Solvability of the Weak Dirichlet Problem in L q with Homogeneous Boundary Conditions for ∆ 2

and ∆

³

In this subsection we cite the important variational inequalities and solvabil-ity theorems which will be used in the following.

Theorem 2.7. M¨uller’s variational inequality in H₀^2,q(G) (see [10], Haupt-satz, page 191):

Let G⊂ Rⁿ be a bounded domain with ∂G∈ C², 1 < q <∞ with q⁰ := _q−1^q . Then there is a constant C_M,q >0 depending only on G and q such that

k∆uk_q ≤C_M,q sup

06=Φ∈H₀^2,q0(G)

h∆u,∆Φi k∆Φk_q0

holds for all u∈H₀^2,q(G).

For a proof, we refer to [10], pages 191-194.

In fact M¨uller proved this variational inequality not just for the case of bounded domains but also for exterior domains. We also have a vector-version of this variational inequality:

Theorem 2.8. M¨uller’s variational inequality in H^2,q₀ (G):

Let G⊂ Rⁿ be a bounded domain with ∂G∈ C², 1 < q <∞ with q⁰ := _q−1^q . Then there is a constant C_M,q >0 depending only on G and q such that

k∆uk_q ≤C_M,q sup

06=Φ∈H^2,q₀ ⁰(G)

h∆u,∆Φi k∆Φk_q0

holds for all u∈H^2,q₀ (G).

Proof. We have for u∈H^2,q₀ (G):

k∆uk_q =

n

X

j=1

k∆u_jk^q_q

!¹_q

and as for j = 1, . . . , nwe have k∆u_jk^q_q ≤C_M,q^q



 sup

06=Φ∈H₀^2,q0(G)

h∆u_j,∆Φi k∆Φk_q0





q

by Theorem 2.7, we find

Validity of M¨uller’s variational inequalities forq and q⁰ are equivalent to the unique solvability of the weak Dirichlet problem for ∆² in L^q and L^q0 with homogeneous boundary conditions. For a proof, we refer to [10], Lemma III.15. on page 164, but for the analogous problem (Theorem 2.17) for ∆³ we will give a proof below, see Theorem 2.17.

Theorem 2.9. Let F be a bounded linear functional ∈

H₀^2,q0(G)∗

. Then there is exactly one u∈H₀^2,q(G) with

h∆u,∆Φi=F(Φ) for all Φ∈H₀^2,q0(G).

Moreover, the solution u satisfies

k∆uk_q < C_∆²kFk^“

H₀^2,q0(G)”∗

with a C_∆² =C_∆²(q, G)>0.

For a proof, see [10], Lemma III.15., page 164.

In [10], Satz IV.1.1., page 195, M¨uller gives the following regularity result in a version for exterior domains. We state the theorem in another version for our domains which we will give a proof for:

Theorem 2.10. Let 1< q, s <∞ and u∈H₀^2,q(G) satisfying sup

Φ∈C₀^∞(G)

h∆u,∆Φi k∆Φk_s0

<∞. (5)

Then u∈H₀^2,s(G).

Proof. In the case 1 < s ≤ q < ∞ the statement u ∈ H₀^2,s(G) is shown easily: For s =q everything is clear and for s < q the statement is merely a consequence of the boundedness of Gand the H¨older inequality. In this case we do not even need the validity of assumption (5).

So look now at the case 1 < q < s <∞. Let first 0 6= Φ ∈ C₀^∞(G). By (5) we see that by

F(Φ) :=h∆u,∆Φi we have

|F(Φ)|=|h∆u,∆Φi|= |h∆u,∆Φi|

k∆Φk_s0

k∆Φk_s0 ≤ sup

06=Ψ∈C₀^∞(G)

h∆u,∆Ψi k∆Ψk_s0

k∆Φk_s0. As C₀^∞(G) is dense in H₀^2,s0(G) with respect to the norm k∆·k_s0, there is an unique linear and continuous extension ˜F ∈

H₀^2,s0(G)∗

of F. By Theorem 2.9 we find an unique v ∈H₀^2,s(G) with

h∆v,∆Φi= ˜F(Φ) for all Φ ∈H₀^2,s0(G).

