L^r$-variational inequality for vector fields and the

Lr -variational

Inequality for Vector Fields and the Helmholtz-Weyl Decomposition in Bounded Domains H IDEO KOZONO & TAKU Y ANAGISAWA

Dedicated to Professor Masatake Miyake on the occasion of his 60th birthday

A BSTRACT. We show that every Lr -vector field on Ω can be uniquely decomposed into two spaces with scalar and vector potentials, and the harmonic vector space via operators rot and div, where Ω is a bounded domain in R3 with the smooth boundary ∂Ω. Our decomposition consists of two kinds of boundary conditions such as u · ν|∂Ω = 0 and u × ν|∂Ω = 0, where ν denotes the unit outward normal to ∂Ω. Our results may be regarded as an extension of the well-known de Rham-Hodge-Kodaira decomposition of C ∞ -forms on compact Riemannian manifolds into Lr -vector fields on Ω. As an application, the generalized Biot-Savart law for the incompressible fluids in Ω is obtained. Furthermore, various bounds of u in Lr for higher derivatives are given by means of rot u and div u.

1. I NTRODUCTION Let Ω be a bounded domain in R3 with C ∞ -boundary ∂Ω. It is well known that every vector field u in Lr (Ω), 1 < r < ∞, can be uniquely represented as (1.1)

u = v + ∇p,

where v ∈ Lr (Ω) with div v = 0 in the sense of distributions in Ω with v · ν = 0 on ∂Ω, and p ∈ W 1,r (Ω). Here and in what follows, ν denotes the unit outer 1853

Indiana University Mathematics Journal c , Vol. 58, No. 4 (2009)

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H IDEO KOZONO & TAKU Y ANAGISAWA

normal to ∂Ω. For smooth vector fields in Ω, Weyl [34] proved such a decomposition as an orthogonal sum in L2 (Ω). The case for more general Lr -vector fields was treated by Fujiwara-Morimoto [12], Solonnikov [27] and Simader-Sohr [24]. It should be noted that (1.1) holds for all u ∈ Lr (Ω), so we can define the projection operator Pr by Pr u = v which plays an important role for investigation into the Navier-Stokes equations. In this paper, we shall prove a more precise decomposition for v in (1.1): (1.2)

v = h + rot w,

¯ satisfies where w ∈ W 1,r (Ω) with w × ν = 0 on ∂Ω, and where h ∈ C ∞ (Ω) rot h = 0, div h = 0 in Ω with h · ν = 0 on ∂Ω. This may be regarded as the generalization of the de Rham-Hodge-Kodaira orthogonal decomposition in L2 for C ∞ p -forms Λp (M) on compact Riemannian n-manifolds (M, g) without boundary (1.3) Λp (M) = H p (M) ⊕ +d(Λp−1 (M)) ⊕ δ(Λp+1 (M)),

p = 1, . . . , n − 1,

where d and δ denote the exterior differentiation and codifferential operator, respectively, and H p (M) = {h ∈ Λp (M) | dh = 0, δh = 0}. Our decomposition (1.2) holds for all u ∈ Lr (Ω) with 1 < r < ∞ and for all smooth bounded domains Ω in R3 . In the case Ω has a certain topological type, similar decomposition to (1.2) in L2 (Ω) was investigated by many authors (see, for example, Friedrichs [10], Morrey [21], Foias-Temam [11], Georgescu [13], BendaliDomingues-Gallic [3], Yoshida-Giga [35] and Schwarz [23]). To prove (1.2), the vector potential w is formally obtained from the boundary value problem ( rot rot w = rot u in Ω, w ×ν =0 on ∂Ω. It should be noted that this is not an elliptic system for w . Hence, to recover ellipticity, we need to impose on w the following additional condition: (1.4)

   rot rot w = rot u div w = 0   w × ν = 0

in Ω, in Ω, on ∂Ω.

Unfortunately, this modified system is not an elliptic boundary value problem in the sense of Agmon-Douglis-Nirenberg [1]. Indeed, if w ∈ W 2,r (Ω) for some 1 < r < ∞, then we may rewrite (1.4) as (1.5)

   −∆w = rot u div w = 0   w × ν = 0

in Ω, on ∂Ω, on ∂Ω,

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which can be treated as an elliptic boundary value problem in the sense of AgmonDouglis-Nirenberg (see Lemma 4.4 (1) below). Since we need to solve (1.4) for an arbitrary given u ∈ Lr (Ω), we can expect only that w ∈ W 1,r (Ω), so the value div w on the boundary ∂Ω in (1.5) cannot be always well-defined. This means that we are not able to apply to (1.4) the fully established theory on existence and regularity of solutions to the elliptic boundary value problems. To get around such difficulty, we shall formulate (1.4) in a weak sense such as to find w ∈ W 1,r (Ω) with div w = 0 satisfying Z

Z

(1.6)

Ω

rot w · rot Φ dx =

Ω

u · rot Φ dx

0

for all Φ ∈ W 1,r (Ω) with div Φ = 0 in Ω, Φ ×ν = 0 on ∂Ω, where r 0 = r /(r − 1). This procedure is similar to that of finding a scalar potential p in (1.1). Indeed, p ∈ W 1,r (Ω) is obtained from the weak solution of the Neumann boundary problem for ∆ in Ω. Z

Z

(1.7)

Ω

∇p · ∇φ dx =

Ω

u · ∇φ dx

0

for all φ ∈ W 1,r (Ω).

