Dedicated to Professor Philippe G. Ciarlet
on his 70th birthday
ON POINCARÉ AND DE RHAM'S THEOREMS
SORIN MARDARE
We prove Poincaré's theorem under general assumptions on the data. Then we
derive the regularity of the solution from a result of Borchers and Sohr [6]. Finally,
we give an elementary proof of the de Rham's theorem in the case of 1-dimensional
ows on the Euclidean space by applying the techniques introduced in the proof
of Poincaré's theorem.
AMS 2000 Subject Classication: Primary 35N10; Secondary 35D05, 58A12.
Key words: Poincaré's lemma, de Rham's theorem, de Rham cohomology.
1. INTRODUCTION
In dierential geometry, the theorems of Poincaré and de Rham give a
characterization of the de Rham cohomology groups. Poincaré's theorem states
that de Rham's cohomology groups of a contractible manifold coincide with
those of a single point. If one is only interested in
cohomology group
1
HdR
1-forms i.e.
in the de Rham
the simple-connectedness of the manifold is enough.
For arbitrary smooth manifolds, de Rham's theorem states that de Rham's
cohomology groups are isomorphic with the singular cohomology groups.
In a partial dierential equations setting, these two theorems solve an
over-determined system of linear partial dierential equations of order one.
The proof of these theorems are by no means simple, especially if they are
stated in their most general setting, i.e., when the data are only distributions.
To this day, there is no proof, to the best knowledge of the author, of the
Poincaré theorem in the general case where the data are distributions in simplyconnected domains. However, we wish to emphasize that the Poincaré theorem
was already proved by Schwartz [16] in the case where the domain is the whole
Euclidean space
Rn and by several authors (see Section 3) in the case where the
data are suciently smooth. By contrast, de Rham's theorem was proved in
its whole generality in [15]. However, an important prerequisite about chains
and ows on dierential manifolds is needed in order to understand the proof
in [15].
REV. ROUMAINE MATH. PURES APPL.,
53
(2008), 56, 523541
524
Sorin Mardare
2
The most relevant case for the partial dierential equations theory is when
the data are
1-forms dened on Euclidean spaces.
For instance, de Rham's the-
orem is used in uid mechanics theory in order to nd the pressure component
of the unknown of the Stokes or Navier-Stokes equations, once the velocity
eld is found.
In this paper, we restrict the presentation to the case of
on Euclidean spaces.
1-forms
dened
More specically, the manifold will be an open subset
of the Euclidean space
Rn .
First, we give elementary proofs of Poincaré and
de Rham's theorems in this setting (Theorems 2.1 and 4.1). Then we study
the regularity of the solution to the Poincaré problem in the setting of Sobolev
spaces. The regularity of the de Rham problem was already studied by Amrouche and Girault in [2]. We will use the same regularity result as in [2] (rst
proved by Borchers and Sohr in [6]) in conjunction with the distributional
Poincaré theorem (Theorem 2.1) in order to derive the regularity of the solution
to the Poincaré problem. Since we want to make our proof as elementary as
possible, in all that follows we will use the terminology of the partial dierential
equations theory. For convenience of the readers, we also restate our results in
the terminology of dierential geometry in remarks following each theorem.
We end this section by recalling two well known theorems in the distribution theory that will be used in the next sections. Throughout the paper,
Ω ⊂ Rn
D(Ω) the space of indenitely dierentiable funcΩ and D0 (Ω) the space of all distribu0
tions over Ω. The gradient of a distribution u is denoted ∇u. If u, v ∈ D (Ω)
and ω ⊂ Ω, we say that u = v in ω if hu, ϕi = hv, ϕi for all ϕ ∈ D(ω) (the
inclusion D(ω) ⊂ D(Ω) is dened by extending the functions in D(ω) by zero).
denotes an open set,
tions with compact support contained in
The rst result below states that, in a connected open set, a distribution
whose gradient vanishes is constant (see also Remark 2.2).
Let Ω ⊂ Rn be a connected open set and let u ∈ D0 (Ω)
such that ∇u = 0 in (D0 (Ω))n , i.e.,
Theorem 1.1.
D
∂ϕ E
u,
=0
∂xi
for all ϕ ∈ D(Ω) and i ∈ {1, 2, . . . , n}.
Then there exists a constant C ∈ R such that u = C .
The second result shows how to reconstruct a distribution over
its restriction to smaller domains covering
Ω.
Ω
from
It has been rst introduced by
Schwartz [16] under the name principe du recollement des morceaux.
Let {Ωi }i∈I be a family of open sets in Rn and Ω :=
i )}i∈I be a family of distributions such that Ti = Tj in
i∈I Ωi . Let {Ti ∈
Ωi ∩ Ωj for all i, j ∈ I satisfying Ωi ∩ Ωj 6= ∅.
Then there exists one and only one distribution T ∈ D0 (Ω) such that
(1.1)
T = Ti on Ωi for all i ∈ I.
Theorem 1.2.
S
D0 (Ω
3
On Poincaré and de Rham's theorems
The idea of the
proof is as follows.
hT, ϕi :=
(1.2)
For any
m
X
ϕ ∈ D(Ω)
525
one denes
hTij , θj ϕi,
j=1
(θj )m
a partition of unity subordinated to a covering
j=1 form
S
Ω for some m ∈ N∗ and i1 , . . . , im ∈ I since
supp ϕ (supp ϕ ⊂ m
j=1
S ij
supp ϕ is compact and supp ϕ ⊂ i∈I Ωi ). Then one proves that the denition
(1.2) is independent of the choice of the couple ((Ωij ), (θj )) satisfying the above
properties. In turn, this independence allows to prove that T is a distribution
over Ω and that this distribution satises (1.1).
where the functions
(Ωij )m
j=1 of
2. POINCARÉ'S THEOREM
Poincaré's theorem, also known in the literature as Poincaré's lemma,
states that on a contractible manifold of dimension
is exact if and only if it is closed.
1-forms
Rn .
on an open subset of the Euclidean space
that the domain
Ω
n,
any
k -form, 1 ≤ k ≤ n,
We consider here the particular case of
The main point here is
can be any simply-connected open set of
Rn ,
regardless of
the regularity of its boundary. The main result of this section is as follows.
Let Ω be a simply-connected open subset of Rn and let
f1 , f2 , . . . , fn ∈ D0 (Ω) such that
Theorem 2.1.