As s > q we find that v ∈H₀^2,q(G), too. For all Φ∈ C₀^∞(G) we find that:

h∆(v−u),∆Φi= 0

and thus by the uniqueness in Theorem 2.9 we concludeu=v ∈H₀^2,s(G).

For the solvability of the analogous problem to Theorem 2.9 for ∆³ we have to refer to [15], Theorems 7.5. and 7.6., which apply not only to ∆³ but to uniformly strongly elliptic regular Dirichlet bilinear forms of given order m in the sense of [15], Definitions 1.3 and 1.4, pages 14-16. This means:

Definition 2.11. Let G ⊂⊂ Rⁿ be open, n, m ∈ N with n ≥ 2 and m ≥ 1. Let for every α, β ∈ (N0)ⁿ with |α|,|β| ≤ m a complex-valued bounded measurable function a_α,β defined in G be given. For Φ,Ψ∈ C₀^∞(G) let

B[Φ,Ψ] := X

|α|≤m

|β|≤m

ha_αβD^αΦ, D^βΨi.

and

L_B := (−1)^m X

|α|=|β|=m

a_αβ(·)D^αD^β.

Then B is called an uniformly strongly elliptic Dirichlet bilinear form of order m in G, if the differential operator L_B is uniformly strongly elliptic of order 2m in G, that is:

• For every fixed (l₁, . . . , ln−1) =: l ∈ Rⁿ⁻¹ \ {0} and every x ∈ G the polynomial in τ ∈C

P(τ, l, x) := X

|α|=|β|=m

a_αβ(x)l^α0+β0τ^αn+βn, α = (α⁰, α_n), β = (β⁰, β_n) has exactly m roots with positive and m roots with negative imaginary part.

• There exists a constant E >0 such that (−1)^mRe X

|α|=|β|=m

a_αβ(x)l^α+β ≥E|l|^2m holds for every x∈G and l∈Rⁿ .

Moreover, by regularity of B, it is meant that the functions a_αβ admit for

|α| = |β| = m a continuous continuation to G and are bounded in G for

|α|=|β|< m.

Remark 2.12. All the bilinear forms we will use in the following are defined for some m ∈N by

B_m[Φ,Ψ] :=

(h∆^m2Φ,∆^m2Ψi for even m

h∇∆^m−12 Φ,∇∆^m−12 Ψi for odd m .

Now it is quickly seen that Bm defines an uniformly strongly elliptic regular Dirichlet bilinear form of order m in the above introduced sense:

We see that we can write

B_m[Φ,Ψ] = X

|α|≤m

|β|≤m

ha_αβD^αΦ, D^βΨi

with all the a_αβ ≥0 constant.

Next, we see that the differential operator L_B associated to the bilinear form B =B_m has the form

L_B = (−1)^m(−1)^m∆^m = ∆^m. The requirements from Definition 2.11 are quickly verified:

• The regularity assumptions on the a_αβ are trivially fulfilled, for they are all constant.

• The polynomial P(τ, l, x) with l ∈Rⁿ⁻¹\ {0}, τ ∈C can be written as (−1)^m

n−1

X

i=1

l²_i +τ²

!m

.

This polynomial has the same zeros τ as the polynomial Pn−1

i=1 l²_i +τ², but withm-times as much multiplicities. As the polynomialPn−1

i=1 l_i²+τ² has real coefficients, the zeros occur in pairs of complex conjugates and as l 6= 0, there can be no real zeros. So the original polynomial must have one zero with positive imaginary part (with multiplicity m) and one zero with negative imaginary part (with multiplicity m).

• Looking at

(−1)^mRe (−1)^m|l|^2m

=|l|^2m, we see that we can choose E = 1.

Having now verified that our B_m[·,·] are admissible for Simader’s theory from [15], we cite the important theorems from there which we are going to use in order to get our solvability statements:

Theorem 2.13. (Compare [15], Theorem 7.5., page 129)

Let m≥1 be an integer and let G⊂Rⁿ (n ≥2) be a bounded open set with boundary ∂G ∈ C^m. Let B[Φ,Ψ] be an uniformly strongly elliptic regular Dirichlet bilinear form of order m andq, q⁰ two real numbers with1< q, q⁰ <

∞ and ¹_q +_q¹0 = 1.

