Generalized Lax-Milgram theorem in Banach spacesand its application to the elliptic system of boundaryvalue problems
Received: 20 April 2012 / Revised: 3 August 2012
Abstract. We generalize the well-known Lax-Milgram theorem on the Hilbert space to thaton the Banach space. Suppose that a(·, ·) is a continuous bilinear form on the product X ×Yof Banach spaces X and Y , where Y is reflexive. If null spaces NX and NY associated witha(·, ·) have complements in X and in Y , respectively, and if a(·, ·) satisfies certain variationalinequalities both in X and in Y , then for every F ∈ N⊥
Y , i.e., F ∈ Y ∗ with F(φ) = 0 for allφ ∈ NY , there exists at least one u ∈ X such that a(u, ϕ) = F(ϕ) holds for all ϕ ∈ Y with‖u‖X ≤ C‖F‖Y ∗ . We apply our result to several existence theorems of Lr -solutions to theelliptic system of boundary value problems appearing in the fluid mechanics.
0. Introduction
Let X be a Hilbert space with the norm ‖ ·‖X and let a(·, ·) be a continuous bilinearform on X × X . If a(·, ·) is coercive, i.e., a(u, u) ≥ δ‖u‖2
X for all u ∈ X with someδ > 0, then for every bounded linear functional F ∈ X∗, there is a unique u ∈ Xsuch that a(u, ϕ) = F(ϕ) holds for all ϕ ∈ X . As an immediate consequence, wehave the estimate ‖u‖X ≤ δ−1‖F‖X∗ . This is a well-known Lax-Milgram theoremfor continuous, coercive bilinear form on the Hilbert space. A typical application ofthe Lax-Milgram theorem is to prove existence of weak solutions of the boundaryvalue problem of the elliptic partial differential equations like{
Lu = f in �,Bu = 0 on ∂�,
(0.1)
where � is a domain in Rn with the smooth boundary ∂�. For instance, if L is a
2mth ordered formally self-adjoint elliptic differential operator in � with the mthordered boundary operator B on ∂�, then we may define a self-adjoint operator A inL2(�) by Au = Lu for u ∈ D(A) with the domain D(A) = {u ∈ H2m(�); Bu =0 on ∂�}. Furthermore, if A is positive definite in L2(�), i.e., (Au, u) ≥ δ‖u‖2 for
all u ∈ D(A)with some δ > 0, then the Hilbert space X is chosen as X = D(
A12
)
and the bilinear form a(·, ·) on X is defined by a(u, ϕ) =(
A12 u, A
12 ϕ
), where
H. Kozono (B): Department of Mathematics, Waseda University, Tokyo 169-8555, Japan.e-mail: [email protected]
T. Yanagisawa: Department of Mathematics, Nara Women’s University, Nara 630-8506,Japan. e-mail: [email protected] Subject Classifications (2000): 35Q40, 35F99
DOI: 10.1007/s00229-012-0586-6
H. Kozono, T. Yanagisawa
(·, ·) and ‖ · ‖ denote the inner product and the norm on L2(�), respectively. Insuch a situation, we can apply the Lax-Milgram theorem to show the existence of
a unique weak solution u ∈ D(A12 ) of the problem (0.1) which means that(
A12 u, A
12 ϕ
)= 〈 f, ϕ〉 for all ϕ ∈ D(A
12 ),
where 〈·, ·〉 denotes the duality pairing between D(A12 )∗ and D(A
12 ). See e.g.,
Agmon [1], Lions-Magenes [22], Miranda [23], Peetre [25] and references therein.Simader [29] considered the 2mth ordered elliptic operator L with the Dirich-
let boundary condition on ∂� associated with (0.1) and proved the existence ofa weak solution in the Sobolev space W m,r
0 (�) for 1 < r < ∞. To this end, he
made use of the generalized Lax-Milgram theorem on W m,r0 (�)× W m,r ′
0 (�) with1/r + 1/r ′ = 1. The coerciveness of the bilinear form a(·, ·) can be replaced bythe following variational inequality
‖u‖W m,r0
≤ C supϕ∈W m,r ′
0
|a(u, ϕ)|‖ϕ‖
W m,r ′0
for all u ∈ W m,r0 (�). (0.2)
It was clarified that such a variational inequality plays an important role in show-ing the existence of Lr -weak solutions to the elliptic boundary value problem. Forexample, based on the similar type of the variational inequality to (0.2), Sima-der-Sohr [30] made it clear that there is an essential difference for solvability inexterior domains� ⊂ R
n between the Dirichlet and the Neumann problems for theLaplace operator�; the former has a unique weak solution in W 1,r
0 (�) if and onlyif n′ < r < n, while the latter has a weak solution in W 1,r (�) for all 1 < r < ∞.
The purpose of this paper is to establish a more general Lax-Milgram theoremon the continuous bilinear form a(·, ·) defined by the product X × Y of Banachspaces X and Y , where Y is assumed to be reflexive. By modifying such a var-iational inequality as in (0.2), we remove the hypothesis on coerciveness of thebilinear form a(·, ·). To this end, we need to characterize the null space NX ≡ {u ∈X; a(u, ϕ) = 0 for all ϕ ∈ Y }. Our result may be regarded as a generalizationof the Fredholm alternative in the Banach space associated with the bilinear forma(·, ·). It is known by Fichera [10] and Faedo [9] that the Lax-Milgram theoremis obtained by the existence of a linear functional associated with two boundedoperators from a normed space to Banach spaces. Our characterization consists inthe technique how to handle the null space NX , and the main theorem does not haveany inclusion relation between their results.
As an application we shall consider elliptic systems of the boundary value prob-lem in Lr . In particular, we are interested in the Helmholtz-Weyl decompositionsfor the vector fields in Lr (�), where� is a bounded domain in R
3 with the smoothboundary ∂�. Although the original result on such an Lr -decomposition was provedin our previous paper [19], we shall establish a systematic treatment in terms ofour new variational inequality. Similar Lr -decompositions to the de Rham-HodgeKodaira theorem for general differential p-forms on the n-dimensional compactRiemannian manifold with the boundary will be discussed. We shall also deal withthe stationary problem of the Stokes equations in exterior domains for the incom-pressible fluid.
Generalized Lax-Milgram theorem
1. Result
First, we formulate our problem in an abstract way. Let X be a Banach space withthe norm ‖ · ‖X and let Y be a reflexive Banach space with the norm ‖ · ‖Y . Supposethat a(·, ·) : X × Y → C is a bilinear form satisfying the following assumption.
Assumption. (i) There is a constant M > 0 such that
|a(u, ϕ)| ≤ M‖u‖X‖ϕ‖Y for all u ∈ X and ϕ ∈ Y. (1.1)
(ii) Let NX ≡ {u ∈ X; a(u, ϕ) = 0,∀ϕ ∈ Y } and let NY ≡ {ϕ ∈ Y ; a(u, ϕ) =0,∀u ∈ X}. There are closed subspaces RX in X and RY in Y such that
X = NX ⊕ RX (direct sum), (1.2)
Y = NY ⊕ RY (direct sum). (1.3)
(iii) There is a constant C > 0 such that
‖u‖X ≤ C
(supϕ∈Y
|a(u, ϕ)|‖ϕ‖Y
+ ‖PX u‖X
)for all u ∈ X, (1.4)
‖ϕ‖Y ≤ C
(supu∈X
|a(u, ϕ)|‖u‖X
+ ‖PYϕ‖Y
)for all ϕ ∈ Y, (1.5)
where PX and PY are the projections from X onto NX along (1.2) and from Y ontoNY along (1.3), respectively.