Simader-Sohr [24] solved (1.7) by introducing a variational inequality in W 1,r (Ω) which is a variant of coercive estimate of the Dirichlet form associated to the operator −∆. Our proof for solvability of (1.4) is also based on the following variational inequality. In fact, we shall show that for every 1 < r < ∞, there is a constant C such that Z    rot w · rot Φ dx 

(1.8) kwkW 1,r (Ω) Ú C sup   

Ω

kΦkW

1,r 0

0

: Φ ∈ W 1,r (Ω),

(Ω)

   

L Z X div Φ = 0 in Ω, Φ × ν = 0 on ∂Ω + w · ψi dx   i=1 Ω

holds for all w ∈ W 1,r (Ω) with div w = 0 in Ω, w × ν = 0 on ∂Ω, where ¯ | {ψ1 , . . . , ψL } is a basis of the finite dimensional space Vhar (Ω) = {ψ ∈ C ∞ (Ω) rot ψ = 0, div ψ = 0 in Ω, ψ × ν|∂Ω = 0}. If ∂Ω consists of L + 1 connected components Γ0 , Γ1 , . . . , ΓL of disjoint surfaces with Γ1 , . . . , ΓL inside of Γ0 , i.e., ∂Ω = SL i=1 Γi , then we have dim Vhar (Ω) = L. A similar investigation into the variational inequality was done by Griesinger [14] in the case when Ω is a star-shaped domain. Compared with our situations, she treated the special case when Vhar (Ω) = {0}. Since she took w in W 1,r (Ω) with w = 0 on ∂Ω, it seems to be an open question whether the complement of the space (1.2) coincides with the vector space ∇p with the scalar potential p ∈ W 1,r (Ω) as in (1.1).

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Furthermore, we shall establish another new kind of decomposition of

Lr -vector fields on Ω, which may be regarded as the dual version to (1.1) and (1.2). Indeed, it will be clarified that every u ∈ Lr (Ω) can be also decomposed as

(1.9)

u = h + rot w + ∇p,

where p ∈ W01,r (Ω), w ∈ W 1,r (Ω) with div w = 0 in Ω and w · ν = 0 on ∂Ω, and h ∈ Vhar (Ω). Compared with the usual decomposition such as (1.1)–(1.2), we may choose the scalar potential p of u so that p = 0 on ∂Ω. The vector potential w in (1.9) is determined by the following system:

(1.10)

   rot rot w = rot u   div w = 0 rot w × ν = u × ν     w · ν = 0

in Ω, in Ω, on ∂Ω, on ∂Ω.

In the same way as we have derived (1.5) from (1.4), to solve (1.10) we need to introduce (1.11)

   −∆w = rot u rot w × ν = u × ν   w · ν = 0

in Ω, on ∂Ω, on ∂Ω,

which is a system of the elliptic boundary value problems in the sense of AgmonDouglis-Nirenberg (see Lemma 4.4 (2) below). On the other hand, the harmonic part h of u should satisfy the boundary condition h × ν = 0 on ∂Ω. Similarly to (1.8), we shall see that the solution w of (1.10) can be derived from the following variational inequality Z    rot w · rot Ψ d x 

(1.12) kwkW 1,r (Ω) Ú C sup   

Ω

kΨ kW

1,r 0

0

: Ψ ∈ W 1,r (Ω),

(Ω)

   

div Ψ = 0 in Ω, Ψ · ν = 0 on ∂Ω +  

N Z X w · ϕi dx i=1

Ω

which holds for all w ∈ W 1,r (Ω) with div w = 0 in Ω, w · ν = 0 on ∂Ω, where ¯ | {ϕ1 , . . . , ϕN } is a basis of the finite dimensional space Xhar (Ω) = {ϕ ∈ C ∞ (Ω) rot ϕ = 0, div ϕ = 0 in Ω, ϕ · S ν|∂Ω = 0}. If there are N cuts Σ1 , . . . , ΣN ˙ = Ω\ N of smooth surfaces with Ω i=1 Σi simply connected, then we see that dim Xhar (Ω) = N .

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Here we should remark that another approach to solve (1.4) and (1.10) by means of (1.5) and (1.11) was fully established by Schwarz [23]. His method is based not on the theory of Agmon-Douglis-Nirenberg but on that of pseudodifferential operators together with the Lopatinski-Sapiro condition. When Ω has ˙ = Ω \ SN such a specially topological type as simply connected domain Ω i=1 Σi SL having the boundary ∂Ω = j=0 Γj , von Wahl [32] gave a representation formula (1.2) and (1.9). However, his method relies on the potential theory. On the other hand, our variational inequalities (1.8) and (1.12) make it possible to treat any bounded domain Ω in R3 . As an application of (1.2) and (1.9), we shall show the generalized Biot-Savart law for the vector field u in W 1,r (Ω) with u · ν = 0 and u × ν = 0 on ∂Ω. In the whole space R3 , if v ∈ W 1,r (R3 ) with div v = 0, then v can be represented as Z v(x) =

R3

K(x − y) × rot v(y) dy,

K(x) = −

1 x 4π |x|3

for all x ∈ R3 . Since ∇K(x) is a Calderon-Zygmund kernel, there holds k∇vkLr (R3 ) Ú Ck rot vkLr (R3 ) ,