∂fj
∂fi
=
∂xj
∂xi
(2.1)
in D0 (Ω)
for all i, j ∈ {1, 2, . . . , n}. Then there exists u ∈ D0 (Ω) such that
∂u
= fi
∂xi
(2.2)
in D0 (Ω)
for all i ∈ {1, 2, . . . , n}.
Remark
.
2.1
(1) In the case where
Ω = Rn ,
Theorem 2.1 was proved
in [16].
(2) If we consider the
1-form
on
Ω
with distributional components
α = f1 dx1 + f2 dx2 + · · · + fn dxn ,
then the above theorem can be restated as follows.
Ω, then α is exact, i.e., there exists a 0-form β ∈
Proof of Theorem
2.1.
If α is a closed 1-form on
such that α = dβ.
Λ0 (Ω)
First, we prove the existence of local solutions
to system (2.2), then we prove the existence of a global solution by using the
simple-connectedness of the set
Ω.
526
Sorin Mardare
4
To prove the existence of local solutions to system (2.2), we follow the
same ideas as in Schwartz [16]. For the sake of completeness, we give below
the whole argument.
ω := (a1 , b1 ) × (a2 , b2 ) × · · · × (an , bn ) ⊂ Ω and ϕ ∈ D(ω). The
ϕ in order to obtain a derivative of another
function in D(ω).
R
For every i ∈ {1, 2, . . . , n} let θi ∈ D((ai , bi )) be such that
R θi (t) dt = 1.
ϕ
Dene the function ψ1 : ω → R by
Z
ϕ
(2.3)
ψ1 (x1 , x2 , . . . , xn ) = ϕ(x1 , x2 , . . . , xn ) − θ1 (x1 ) ϕ(s, x2 , . . . , xn ) ds
Let
idea is to modify the test function
R
ϕ
and note that ψ1
=
∂Ψϕ
1
∂x1 , where the function
Ψϕ
1 (x1 , x2 , . . . , xn ) =
(2.4)
x1
Z
Ψϕ
1 is dened by
ψ1ϕ (t, x2 , . . . , xn ) dt.
−∞
ϕ
It is easy to check that Ψ1 ∈ D(ω). Hence a distribution
∂u
0
satises
∂x1 = f1 in D (ω) must satisfy the relation
u ∈ D0 (ω)
that
hu, ψ1ϕ i = −hf1 , Ψϕ
1i
or, equivalently, the relation (the variables
x1 , x2 , . . . , xn appearing in the right-
hand side are mute)
Z
E
D
ϕ(s,
x
,
.
.
.
,
x
)
ds
,
hu, ϕi = −hf1 , Ψϕ
i
+
u,
θ
(x
)
2
n
1 1
1
R
ϕ
Ψ1 are the functions dened by (2.3) and (2.4), respectively.
ϕ
ϕ
We apply the same method to construct the functions ψ2 and Ψ2 , but
ϕ
where ψ1 and
this time we start with the test function
Z
(x1 , x2 , . . . , xn ) 7→ θ1 (x1 )
ϕ(s, x2 , . . . , xn ) ds
R
(instead of
After
ϕ) and consider
n iterations of
the derivative with respect to the variable
the formula
ϕ
ϕ
hu, ϕi = −hf1 , Ψϕ
1 i − hf2 , Ψ2 i − · · · − hfn , Ψn i + hC, ϕi,
(2.5)
where
(2.6)
C
is a constant and the functions
Ψϕ
i (x1 , . . . , xi , . . . , xn ) :=
Z
ϕ
Ψϕ
1 , . . . , Ψn
xi
−∞
are dened by
ψiϕ (x1 , . . . , t, . . . , xn ) dt
with
(2.7)
x2 .
this argument, we obtain for the distribution
ϕ
ψiϕ = ηi−1
− ηiϕ
u
5
On Poincaré and de Rham's theorems
527
and
η0ϕ = ϕ,
Z Z
ηiϕ (x1 , . . . , xn ) := θ1 (x1 ) . . . θi (xi ) · · · ϕ(s1 , . . . , si , xi+1 , . . . , xn ) ds1 . . . dsi .
(2.8)
R
R
u dened by (2.5) belongs to the space D0 (ω).
Indeed, it is clear that u in linear with respect to ϕ. It is also easy to check
that if a sequence (ϕm )m∈N of test functions satises ϕm → ϕ in D(ω), then
Ψiϕm → Ψϕ
i in D(ω) for all i ∈ {1, 2, . . . , n}. Then the continuity of u (with
respect to the usual topology of the space D(ω)) follows from the continuity
of the mappings fi : D(Ω) → R (recall that fi are distributions).
Now, we prove that the distribution u given by (2.5) satises the equations
We claim that the mapping
∂u
= fk
∂xk
in
D0 (ω)
for all
k ∈ {1, 2, . . . , n}.
∂ ϕ̃
∂xk for some ϕ̃ ∈ D(ω). Then a straightforward
ϕ
computation shows that the functions ψi introduced above satisfy
Let
k ∈ {1, 2, . . . , n}
and
ϕ=
ψiϕ =


∂ψiϕ̃




∂xk



ϕ̃
∂ηk−1




∂xk



0
if
i < k,
if
i = k,
if
i > k,
whence we deduce that
Ψϕ
i =

ϕ̃

 ∂Ψi



 ∂xk
ϕ̃

ηk−1




0
if
i < k,
if
i = k,
if
i > k.
It then follows from formula (2.5) and equations (2.1) that
D
k−1
u,
X D ∂Ψϕ̃ E
∂ ϕ̃ E
ϕ̃
i
=−
fi ,
− hfk , ηk−1
i+C
∂xk
∂xk
=−
i=1
k−1 D
X
ω
∂ ϕ̃
dx
∂xk
k−1
fk ,
i=1
D
= − fk ,
Z
E
X
∂Ψϕ̃
ϕ̃
ϕ̃
i
− hfk , ηk−1
i=−
hfk , ψiϕ̃ i − hfk , ηk−1
i
∂xi
i=1
k−1
X
i=1
E
ϕ̃
ϕ̃
(ηi−1
− ηiϕ̃ ) + ηk−1
= −hfk , η0ϕ̃ i = −hfk , ϕ̃i.
528
Sorin Mardare
6
This ends the proof of the existence of local solutions to Poincaré's system
(2.2). Note that if
can take
C
u
satises (2.2), then
u+C
also satises (2.2), hence we
to be any real constant in formula (2.5).