Our generalization of the Lax-Milgram theorem now reads
Theorem 1.1. Let X be a Banach space with the norm ‖·‖X and let Y be a reflexiveBanach space with the norm ‖ · ‖Y . Suppose that a(·, ·) is a bilinear form on X ×Ysatisfying (i), (ii) and (iii) in the Assumption. Then for every F ∈ N⊥
Y , i.e., F ∈ Y ∗with F(φ) = 0 for all φ ∈ NY there exists w ∈ X such that
a(w, ϕ) = F(ϕ) for all ϕ ∈ Y. (1.6)
Such w is subject to the estimate
‖w‖X ≤ C‖F‖Y ∗ , (1.7)
where C is a constant independent of w and F.
Remark. (i) Our theorem includes the usual Lax-Milgram theorem for the contin-uous, coercive bilinear form a(·, ·) : H ×H → C on the Hilbert space H . Indeed, ifthere is some δ > 0 such that a(u, u) ≥ δ‖u‖2
H for all u ∈ H , then we may chooseX = Y = H in Theorem 1.1. Obviously in such a case, we have NX = NY = {0}with
‖u‖H ≤ δ−1a(u, u/‖u‖H ) ≤ δ−1 supϕ∈H
a(u, ϕ)
‖ϕ‖Hfor all u ∈ H,
H. Kozono, T. Yanagisawa
which implies (1.4) and (1.5) with PX = PY = 0.(ii) The closed subspaces NX and NY may be regarded as the null spaces associatedwith the bilinear form a(·, ·). If a(·, ·) is a bilinear form defined by the self-adjointelliptic operator L as in (0.1), then we have NX = NY and it holds that NX is offinite dimension. In case NX and NY are of finite dimensions, we have direct sums(1.2) and (1.3).(iii) It is unnecessary that X is reflexive. If the bilinear form a(·, ·) is defined bythe elliptic operator L as in (0.1), then it holds that NY coincides with the kernel ofthe adjoint operator L∗ of L . Hence, Theorem 1.1 includes the Fredholm alternativein Banach spaces.(iv) Concerning the Dirichelt problem of elliptic equations of the 2mth order,Simader [29] originally established the generalized Lax-Milgram theorem withX = W m,r
0 (�) and Y = W m,r ′0 (�), where 1/r + 1/r ′ = 1. However, he did not
mention how to treat the null spaces NX and NY . Our theorem makes it clear thatif we add PX u and PYϕ on the right hand side of the variational inequalities (1.4)and (1.5), respectively, then we can obtain the same existence result as in the caseof the positive definite bilinear form.(v) Another type of the generalization of the Lax-Milgram theorem was estab-lished by Fichera [10]; Let V be a normed space, and let X and Y be Banach spaceswith norms ‖·‖X and ‖·‖Y , respectively. Suppose that S : V → X and T : V → Yare bounded linear operators. Then for every F ∈ Y ∗, there exists a unique� ∈ X∗such that �(Sϕ) = F(Tϕ) for all ϕ ∈ V if and only if there is M > 0 such that‖Tϕ‖Y ≤ M‖Sϕ‖X for all ϕ ∈ V . By an easy observation, we see that such atheorem includes the Lax-Milgram theorem. Further generalization was developedby Faedo [9] and Ramaswamy [26,27]. Our theorem and their results do not haveany inclusion relation to each other.
Proof of Theorem 1.1. Since RX and RY are closed subspaces of X and Y , weregard RX and RY as Banach spaces with the same norms ‖ · ‖X and ‖ · ‖Y , respec-tively. Let us consider the linear mapping T : RX → R∗
Y defined by
〈Tw,ψ〉 ≡ a(w,ψ), w ∈ RX , ψ ∈ RY ,
where 〈·, ·〉 denotes the duality pairing between RY and R∗Y . By (1.1) we see that
‖Tw‖R∗Y
≤ M‖w‖X for all w ∈ RX , which implies that T is a bounded lin-ear operator from RX to R∗
Y . Moreover, the range R(T ) of T is closed in R∗Y .
Indeed, assume that the sequence {w j }∞j=1 ⊂ RX satisfies Tw j → v in R∗Y . Since
PXw j = 0, j = 1, 2, · · · , and since a(u, PYϕ) = 0, (1 − PY )ϕ ∈ RY with‖(1 − PY )ϕ‖Y ≤ C‖ϕ‖Y for all u ∈ X and all ϕ ∈ Y , it follows from (1.4) that
‖w j − wk‖X ≤ C supϕ∈Y
|a(w j − wk, ϕ)|‖ϕ‖Y
≤ C supϕ∈Y
|a(w j − wk, (1 − PY )ϕ)|‖(1 − PY )ϕ‖Y
= C supψ∈RY
|a(w j − wk, ψ)|‖ψ‖Y
Generalized Lax-Milgram theorem
= C supψ∈RY
|〈T (w j − wk), ψ〉|‖ψ‖Y
= C‖T (w j − wk)‖R∗Y
→ 0 as j, k → ∞,
which yields thatw j → w in RX for somew ∈ RX . Since T is a bounded operatorfrom RX to R∗
Y , we obtain Tw = v and hence v ∈ R(T ). This implies that R(T )is closed in R∗
Y .We next show that R(T ) = R∗
Y . Suppose that R(T ) ⊂ R∗Y with R(T ) �= R∗
YThen there is a g ∈ R∗∗
Y with g �= 0 such that g(Tw) ≡ 0 for all w ∈ RX . Since Yis reflexive, so is its closed subspace RY . Hence, there is a uniqueψ ∈ RY such thatg(v) = 〈v,ψ〉 for all v ∈ R∗
Y with ‖g‖R∗∗Y
= ‖ψ‖Y . In particular, taking v = Tw,we have that
0 = g(Tw) = 〈Tw,ψ〉 = a(w,ψ) for all w ∈ RY .
Since PYψ = 0 and since a(PX u, ψ) = 0, ‖(1− PX )u‖X ≤ C‖u‖X for all u ∈ X ,it follows from (1.5) that
‖ψ‖Y ≤ C supu∈X
|a(u, ψ)|‖u‖X
≤ C supu∈X
|a((1 − PX )u, ψ)|‖(1 − PX )u‖X
= C supw∈RX
|a(w,ψ)|‖w‖X
= 0,
which contradicts g �= 0.Let F ∈ N⊥
Y , i.e., F ∈ Y ∗ satisfying F(φ) = 0 for all φ ∈ NY . Since we mayregard F ∈ R∗
Y and since R(T ) = R∗Y , there exists a w ∈ RX such that Tw = F ,
which implies that
〈Tw,ψ〉 = F(ψ) for all ψ ∈ RY . (1.8)
For every ϕ ∈ Y , we have (1 − PY )ϕ ∈ RY and F(ϕ) = F((1 − PY )ϕ), and henceit follows from (1.8) that
This implies (1.6). Moreover, since PXw = 0, we have by (1.4) and (1.6) that
‖w‖X ≤ C supϕ∈Y
|a(w, ϕ)|‖ϕ‖Y
= C supϕ∈Y
|F(ϕ)|‖ϕ‖Y
= ‖F‖Y ∗ ,
which yields (1.7). This completes the proof of Theorem 1.1.
2. Applications
In this section, we consider applications of Theorem 1.1 to the boundary valueproblem to the elliptic system.
H. Kozono, T. Yanagisawa
2.1. Stokes equations in exterior domains
We first treat the stationary Stokes equations in an exterior domain. The existenceand uniqueness in Lr of the exterior Stokes problems have been intensively investi-gated for a longer time by many authors. See e.g., Chang-Finn [7], Finn [11], Giga-Sohr [15], Borchers-Miyakawa [3], Galdi-Simader [13], Galdi [12], Kozono-Sohr[17]. Although our theorem below is not altogether new, based on the generalizedLax-Milgarm theorem, we shall establish a systematic treatment of questions onLr -solvability of the exterior Stokes problems.