1 < r < ∞, k∇vkL∞ (R3 ) Ú C{1 + k rot vkbmo log(e + kvkH 3 (R3 ) )}. See e.g., Beale-Kato-Majda [2] and Kozono-Taniuchi [15]. In bounded domains Ω in R3 , our decompositions (1.2) and (1.9) give the corresponding estimates (1.13) (1.14)

k∇ukLr (Ω) Ú C(k div ukLr (Ω) + k rot ukLr (Ω) + kukL1 (Ω) ), n k∇ukL∞ (Ω) Ú C 1 + kukLr (Ω) + (k div ukbmo + k rot ukbmo ) o × log(e + kukW s,r (Ω) ) , s > 1 + 3/r

for 1 < r < ∞ provided u satisfies u · ν = 0 or u × ν = 0 on ∂Ω. By means of the representation formula for u ∈ W 1,r (Ω) given by Kress [17], von Wahl [33] obtained (1.13) without kukL1 (Ω) on the right hand side when h always vanishes in (1.2) and (1.9). On the other hand, if u ∈ W 1,r (Ω) with u·ν = 0 or u×ν = 0 on ∂Ω, then our decompositions (1.2) and (1.9) yield necessarily such a representation formula as we can deduce (1.13) immediately. Since we need not impose any assumption on Ω, our estimate (1.13) may be regarded as a generalization of von Wahl [33]. Furthermore, we shall show the corresponding estimate for u in higher order Sobolev space W s,r (Ω) via div u and rot u in W s−1,r (Ω) even though u · ν or u × ν does not vanish on ∂Ω. Concerning (1.14), by introducing a different elliptic system from (1.5), Ferrari [9], Shirota-Yanagisawa [26] and Ogawa-Taniuchi [18] gave the proof for u with div u = 0, u · ν|∂Ω = 0. On the other hand, we shall see that these estimates can be derived directly from the regularity theorem of weak solutions to (1.4) and

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(1.10). Since we need to impose neither assumption on Ω nor div u = 0, our estimates (1.13) and (1.14) may be regarded as generalization of [33], [9], [26] and [18]. This paper is organized as follows. In Section 2, we shall state our main theorems. Since we deal with all Lr -vector fields u on Ω, it is necessary to consider the trace of u · ν and u × ν on ∂Ω in the generalized sense. So, the derivatives of functions related to the operators div and rot should be defined in the dense of distributions in Ω, and their generalized trace will be discussed in connection with the Stokes integral formula. Section 3 is devoted to the proof of two kinds of variational inequalities such as (1.8) and (1.12). First, we shall prove the corresponding inequality in the half space R3+ . Then we regard (1.8) and (1.12) as the perturbation of those from R3+ . To this end, the technique of partition of unity from the view point of differential geometry will be fully used. In Section 4, we shall prove our main theorems. Besides the variational inequalities, we shall show that the boundary value problem (1.5) and its adjoint system fulfil the complementing condition in the sense of Agmon-Doglus-Nirenberg [1]. As a result, we shall establish the W s,r -bound of u via div u and rot u as well as Lr -decompositions like (1.2) and (1.9). Finally in the Appendix, we shall construct bases {ϕ1 , . . . , ϕN } in Xhar (Ω) and {ψ1 , . . . , ψL } in Vhar (Ω) provided Ω has the first and the second Betti numbers N and L, respectively. 2. R ESULTS Let us first impose the following assumption on the domain Ω: Assumption. Ω is a bounded domain in R3 with the C ∞ -boundary ∂Ω. Before stating our results, we introduce some function spaces. Let C0∞,σ (Ω) denote the set of all C ∞ -vector functions ϕ = (ϕ1 , ϕ2 , ϕ3 ) with compact support in Ω, such that div ϕ = 0. Lrσ (Ω) is the closure of C0∞,σ (Ω) with respect to the 0 Lr -norm k· kr ; (· , · ) denotes the duality pairing between Lr (Ω) and Lr (Ω), where 1/r + 1/r 0 = 1. Lr (Ω) stands for the usual (vector-valued) Lr -space over Ω, 1 < r < ∞. Let us recall the generalized trace theorem for u · ν and u × ν on r r ∂Ω defined on the spaces Ediv (Ω) and Erot (Ω), respectively. r Ediv (Ω) ≡ {u ∈ Lr (Ω) | div u ∈ Lr (Ω)} r = kuk + k div uk , with the norm kukEdiv r r r Erot (Ω) ≡ {u ∈ L (Ω) | rot u ∈ L (Ω)} r = kukr + k rot ukr . with the norm kukErot

r

r

r r It is known that there are bounded operators γν and τν on Ediv (Ω) and Erot (Ω) with properties that r γν : u ∈ Ediv (Ω) , γν u ∈ W 1−1/r

0 ,r 0

(∂Ω)∗ ,

r (Ω) , τν u ∈ W 1−1/r τν : u ∈ Erot

0 ,r 0

(∂Ω)∗ ,

¯ γν u = u · ν ∂Ω if u ∈ C 1 (Ω), ¯ τν u = u × ν ∂Ω if u ∈ C 1 (Ω),

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respectively. We have the following generalized Stokes formula (2.1)

(u, ∇p) + (div u, p) = hγν u, γ0 pi∂Ω

(2.2)

r (Ω) and all p ∈ W 1,r (Ω), for all u ∈ Ediv (u, rot φ) = (rot u, φ) + hτν u, γ0 φi∂Ω

0

0

r for all u ∈ Erot (Ω) and all φ ∈ W 1,r (Ω), 0

0

0

where γ0 denotes the usual trace operator from W 1,r (Ω) onto W 1−1/r ,r (∂Ω), 0 0 0 0 and h· , · i∂Ω is the duality paring between W 1−1/r ,r (∂Ω)∗ and W 1−1/r ,r (∂Ω). r r Notice that Lσ (Ω) = {u ∈ L (Ω) | div u = 0 in Ω with γν u = 0}. Let us define two spaces X r (Ω) and V r (Ω) for 1 < r < ∞ by (2.3) (2.4)

n o X r (Ω) ≡ u ∈ Lr (Ω) | div u ∈ Lr (Ω), rot u ∈ Lr (Ω), γν u = 0 , n o V r (Ω) ≡ u ∈ Lr (Ω) | div u ∈ Lr (Ω), rot u ∈ Lr (Ω), τν u = 0 .