Now, we will construct a global solution to system (2.2) by using the
simple-connectedness of the set
Ω.
To this end, we rst note that the local
existence result we just proved insures the existence of local solutions dened
on open balls that can be included in cubes contained in
if
x
is any point in
Ω,
Ω.
More specically,
then there exists a local solution of system (2.2) in any
r ≤ √1n dist(x, Ωc ), where Ωc := Rn \ Ω.
Let there be given a point x0 ∈ Ω and an open ball B(x0 , r0 ) with r0 ≤
√1 dist(x0 , Ωc ). The above local existence result yields a distribution u0 ∈
n
D0 (B(x0 , r0 )) that satises
open ball
B(x, r)
with radius
∂u0
= fk
∂xk
in
D0 (B(x0 , r0 )),
k ∈ {1, 2, . . . , n}.
x ∈ Ω and consider a triple (γ, ∆, (Bj )), where γ : [0, 1] → Ω is a continuous
x0 to x, i.e., γ(0) = x0 and γ(1) = x, ∆ = (0 = t0 < t1 < · · · <
tN < tN +1 = 1) is a division of the interval [0, 1], and (Bj )N
j=0 are open balls
contained in cubes that are themselves contained in Ω and satisfy
Let
path joining
(2.9)
B0 = B(x0 , r0 ),
γ([tj , tj+1 ]) ⊂ Bj
for all
j ∈ {1, 2, . . . , N }.
x ∈ Ω. Indeed, for any continuous path γ
x0 to x, the triple (γ, ∆, (Bj )) with B0 = B(x0 , r0 ) and Bj = B(γ(tj ), r)
1
c
for j ∈ {1, 2, . . . , N }, where r < √ dist(γ([0, 1]), Ω ) and |tj+1 − tj | ≤ δr for
n
all j ∈ {1, 2, . . . , N }, satises the conditions above if δr is chosen in such a
way that |γ(t) − γ(s)| < min{r, r0 } whenever |t − s| ≤ δr (the existence of δr
is given by the uniform continuity of γ ).
We construct o global solution to system (2.2) as follows. For x ∈ Ω and
(γ, ∆, (Bj )) satisfying the conditions above, we dene recursively the distribuj
0
tions u ∈ D (Bj ), j = 1, 2, . . . , N , such that
Note that such triples do exist for any
joining
(2.10)
∂uj
= fk
∂xk
uj = uj−1
in
D0 (Bj ),
in
Bj ∩ Bj−1 .
Bj ∩ Bj−1 is a
uj−1 and uj respectively satisfy
∂uj−1
∂uj
0
0
j−1 −
∂xk = fk in D (Bj−1 ) and ∂xk = fk in D (Bj ), then the distribution (u
uj ) is constant in Bj−1 ∩ Bj and so we can add if necessary a constant to uj
j
j−1 in B ∩ B
in order to meet the condition that u = u
j
j−1 .
Note that this construction is possible since the intersection
convex set, in particular connected; hence, if
7
On Poincaré and de Rham's theorems
529
From this construction we keep only the last distribution
N is independent of the triple
that u
(γ, ∆, (Bj ))
uN
and claim
in the following sense. If
(γ, ∆ := (0 = t0 < t1 < · · · < tN < tN +1 = 1), (Bj )N
j=0 ),
Ñ
˜ := (0 = t̃0 < t̃1 < · · · < t̃ < t̃
(γ̃, ∆
Ñ
Ñ +1 = 1), (B̃j )j=0 )
are two triples associated with
x,
then
ũÑ = uN
in
N is independent of
Let us rst prove that u
γ
and
∆
and consider two sequences of open balls
BN ∩ B̃Ñ .
(Bj ). To this end, we x
Ñ
(Bj )N
j=0 and (B̃j )j=0 , both
satisfying (2.9). We apply an induction argument. First, we have
u0 = ũ0
j
Assume that u
=
ũj in
in
Bj ∩ B̃j .
D0 (B0 )
(by denition).
Then from (2.10) we have
uj+1 = uj = ũj = ũj+1
in
Bj ∩ B̃j ∩ Bj+1 ∩ B̃j+1 .
Bj ∩ B̃j ∩ Bj+1 ∩ B̃j+1 is a nonempty open set, since it contains
γ(tj+1 ). It follows that uj+1 = ũj+1 in Bj+1 ∩ B̃j+1 , since the set Bj+1 ∩ B̃j+1
Note that
is a connected open set.
uN is independent of ∆. To this end, we rst consider
˜ of the form ∆ = (0 = t0 < t1 < · · · < tN +1 =
the case of two divisions ∆ and ∆
˜ = (0 = t0 < t1 < · · · < tk < t∗ < tk+1 < · · · < tN +1 = 1). Let
1) and ∆
˜
(Bj )N
j=0 be a family of open balls satisfying (2.9). For ∆, we consider the
N +1
family (B̃j )j=0 dened by
Now, we prove that
B̃j = Bj
for all
j ∈ {0, 1, . . . , k},
B̃j+1 = Bj
for all
j ∈ {k, k + 1, . . . , N }.
+1
˜
(B̃j )N
j=0 associated with the division ∆ satises (2.9).
j
j
0
We also have ũ = u in D (Bj ) for all j ≤ k . By (2.10), we then have
k+1
k
k
ũ
= ũ = u in B̃k+1 ∩ B̃k = Bk . Using again system (2.10), we obtain by
j+1 = uj in B for all j ≥ k and in particular for j = N .
induction that ũ
j
˜ = (0 = t̃0 <
Let now ∆ = (0 = t0 < t1 < · · · < tN +1 = 1) and ∆
t̃1 < · · · < t̃Ñ +1 = 1) be two arbitrary divisions with the respective associated
Obviously, the family
Ñ
(Bj )N
j=0 and (B̃j )j=0 satisfying (2.9).
∆ := (0 = s0 < s1 < · · · < sM +1 = 1) dened by
families of open balls
division
Consider the joint
{s0 , s1 , . . . , sM +1 } = {t0 , t1 , . . . , tN +1 } ∪ {t̃0 , t̃1 , . . . , t̃Ñ +1 },
Starting with
∆
and applying
(M − N )
M ≤ N + Ñ .
times the above argument about divi-
sions diering at one single point, we obtain (with self-explanatory notation)
uM = uN
in
BN .