Let � be an exterior domain in Rn , n ≥ 2 with the C2+μ-compact boundary
∂� (μ > 0). We consider the following stationary Stokes equations
⎧⎨⎩
−�v + ∇ p = div F in �,div v = g in �,v = 0 on ∂�,
(S)
where v = v(x) = (v1(x), . . . , vn(x)) and p(x) denote the unknown velocity vec-tor and the unknown pressure at the point x ∈ �, respectively, while F = F(x) ={Fi j (x)}1≤i, j≤n and g = g(x) denote the given n × n-matrix valued function and
the given scalar function, respectively. For F ∈ Lr (�)n2
and g ∈ Lr (�) with1 < r < ∞, we discuss the existence and uniqueness of the solution {v, p} of (S).
Before stating our result, let us introduce some notations and function spaces.H1,r
0 (�) is the completion of C∞0 (�)with respect to the homogeneous norm ‖∇·‖r .
Here and in what follows, ‖ · ‖r denotes the Lr -norm on �; (·, ·) is the dual-ity paring between Lr (�) and Lr ′
(�), where r ′ is the Hölder conjugate expo-nent, i.e., 1/r + 1/r ′ = 1. For the concrete characterization of the homogeneousSobolev space H1,r
0 (�), we refer to [17, Lemma 2.2]. We denote by C∞0,σ (�)
the set of all ϕ ∈ C∞0 (�)
n such that div ϕ = 0. Let H1,r0,σ (�) be the comple-
tion of C∞0,σ (�) with respect to the homogeneous norm ‖∇ · ‖r . It is known that
H1,r0,σ (�) = {u ∈ H1,r
0 (�)n; div u = 0}. See e.g, Kozono-Sohr [18]. We denote byC various positive constants which may change from the line to line. In particular,C = C(∗, . . . , ∗) denotes a constant depending only on the quantities appearing inthe parentheses.
Our result on existence and uniqueness of solutions to (S) now reads
Theorem 2.1. (1) There are n linearly independent pairs {ϕ1, π1}, . . . , {ϕn, πn} offunctions on � such that the following properties (i), (ii) and (iii) are satisfied.
(i) ϕi ∈ C∞(�)n ∩ H1,q0,σ (�), πi ∈ C∞(�) ∩ Lq(�) for all n′ < q < ∞ with
−�ϕi + ∇πi = 0, div ϕi = 0 in �, i = 1, . . . , n; (2.1)
(ii) Let 1 < r ≤ n′ for n ≥ 3 and 1 < r < 2 for n = 2. If F ∈ Lr (�)n2
andg ∈ Lr (�) satisfy
(F,∇ϕi )− (g, πi ) = 0, i = 1, . . . , n, (2.2)
Generalized Lax-Milgram theorem
then there exists a unique pair {v, p} ∈ H1,r0 (�)n × Lr (�) such that
(∇v,∇�)− (p, div �) = (F,∇�) for all � ∈ C∞0 (�)
n, (2.3)
div v = g in �. (2.4)
Moreover, such a pair {v, p} is subject to the estimate
‖∇v‖r + ‖p‖r ≤ C(‖F‖r + ‖g‖r ), (2.5)
where C = C(�, n, r).(iii) Let n ≤ r < ∞ for n ≥ 3 and 2 < r < ∞ for n = 2. For every F ∈ Lr (�)n
2
and g ∈ Lr (�), there exists at least one pair {v, p} ∈ H1,r0 (�)n × Lr (�) such that
(2.3) and (2.4) hold. If another pair {v′, p′} ∈ H1,r0 (�)n × Lr (�) satisfies (2.3)
and (2.4), then it holds that
v(x) = v′(x)+n∑
i=1
ciϕi (x), p(x) = p′(x)+n∑
i=1
ciπi (x) for all x ∈ � (2.6)
with some constants c1, . . . , cn in R.(2) Let n′ < r < n for n ≥ 3 and let r = 2 for n = 2. Then for every
F ∈ Lr (�)n2
and g ∈ Lr (�) there is a unique pair {v, p} ∈ H1,r0 (�)n × Lr (�)
such that the identities (2.3) and (2.4) are satisfied. Moreover, such a pair {v, p} issubject to the estimate (2.5).
To prove Theorem 2.1, we first reduce the equations (S) to those under the sole-noidal vector field. Indeed, it follows from Borchers-Sohr [4, Theorem 3.3] that forevery g ∈ Lr (�) there is a vector fields w ∈ H1,r
0 (�)n such that div w = g in �with the estimate ‖∇w‖r ≤ C‖g‖r , where C = C(�, n, r). Defining u ≡ v − w,we see that the original equations (S) can be rewritten to the following equations(S′).
⎧⎨⎩
−�u + ∇ p = div F +�w in �,div u = 0 in �,u = 0 on ∂�,
(S′)
where u = u(x) = (u1(x), . . . , un(x)) is the new unknown vector function.Our definition of a weak solution of (S′) reads as follows.
Definition. Let 1 < r < ∞. Suppose that F ∈ Lr (�)n2
and w ∈ H1,r0 (�)n with
div w = g. A measurable function u in H1,r0,σ (�) is called a weak solution of (S′)
if it holds
(∇u,∇ϕ) = −(F + ∇w,∇ϕ) for all ϕ ∈ C∞0,σ (�). (2.7)
Concerning the existence of the pressure p in (S), we have the following prop-osition.
H. Kozono, T. Yanagisawa
Proposition 2.1. Let 1 < r < ∞. Suppose that F ∈ Lr (�)n2
and w ∈ H1,r0 (�)n
with div w = g. If u ∈ H1,r0,σ (�) is a weak solution of (S′) in the sense of the
Definition, then there exists a unique scalar function p ∈ Lr (�) such that
(∇u,∇�)− (p, div �) = −(F + ∇w,∇�) (2.8)
holds for all � ∈ C∞0 (�)
n. Such p is subject to the estimate
‖p‖r ≤ C(‖∇u‖r + ‖F‖r + ‖g‖r ), (2.9)
where C = C(�, n, r).
Proof. Defining G = ∇u + F + ∇w, we have that div G ∈ H−1,r (�)n , whereH−1,r (�) ≡ H1,r ′
0 (�)∗. Since C∞0,σ (�) is dense in H1,r ′
0,σ (�), we obtain from (2.7)that
〈div G,�〉 = −(G,∇�) = 0 for all � ∈ H1,r ′0,σ (�),
where 〈·, ·〉 denotes the duality pairing between H−1,r (�)n and H1,r ′0 (�)n . Then it
follows from Giga-Sohr [15, Corollary 2.2] that there exists a unique scalar functionp ∈ Lr (�) such that div G = ∇ p in � in the sense of distributions in �, whichimplies (2.8). Moreover, such p is subject to the estimate
‖p‖r ≤ C‖G‖r ,
which shows (2.9). This proves Proposition 2.1.
Proof of Theorem 2.1. (1) By Proposition 2.1, it suffices to show unique solvabil-ity of the equations (2.7) for the given F ∈ Lr (�)n
2and w ∈ H1,r
0 (�)n with
div w = g. To this end, we make use of Theorem 1.1. Indeed, let X = H1,r0,σ (�)
and Y = H1,r ′0,σ (�). We define the bilinear form a(·, ·) on X × Y by
a(u, ϕ) = (∇u,∇ϕ) for u ∈ X and ϕ ∈ Y.
Obviously, Assumption (i) holds with M = 1.Let us first consider the case 1 < r ≤ n′ for n ≥ 3 and 1 < r < 2 for n = 2.
In such a case, we have n ≤ r ′ < ∞ for n ≥ 3 and 2 < r < ∞ for n = 2,respectively. Then it follows from [17, Lemma 2.2 (i), Theorem B] that
Hence, we see that the decompositions (1.2) and (1.3) in Assumption (ii) are satis-fied.
By Proposition 2.1, there are n scalar functions πi ∈ Lr (�), i = 1, . . . , n suchthat
(∇ϕi ,∇�)− (πi , div �) = 0 for all � ∈ C∞0 (�)
n, i = 1, . . . , n.