Equipped with the norms kukX r and kukV r (2.5)

kukX r , kukV r ≡ k div ukr + k rot ukr + kukr ,

we may regard X r (Ω) and V r (Ω) as Banach spaces. Indeed, in Theorem 2.4 below, we shall see that both X r (Ω) and V r (Ω) are closed subspaces in W 1,r (Ω) since it holds that (2.6a) (2.6b)

k∇ukr Ú CkukX r k∇ukr Ú CkukV r

for all u ∈ X r (Ω), for all u ∈ V r (Ω),

respectively, where C = C(r ) is a constant depending only on r . Furthermore, we define two spaces Xσr (Ω) and Vσr (Ω) by (2.7a) (2.7b)

n o Xσr (Ω) ≡ u ∈ X r (Ω) | div u = 0 in Ω , n o Vσr (Ω) ≡ u ∈ V r (Ω) | div u = 0 in Ω .

r r (Ω) and Vhar (Ω) the Lr -spaces of harmonic vector fields Finally, we denote by Xhar on Ω as

(2.8a) (2.8b)

n o r Xhar (Ω) ≡ u ∈ Xσr (Ω) | rot u = 0 , n o r Vhar (Ω) ≡ u ∈ Vσr (Ω) | rot u = 0 .

Our main result now reads as follows.

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Theorem 2.1. Let Ω be as in the Assumption. Suppose that 1 < r < ∞. (1) It holds that n o r ¯ | div h = 0, rot h = 0 in Ω with h · ν = 0 on ∂Ω Xhar (Ω) = h ∈ C ∞ (Ω) ( ≡ Xhar (Ω)), n o r ¯ | div h = 0, rot h = 0 in Ω with h × ν = 0 on ∂Ω (Ω) = h ∈ C ∞ (Ω) Vhar ( ≡ Vhar (Ω)).

Both Xhar (Ω) and Vhar (Ω) are finite dimensional vector spaces. (2) For every u ∈ Lr (Ω), there are p ∈ W 1,r (Ω), w ∈ Vσr (Ω) and h ∈ Xhar (Ω) such that u can be represented as u = h + rot w + ∇p.

(2.9)

Such a triplet {p, w, h} is subordinate to the estimate (2.10)

k∇pkr + kwkV r + khkr Ú Ckukr

with the constant C = C(r ) independent of u. The above decomposition (2.9) is unique. In fact, if u has another expression ˜ + rot w˜ + ∇p˜ u=h for p˜ ∈ W 1,r (Ω), w˜ ∈ Vσr (Ω) and h˜ ∈ Xhar (Ω), then we have (2.11)

˜ h = h,

˜ rot w = rot w,

˜ ∇p = ∇p.

(3) For every u ∈ Lr (Ω), there are p ∈ W01,r (Ω), w ∈ Xσr (Ω) and h ∈ Vhar (Ω) such that u can be represented as u = h + rot w + ∇p.

(2.12)

Such a triplet {p, w, h} is subordinate to the estimate (2.13)

k∇pkr + kwkX r + khkr Ú Ckukr

with the constant C = C(r ) independent of u. The above decomposition (2.12) is unique. In fact, if u has another expression ˜ + rot w˜ + ∇p˜ u=h for p˜ ∈ W01,r (Ω), w˜ ∈ Xσr (Ω) and h˜ ∈ Vhar (Ω), then we have (2.14)

˜ h = h,

˜ rot w = rot w,

˜ ∇p = ∇p.

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An immediate consequence of the above theorem is the following result. Corollary 2.2. Let Ω be as in the Assumption. (1) By the unique decompositions (2.9) and (2.12) we have two kinds of direct sums (2.15)

Lr (Ω) = Xhar (Ω) ⊕ rot Vσr (Ω) ⊕ ∇W 1,r (Ω),

(2.16)

Lr (Ω) = Vhar (Ω) ⊕ rot Xσr (Ω) ⊕ ∇W0 (Ω),

1,r

for 1 < r < ∞. (2) Let Sr , Rr and Qr be projection operators associated to both (2.9) and (2.12) from Lr (Ω) onto Xhar (Ω), rot Vσr (Ω) and ∇W 1,r (Ω), and from Lr (Ω) onto 1,r Vhar (Ω), rot Xσr (Ω) and ∇W0 (Ω), respectively, i.e., (2.17)

Sr u ≡ h,

Rr u ≡ rot w,

Qr u ≡ ∇p.

kRr ukr Ú Ckukr ,

kQr ukr Ú Ckukr

Then we have (2.18)

kSr ukr Ú Ckukr ,

for all u ∈ Lr (Ω), where C = C(r ) is a constant depending only on 1 < r < ∞. Moreover, there holds

(2.19)

  Sr2 = Sr ,     Rr2 = Rr ,     Q 2 = Q , r r

Sr∗ = Sr 0 , Rr∗ = Rr 0 , Qr∗ = Qr 0 , 0

where Sr∗ , Rr∗ and Qr∗ denote the adjoint operators on Lr (Ω) of Sr , Rr and Qr , respectively. Remark 2.3. (1) It is known that (2.20)

Lr (Ω) = Lrσ (Ω) ⊕ ∇W 1,r (Ω),

1 < r < ∞ (direct sum).

See Fujiwara-Morimoto [12], Solonnikov [27] and Simader-Sohr [24]. Our decomposition (2.15) gives a more precise direct sum of Lrσ (Ω) such as (2.21)

Lrσ (Ω) = Xhar (Ω) ⊕ rot Vσr (Ω),

1 < r < ∞ (direct sum).

On the other hand, our new decomposition (2.16) imposes on p the homogeneous boundary condition on ∂Ω. Compared with Lrσ (Ω) in (2.21), any boundary condition on ∂Ω cannot be prescribed on the vector field v ≡ h + rot w in (2.12). If u ∈ W 1,r (Ω), then we have u × ν = v × ν on ∂Ω.