530
Sorin Mardare
Similarly, but starting with
8
˜ and applying (M − Ñ ) times the same argument,
∆
we obtain
uM = ũÑ
in
B̃Ñ .
Combining the last two equalities and using the fact that
of
(Bj ),
uM
is independent
we get
uN = ũÑ
Finally, we prove that
uN
in
BN ∩ B̃Ñ .
˜ (B̃j )Ñ )
(γ̃, ∆,
j=0
γ . To this end,
(γ, ∆, (Bj )N
j=0 ) and
is independent of the path
we use, as expected, the simple-connectedness of
Ω.
Let
x ∈ Ω. Since Ω is
H : [0, 1] × [0, 1] → Ω
be two triples associated with the point
simply connected, there exists a continuous function
such that
H(0, ·) = γ,
H(1, ·) = γ̃,
H(· , 0) = x0 ,
H(· , 1) = x.
s
s ∈ [0, 1], set γs := H(s, ·) and consider a triple (γs , ∆s , (Bjs )N
j=0 )
N
associated with x. For s = 0 we choose the triple (γ, ∆, (Bj )j=0 ), while for
˜ (B̃j )Ñ ).
s = 1 we choose the triple (γ̃, ∆,
For each
j=0
Let (with self-explanatory notation)
s∗ := sup{s ∈ [0, 1]; us,Ns = uN := u0,N0
in
s
BN
∩ BN }.
s
We wish to prove that
1 = s∗ = max{s ∈ [0, 1]; us,Ns = uN
First, we prove that
s∗ > 0.
in
s
BN
∩ BN }.
s
To this end, we show that for
s
su-
ciently small,
us,Ns = uN
in
s
BN
∩ BN .
s
In order to prove this relation, we begin by showing that the triple
(γs , ∆, (Bj )N
j=0 )
associated with
x
Bj
s is suciently
j ∈ {0, 1, . . . , N }, γs ([tj , tj+1 ]) ⊂
is admissible provided that
small. It is enough to prove that, for any xed
s. We argue by contradiction. Suppose that there
sm → 0 and tm ∈ [tj , tj+1 ] (m ∈ N is an index going to inm
m m
nity) such that γsm (t ) = H(s , t ) 6∈ Bj . Since [tj , tj+1 ] is compact, there
m → t ∈ [t , t
exists a subsequence of (tm ), still denoted (tm ), such that t
j j+1 ].
m
m
The function H being continuous, we have H(s , t ) → H(0, t) = γ(t) ∈ Bj ,
m m
which contradicts the relation H(s , t ) 6∈ Bj for all m ∈ N (the contradiction
follows from the fact that Bj is an open set).
∗
Next, we prove that s is a maximum. The above contradiction argument shows that for α suciently small and for all 0 ≤ ε < α, the triple
for any suciently small
exist sequences
9
On Poincaré and de Rham's theorems
∗
N
∗
s
(γs∗ −ε , ∆s∗ , (Bjs )j=0
) is admissible for x.
respect to ∆ and (Bj ), this implies that
us
for all
ε ∈ [0, α).
∗ ,N ∗
s
Since
u
for some
δ ∈ [0, α).
s∗
= us
∗ −ε,N ∗
s −ε
531
By the independence of
on
∗ ,N ∗
s
Finally, we prove that
∗
∗
s −ε
s
BN
∗ ∩ B
Ns∗ −ε
= uN
on
∗
s −δ
BN
∩ BN
s∗ −δ
= uN
s∗ = 1
on
∗
s
BN
∩ BN .
s∗
by a contradiction argument.
then using once again the previous argument shows that for
s∗ .
with
is a supremum, we have
s∗ −δ,Ns∗ −δ
us
denition of
∗ ,N ∗
s
By combining the last two equalities, we get
(2.11)
small, the triple
us
∗
N
∗
s
(γs∗ +ε , ∆s∗ , (Bjs )j=1
) is admissible for x.
Therefore
s∗ = 1
and
ε>0
If
s∗ < 1
suciently
This contradicts the
uN = u0,N0 = u1,N1 = ũÑ .
Now, we are in a position to dene a global solution to the Poincaré
x ∈ Ω, let us choose Bx ⊂ Ω to be an admissible nal
Bx = BN for some admissible triple (γ, ∆, (Bj )N
j=1 ) associated with
x
N
N
0
x. Denote u = u , where the distribution u ∈ D (Bx ) is constructed as
above. Let us prove that for any x, y ∈ Ω such that Bx ∩ By 6= ∅ we have
system (2.2). For any
ball, i.e.,
ux = uy
(2.12)
on
Bx ∩ By .
Ñ
˜
(γ, ∆, (Bj )N
j=1 ) and (γ̃, ∆, (B̃j )j=1 ) be two admissible triples for x and
y , respectively. Let z ∈ Bz ∩ By . We consider the path obtained by joining γ
with the segment [x, z], parameterized for instance by (prime is not a symbol
for the derivative with respect to t)
(
1
γ(2t)
if t ∈ [0, ],
2
0
γ (t) :=
1
(2 − 2t)x + (2t − 1)z if t ∈ ( 2 , 1].
Let
t
+1
(γ 0 , ∆0 := (0 = t20 < t21 < · · · < N2+1 < tN +2 = 1), (Bj0 )N
j=1 ),
0
0
where Bj = Bj for all j ≤ N and BN +1 = BN = Bx , is admissible for
z (since Bx is a convex set, γ 0 ([ 21 , 1]) = [x, z] ⊂ Bx ). We make a similar
construction for γ̃ to nd another admissible triple for z . Then one can see
∗
that the relation (2.12) is a consequence of the relation (2.11) with s = 1,
0N +1 = uN = ux in B and ũ0N +1 = ũN = uy in B .
since u
x
y
Then the triple
The property (2.12) allows us to construct a distribution on the set
S
x∈Ω Bx
(2.13)
=Ω
by letting, for any given test function
hu, ϕi :=
m
X
i=1
huxi , θi ϕi,
ϕ ∈ D(Ω),
532
Sorin Mardare
10
Sm
supp ϕ ⊂
i=1 Bxi (such a nite covering exists because the support
Ω) and the family (θi )m
i=1 is a partition of unity
m
subordinated to the covering (Bxi )i=1 of supp ϕ (this means that θi ∈ D(Bxi )
Pm
for all i and
i=1 θi = 1 on supp ϕ). By Theorem 1.2, the relation (2.13)
x
denes a distribution u over Ω that satises u = u in Bx for all x ∈ Ω.
where
ϕ
of
is a compact subset of
u dened by (2.13)
satises the system
∂ϕ
∈ D(Ω) and supp ∂xk ⊂ supp ϕ. Let K ⊂ Ω
(2.2). Let ϕ ∈
m
be a compact neighborhood of supp ϕ. Consider a family of open balls (Bxi )i=1
Sm
such that K ⊂
(θi )m
i=1 subordinated to the
i=1 Bxi and a partition of unity
Pm
m
covering (Bxi )i=1 . In particular, (θi ) satises
θ
i=1 i = 1 in K ⊃ supp Ω.