Let

N_q :=n

w∈H₀^m,q(G) : B[w,Φ] = 0 for every Φ∈H₀^m,q0(G)o and let

N_q⁰ :=n

z∈H₀^m,q0(G) : B[Ψ, z] = 0 for every Ψ∈H₀^m,q(G)o . Then dimNq = dimNq⁰ = d < ∞. For F ∈

H₀^m,q0(G) ?

the functional equation

B[u,Φ] =F(Φ) for every Φ∈H₀^m,q0(G)

has a solution u∈H₀^m,q(G) if and only if F(z) = 0 for every z ∈Nq⁰. Particularly, in case of d= 0, the equation is uniquely solvable for arbitrary F ∈

H₀^m,q0(G)?

.

To show in our case that d= 0, we use the following theorem:

Theorem 2.14. (Compare [15], Theorem 7.6., page 131) Let m ≥ 1 be an integer and let G⊂ Rⁿ (n ≥2) be a bounded open set with boundary ∂G ∈ C^m. Let B[Φ,Ψ] be an uniformly strongly elliptic regular Dirichlet bilinear form of order m, let q be a real number with 1 < q < ∞ and u ∈ H₀^m,q(G) such that

B[u,Φ] = 0 for every Φ∈ C₀^∞(G).

Then u∈H₀^m,r(G) for every 1< r <∞.

Remark 2.15. So in our case, where m := 3, B[Ψ,Φ] := h∇∆Ψ,∇∆Φi, look at an u∈N_q. Then for every Φ∈ C₀^∞(G) we find

h∇∆u,∇∆Φi= 0

and with Theorem 2.14 we conclude that u ∈ H₀^3,2(G) and u ∈ N₂(G), too.

Thus taking u itself as a testing function (which can be justfied by approxi-mating u in the H₀^3,2(G)-sense by C₀^∞(G)-functions), we see that

h∇∆u,∇∆ui= 0 and thus u= 0.

This leads us to the following solvability theorem:

Theorem 2.16. Let F be a bounded linear functional ∈

H₀^3,q0(G) ∗

. Then there is exactly one u∈H₀^3,q(G) satisfying

h∇∆u,∇∆Φi=F(Φ) for all Φ∈H₀^3,q0(G). (6) Moreover, there is aC_∆³ =C_∆³(q, G)>0such that for everyF ∈

H₀^3,q0(G)^∗ and u with (6) we have the estimate

k∇∆uk_q ≤C_∆³kFk^“

H₀^3,q0(G)”∗. (7)

Proof. The existence of an unique u∈H₀^3,q(G) satisfying

h∇∆u,∇∆Φi=F(Φ) for all Φ∈H₀^3,q0(G) (8) is a direct consequence of Theorem 2.13 and Remark 2.15. The only thing that remains to be shown is the existence of a C_∆³ =C_∆³(q, G) such that

k∇∆uk_q ≤C_∆³kFk^“

H₀^3,q0(G)

”∗.

With Theorem 9.11 in [15] (see our Theorem 6.1 (with m = 3, j = 0)) we get with (8) the estimate

kuk_3,q ≤γ

kFk^“

H₀^3,q0(G)”∗ +kuk_q

, (9)

where γ is dependent only on q and G(note that m, j and B, as they were called in Theorem 6.1 are fixed here and n is already coded in G). In view of the equivalence of the normsk∇∆·k_q and k·k_3,q onH₀^3,q(G), it is sufficient to show that for u we have an estimate of the form

kuk_q ≤CkFk^“

H₀^3,q0(G)”∗ (10)

with a C =C(q, G)>0. As we have

kuk_q ≤ kuk_3,q,

it suffices to show validity of an estimate of the form kuk_3,q ≤CkFk^“

H₀^3,q0(G)”∗ (11)

with a C =C(q, G)>0. Then estimate (7) follows easily with (9).

Assume (11) were false. Then we could find a sequence (F_ν)_ν∈

N ⊂

H₀^3,q0(G)∗

and (u_ν)_ν∈

N ⊂H₀^3,q(G) with

h∇∆u_ν,∇∆Φi=F_ν(Φ) for all Φ∈ C₀^∞(G), ν∈N with

kuνk_3,q = 1 (12)

and

ku_νk_3,q > νkF_νk^“

H₀^3,q0(G)

”∗. (13)

With (12) and (13) we conclude F_ν

“

H₀^3,q0(G)”∗

−−−−−−−→0.