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(2) Suppose that the boundary ∂Ω has L+1 connected components Γ0 , Γ1 , . . . , ΓL of C ∞ -surfaces such that Γ1 , . . . , ΓL lie inside of Γ0 with Γi ∩ Γj = φ for i , j , and such that ∂Ω =

(2.22)

L [

Γj .

j=0

Moreover, we assume that there are N C ∞ -surfaces Σ1 , . . . , ΣN transversal to ∂Ω such that Σi ∩ Σj = φ for i , j , and such that (2.23)

˙ ≡Ω\Σ Ω

is a simply connected domain, where Σ ≡

N [

Σj .

j=1

Then Foias-Temam [11] showed that (2.24)

dim Xhar (Ω) = N.

They [11] also gave an orthogonal decomposition of L2σ (Ω) such as L2σ (Ω) = Xhar (Ω) ⊕ H1 (Ω)

(orthogonal sum in L2 (Ω)),

where Z  H1 (Ω) ≡ u ∈ L2σ (Ω) |

Σj

 u · ν dS = 0 for all j = 1, . . . , N .

Yoshida-Giga [35] investigated the operator rot with its domain D(rot) = {u ∈ H1 (Ω) | rot u ∈ H1 (Ω)} and showed that H1 (Ω) = rot Vσ2 (Ω). Furthermore, they [35] gave another type of orthogonal L2 -decomposition of vector fields u ∈ D(rot). From our decomposition (2.21) with r = 2, it follows also that H1 (Ω) = rot Vσ2 (Ω). (3) In the case when Ω is a star-shaped domain, Griesinger [14] gave a similar decomposition in Lr (Ω) for 1 < r < ∞. In her case, it holds N = 0. Since she took the space W01,r (Ω), smaller than our space V r (Ω), it seems to be an open question whether, in the same way as in (2.15), the annihilator rot W01,r (Ω)⊥ of 0 0 rot W01,r (Ω) in Lr (Ω) coincides with ∇W 1,r (Ω). (4) If Ω has the boundary ∂Ω such as (2.22), then we shall show in the Appendix that (2.25)

dim Vhar (Ω) = L.

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Moreover, it holds (2.26) {rot w | w ∈ W 2,r (Ω) ∩ Xσr (Ω)}

Z   = v ∈ W 1,r (Ω) | div v = 0 in Ω, v · ν dS = 0 for all j = 0, 1, . . . , L . Γj

(5) If Ω has such a topological type as (2.22) and (2.23), then von Wahl [32] gave also a representation formula like (2.9) and (2.12) by means of the potential theory. Our theorem does not need any restriction on the topological type of Ω, which seems to be an advantage for the use of Lr -variational inequalities (1.8) and (1.12). In the more general case when Ω is an n-dimensional C ∞ -manifold with the boundary, Schwarz [23] established an orthogonal decomposition of p -forms in L2 (Ω) and in W s,r (Ω) for s Û 1 and 2 Ú r < ∞. However, his method depends on the theory of pseudo-differential operators, which is different from our Lr -variational approach based on the theory of Agmon-Douglis-Nirenberg. As an application of our decomposition, we have the following gradient and higher order estimates of vector fields via div and rot. Theorem 2.4. Let Ω be as in the Assumption. Suppose that 1 < r < ∞. (1) (prescribed γν u) Let dim Xhar (Ω) = N and let {ϕ1 , . . . , ϕN } be a basis of Xhar (Ω). (i) It holds X r (Ω) ⊂ W 1,r (Ω) with the estimate 

(2.27) k∇ukr + kukr Ú C k div ukr + k rot ukr +

N X

|(u, ϕj )|



j=1

for all u ∈ X r (Ω), where C = C(Ω, r ). (ii) Let s ≥ 1. Suppose that u ∈ Lr (Ω) with div u ∈ W s−1,r (Ω), rot u ∈ W s−1,r (Ω) and γν u ∈ W s−1/r ,r (∂Ω). Then we have u ∈ W s,r (Ω) with the estimate 

(2.28) kukW s,r (Ω) Ú C k div ukW s−1,r (Ω) + k rot ukW s−1,r (Ω) + kγν ukW s−1/r ,r (∂Ω) +

N X

 |(u, ϕj )| ,

j=1

where C = C(Ω, r ). (2) (prescribed τν u) Let dim Vhar (Ω) = L and let {ψ1 , . . . ψL } be a basis of Vhar (Ω).

1864

H IDEO KOZONO & TAKU Y ANAGISAWA (i) It holds V r (Ω) ⊂ W 1,r (Ω) with the estimate 

(2.29) k∇ukr + kukr Ú C k div ukr + k rot ukr +

L X

|(u, ψj )|



j=1

for all u ∈ V r (Ω), where C = C(Ω, r ). (ii) Let s ≥ 1. Suppose that u ∈ Lr (Ω) with div u ∈ W s−1,r (Ω), rot u ∈ W s−1,r (Ω) and τν u ∈ W s−1/r ,r (∂Ω). Then we have u ∈ W s,r (Ω) with the estimate 

(2.30) kukW s,r (Ω) Ú C k div ukW s−1,r (Ω) + k rot ukW s−1,r (Ω) + kτν uk

W s−1/r ,r (∂Ω)

+

L X

 |(u, ψj )| ,

j=1

where C = C(Ω, r ). (3) (L∞ -gradient bound) Let u ∈ W s,r (Ω) for s > 1 + 3/r with u · ν|∂Ω = 0 or u × ν|∂Ω = 0. Then we have ∇u ∈ L∞ with the estimate n