We now prove that the distribution
∂ϕ
D(Ω). Then ∂x
k
Then we have
D
=
m D
X
i=1
u,
m
m
i=1
i=1
∂ϕ E X D xi
∂ϕ E XD xi ∂
∂θi E
=
=
(θi ϕ) −
ϕ
u , θi
u ,
∂xk
∂xk
∂xk
∂xk
m
m
E D ∂θ E X
D X
∂θi E
∂
i
xi
(θi ϕ) − u,
ϕ =
ϕ
− hfk , θi ϕi − u,
u ,
∂xk
∂xk
∂xk
i=1
i=1
P
m
m
D X
E D ∂
E
θ
i=1 i
= − fk ,
θi ϕ − u,
ϕ = −hfk , ϕi.
∂xk
i=1
Note that we used in the third equality the fact that
u = u xi
∂uxi
in
Bxi .
In
∂xk = fk in Bxi . In the last equality,
Pm
we used the fact that
i=1 θi = 1 in the neighborhood
K of supp ϕ, which
Pm
Pm
∂
θ
∂
θ
i
i
i=1
i=1
implies that
= 0 on supp ϕ, hence
ϕ = 0 in Ω. This
∂xk
∂xk
completes the proof.
the forth equality, we used the relation
Remark
.
2.2
By following the argument used to construct a local solution
to system (2.2) (see formula (2.5)), we can see that any
ω
is given by the formula (the variables
x1 , . . . , xn
u
satisfying (2.2) in
appearing in the formula
below are mute)
hu, ϕi =
−hf1 , Ψϕ
1i
− ··· −
hfn , Ψϕ
ni
Z
+ hu, θ1 (x1 ) . . . θn (xn )i
ϕ(x) dx,
ω
ϕ
Ψϕ
1 , . . . , Ψn are constructed as in the proof of Theorem 2.1. In particular,
if f1 = f2 = · · · = fn = 0, then u is a constant (equal to hu, θ1 (x1 ) . . . θn (xn )i).
where
This argument proves that on a connected open set, a distribution has null
partial derivatives if and only if it is a constant. We have used this well-known
uniqueness up to a constant result (Theorem 1.1) in the construction of a
global solution
u.
See (2.10).
11
On Poincaré and de Rham's theorems
533
3. REGULARITY OF THE SOLUTION
TO POINCARÉ'S SYSTEM
u to system
f1 , f2 , . . . , fn all belong to some Sobolev
whether the solution u dened in the space of
Now we address the question of regularity for the solution
(2.2). More specically, assuming that
W m,p (Ω),
space
we investigate
distributions by Theorem 2.1 possesses some regularity properties. Intuitively,
we expect
u
to be of class
W m+1,p (Ω),
at least if the boundary of
Ω
is su-
ciently smooth.
If
m≥0
to system (2.2)
Ω,
set
p ∈ [1, ∞], it is well known that any solution u ∈ D0 (Ω)
m+1,p
satises u ∈ Wloc
(Ω). This result is valid for any open
and
irrespectively of the regularity of its boundary (see, e.g., Maz'ja [11,
Theorem at page 7]). The usual proof of this regularity theorem consists in
proving that
Ω.
u
is of class
W m+1,p
in any open ball or hypercube contained in
In order to prove this local regularity property of
u,
one could use either
a regularizing argument (as in [4]), or an integral formula of
sections of a hypercube contained in
Ω
(as in [12]).
u
on well chosen
Ω is a
u ∈ W m+1,p (Ω).
If in addition
bounded domain with a Lipschitz-continuous boundary, then
This global regularity result is a consequence of the formula (obtained after a
change of variables, if necessary)
u(x0 , xn ) = u(x0 , 0) +
Z
xn
0
which is valid for almost all points
good subset of
Ω
(x0 , xn ),
∂u 0
(x , t) dt,
∂xn
where
x0 := (x1 , . . . , xn−1 ),
in a
whose precise denition depends on the regularity of
Ω.
Ω is
m ≥ 0). For instance, in the case where m ≥ 0 and
p ∈ [1, ∞), it is enough that Ω be a bounded domain with continuous boundary
such that Ω lies locally on the same side of its boundary. Note that the
In fact, one can see from this argument that a much weaker regularity of
really needed (we recall that
regularity assumption on the boundary cannot be dropped altogether, because
counterexamples (see Maz'ja [11]) show that the global regularity result fails
if
Ω
is an arbitrary bounded domain.
The regularity of
u
in the case
m < 0
is much more dicult to study.
The one dimensional case (n
= 1) is however easy to study thanks to formula
Ω = (a, b) be a bounded open interval in R and assume
m,p (Ω), m < 0, p ∈ (1, ∞). Then for any ϕ ∈ D(Ω), we have
that f1 ∈ W
(without any loss in generality, we choose C = 0 in (2.5))
(2.5). To see this, let
ϕ
|hu, ϕi| = |hf1 , Ψϕ
1 i| ≤ kf1 kW m,p (Ω) kΨ1 kW −m,q (Ω) ,
0
534
Sorin Mardare
where
q
is dened by
12
1
p
explicit denition (2.4)
+ 1q = 1. Then Poincaré's
ϕ
of Ψ1 ) implies that
inequality (or simply the
0
|hu, ϕi| ≤ Ckf1 kW m,p (Ω) k(Ψϕ
1 ) kW0−m−1,q (Ω) .
R
ϕ 0
ϕ
Or (Ψ1 ) (x1 ) = ψ1 (x1 ) = ϕ(x1 ) − θ1 (x1 )
R ϕ dx1 for some xed function
R
θ1 ∈ D(Ω) such that R θ1 dx1 = 1. Hence
Z
(3.1) |hu, ϕi| ≤ Ckf1 kW m,p (Ω) kϕk −m−1,q
ϕ
dx
+
kθ
k
−m−1,q
1
1
(Ω)
(Ω)
W0
W0
R
≤ Ckf1 kW m,p (Ω) 1 + kθ1 kW −m−1,q (Ω) kϕkW −m−1,q (Ω) ,
0
0
the last inequality being a consequence of Hölder's inequality
Z
Z
1
ϕ dx1 = ϕ dx1 ≤ (b − a) p kϕkLq (Ω) .