By (12) the sequence (u_ν)_ν∈

N ⊂ H₀^3,q(G) is bounded in H₀^3,q(G) and we can assume (by passing to a subsequence) without loss of generality that there is an u∈H₀^3,q(G) with

u_ν ^{weakly inH}

3,q 0 (G)

−−−−−−−−−−→u

and

u_ν strongly inH₀^2,q(G)

−−−−−−−−−−−→u

by Rellich’s compact embedding from H₀^3,q(G) into H₀^2,q(G), see for example [2], A6.1, pages 256, 257.

By u_ν ^{weakly inH}

3,q 0 (G)

−−−−−−−−−−→uand F_ν

“ H^3,q₀ ⁰(G)

”∗

−−−−−−−→0 we see easily that h∇∆u,∇∆Φi= 0

for all Φ ∈ H₀^3,q0(G) and by the unique solvability already verified, we see u= 0. By the inequality (9) and the convergence ofu_ν inL^q(G) to 0, we see that

ku_ν −u_µk_3,q ≤γ

kF_ν−F_µk^“

H₀^3,q0(G)”∗ +ku_ν −u_µk_q

≤

≤γ

kF_νk^“

H₀^3,q0(G)”∗+ F_µ

^“

H₀^3,q0(G)”∗+ku_νk_q+ku_µk_q

µ,ν→∞

−−−−→0 and so (u_ν) is a Cauchy-sequence inH₀^3,q(G) and thus has a limitv ∈H₀^3,q(G).

But then (u_ν) converges also weakly in H₀^3,q(G) to v and by uniqueness of the weak limit we have u = v and thus (u_ν) converges strongly to u = 0.

This, however, is a contradiction to ku_νk_3,q = 1 for all ν ∈N. With Theorem 2.16 we also get a variational inequality:

Theorem 2.17. 1. There is a CV = CV(q, G) > 0 such that for all u ∈ H₀^3,q(G) the following inequality is valid:

k∇∆uk_q≤C_V sup

06=Φ∈H₀^3,q0(G)

h∇∆u,∇∆Φi k∇∆Φk_q0

2. The validity of this variational inequality is equivalent to our solvability Theorem 2.16 in the following sense: If G⊂Rⁿ is a domain such that the statement of the variational inequality is valid for 1< q < ∞ and q⁰ with ¹_q+_q¹0 = 1 then also the solvability theorem is valid for q and q⁰ and vice versa.

Proof. At first we will prove the statement of the variational inequality using the solvability Theorem 2.16, thus showing 1. and one part of the equivalence in 2.:

Let u∈H₀^3,q(G) be arbitrary. Then by setting for Φ∈H₀^3,q0(G) F(Φ) :=h∇∆u,∇∆Φi,

a bounded linear functional F ∈

H₀^3,q0(G)∗

is defined. By definition, we have

h∇∆u,∇∆Φi=F(Φ) for all Φ∈H₀^3,q0(G) and thus by Theorem 2.16 we have

k∇∆uk_q ≤C_∆³kFk^“

So, validity of our solvability statement forqimplies validity of our variational inequality for q.

To show the other implication in 2., we assume validity of our variational inequality for q and q⁰. Take a look at the set T := By the variational inequality, we see that T ⊂

H₀^3,q0(G)^∗

existing by definition of T, we see by the variational inequality forq applied to Cauchy differences, that

k∇∆(u_ν −u_µ)k_q ≤C_V sup

06=Φ∈H₀^3,q0(G)

h∇∆(u_ν −u_µ),∇∆Φi k∇∆Φk_q0

=

=CV sup

06=Φ∈H₀^3,q0(G)

(Fν −Fµ)(Φ) k∇∆Φk_q0

µ,ν→∞

−−−−→ 0.

So, the sequence (uν) converges in H₀^3,q(G) towards an element u. For any Φ∈ C₀^∞(G), we have

F(Φ) = lim

ν→∞F_ν(Φ) = lim

ν→∞h∇∆u_ν,∇∆Φi=h∇∆u,∇∆Φi and thus F ∈T.