(2.31) k∇uk∞ Ú C 1 + kukr + (k div ukbmo + k rot ukbmo )

o × log(e + kukW s,r (Ω) ) ,

where C = C(s, r ) is a constant independent of u. For definition of the bmonorm, see Remark 2.5 below. Remark 2.5. (1) Let us recall the bmo-norm in Ω. For f ∈ L1loc (R3 ), we define kf kbmo(R3 ) by kf kbmo(R3 ) =

sup x∈R3 , 0 γ(x 0 ), x 0 = (x 1 , x 2 ) ∈ R2 ,

where γ ∈ C0∞ (R2 ). We may treat Dγ as a domain perturbed from the half space R3+ provided maxx0 ∈R2 |∇0 γ(x 0 )| is sufficiently small, where ∇0 γ = (∂γ/∂x 1 , ∂γ/∂x 2 ). To this end, let us introduce a diffeomorphism Θ : x = (x 1 , x 2 , x 3 ) ∈ Dγ , Θ(x) = y = (y 1 , y 2 , y 3 ) ∈ R3+ by (3.23)

y 1 = x1,

y 2 = x2,

y 3 = x 3 − γ(x 1 , x 2 ).

Let us choose a basis {e1 , e2 , e3 } of the moving coordinate system in R3+ as follows.   ∂γ , e1 = 1, 0, ∂x 1

  ∂γ e2 = 0, 1, , ∂x 2

  ∂γ ∂γ e3 = − 1 , − 2 , 1 . ∂x ∂x

It is easy to see that {e1 , e2 , e3 } is linearly independent with e1 · e3 = e2 · e3 = 0. Notice that the unit outer normal ν on

H IDEO KOZONO & TAKU Y ANAGISAWA

1872

∂Dγ ≡ {x = (x 0 , x 3 ) | x 0 ∈ R2 , x 3 = γ(x 0 )}

can be written as ν = (1 + |∇0 γ|2 )−1/2 e3 . For the vector field u = (u1 (x), u2 (x), u3 (x)) =

3 X

ui (x)ji

i=1

on Dγ ({ji }i=1,2,3 : the canonical basis in R3 ), we have the following representation of u in terms of {e1 , e2 , e3 }: 3 X

u=

(3.24)

˜ i (y)ei u

on R3+ ,

i=1

where ˜ i (y) = u

3 X

αji uj x=Θ−1 (y) ,

y ∈ R3+ , i = 1, 2, 3

j=1

1,2,3 with (αji )i→ j↓1,2,3 such that



  ∂γ 2 1 +  ∂x 2  
i→1,2,3 1 ∂γ ∂γ  αji j↓1,2,3 = − 1 + |∇0 γ|2   ∂x 1 ∂x 2   ∂γ ∂x 1

∂γ ∂γ ∂x 1 ∂x 2 2  ∂γ 1+ ∂x 1 ∂γ ∂x 2 −

 ∂γ  ∂x 1   ∂γ   − 2. ∂x    −

1

We see that

(3.25) u × ν ∂D = 0 implies

˜ 1 (y 0 , 0) = u˜ 2 (y 0 , 0) = 0, u ∀ y 0 = (y 1 , y 2 ) ∈ R2 .

Indeed, u × ν|∂Dγ = 0 yields that u × e3 = u˜ 1 e1 × e3 + u˜ 2 e2 × e3 = 0 on R2 . Since e1 × e3 and e2 × e3 are linearly independent, we obtain (3.25). Moreover, since e1 · e3 = e2 · e3 = 0, we see that (3.26)

u · ν ∂D = 0

implies

˜ 3 (y 0 , 0) = 0 u

for all y 0 = (y 1 , y 2 ) ∈ R2 .

Concerning the scalar product u · φ(x) at each point of x ∈ Dγ of two vectors u and φ, there holds u · φ(x) =

3 X

ui (x)φi (x)

i=1

=

3 X

˜ j (y)φ˜ k (y), gjk (Θ−1 (y))u

j,k=1

y = Θ−1 (x) ∈ R3+ ,

Helmholtz-Weyl Decomposition in Lr

1873

where gjk ≡ ej · ek (j , k = 1, 2, 3) have the components 

(3.27)

2 ∂γ 1 +  ∂x 1   (gjk )j,k=1,2,3 =   ∂γ ∂γ  ∂x 1 ∂x 2   



∂γ ∂γ ∂x 1 ∂x 2   ∂γ 2 1+ ∂x 2

0

0 0 1 + |∇0 γ|2

0

    .    

The eigenvalues λ1 Ú λ2 Ú λ3 of the matrix (gjk )j,k=1,2,3 are expressed as λ1 = 1,

(3.28)

λ2 = λ3 = 1 + |∇0 γ|2 .

Furthermore, we need to represent ∇u · ∇φ(x) at each point x ∈ Dγ in terms of coordinate y = Θ−1 (x) ∈ R3+ : (3.29)

∇u · ∇φ(x) =

=

3 X ∂φj ∂uj (x) (x) ∂x i ∂x i i,j=1 3 3 X X

gjk (y)

i=1 j,k=1

3 ˜k ˜ j ∂ φ˜ k X ∂u j ` ∂φ ˜ + Γ u `i ∂y i ∂y i ∂y i `=1

+

3 X

k ˜` Γ`i φ

`=1

! 3 X ˜j ∂u j ` k ˜p ˜ (y), + Γ u Γ φ pi `i ∂y i `,p=1

where !

` Γjk

3 ∂gpk ∂gjk ∂gpj 1 X , = g `p + − 2 p=1 ∂y k ∂y j ∂y p

j, k, ` = 1, 2, 3

−1 }`,p=1,2,3 . Note that denotes the Christoffel symbol with {g `p }`,p=1,2,3 = {g`p



(3.30)

  ∂γ 2 ∂γ ∂γ 1 + − 1 2  ∂x ∂x ∂x 2   1    (g `p )`,p=1,2,3 = ∂γ ∂γ ∂γ 2 1 + |∇0 γ|2  − 1 +   ∂x 1 ∂x 2 ∂x 1 

0

0



0 

  . 0   

1

By the explicit representations of the eigenvalues {λj }j=1,2,3 with (3.28) and the ` Christoffel symbol {Γjk }j,k,`=1,2,3 with (3.27) and (3.30) we can control their sizes 0 3 by means of maxx ∈R |∇0 γ(x 0 )|. Note that supx0 ∈R2 |(∇0 )2 γ(x 0 )| < ∞ since γ ∈ C0∞ (R2 ).