R
Ω
u is a continuous linear functional over the
W0−m−1,q (Ω) and therefore u ∈ W m+1,p (Ω). Moreover, the chosen distribution u, i.e., the one dened by (2.5) with C = 0 (in other words, the solution
to Poincaré's system (2.2) satisfying hu, θ1 i = 0), satises the inequality
Inequality (3.1) implies that
space
kukW m+1,p (Ω) ≤ Ckf1 kW m,p (Ω)
for some constant
C
depending only on
Ω.
Note that the above argument does not apply in higher dimensions (n
2), since the term k∇Ψϕ
i kW0−m−1,q (Ω) cannot be controlled by
To see this, consider for instance the case
m = −1.
≥
kϕkW −m−1,q (Ω) .
0
Then
Z x1
∂ψ1ϕ
∂Ψϕ
1
(x1 , x2 , . . . , xn ) =
(t, x2 , . . . , xn ) dt
∂x2
−∞ ∂x2
Z
Z x1 ∂ϕ
∂ϕ
(t, x2 , . . . , xn ) − θ1 (t)
(s, x2 , . . . , xn ) ds dt,
=
R ∂x2
−∞ ∂x2
which means that in order to control the norm
derivatives of
ϕ.
k∇Ψϕ
1 kLq (Ω) ,
one needs partial
On the other hand, this argument does not take into account
(fi ), because
n-tuple (f1 , . . . , fn ).
n ≥ 2.
the compatibility conditions satised by the functions
it relies
only on formula (2.5), which is valid for any
Thus a
dierent argument is needed in dimension
Our aim is to obtain a generalization of Poincaré's lemma in Sobolev
spaces
W m,p (Ω)
valid for any integer
m
(not necessarily nonnegative) and for
m = 0 and p = 2 (i.e.,
f1 , . . . , fn ∈ W 0,2 (Ω) = L2 (Ω)), this was done by Girault and Raviart [9]
in dimension three (n = 3) and by Bourgain, Brezis and Mironescu [4] in
arbitrary dimension under an additional regularity assumption on Ω. The case
m = −1, p = 2 was studied by Ciarlet and Ciarlet, Jr. [7] in dimension
any simply-connected Lipschitz domains. In the case
if
13
On Poincaré and de Rham's theorems
three (n
= 3)
535
and was generalized by Kesavan [10] in arbitrary dimension.
More recently, Amrouche, Ciarlet and Ciarlet, Jr. [1] solved the case
and
m ∈ Z
arbitrary, for three-dimensional domains.
p = 2
As noted in [10], in a
Ω with a Lipschitz-continuous boundary,
W −1,2 (Ω) is equivalent to a well-known Lions lemma
f ∈ D0 (Ω) and ∇f ∈ (W −1,2 (Ω))n implies f ∈ L2 (Ω). This
simply-connected bounded domain
the Poincaré lemma in
stating that
lemma was proved in Duvaut and Lions [8] in the case of smooth domains and
in Borchers and Sohr [6] (see also Amrouche and Girault [2, Proposition 2.10])
in the case of Lipschitz domains. In fact, Amrouche and Girault [2] proved the
following generalization of Lions' lemma.
Let Ω ⊂ Rn be a bounded domain
with a Lipschitz-continuous boundary. If the distribution f ∈ D0 (Ω) satises
∇f ∈ (W m,p (Ω))n with m ∈ Z and p ∈ (1, ∞), then f ∈ W m+1,p (Ω).
Lemma 3.1 (Proposition 2.10 in [2]).
Now, we establish the Poincaré theorem in the setting of Sobolev spaces
by combining Lemma 3.1 with Theorem 2.1.
Furthermore, we establish an
estimate of the solution to the Poincaré system (2.2) in the corresponding
Sobolev norm (see Theorem 3.1).
u ∈ W m,p (Ω), we denote
m,p (Ω)/R and dene its norm by
by û the equivalence class of u in the space W
To begin with, we introduce some notation. If
kûkW m,p (Ω)/R := inf ku + ckW m,p (Ω) .
c∈R
Note that if
u
is a solution to system (2.2), then any element of the class
û
is
also solution to this system.
The announced result is as follows.
Let Ω be a simply-connected open set in Rn and let f1 , f2 ,
. . . , fn ∈ W m,p (Ω) for some m ∈ Z and p ∈ (1, ∞). Assume that the functions
m+1,p
(Ω) such that ∇u =
(fi ) satisfy equations (2.1). Then there exists u ∈ Wloc
0
f := (f1 , . . . , fn ) in D (Ω).
If in addition Ω is connected, bounded, with a Lipschitz-continuous boundary, then u ∈ W m+1,p (Ω) and there exists a constant C depending only on Ω
such that
Theorem 3.1.
kûkW m,p (Ω)/R ≤ Ckf kW m,p (Ω) .
Remark
.
3.1
If
m≥0
then Theorem 3.1 also holds for
p=1
and
p=∞
(see the comments at the beginning of this section).
4. DE RHAM'S THEOREM FOR
1-DIMENSIONAL
FLOWS
We give an elementary proof of de Rham's theorem in the case of homogeneous ows (or currents) of dimension one on a Euclidean space (for the
536
Sorin Mardare
14
denition of these notions, see, e.g., de Rham [15]). Since an Euclidean space is
oriented, we will only consider positively oriented local charts, so the problem
of the parity (for forms or ows) is not posed here.
manifold an open set
Ω⊂
In fact, by choosing as
Rn , we can always take the identity as local chart
around every point. In this setting, we are able to prove the pure analytic form
of the de Rham's theorem below.
Let Ω ⊂ Rn be a connected open set and let f = (f1 , . . . ,
be a vector eld that satises
Theorem 4.1.
fn ) ∈
(D0 (Ω)n
hf , ϕi :=
(4.1)
n
X
hfi , ϕi i = 0
i=1
for all ϕ = (ϕ1 , . . . , ϕn ) ∈
u ∈ D0 (Ω) such that f = ∇u, i.e.,
(D(Ω))n
satisfying divϕ = 0. Then there exists
∂u
= fk in D0 (Ω) for all k ∈ {1, . . . , n}.