We now want to show that T =

H₀^3,q0(G)∗

: Assume that this were not so. Then by a consequence of the Hahn-Banach Theorem, we could find a functional 0 6= H ∈

H₀^3,q0(G)∗∗

with H(F) = 0 for all F ∈ T. But as H₀^3,q0(G) is reflexive, we find an element v ∈H₀^3,q0(G) with H(F) = F(v) for all F ∈

H₀^3,q0(G) ∗

. We find that for v we have therewith h∇∆v,∇∆Φi= 0 for all Φ∈H₀^3,q(G), as every Φ∈H₀^3,q(G) defines an element FΦ ∈

H₀^3,q0(G) ∗

by FΦ(Ψ) :=h∇∆Φ,∇∆Ψifor all Ψ∈H₀^3,q0(G) and F_Φ ∈T. Then we have

h∇∆v,∇∆Φi=F_Φ(v) =H(F_Φ) = 0 and thus by the variational inequality for q⁰ we find:

k∇∆vk_q0 ≤C˜_V sup

06=Φ∈H₀^3,q(G)

h∇∆v,∇∆Φi k∇∆Φk_q = 0,

so v = 0 and thus H = 0, a contradiction. The uniqueness of the solution is shown easily: Assume that u∈H₀^3,q(G) with

h∇∆u,∇∆Φi= 0 for all Φ∈H₀^3,q0(G).

Then as above with the variational inequality for q we find that u = 0, so the solution must be uniquely determined in H₀^3,q(G). The continuity of the solution process is a direct consequence of the variational inequality.

Remark 2.18. The foregoing account to solvability and validity of a vari-ational inequality for the problem related to ∆³ could also have been used without problems for the problem of ∆². However, the cited theorems for the case of ∆² were used due to their generality (they also apply to the case of exterior domains) and the fact that their proof is more elementary than the proof of the theorems from Simader’s Theory.

2.5 A Decomposition of H

₀^1,q

(G)

Definition 2.19. Between the spaces from Definition 2.1 (equipped with the respective norms) we have the following continuous linear mappings:

• div :H^2,q₀ (G)→H_0,0^1,q(G), v = (v₁, . . . , v_n)7→Pn i=1∂_iv_i.

• T_q : H_0,0^1,q(G) → H^2,q₀ (G), p 7→ v where v is the unique element in H^2,q₀ (G) satisfying

h∆v,∆Φi=h∇p,∇div Φi ∀Φ∈H^2,q₀ ⁰(G). (14) The solvability of (14), the uniqueness of v and the continuity of T_q are guaranteed by Theorem 2.9.

• By Z_q :H_0,0^1,q(G)→H_0,0^1,q(G) we denote the composition Z_q = div◦T_q. We state a generalization of Weyl’s Lemma which is valid for arbitrary open sets G⊂Rⁿ:

Weyl’s Lemma 2.20. Assume f ∈L¹_loc(G) satisfies

hf,∆^mΦi= 0 for every Φ∈ C₀^∞(G). (15) Then f ∈ C^∞(G) and ∆^mf = 0.

Proof. A very elementary proof for the cases m = 1,2 making big use of Friedrichs’ mollification can be found in [14] (see Lemma 2.7, page 767 and Theorem 3.4, page 770). It can easily be generalized to m ∈ N by a simple induction argument, the first part already being done. Let m ∈ N, m > 1 and the assumption hold for m −1 and m = 1. As being C^∞ is a local property we can look at x ∈ G arbitrary and it suffices to show that f is C^∞ in an open ball B_r(x)⊂ G. So let now x ∈ G be arbitrary, r >0 be so small thatB_r(x)⊂⊂Gand ε >0 be so small that B_r+2ε(x)⊂⊂G. Because Φz(y) := jε(y−z) is for fixedz ∈Br(x) a function with compact support in G, we see with equation (15) that we have

0 = Z

G

f(y)∆^m_y j_ε(y−z)dy = Z

G

f(y)∆^m_z j_ε(y−z)dy=

= ∆^m Z

G

j_ε(y−z)f(y)dy= ∆^mf_ε(z)

and thus ∆^mf_ε = 0 in B_r(x). So we have shown for every sufficiently small ε > 0 that ∆^m−1f_ε is harmonic in B_r(x). By the property of harmonic functions that they stay invariant under mollification with an only radially depending kernel (to be more precise: For h harmonic in G we haveh(z) = h_ε(z) for all z ∈Gwith dist(z, ∂G)< ε) and the fact that for z ∈B_r(x) and 0< δ < ε we have