H IDEO KOZONO & TAKU Y ANAGISAWA

1874

Proposition 3.4. For every ε > 0, there is a constant δ > 0 such that if maxx0 ∈R2 |∇0 γ(x 0 )| < δ, then it holds λ1 (Θ−1 (y)) ≡ 1 < λ2 (Θ−1 (y)) = λ3 (Θ−1 (y)) < 1 + ε,

max |gjk (Θ−1 (y)) − δjk | < ε,

j,k=1,2,3

max

j,k,`=1,2,3

` |Γjk (y)| < ε,

for all y ∈ R3+ , where δjk denotes the Kronecker symbol. Similarly to (3.1) and (3.2), we define function spaces Vˆ (Dγ ), Vˆ r (Dγ ), ˆ γ ) and Xˆ r (Dγ ) by X(D n o ¯ γ ) | φ × ν = 0 on ∂Dγ , Vˆ (Dγ ) ≡ φ ∈ C0∞ (D n o Vˆ r (Dγ ) ≡ u ∈ W 1,r (Dγ ) | u × ν = 0 on ∂Dγ , n o ˆ γ ) ≡ φ ∈ C0∞ (D¯ γ ) | φ · ν = 0 on ∂Dγ , X(D n o Xˆ r (Dγ ) ≡ u ∈ W 1,r (Dγ ) | u · ν = 0 on ∂Dγ ,

for 1 < r < ∞, respectively. Our variational inequalities on Vˆ r (Dγ ) and Xˆ r (Dγ ) now read as follows. Lemma 3.5. (1) For every 1 < r < ∞ there are positive constants δ = δ(r ) and C = C(r ) such that if |∇0 γ(x 0 )| < δ, max 0 2 x ∈R

then we have the following two estimates (i) and (ii): (i) For every u ∈ Vˆ r (Dγ ) it holds (3.31)

k∇ukLr (Dγ ) + kukLr (Dγ ) Ú C

sup φ∈Vˆ (Dγ )

|(∇u, ∇φ)Dγ + (u, φ)Dγ | k∇φkLr 0 (Dγ ) + kφkLr 0 (Dγ )

.

(ii) For every u ∈ Xˆ r (Dγ ) it holds (3.32)

k∇ukLr (Dγ ) + kukLr (Dγ ) Ú C

sup ˆ γ) φ∈X(D

|(∇u, ∇φ)Dγ + (u, φ)Dγ | k∇φkLr 0 (Dγ ) + kφkLr 0 (Dγ )

.

(2) For every 1 < q, r < ∞ there is a constant δ = δ(q, r ) such that under the hypothesis |∇0 γ(x 0 )| < δ max 0 2 x ∈R

we have the following properties (iii) and (iv):

Helmholtz-Weyl Decomposition in Lr

1875

(iii) If u ∈ Vˆ q (Dγ ) satisfies (3.33)

|(∇u, ∇φ)Dγ + (u, φ)Dγ |

sup φ∈Vˆ (Dγ )

k∇φkLr 0 (Dγ ) + kφkLr 0 (Dγ )

< ∞,

then we have u ∈ Vˆ r (Dγ ) with the estimate (3.31). (iv) If u ∈ Xˆ q (Dγ ) satisfies (3.34)

|(∇u, ∇φ)Dγ + (u, φ)Dγ |

sup ˆ γ) φ∈X(D

k∇φkLr 0 (Dγ ) + kφkLr 0 (Dγ )

< ∞,

then we have u ∈ Xˆ r (Dγ ) with the estimate (3.32). Here (· , · )Dγ and k· kLr (Dγ ) denote the inner product and the Lr -norm on Dγ , respectively. Proof. (1) We reduce our problem on Dγ to that on R3+ in Lemma 3.3. Since the proofs for Vˆ r (Dγ ) and Xˆ r (Dγ ) are completely the same, we will show (3.31) ˆ γ ) and Yˆ r (Dγ ) by and (3.32) simultaneously. To this end, we define Y(D ˆ γ ) ≡ Vˆ (Dγ ), X(D ˆ γ) Y(D

Yˆ r (Dγ ) ≡ Vˆ r (Dγ ), Xˆ r (Dγ ),

and

respectively. By Proposition 3.4 we may assume that 1 = λ1 (Θ−1 (y)) < λ2 (Θ−1 (y)) = λ3 (Θ−1 (y)) < 2 for all y ∈ R3+ . Hence we have 2 2 ˜ ˜ µ(|∂ u(y)| + |u(y)| ) Ú |∇u(x)|2 + |u(x)|2 2 2 ˜ ˜ Ú C(|∂ u(y)| + |u(y)| ) for all x ∈ Dγ with y = Θ(x)

with some constants µ and C , where ˜ 2= |∂ u|

3 X i,j=1

˜j ∂u ∂y i

!2 ,

˜ 2= |u|

3 X

˜ j )2 . (u

j=1

Since x = Θ−1 (y) satisfies det(∂x i /∂y j )i,j=1,2,3 = 1, it holds dx = dy , and hence we obtain from the estimate above




