.4. Let G⊂Rⁿ be open and bounded. Letu∈H^1,q(G) (strictly: let ube a representative) and

Z(u) := {x∈G : u(x) = 0}

Then

∂iu(x) = 0 for almost all x∈Z(u) and for alli= 1, . . . , n

Proof. see [SiDGL, Satz 6.15]

Theorem 5.5. Let 1≤q <∞and let G⊂Rⁿ be open and bounded. Suppose u∈C⁰(G)∩H^1,q(G) andu

∂G = 0. Then u∈H₀^1,q(G)

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Proof. (a) Choose

ϕ∈C^∞(R), 0≤ϕ≤1, ϕ(t) =ϕ(−t), ϕ(t) =

0 ,if |t| ≤1 1 ,if |t| ≥2 Fork∈Nlet

f_k(t) :=

Z t 0

ϕ(ks)ds Then

fk(t) = 0 ∀ |t| ≤ 1 k and

|t−f_k(t)| ≤ Z |t|

0

(1−ϕ(ks))ds≤min

|t|,2 k

Furthermore

f_k⁰(t) =ϕ(kt) −→

(k→∞)

0 ,if t= 0 1 ,if t6= 0 (b) Let

u_k(x) :=f_k(u(x)) By Lemma 5.3 holds

u_k ∈H^1,q(G) Becauseu

∂G= 0, u∈C⁰(G) and ∂G compact, there areGk⊂⊂G with|u(x)| ≤ ¹_k for allx∈G\G_k

Therefore

u_k(x) = 0 ∀x∈G\G_k and

u_k ∈H₀^1,q(G) (c) Now

|u_k(x)−u(x)| = |f_k(u(x))−u(x)| ≤

(a)

min

|u(x)|,2 k

→0 for allx∈Gand therefore by the dominated convergence theorem

ku_k−uk_q;G→0 (k→ ∞)

(d) By Lemma 5.4 for almost everyx∈Z(u) holds

∂_iu(x)−∂_iu_k(x) =∂_iu(x)h

1−ϕ(k u(x))i

= 0 Forx∈G\Z(u) we get by (a):

∂_iu(x)−∂_iu_k(x)→0

Lq-solutions to the Cosserat spectrum in bounded and exterior domains

L

-solutions to the Cosserat spectrum in bounded and exterior domains

Contents

Abstract

Part I: Preliminaries

1 Notations

2 The space H b

(G)

3 The space H b

(G)

4 The spaces A

(G) and B

(G)

5 Theorems about differentiable functions