(f_ε)_δ(z) = (f_δ)_ε(z) and

∂i(gε) (z) = (∂ig)_ε(z) forg ∈ C¹(G) and i∈ {1, . . . , n}

used iteratively, we see that for z ∈ B_r(x) we have with 0 < δ < ε since

∆^m−1(f_δ) and ∆^m−1(f_ε) are harmonic

∆^m−1f_ε(z) = ∆^m−1f_ε

δ(z) = ∆^m−1(f_ε,δ) (z) = ∆^m−1(f_δ,ε) (z) =

= ∆^m−1fδ

ε(z) = ∆^m−1fδ(z).

So we find for all 0 < δ < ε:

∆^m−1(f_δ) = ∆^m−1(f_ε) on B_r(x)

Defining g := ∆^m−1f_ε, we see that g is harmonic on B_r(x) and that it is no restriction to assume g ∈ C^∞(Br(x)). We can find a h ∈ C^∞(Br(x)) with

∆^m−1h = g: By classical theory we find a h₁ ∈ C^∞(B_r(x)) with ∆h₁ = g with the representation formula

h1(y) := − Z

Br(x)

S(y−z)g(z)dz,

where S denotes the fundamental solution to the Laplacian, see for example [16], Satz 4.5, page 102. Iterating this process, we finally reach our sought after h. Taking now a close look at f −h, we see that for Φ ∈ C₀^∞(G) we have

hf−h,∆^m−1Φi=hf,∆^m−1Φi−hh,∆^m−1Φi= lim

ε→0hf_ε,∆^m−1Φi−h∆^m−1h,Φi=

= lim

ε→0hf_ε,∆^m−1Φi − hg,Φi= lim

ε→0 hf_ε,∆^m−1Φi − h∆^m−1f_ε,Φi

= 0, and thus by the induction hypothesis we conclude f −h ∈ C^∞(Br(x)) and as h∈ C^∞(B_r(x)) we also findf ∈ C^∞(B_r(x)).

Definition 2.21. We introduce the spaces

• A^q(G) :=

In view of Weyl’s Lemma 2.20 above we readily see that the space B^q(G) is consisting exactly of the biharmonic H₀^1,q(G)-functions. In particular, every h∈B^q(G) fulfills h∈ C^∞(G).

Gp_νdx. Then with the H¨older inequality we get

The weak solvability of the Dirichlet problem for ∆³ with zero boundary data from Theorem 2.16 gives rise to a direct (if q= 2 orthogonal) decomposition of H₀^1,q(G) and H_0,0^1,q(G) similar to the decomposition of L^q(G) obtained by M¨uller, see [10], Satz IV.2.1, page 201:

Theorem 2.23. We have the direct decompositions

H₀^1,q(G) = A^q(G)⊕B^q(G) (16) H_0,0^1,q(G) =A^q₀(G)⊕B^q₀(G) (17) These decompositions are orthogonal if q= 2.

If p= ∆s+h according to this decomposition we find the estimate:

k∇∆sk_q+k∇hk_q ≤C_Dk∇pk_q (18) with a constant C_D >0 depending only on G and q.

Proof. Forp∈H₀^1,q(G) there exists according to our Theorem 2.16 an unique s ∈H₀^3,q(G) satisfying

h∇∆s,∇∆Φi=h∇p,∇∆Φi ∀Φ∈H₀^3,q0(G) (19) and we have a constantC_∆³ =C_∆³(q, G) withk∇∆sk_q ≤C_∆³k∇pk_q. Define h:=p−∆s ∈H₀^1,q(G). Then we have for all Φ∈ C₀^∞(G):

hh,∆²Φi=−h∇h,∇∆Φi=h∇p− ∇∆s,∇∆Φi= 0

that is h ∈ B^q(G) and we have a representation p = ∆s +h as desired.