Generalized Lax-Milgram theorem
By [17, Theorem 3.1] and the standard regularity theorem for the Stokes equations(see e.g., Cattabriga [6]), it holds that
ϕi ∈ C∞(�)n ∩ H1,q0,σ (�), πi ∈ C∞(�) ∩ Lq(�) for all q > n′, i = 1, . . . , n,
and that {ϕi , πi }i=1,...,n satisfy (2.1) in the usual classical sense.Next, we shall show that such {ϕi , πi }i=1,...,n yield the weak solution u ∈
H1,r0,σ (�) of (2.7) provided the condition (2.2) is satisfied. For that purpose, we may
prove the variational inequalities (1.4) and (1.5). It follows from Kozono-Sohr [17,Theorem 3.3] and Borchers-Miyakawa [3, Theorem 3.5] that
‖∇u‖r ≤ C supϕ∈H1,r ′
0,σ (�)
|(∇u,∇ϕ)|‖∇ϕ‖r ′
for all u ∈ H1,r0,σ (�)
with C = C(�, r, n), which implies (1.4). The proof of (1.5) is done by contradic-tion. Suppose that (1.5) fails. Then there exists a sequence {φm}∞m=1 ⊂ H1,r ′
0,σ (�)
such that
‖∇φm‖r ′ ≡ 1, m = 1, 2, . . . , (2.10)
εm ≡ supu∈H1,r
0,σ (�)
|(∇u,∇φm)|‖∇u‖r
→ 0, (φm, ϕi ) → 0 i = 1, 2, . . . , n as m → ∞.
(2.11)
By the similar argument to the proof of Borchers-Miyakawa [3, Theorem 3.5],it holds that for every m = 1, 2, . . ., there is a function pm ∈ Lr ′
(�) such that−�φm + ∇ pm ∈ H−1,r ′
(�)n satisfying
〈−�φm + ∇ pm, u〉 = (∇u,∇φm) for all u ∈ H1,r0,σ (�), (2.12)
‖ −�φm + ∇ pm‖H−1,r ′ = εm,
where 〈·, ·〉 denotes the duality between H−1,r ′(�)n and H1,r
for all m = 1, 2, . . . with C = C(�, n, r), we see that the sequence {pm}∞m=1 isbounded in Lr ′
(�). On the other hand, it follows from Kozono-Sohr [17, Lemma3.2] that
H. Kozono, T. Yanagisawa
‖∇φm − ∇φk‖r ′ + ‖pm − pk‖r ′ ≤ C
⎛⎜⎝εm + εk + ‖φm − φk‖Lr ′
(�R)
+‖pm − pk‖H−1,r ′(�R)
+
∣∣∣∣∣∣∣∫�R
ψ(x)(pm(x)− pk(x))dx
∣∣∣∣∣∣∣
⎞⎟⎠
for all m, k = 1, 2, . . . with C = C(�, n, r, R), where �R = � ∩ BR with∂� ⊂ BR(BR ; the ball centered at the origin with the radius R) and ψ ∈ C∞
0 (BR)
is the cut-off function with the property that ψ(x) = 1 in the neighbourhood of∂�. Since the embeddings H1,r ′
0 (�) ⊂ Lr ′(�R) and Lr ′
(�) ⊂ H−1,r ′(�R) are
both compact, we see from (2.11) and the above estimate that there exist subse-quences of {φm}∞m=1 and {pm}∞m=1, which we denote by themselves for simplicity,
and functions φ ∈ H1,r ′0 (�), p ∈ Lr ′
(�) such that
∇φm → ∇φ in Lr ′(�)n
2, pm → p in Lr ′
(�) as m → ∞. (2.13)
Now letting m → ∞ in (2.12), we have by (2.11) and (2.13) that
−�φ + ∇ p = 0 in H−1,r ′(�), div φ = 0 in �,
which implies that φ ∈ NY . Since (φ, ϕi ) = 0 for all i = 1, . . . ,m, it holds thatφ = 0. Then from (2.13) it follows that ∇φm → 0 in Lr ′
(�)n2, which contradicts
(2.10).Since all hypotheses of the Assumption (i), (ii) and (iii) are fulfilled, it follows
from Theorem 1.1 that there exists a solution u ∈ H1,r0,σ (�) of (2.7) with the estimate
‖∇u‖r ≤ C(‖∇F‖r + ‖g‖r ) provided
(F + ∇w,∇ϕi ) = 0, i = 1, 2, . . . , n. (2.14)
Indeed, we see that the above relation (2.14) is equivalent to (2.2) because it holdsthat
(∇w,∇ϕi ) = −(g, πi ), i = 1, 2, . . . , n. (2.15)
The identity (2.15) is formally obtained from (2.1) and integration by parts. For read-ers’ convenience, we give here its proof. Let us first consider the case n ≥ 3. Since∇ϕi ∈ Lq(�)n
2, pi ∈ Lq(�) for all q > n′, i = 1, . . . , n and sincew ∈ L
nrn−r (�)n
(see [17, Lemma 2.2 (ii)]), by taking q = n we have w ⊗ ∇ϕi ∈ Lr (�)n2, piw ∈
Lr (�)n , i = 1, . . . , n. Since 1 < r ≤ n′ for n ≥ 3, it follows from Kozono [16,Lemma 2.1] that there is a sequence R1 < R2 < · · · < Rm < Rm+1 < · · · → ∞such that ∫
|x |=Rm
(|w ⊗ ∇ϕi | + |πiw|)d S → 0 as m → ∞, (2.16)
Generalized Lax-Milgram theorem
where d S denotes the surface element of the sphere. Hence by (2.1) and integrationby parts, we have
(∇w,∇ϕi )
= limm→∞
∫�∩BRm
∇w · ∇ϕi ax
= limm→∞
⎛⎜⎝−
∫�∩|x |=Rm
w ·�ϕi dx +∫
|x |=Rm
w · ν · ∇ϕi d S
⎞⎟⎠
= limm→∞
⎛⎜⎝
∫�∩|x |=Rm
w · ∇πi dx +∫
|x |=Rm
w · ν · ∇ϕi d S
⎞⎟⎠
= limm→∞
⎛⎜⎝−
∫�∩|x |=Rm
πi div wdx +∫
|x=Rm |πiw · νd S +
∫|x |=Rm
w · ν · ∇ϕi d S
⎞⎟⎠
= −(g, πi ), i = 1, 2, . . . ,
where ν is the unit outer normal to the sphere |x | = Rm . This implies (2.15) forn ≥ 3.
In the case n = 2, the above argument is not directly applicable since ∇ϕi /∈L2(�)2
2, πi /∈ L2(�), i = 1, 2. However, it follows from Chang-Finn [7, Theorem
1] that
∂ϕi
∂x j− ci j
|x | ∈ L2(�)2, πi − ci
|x | ∈ L2(�), i, j = 1, 2, (2.17)
with some constants {ci j }i, j=1,2 and {ci }i=1,2 in R. Since ∇w ∈ Lr (�)2 withw|∂� = 0 and since 1 < r < 2, we have by the Hardy inequality that
w
|x | ∈ Lr (�)2.
Hence the limit (2.16) remains true even for n = 2 and we obtain the identity (2.15)for all n ≥ 2. This proves Theorem 2.1 (1) (i) and (ii).
We next show (iii), i.e., the case for n ≤ r < ∞ for n ≥ 3 and 2 < r < ∞ forn = 2. In such a case, we have by duality that
NX = Span {ϕ1, . . . , ϕn}, NY = {0}with the variational inequalities
‖∇u‖r ≤ C supϕ∈H1,r ′
0,σ (�)
|(∇u,∇ϕ)|‖∇ϕ‖r ′
+ Cn∑
i=1
|(u, ϕi )| for all u ∈ H1,r0,σ (�),
‖∇ϕ‖r ′ ≤ C supu∈H1,r
0,σ (�)
|(∇u,∇ϕ)|‖∇u‖r
for all ϕ ∈ H1,r ′0,σ (�),
H. Kozono, T. Yanagisawa
where C = C(�, n, r). Then the desired result with such a uniqueness assertionas in (2.6) follows from Theorem 1.1.