∂xk
(4.2)
Remark
.
4.1
(1) In dierential geometry terms, Theorem 4.1 can be re-
stated as follows:
Let T be a homogeneous ow of dimension one and assume that
hT, ϕi = 0
for all compactly supported 1-forms ϕ of class C ∞ satisfying δϕ = 0. Then
there exists a homogeneous ow S of dimension zero (i.e., a distribution on Ω)
such that
dS = T,
where dS is dened by hdS, ϕi := hS, δSi for all 1-forms ϕ ∈ Λ1 (Ω) of class
C ∞ with compact support in Ω.
This statement holds verbatim in the case of oriented Riemannian manifolds. The operator
δ : Λk (Ω) → Λk−1 (Ω) is the codierential operator and can
be dened on any oriented Riemannian manifold; see, e.g., Bleecker [5]. In the
particular case of Euclidean spaces, this denition reduces to the expression
δ(ϕ1 dx1 + ϕ2 dx2 + · · · + ϕn dxn ) = −divϕ,
where
ϕ := (ϕ1 , ϕ2 , . . . , ϕn ).
The de Rham theorem in the most general setting i.e., for ows on
manifolds which are not necessarily Riemannian can be found in [15].
(2) Unlike Poincaré's theorem, Theorem 4.1 holds in any connected open
set
Ω,
not necessarily simply-connected. This weaker hypothesis on
Ω
is pos-
sible because condition (4.1) is stronger than condition (2.1). Indeed, we will
see in the proof of Theorem 4.1 that (4.1) implies (2.1). The converse is false
because otherwise Poincaré's theorem would apply in domains that are not
15
On Poincaré and de Rham's theorems
537
simply-connected, which is known to be false. Therefore, Theorem 4.1 is an
immediate consequence of Theorem 2.1 in the case of simply-connected sets.
Proof of Theorem
4.1.
satises equations (2.1). For
We begin by showing that the vector eld
ϕ ∈ D(Ω)
and
i < j,
f
let
∂ϕ
∂ϕ
ϕ := 0, . . . , 0,
, 0, . . . , 0, −
, 0, . . . , 0 ,
∂xj
∂xi
ith and j th positions. It is
div ϕ = 0 and therefore equation (4.1) implies that hf , ϕi = 0. Since
D
∂ϕ E D
∂ϕ E
hf , ϕi = fi ,
− fj ,
,
∂xj
∂xi
where the nonzero components are placed on the
clear that
equation (2.1) follows.
Thus, we can construct
u
locally as in the proof of Theorem 2.1. Note
u
that formula (2.5), which denes
in a hypercube contained in
Ω,
can be
rewritten as
where
C
is a constant
hu, ϕi = −hf , Ψϕ i + hC, φi,
ϕ
ϕ
ϕ
and Ψ = (Ψ1 , . . . , Ψn ) is dened
as in the proof of
Theorem 2.1.
Then we dene a global solution to (4.2) as follows. Since
Ω is connected,
we use the construction of the global solution described in the proof of Theorem 2.1 (see (2.10)) to associate with every triple
uN satisfying equations (4.2) in the open ball
(γ, ∆, (Bj )N
j=0 ) a distribution
BN .
Recall that in the proof of Theorem 2.1 the simple-connectedness of the
domain
Ω
has only been used to prove that
γ . We now have to prove the independence
(γ, ∆, (Bj )N
j=0 ) by other means. Specically,
uN is independent of the path
N with respect to the triple
of u
this will be done by using equa-
tion (4.1) in its full generality. In fact, we will establish an explicit formula
for
uN
(see (4.6)(4.8)) in terms of the triple
(γ, ∆, (Bj )N
j=0 )
from which the
independence property can be easily deduced.
Let
B0
and
u0
be xed as in the proof of Theorem 2.1. In particular, the
0
distribution u is dened by the formula (taking
(4.3)
hu0 , ϕi =
−hf1 , Ψ0,ϕ
1 i
hf2 , Ψ0,ϕ
2 i
−
C=0
in (2.5))
− · · · − hfn , Ψ0,ϕ
n i,
where (see (2.6)(2.8))
(4.4)
Ψ0,ϕ
k (x1 , . . . , xk , . . . , xn ) :=
ψk0,ϕ
:=
0,ϕ
ηk−1
−
Z
ηk0,ϕ (x1 , . . . , xn ) := θ10 (x1 ) . . . θk0 (xk )
Z
xk
ψk0,ϕ (x1 , . . . , t, . . . , xn ) dt,
−∞
ηk0,ϕ ,
η00,ϕ := ϕ,
Z
. . . ϕ(s1 , . . . , sk , xk+1 , . . . , xn )ds1 . . . dsk .
R
R
538
Sorin Mardare
16
R
(θi0 )ni=1 are given functions that satisfy θi0 ∈ D((ai , bi )) and R θi0 (t) dt =
1, where (a1 , b1 ) × · · · × (an , bn ) ⊂ Ω is any given cube containing B0 .
N
Now, let x ∈ Ω and let (γ, ∆, (Bj )j=0 ) be an admissible triple for x.
j n
For each j ∈ {1, 2, . . . , n}, we choose an n-tuple (θi )i=1 associated with a
j
hypercube contained in Ω and containing Bj , whose components θi satisfy
0
j
the same properties as the functions θi . Next, we dene the distributions u ,
j = 1, 2, . . . , N , by
Here,
j,ϕ
j,ϕ
huj , ϕi = hCj , ϕi − hf1 , Ψj,ϕ
1 i − hf2 , Ψ2 i − · · · − hfn , Ψn i,
(4.5)
Cj ∈ R are constants insuring that uj = uj−1 on Bj ∩ Bj−1 and the
j,ϕ
j,ϕ
j,ϕ
functions Ψk , ψk
and ηk
are dened as in (4.4). In what follows, these
j
j
j
functions are renamed Ψk (θ , ϕ), ψk (θ , ϕ) and ηk (θ , ϕ) to emphasize the
j
j
j
fact that they are completely determined by θ = (θ1 , . . . , θn ) and ϕ ∈ D(Bj ).