The uniqueness of the representation of p = ∆s+h with s ∈ H₀^3,q(G) and h∈B^q(G) is due to the unique solvability of (19): Assume thatp= ∆s₁+h₁ andp= ∆s₂+h₂. Then we have ∆(s₁−s₂) = h₂−h₁ ∈B^q(G) is biharmonic and therewith

h∇∆(s₁−s₂),∇∆Φi=−h∆(s₁ −s₂),∆²Φi= 0 ∀Φ∈ C₀^∞(G)

so s₁ =s₂ and then h₁ =p−∆s₁ =p−∆s₂ =h₂. So the decomposition is direct and we have shown (16).

Furthermore, we see that for every p ∈ H₀^1,q(G) we have ∆s ∈H_0,0^1,q(G) and thus h∈H_0,0^1,q(G) if and only if p∈H_0,0^1,q(G) yielding (17).

To see that this decomposition is orthogonal in caseq= 2 we note that ifh∈ B²(G), ∆s∈A²(G) we findh∇h,∇∆si= 0 (throughH₀^3,2(G)-approximation of s by C₀^∞(G)-functions and partially integrating).

Further we have for a given p∈H₀^1,q(G) and p= ∆s+h:

k∇∆sk_q ≤C_∆³k∇pk_q (20) and

k∇hk_q =k∇p− ∇∆sk_q ≤ k∇pk_q+k∇∆sk_q ≤(C_∆³ + 1)k∇pk_q (21) Sticking (20) and (21) together we get:

k∇∆sk_q+k∇hk_q ≤(2C_∆³ + 1)

| {z }

=:CD

k∇pk_q.

The decomposition (17) and the operator Z_q defined in (2.19) are closely related and so (17) plays an important role in the study of Z_q. As a first insight we have:

Theorem 2.24. Regarding the restrictions of Z_q toA^q₀(G)and to B₀^q(G), we get

Z_q|_A^q

0(G) :A^q₀(G)→A^q₀(G) and Z_q(p) =p ∀p∈A^q₀(G) Z_q|_B^q

0(G) :B₀^q(G)→B₀^q(G)

Proof. Let p ∈ A^q₀(G). Then p = ∆s for an s ∈ H₀^3,q(G) and we find T_q(p) =∇s, for we have

h∆∇s,∆Φi=h∇∆s,∆Φi=h∇p,∆Φi for all Φ∈H^2,q₀ ⁰(G) and so by uniqueness of the solution (Theorem 2.16) we have

T_q(p) =∇s, Z_q(p) = div∇s= ∆s=p For p∈B₀^q(G) we have for Φ∈ C₀^∞(G)

hZ_q(p),∆²Φi=hdivT_q(p),∆²Φi=−h∆T_q(p),∇∆Φi=−h∇p,∆∇Φi=

=hp,∆²Φi= 0 and so we conclude Z_q(p)∈B₀^q(G).

Regarding eigenvalues of Z_q we have due to our direct decomposition from Theorem 2.23 and Theorem 2.24 the following easy fact:

Theorem 2.25. Suppose λ ∈R and p ∈ H_0,0^1,q(G) suffice Z_q(p) = λp. Then we have λ= 1 or p∈B₀^q(G).

Proof. Assume that Z_q(p) = λp. Applying the decomposition (17) from Theorem 2.23 to p we get p= ∆s+h with s ∈H₀^3,q(G) and h∈ B₀^q(G). So we have on the one hand

Z_q(p) = λp=λ(∆s+h) = λ∆s

|{z}

∈A^q₀(G)

+ λh

|{z}

∈B₀^q(G)

and on the other hand using Theorem 2.24 we have Z_q(p) =Z_q(∆s+h) = Z_q(∆s) +Z_q(h) = ∆s

|{z}

∈A^q₀(G)

+Z_q(h)

| {z }

∈B^q₀(G)

.

So by the directness of the decomposition (2.23) we have:

λ∆s= ∆s and λh=Zq(h)

The first of these two equalities can only be satisfied if λ = 1 or s = 0. So we have shown: λ= 1 or p=h∈B^q₀(G).

Note that the “or” in (2.25) is not an exclusive one. The question whether there are p ∈ B₀^q(G) with Z_q(p) = p will be examined later (see subsection 8.2). It will show up that there is a finite dimensional subspace of B₀^q(G) of such elements and that the dimension of this subspace is only dependent on topological properties of G.

Im Dokument Cosserat Operators of Higher Order and Applications (Seite 23-38)