(2) Finally, we deal with the case n′ < r < n for n ≥ 3 and r = 2 for n = 2.In this case, we have also n′ < r ′ < n for n ≥ 3 and r ′ = 2 for n = 2. Again byduality, it holds that
NX = NY = {0}with the variational inequalities
‖∇u‖r ≤ C supϕ∈H1,r ′
0,σ (�)
|(∇u,∇ϕ)|‖∇ϕ‖r ′
, for all u ∈ H1,r0,σ (�)
‖∇ϕ‖r ′ ≤ C supu∈H1,r
0,σ (�)
|(∇u,∇ϕ)|‖∇u‖r
for all ϕ ∈ H1,r ′0,σ (�),
where C = C(�, n, r). Then the desired unique existence result without any restric-tion on the given F ∈ Lr (�)n
2and g ∈ Lr (�) is a consequence of Theorem 1.1.
This proves Theorem 2.1.
2.2. Lr -Helmholtz-Weyl decomposition in 3D bounded domains
In this subsection, we consider the Helmholtz-Weyl decomposition of vector fieldsin Lr in three dimensional bounded domains. Although such a decomposition hasbeen shown by our previous paper [19], we give here a more simplified proof whichis based on Theorem 1.1.
Let � be a bounded domain in R3 with the smooth boundary ∂�. We first
introduce two function spaces Xhar and Vhar of harmonic vector fields in � withdifferent boundary conditions on ∂� defined by
Xhar ≡ {h ∈ C∞(�)3; div h = 0, rot h = 0 in �, h · ν|∂� = 0},Vhar ≡ {h ∈ C∞(�)3; div h = 0, rot h = 0 in �, h × ν|∂� = 0},
where ν denotes the unit outer normal to ∂�. Corresponding to the boundary con-ditions of Xhar and Vhar , for 1 < r < ∞ we define the solenoidal vector fields Xr
σ
and V rσ by
Xrσ ≡ {w ∈ W 1,r (�)3; div w = 0 in �, w · ν|∂� = 0},
V rσ ≡ {w ∈ W 1,r (�)3; div w = 0 in �, w × ν|∂� = 0}.
Then we have the following decompositions for all vector fields in Lr (�)3, 1 <r < ∞.
Theorem 2.2. (1) The spaces Xhar and Vhar are both finite dimensional vectorspaces.(2) Let 1 < r < ∞. For every u ∈ Lr (�)3, there exist h ∈ Xhar , w ∈ V r
σ andp ∈ W 1,r (�) such that u can be represented as
u = h + rot w + ∇ p (2.18)
Generalized Lax-Milgram theorem
with the estimates
‖h‖r + ‖w‖W 1,r + ‖p‖W 1,r ≤ C‖u‖r , (2.19)
where C = C(�, r) is independent of u. If there is another triplet {h, w, p} ∈Xhar × V r
σ × W 1,r (�) such that u = h + rot w + ∇ p, then it holds that
h = h, rot w = rot w, ∇ p = ∇ p, (2.20)
which implies that the representation (2.18) of u is unique.(3) Let 1 < r < ∞. For every u ∈ Lr (�)3, there exist k ∈ Vhar , v ∈ Xr
σ andq ∈ W 1,r
0 (�) such that u can be represented as
u = k + rot v + ∇q (2.21)
with the estimates
‖k‖r + ‖v‖W 1,r + ‖q‖W 1,r0
≤ C‖u‖r , (2.22)
where C = C(�, r) is independent of u. If there is another triplet {k, v, q} ∈Vhar × Xr
σ × W 1,r0 (�) such that u = k + rot v + ∇q , then it holds that
k = k, rot v = rot v, ∇q = ∇q, (2.23)
which implies that the representation (2.21) of u is unique.
Remark. Theorem 2.2 may be regarded as the de Rham-Hodge-Kodaira decompo-sition of general vector fields in Lr on �. There are several corresponding resultsto ours such as von Wahl [32,33] and Schwarz [28]. The numbers of dimension ofXhar and Vhar are called the first and the second Betti numbers, respectively, whichare closely related to the topological invariance of the geometry of the domain �.See e.g., [19, Remark 1] and von Wahl [32,33].
Proof of Theorem 2.2. (1) It follows from Duvaut-Lions [8, Chap. VII, Theorem6.1], Bourguignon-Brezis [5] and [19, Lemmata 3.5, 4.4] that
for all u ∈ W s,r (�)3 with u · ν|∂� = 0, or u × ν|∂� = 0, where C = C(�, s, r).Hence from (2.24) and the Rellich theorem, we see that the unit spheres of Xhar andVhar are both compact in Lr (�)3, which necessarily implies that dim.Xhar < ∞and dim.Vhar < ∞.(2) For the given u ∈ Lr (�)3, its scalar potential p ∈ W 1,r (�) and vector po-tential w ∈ V r
σ are determined by the following weak formulation of the quadraticforms;
(∇ p,∇ϕ) = (u,∇ϕ) for all ϕ ∈ W 1,r ′(�) (2.25)
H. Kozono, T. Yanagisawa
with the estimate
‖∇ p‖r + ‖p‖r ≤ C‖u‖r . (2.26)
(rot w, rot �) = (u, rot �) for all � ∈ V r ′σ (2.27)
with the estimate
‖∇w‖r + ‖w‖r ≤ C‖u‖r . (2.28)
Here C = C(�, r) is independent of u. To prove existence of p satisfying (2.25)and (2.26), let us take X = W 1,r (�), Y = W 1,r ′
(�) and define the bilinear forma(·, ·) on X × Y by
a(p, ϕ) = (∇ p,∇ϕ), p ∈ X, ϕ ∈ Y.
Obviously, the bound (1.1) in Assumption (i) is fulfilled with M = 1. Since NX =NY = R, we see that the direct sums of (1.2) and (1.3) hold true. The variationalinequalities (1.4) and (1.5) in Assumption (iii) follow from Simader-Sohr [30]which states
‖∇ p‖γ + ‖p‖γ ≤ C
⎛⎝ supϕ∈W 1,γ ′
|(∇ p,∇ϕ)|‖∇ϕ‖γ ′ + ‖ϕ‖γ ′
+∣∣∣∣∣∣∫�
p(x)dx
∣∣∣∣∣∣⎞⎠, 1 < γ < ∞
for all p ∈ W 1,γ (�) with C = C(�, γ ). Hence all hypotheses (i), (ii) and (iii) inthe Assumption are satisfied.
Now for every u ∈ Lr (�)3, we define Fu ∈ Y ∗ = W 1,r ′(�)∗ by
Fu(ϕ) = (u,∇ϕ) for ϕ ∈ Y.
Since NY = R, it is easy to see that Fu ∈ N⊥Y , i.e., Fu(φ) = 0 for all φ ∈ NY . Since
‖Fu‖Y ∗ ≤ ‖u‖r , it follows from Theorem 1.1 that there exists a p ∈ W 1,r (�) suchthat (2.25) and (2.26) are satisfied.
We next show the existence of w ∈ V rσ satisfying (2.27) and (2.28). For that
purpose, we introduce X = V rσ , Y = V r ′
σ and define the bilinear form a(·, ·) onX × Y by
a(w,�) = (rot w, rot �) for w ∈ X, � ∈ Y.