˜ Lr (R3+ ) + |u| ˜ Lr (R3+ ) Ú k∇ukLr (Dγ ) + kukLr (Dγ ) (3.35) µ |∂ u|




˜ Lr (R3+ ) + |u| ˜ Lr (R3+ ) Ú C |∂ u|

H IDEO KOZONO & TAKU Y ANAGISAWA

1876

for all u ∈ Yˆ r (Dγ ). By (3.29) we have that Z

(3.36) (∇u, ∇φ)Dγ + (u, φ)Dγ = Z

 X  3 3 ˜ j ∂ φ˜ j X ∂u j ˜j ˜ + u φ dy ∂y i ∂y i j=1 R3+ i,j=1

X  3 ˜ j ∂ φ˜ k ∂u j ˜k ˜ (g − δ ) + u φ dy jk jk ∂y i ∂y i R3+ j,k=1 i=1

+

Z

3 X

X  3 3 3 ˜k X ˜j X j j k ˜ ` ∂u ` ∂φ ` k ˜p ˜ ˜ + 3 gjk (y) Γ`i u + Γ`i φ + Γ`i u Γpi φ dy. ∂y i ∂y i R+ i=1 j,k=1 `=1 `=1 `,p=1 3 3 X X

If maxx0 ∈R2 |∇0 γ(x 0 )| < δ, then we see by Proposition 3.4 (3.37)

Z

X  3 ˜ j ∂ φ˜ k ∂u j ˜k ˜ (gjk − δjk ) + u φ dy i ∂y i 3 ∂y R+ j,k=1 i=1 3 X

 



˜

r 0 3 +

|u| ˜

r 0 3 , ˜ Lr (R3+ ) |∂ φ| ˜ Lr (R3+ ) |φ| Ú ε |∂ u| L (R+ ) L (R+ )

(3.38)

Z 3 3 3 X ˜k X X ∂φ j ˜` i g (y) Γ`i u jk R3+ ∂y i=1 j,k=1

`=1

+

3 X



3  j X j ` k ˜p ˜ ∂ u˜ ˜ Γ u Γ φ d y qi `i i

k ` Γ`i φ

`=1



˜ Ú ε |∂ u|

Lr (R3+ )

˜ + |u|

∂y

Lr (R3+ )

`,p=1

 ˜

|∂ φ|

0

Lr (R3+ )



˜

r 0 3 . + |φ| L (R+ )

By (3.36), (3.37) and (3.38), we have Z

 X  3 3 ˜ j ∂ φ˜ j X ∂u j ˜j ˜ + u φ d y i i 3 ∂y ∂y R+ i,j=1 j=1   

˜

r 0 3 +

|φ| ˜

r 0 3 ˜ Lr (R3+ ) + |u| ˜ Lr (R3+ ) |∂ φ| − ε |∂ u| L (R+ ) L (R+ )

(3.39) (∇u, ∇φ)Dγ + (u, φ)Dγ ≥

ˆ γ ) provided maxx0 ∈R2 |∇0 γ(x 0 )| < δ. for all u ∈ Yˆ r (Dγ ) and all φ ∈ Y(D ˆ γ ) , φ˜ = (φ˜ 1 , φ˜ 2 , φ˜ 3 ) ∈ Y(R3+ ) ≡ By (3.25) and (3.26) the map φ ∈ Y(D V (R3+ ), X(R3+ ) is surjective. Hence we see from (3.35), (3.39) and Lemma 3.3 that if u ∈ Yˆ r (Dγ ) satisfies (3.31) and (3.32), then it holds

Helmholtz-Weyl Decomposition in Lr |(∇u, ∇φ)Dγ + (u, φ)Dγ |

sup ˆ γ) φ∈Y(D

≥C sup

1877

k∇φkLr 0 (Dγ ) + kφkLr 0 (Dγ )

Z  X  3 3 ˜ j ∂ φ˜ j X ∂u j ˜j ˜ + u φ d y i i ∂y ∂y i,j=1 j=1 R3+

˜ Y(R ˆ 3+ ) φ∈

˜

|∂ φ|

0

Lr (R3+ )

˜

+ |φ|

0

Lr (R3+ )



˜ Lr (R3+ ) − ε |u| ˜ Lr (R3+ ) − ε |∂ u|




˜ Lr (R3+ ) + |u| ˜ Lr (R3+ ) ≥(C − ε) |∂ u| ≥C(C − ε)(k∇ukLr (Dγ ) + kukLr (Dγ ) ).

Taking ε > 0 sufficiently small in Proposition 3.4, we have (3.40)

k∇ukLr (Dγ ) + kukLr (Dγ ) Ú C

sup ˆ γ) φ∈Y(D

|(∇u, ∇φ)Dγ + (u, φ)Dγ | k∇φkLr 0 (Dγ ) + kφkLr 0 (Dγ )

provided maxx0 ∈R2 |∇0 γ(x 0 )| < δ. This implies (3.31) and (3.32). (2) Let us first consider the case 1 < r Ú q < ∞. Suppose that u ∈ Yˆ q (Dγ ) satisfies (3.33) and (3.34). We take R > 0 so large that supp γ ⊂ {x 0 ∈ R2 : |x 0 | < R}. We choose a cut-off function η ∈ C ∞ (R3 ) so that η(x) = 0 for |x| Ú R , η(x) = 1 ˆ 3+ ). Since for |x| Û 2R . Then we see that ηu ∈ Yˆ q (R3+ ). Let us take ψ ∈ Y(R 3 0 0 supp ηψ ⊂ {x ∈ R+ : |x| > R} and since q Ú r , we have |(∇(ηu), ∇ψ)R3+ + (ηu, ψ)R3+ | = |(∇u, ∇(ηψ))R3+ + (u, ηψ)R3+ − (∇η · ∇u, ψ)R3+ + (u, ∇η · ∇ψ)R3+ | Ú |(∇u, ∇(ηψ))Dγ + (u, ηψ)Dγ | + C(k∇ukLq ({x∈R3+ :R