Obviously, the bound (1.1) in Assumption (i) is fulfilled with M = 1. By (2.24) andthe Sobolev embedding, it holds that NX = NY = Vhar , and hence by the above(1), we see that the direct sums (1.2) and (1.3) are true. Moreover, it follows from[19, Lemma 4.1] that
‖∇w‖γ + ‖w‖γ ≤ C
⎛⎝ sup�∈V γ ′
σ
|(rot w, rot �)|‖∇�‖γ ′ + ‖�‖γ ′
+L∑
i=1
|(w,�i )|⎞⎠, 1 < γ < ∞
Generalized Lax-Milgram theorem
for all w ∈ V γσ with C = C(�, γ ), where {�1, . . . , �L} is the orthogonal basis in
L2(�)3 of Vhar . Then we see that all hypotheses (i), (ii) and (iii) in the Assumptionare satisfied. For every u ∈ Lr (�)3, we define Fu ∈ Y ∗ by
Fu(�) ≡ (u, rot �) for � ∈ Y.
Since rot � = 0 for all � ∈ NY , we have Fu ∈ N⊥Y with the estimate ‖Fu‖Y ∗ ≤
‖u‖r . Hence, it follows from Theorem 1.1 that there exists a w ∈ V rσ such that
(2.27) and (2.28) hold.It remains to prove that h ≡ u − ∇ p − rot w ∈ Xhar together with such a
uniqueness assertion as (2.20). To this end, let us recall the following generalizedStokes integral formula;
(U,∇ϕ)+ (div U, ϕ) = 〈γνU, γ0ϕ〉∂� (2.29)
for U ∈ Lr (�)3 with div U ∈ Lr (�) and ϕ ∈ W 1,r ′(�) and
(V, rot �)− (rot V,�) = 〈τνV, γ0�〉∂� (2.30)
for V ∈ Lr (�)3 with rot V ∈ Lr (�)3 and� ∈ W 1,r ′(�)3 where γ0 : W 1,r ′
(�) →W 1−1/r ′,r ′
(∂�) denotes the usual trace operator and 〈·, ·〉∂� represents the dualitypairing between W 1−1/r ′,r ′
(∂�)∗ and W 1−1/r ′,r ′(∂�). Note that γνU ∈ W 1−1/r ′,r ′
(∂�)∗ is the generalized normal component of U on the boundary ∂� since it holdsthat γνU = U · ν|∂� for U ∈ C1(�)3. Similarly, τνV ∈ (W 1−1/r ′,r ′
(∂�)3)∗ isthe generalized tangential competent of V on the boundary ∂� since it holds thatτνV = V × ν|∂� for V ∈ C1(�)3.
We first show that
div h = 0 in the sense of distributions in � with γνh = 0. (2.31)
Indeed, by (2.25) it holds that div (u − ∇ p) = 0 in the sense of distributions in �.Taking U = u − ∇ p in (2.29), we have by (2.25) that
〈γν(u − ∇ p), γ0ϕ〉∂� = 0 for all ϕ ∈ W 1,r ′(�).
Since the trace operator γ0 : W 1,r ′(�) → W 1−1/r ′,r ′
(∂�) is surjective, this iden-tity implies that γν(u−∇ p) = 0. Sincew ∈ V r
σ , it holds that τνw = w×ν|∂� = 0,so by taking V = w, � = ∇ϕ in (2.30), we have
(rot w,∇ϕ) = (w, rot (∇ϕ))− 〈τνw, γ0(∇ϕ)〉∂� = 0 for all ϕ ∈ C∞(�).(2.32)
In particular, this yields that
div (rot w) = 0 in the sense of distributions in �.
On the other hand, since C∞(�) is dense in W 1,r ′(�), taking U = rot w in (2.29)
we have by (2.32) that
0 = (rot w,∇ϕ) = 〈γν(rot w), γ0ϕ〉∂� for all ϕ ∈ W 1,r ′(�).
H. Kozono, T. Yanagisawa
Again by surjectivity of the trace operator γ0 : W 1,r ′(�) → W 1−1/r ′,r ′
(∂�), theabove identity implies γν(rot w) = 0, and hence we obtain (2.31).
We next show that
rot h = 0 in the sense of distributions in �. (2.33)
By replacing r by r ′ and then taking ϕ = p and U = rot � in (2.29), we have
(∇ p, rot �) = −(p, div (rot �))+ 〈γ0 p, γν(rot �)〉∂� = 0 for all � ∈ C∞0 (�)
3,
which implies that rot (∇ p) = 0 in the sense of distributions in �. For every� ∈ C∞
0 (�)3, we consider � = �− ∇g with g ∈ C∞(�) satisfying
�g = div � in �, g = 0 on ∂�.
Since ∇g is parallel to ν on ∂�, we have div � = 0 in�,�× ν = 0 on ∂� whichmeans that � ∈ V r ′
σ with rot � = rot �. Hence it follows from (2.27) that
(u − rot w, rot �) = (u − rot w, rot �) = 0 for all � ∈ C∞0 (�)
3,
which implies that
rot (u − rot w) = 0 in the sense of distributions in �,
and we obtain (2.33). Now the fact that the function h ∈ Lr (�)3 with the properties(2.31) and (2.33), in fact, belongs to Xhar is a consequence of (2.24).
Concerning the unique representation of u such as (2.20), we have that
h = h =N∑
i=1
(u,�i )�i , (2.34)
where {�1, . . . , �N } is the orthogonal basis in L2(�)3 of Xhar . Notice that by(2.29) and (2.30) we have
(rot w,�i ) = (w, rot �i )− 〈τνw, γ0�i 〉∂� = 0,
(∇ p,�i ) = (p, div �i )+ 〈γν�i , γ0 p〉∂� = 0, i = 1, . . . , N .
By (2.34) it holds that ∇(p − p) = rot (w − w), and hence it follows from (2.29)that
(∇(p − p),∇ϕ) = (rot (w − w),∇ϕ)= (w − w, rot (∇ϕ))− 〈τν(w − w), γ0(∇ϕ)〉∂� = 0
for all ϕ ∈ C∞(�), which implies that �(p − p) = 0 in the sense of distributionsin�. Furthermore, taking U = ∇(p − p) in (2.29), we see that γν(∇(p − p) = 0,so it follows from (2.24) that p − p ∈ W 2,r (�). A well-known uniqueness resulton the harmonic function with the homogeneous Neumann condition states thatp − p ≡ const. on �, and we obtain that
∇ p = ∇ p, rot w = rot w,
Generalized Lax-Milgram theorem
from which and (2.34) the desired uniqueness (2.20) follows.(3) The proof of (2.21) is quite similar to that of (2.18). Indeed, as we have seen
in (2.25) – (2.28), for every given u ∈ Lr (�)3, the scalar potential q ∈ W 1,r0 (�) and
the vector potential v ∈ Xrσ are determined by the weak solutions of the following
quadratic forms;
(∇q,∇ψ) = (u,∇ψ) for all ψ ∈ W 1,r ′0 (�) (2.35)
with the estimate
‖∇q‖r + ‖q‖r ≤ C‖u‖r . (2.36)
(rot v, rot �) = (u, rot �) for all � ∈ Xr ′σ (2.37)
with the estimate
‖∇v‖r + ‖v‖r ≤ C‖u‖r . (2.38)
Here C = C(�, r) is independent of u. The existence of q ∈ W 1,r0 (�) satisfying
(2.35) and (2.36) follows from Theorem 1.1 with the aid of the variational inequality
‖∇q‖γ ≤ C supψ∈W 1,γ ′
0
|(∇q,∇ψ)|‖∇ψ‖γ ′
1 < γ < ∞,
where C = C(�, γ ). See, e.g., Simader-Sohr [31]. Concerning the existence ofv ∈ Xr
σ , by taking X = Xrσ , Y = Xr ′
σ and a(v,�) = (rot v, rot �) for v ∈ X and� ∈ Y in the Assumption, we have that NX = NY = Span.{�1, . . . , �N }. Thevariational inequalities (1.4) and (1.5) follows from [19, Lemma 2.1]
‖∇v‖γ + ‖v‖γ ≤ C
⎛⎝ sup�∈Xγ
′σ
|(rot v, rot �)|‖∇�‖γ ′ + ‖�‖γ ′
+N∑
i=1
|(v,�i )|⎞⎠ , 1 < γ < ∞
for all v ∈ Xγσ with C = C(�, γ ). Since it holds rot � = 0 for all � ∈ NY , wesee that Fu ∈ N⊥
Y , where Fu ∈ Y ∗ with u ∈ X is defined by Fu(�) ≡ (u, rot �)for � ∈ Y . Since ‖F‖Y ∗ ≤ ‖u‖r , it follows from Theorem 1.1 that there exists av ∈ Xr
σ satisfying (2.37) and (2.38).Now the proofs that k = u − ∇q − rot v ∈ Vhar and that unique representation
(2.23) are quite parallel to those of the above case (2), so we may omit them. Thisproves Theorem 2.2.
H. Kozono, T. Yanagisawa
2.3. Lr -de Rham-Hodge-Kodaira decomposition on Riemannian manifolds
In this subsection, we consider the de Rham-Hodge-Kodaira decomposition forgeneral l-forms in Lr on n-compact Riemannian manifolds with the boundary. Let(�, g) be a compact n-dimensional Riemannian manifold with smooth boundary∂�. We regard ∂� as a C∞-submanifold of �. Then there is a canonical inclu-sion
∧Tx (∂�) ↪→ ∧
Tx �, where Tx M is the tangent space of the manifold M atx ∈ M , and where
∧Tx M ≡ ⊕n
l=0
∧l Tx M . Notice that∧l Tx M is the l-exterior
product of Tx M . For each x ∈ ∂�, let us denote by νx the vector in Tx � whichis orthogonal to Tx (∂�) and oriented toward the exterior of �, and which has thenorm 1. For every l-form u on �, i.e., u ∈ ∧l
(T �), we define its tangential partτu and its normal part νu as
τu = ν�(ν ∧ u), νu = ν�u, (2.39)
where ν� : ∧l(T �) → ∧l−1
(T �), l = 1, . . . , n, is the interior product definedby
(T �), l = 0, 1, . . . , n−1, the exterior deriv-ative and by ∗ : ∧l
(T �) → ∧n−l(T �), l = 0, 1, . . . , n, the Hodge star operator,
respectively. We define the co-differential operator δ : ∧l(T �) → ∧l−1
(T �),l = 1, . . . , n, by δ = (−1)n+1 ∗ d ∗ χn , where χu = (−1)lu for u ∈ ∧l
(T �).It is known that
∧l(T �), l = 0, 1, . . . , n, has a Banach structure with the scalar
product (·, ·) such as
(u, v) ≡∫�
u ∧ ∗v, for u, v ∈∧l
(T �). (2.40)
Based on this scalar product on∧l(T �), we may define the Lebesgue space Lr (�)l
and the Sobolev space W s,r (�)l . See, e.g., Morrey [24].We next consider the generalized Stokes formula on (�, g) corresponding to
(2.29) and (2.30). Let us introduce two spaces Erd(�)
l−1 and Er ′δ (�)
l for 1 < r <∞, l = 1, . . . , n by
Erd(�)
l−1 ≡{
u ∈ Lr (�)l−1; du ∈ Lr (�)l},
Er ′δ (�)
l ≡{v ∈ Lr ′
(�)l; δv ∈ Lr ′(�)l−1
}. (2.41)
Generalized Lax-Milgram theorem
Then the boundary operators τ and ν defined by (2.39) can be extended uniquelyas continuous linear operators
τ : u ∈ Erd(�)
l−1(�) → τu ∈ W −1/r,r (∂�)l−1 ≡ (W 1−1/r ′,r ′(∂�)l−1)∗,
(2.42)
ν : v ∈ Er ′δ (�)
l → νv ∈ W −1/r ′,r ′(∂�)l−1 ≡ (W 1−1/r,r (∂�)l−1)∗, (2.43)
with the generalized Stokes formula
(du, v)− (u, δv) = 〈τu, νv〉∂�, l = 1, . . . , n (2.44)
for all {u, v} ∈ Erd(�)
l−1 × W 1,r ′(�)l and {u, v} ∈ W 1,r (�)l−1 × Er ′
δ (�)l , where
〈·, ·〉∂� denotes the duality pairing between W −1/r,r (∂�)l−1 and W 1−1/r ′,r ′(∂�)l−1
and between W 1−1/r,r (∂�)l−1 and W −1/r ′,r ′(∂�)l−1, respectively. For details, we
refer to Morrey [24, Lemma 7.5.3] and Georgescu [14, Theorem 4.1.8]. Corre-sponding to Xr
σ and V rσ in Subsection 2.2, for 1 < r < ∞ and l = 1, . . . , n − 1,
we define two spaces Xrd(�)
l+1 and V rδ (�)
l−1 by
Xrd(�)
l+1 ≡ {α ∈ W 1,r (�)l+1; dα = 0 in �, να = 0 on ∂�},V rδ (�)
l−1 ≡ {β ∈ W 1,r (�)l−1; δβ = 0 in �, τβ = 0 on ∂�}.Our result on the Lr -decomposition of l-differential forms on (�, g) now reads
Theorem 2.3. Let (�, g) be a compact n-dimensional Riemannian manifold withsmooth boundary ∂�. Let 1 < r < ∞ and l = 1, · · · , n−1. For everyω ∈ Lr (�)l ,there exist α ∈ Xr
d(�)l+1, β ∈ V r
δ (�)l−1 and h ∈ C∞(�)l ∩ Lr (�)l with dh = 0,
δh = 0 such that
ω = δα + dβ + h (2.45)
with the estimate
‖α‖W 1,r + ‖β‖W 1,r + ‖h‖r ≤ C‖ω‖r , (2.46)
where C = C(�, n, r, l). If there are α′ ∈ Xrd(�)
l+1, β ′ ∈ V rδ (�)
l−1 and h′ ∈C∞(�)l ∩ Lr (�)l with dh′ = 0, δh′ = 0 such that ω = δα′ + dβ ′ + h′, then itholds that
δα = δα′, dβ = dβ ′, h = h′. (2.47)
If ω ∈ W s,r (�)l for s ≥ 1, then we have α ∈ Xrd(�)
l+1 ∩ W s+1,r (�)l+1, β ∈V rδ (�)
l−1 ∩ W s+1,r (�)l−1 and h ∈ C∞(�) ∩ W s,r (�)l with the estimate
The proof of Theorem 2.3 is based on the following key lemma.
H. Kozono, T. Yanagisawa
Lemma 2.1. Let (�, g) be a compact n-dimensional Riemannian manifold withsmooth boundary ∂�. Let 1 < r < ∞ and l = 1, . . . , n − 1.(1) For every ω ∈ Lr (�)l , there exists α ∈ Xr
d(�)l+1 such that
(δα, δ�) = (ω, δ�) for all � ∈ Xr ′d (�)
l+1 (2.49)
with the estimate
‖α‖W 1,r ≤ C‖ω‖r , (2.50)
where C = C(�, n, r, l). In addition, if ω ∈ W s,r (�)l for s ≥ 1, then α can bechoose as α ∈ Xr
d(�)l+1 ∩ W s+1,r (�)l+1 with the estimate
‖α‖W s+1,r ≤ C‖ω‖W s,r , (2.51)
where C = C(�, n, s, r, l).(2) For every ω ∈ Lr (�)l , there exists β ∈ V r
δ (�)l−1 such that
(dβ, dψ) = (ω, dψ) for all ψ ∈ V r ′δ (�)
l−1 (2.52)
with the estimate
‖β‖W 1,r ≤ C‖ω‖r , (2.53)
where C = C(�, n, r, l). In addition, if ω ∈ W s,r (�)l for s ≥ 1, then β can bechoose as β ∈ V r
δ (�)l−1 ∩ W s+1,r (�)l−1 with the estimate
‖β‖W s+1,r ≤ C‖ω‖W s,r , (2.54)
where C = C(�, n, s, r, l).
Proof. (1) Similarly to the proof of Theorem 2.2, we make use of Theorem 1.1 bytaking X ≡ Xr
d(�)l+1, Y ≡ Xr ′
d (�)l+1 and a(·, ·) : X × Y → C as
a(α,�) ≡ (δα, δ�) for α ∈ X, � ∈ Y.
Obviously, the estimate (1.1) holds with M = 1. Since it holds
for all u ∈ W k,r (�)l+1(see e.g., Georgesgue [14, Corollary 4.2.3]), we see that
NX = NY = Xhar (�)l+1 ≡ {H ∈ C∞(�)l+1; d H = 0, δH = 0 in �,
νH = 0 on ∂�}with dim.Xl+1
har (�) < ∞. Hence, the direct sums (1.2) and (1.3) hold. The varia-tional inequalities (1.4) and (1.5) are a consequence of
‖α‖W 1,γ ≤ C
⎛⎝ sup�∈Xγ
′d (�)
l+1
|(δα, δ�)|‖�‖W 1,γ ′
+N∑
i=1
|(α,�i )|⎞⎠ , 1 < γ < ∞
(2.56)
Generalized Lax-Milgram theorem
for all α ∈ Xγd (�)l+1 with C = C(�, n, γ ), where {�1, . . . , �N } is the orthogo-
nal basis of Xhar (�)l+1. Validity of (2.56) follows from the estimate of the elliptic
system of boundary-value problems. For details, we refer to [20,21].For every ω ∈ Lr (�)l , we define Fω ∈ Y ∗ by
Fω(�) = (ω, δ�) for � ∈ Y.
Obviously, it holds that ‖F‖Y ∗ ≤ ‖ω‖r . Since δφ = 0 for all φ ∈ NY , wehave Fω ∈ N⊥
Y . Now, we see that the desired result (2.49) with (2.50) follows fromTheorem 1.1.
In addition, if ω ∈ W s,r (�)l+1 for s ≥ 1, then we see by (2.49) that α ∈Xr
d(�)l+1 is characterized by the equations
⎧⎨⎩�α = dω in �,dα = 0 in �,να = 0 on ∂�.
(2.57)
Since (2.57) is an elliptic system satisfying the complementing condition in thesense of Agmon-Douglis-Nirenberg [2, Theorem 10.5], we obtain (2.51) from thewell-known regularity theorem.
(2) The proof of (2.52)–(2.54) is quite similar to that of (2.49)–(2.51). In theAssumption, we may take X ≡ V r
for all � ∈ Cl+10 (�)l+1. We choose q ∈ C∞(�)l+2 so that⎧⎨
⎩�q = d� in �,dq = 0 in �,νq = 0 on ∂�.
(2.63)
Existence of such q as in (2.63) follows from Lemma 2.1 (1) with ω and l replacedby� and l + 1, respectively with the aid of regularity theorem for elliptic systems.Taking � = � − δq, we have � ∈ C∞(�)l+1 with d� = 0 in � and ν� = 0on ∂�. Notice that ν(δq) = 0 on ∂� since νq = 0 on ∂� (see Morrey [24, p.311 (7.7.6)] and [20, Proposition 4.2 (2)]). This means that � ∈ Xr ′
d (�)l+1 with
δ� = δ�. Hence it follows from (2.49) and (2.62) that
(h, δ�) = (ω − δα, δ�) = 0 for all � ∈ C∞0 (�)
l+1,
which implies that dh = 0 in the sense of distributions in �.Similarly it follows from (2.44) that
l−1. We choose p ∈ C∞(�)l−2 so that⎧⎨⎩�p = δφ in �,δp = 0 in �,τp = 0 on ∂�.
(2.65)
Generalized Lax-Milgram theorem
Existence of such p as in (2.65) follows from Lemma 2.1 (2) with ω and l replacedby φ and l − 1, respectively with the aid of the regularity theorem for elliptic sys-tems. Taking ψ ≡ φ− dp, we have ψ ∈ C∞(�)l−1 with δψ = 0 in�, τψ = 0 on∂�. Notice that τdp|∂� = 0 since τp|∂� = 0(see [20, Proposition 4.2(1)]). Thismeans thatψ ∈ V r ′
δ (�)l−1 with dψ = dφ. Hence it follows from (2.52) and (2.64)
that
(h, dφ) = (ω − dβ, dψ) = 0 for all φ ∈ C∞0 (�)
l−1,
which implies that δh = 0 in the sense of distributions in �. Note that the factthat h ∈ C∞(�) ∩ Lr (�)l follows from the interior regularity theorem for theLaplace operator. Obliviously, the bound (2.46) and the higher regularity (2.28) areconsequences of (2.50), (2.53) and (2.51), (2.54), respectively.
Finally, it remains to show the unique representation of ω as in (2.47). Supposethat ω has another representation such as
ω = δα′ + dβ ′ + h′
for α′ ∈ Xrd(�)
l+1, β ′ ∈ V rδ (�)
l−1 and h ∈ C∞(�)l ∩ Lr (�)l with dh′ = 0,δh′ = 0. Since δ(α − α′) = d(β ′ − β)+ h′ − h and since dh = dh′ = 0, we haveby (2.44) that
l+1, which yields that�(α−α′) = 0 in the sense of distributionsin �. Since ν(α − α′) = 0 on ∂�, the regularity theorem for the boundary valueproblem for harmonic forms states that α−α′ ∈ C∞(�)l+1. Since d(α−α′) = 0,it follows from (2.44) that
Hence we have d(β − β ′) = h′ − h, and it holds that
(d(β − β ′), dφ) = (h′ − h, dφ) = 0 for all φ ∈ C∞0 (�)
l−1.
Since τ(β − β ′) = 0 on ∂�, again by the regularity theorem for the boundaryvalue problem for harmonic forms, we see that β − β ′ ∈ C∞(�)l−1 and satisfies
H. Kozono, T. Yanagisawa
�(β − β ′) = 0 in the usual classical sense in �. Since δ(β − β ′) = 0, it followsfrom (2.44) that
Now we see that the desired unique representation (2.47) is a consequence of (2.66)and (2.67). This completes the proof of Theorem 2.3.
Remark. Theorem 2.3 covers Theorem 2.2. Indeed, if n = 3, then the vector fieldson � can be identified with 1 or 2-differential forms on it. The decomposition(2.18) for u is a consequence of (2.45) for l = 2 with ω identified with u. Thescalar function p ∈ W 1,r (�) determined by (2.25) is given by α ∈ Xr
d(�)3 which
is associated with (2.49) and (2.57) for l = 2. The vector potential w ∈ V rσ deter-
mined by (2.27) is given by β ∈ V rδ (�)
1 which is associated with (2.52) and (2.60)for l = 2. Since the equation (2.57) yields necessarily that γν(∇ p − u) = 0, theboundary condition of h on ∂� in (2.45) is uniquely determined in such a way thath · ν|∂� = 0. This is the reason why the boundary condition of h in (2.18) is fixedindependently of the given u ∈ Lr (�)3.
Similarly, the decomposition (2.21) for u is a consequence of (2.45) for l = 1with ω identified with u. The scalar function q ∈ W 1,r
0 (�) determined by (2.35)is given by β ∈ V r
δ (�)0 which is associated with (2.52) and (2.60) for l = 1. The
vector potential v ∈ Xrσ determined by (2.37) is given by α ∈ Xr
d(�)2 which is
associated with (2.49) and (2.57) for l = 1. Since the Eq. (2.57) yields necessarilythat τν(rot v − u) = 0, the boundary condition of h on ∂� in (2.45) is uniquelydetermined in such a way that k × ν|∂� = 0. This is the reason why the boundarycondition of k in (2.21) is fixed independently of the given u ∈ Lr (�)3.
Acknowledgments. The authors would like to express their sincere thanks to Prof. JochimNaumann and Prof. Paolo Maremonti for their valuable advices.
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