By relations (4.3) and (4.5), for all ϕ ∈ D(B0 ∩ B1 ) we have
Z
1
1
ϕ(x) dx = hu0 , ϕi = −hf , Ψ(θ 0 , ϕ)i,
hu , ϕi = −hf , Ψ(θ , ϕ)i + C1
where
Ω
By choosing a function
explicit value of
C1 ,
ζ1 ∈ D(B0 ∩ B1 )
such that
R
Ω ζ1 (x) dx
= 1,
we get an
namely
C1 = hf , Ψ(θ 1 , ζ1 ) − Ψ(θ 0 , ζ1 )i.
To sum up, we obtained for
1
D
u1
the formula
1
1
0
hu , ϕi = f , −Ψ(θ , ϕ) + Ψ(θ , ζ1 ) − Ψ(θ , ζ1 )
Z
E
ϕ(x) dx
Ω
ϕ ∈ D(B1 ).
Iterating N times
for all
the previous argument for the construction of
u1 ,
for
uN we obtain the formula
(4.6)
huN , ϕi = f , Ψ((γ, ∆, (Bj )), ϕ)
for all
ϕ ∈ D(BN ),
where
(4.7)
Z
N
X
i
i−1
Ψ(θ , ζi )−Ψ(θ , ζi )
Ψ((γ, ∆, (Bj )), ϕ) := −Ψ(θ , ϕ)+
ϕ(x) dx
N
Ω
i=1
and
Z
(4.8)
ζi ∈ D(Bi ∩ Bi−1 )
ζi (x) dx = 1
and
Ω
for all
i ∈ {1, . . . , N }.
17
On Poincaré and de Rham's theorems
On the other hand, note that for any function
θ = (θ1 , . . . , θn )
associated with a hypercube
div Ψ(θ, ζ) =
(4.9)
n
X
ψk (θ, ζ) =
k=1
n
X
ω
539
ζ ∈ D(Ω) and any n-tuple
Ω, we have
contained in
ηk−1 (θ, ζ) − ηk (θ, ζ)
k=1
Z
= η0 (θ, ζ) − ηn (θ, ζ) = ζ − θ1 (x1 ) . . . θn (xn )
ζ(x) dx.
Ω
Then it follows from relations (4.8) and (4.9) that
div Ψ((γ, ∆, (Bj )), ϕ) = −ϕ +
θ1N (x1 ) . . . θnN (xn )
Z
ϕ(x) dx
Ω
+
n
X
−
θ1i (x1 ) . . . θni (xn )
+
θ1i−1 (x1 ) . . . θni−1 (xn )
Z
ϕ(x) dx
Ω
i=1
= −ϕ + θ10 (x1 ) . . . θn0 (xn )
Z
ϕ(x) dx
Ω
for all
ϕ ∈ D(BN ).
Note that the right-hand side of the last equality depends only on
ϕ∈
0
D(Bn ) and on the n-tuple θ which has been xed once for all at the beginning
˜ (B̃j )Ñ ) is another admissible triple for x, then
of the proof. Hence, if (γ̃, ∆,
j=0
Z
0
0
˜
div Ψ((γ̃, ∆, (B̃j )), ϕ) = −ϕ + θ1 (x1 ) . . . θn (xn ) ϕ(x) dx
Ω
= div Ψ((γ, ∆, (Bj )), ϕ).
for all
ϕ ∈ D(BN ∩ B̃Ñ ).
Now, we use assumption (4.1). Combined with the previous relation, it
shows that
for all
˜ (B̃j )), ϕ) = 0
f , Ψ((γ, ∆, (Bj )), ϕ) − Ψ((γ̃, ∆,
ϕ ∈ D(BN ∩ B̃Ñ ).
Hence, by (4.6) we have
huN − ũÑ , ϕi = 0
uN
This proves that
for all
ϕ ∈ D(BN ∩ B̃Ñ ).
is independent of the choice of the triple
(γ, ∆, (Bj )).
The remaining part of the proof is identical with the last part of the proof of
Theorem 2.1.
Remark
.
4.2
If the distribution
f
appearing in the statement of Theo-
rem 4.1 belongs to some Sobolev space, then Theorem 4.1 is a consequence of a
well-known result in functional analysis. More specically, one can prove that
m ≤ −1, p ∈ (1, ∞), and f ∈ (W m,p (Ω))n satises (4.1), then there exists
u ∈ W m+1,p (Ω) such that ∇u = f . It is enough to prove that the gradient
m+1,p (Ω) → (W m,p (Ω))n has closed range. Combined with
operator ∇ : W
if
540
Sorin Mardare
the fact that
−∇
18
is the dual operator of the operator
W0−m−1,q (Ω), where
q
is dened by
1/p + 1/q = 1,
div : (W0−m,q (Ω))n →
this implies that (see e.g.
Brezis [3, Theorem II.18])
Im(∇) = (ker(div))⊥ ,
which is exactly the desired result. The dicult part of the proof is to show
that
Im(∇)
is a closed subspace of
(W m,p (Ω))n .
This result, whose proof
can be found in Amrouche and Girault [2, Corollary 2.5], relies on a theorem
of Peetre and Tartar (see [14] and [17]) and on an inequality due to Ne£as
[13]. The Ne£as inequality states that in a bounded Lipschitz domain, one can
control the
W m,p -norm
of a function by the
gradient. This is obvious if
m ≥ 1,
W m−1,p -norms
of itself and of its
but it is by no means trivial in the case
m ≤ 0.
As in Section 3, we can recover de Rham's theorem in Sobolev spaces by
combining Theorem 4.1 and Lemma 3.1 (as mentioned earlier, this lemma is
due to Borchers and Sohr [6] and can be found in Amrouche and Girault [2,
Proposition 2.10]):
Let Ω be an open set in Rn and let f ∈ (W m,p (Ω))n for
some m ∈ Z, p ∈ (1, ∞) if m < 0, p ∈ [1, ∞] if m ≥ 0. Assume that f satisfy
m+1,p
(Ω) such that ∇u = f in D0 (Ω).
equations (4.1). Then there exists u ∈ Wloc
If in addition Ω is connected, bounded, with a Lipschitz-continuous boundary, then u ∈ W m+1,p (Ω) and there exists a constant C depending only on Ω
such that
Theorem 4.2.
kûkW m,p (Ω)/R ≤ Ckf kW m,p (Ω) .
Remark
.
4.3
The proof of Theorem 4.2 based on the idea described in
Remark 4.2 can be found in Amrouche and Girault [2, Theorem 2.8].
The author have been supported by the Swiss National Science
Foundation under Contracts #20-113287/1 and #20-117614/1.
Acknowledgements.
REFERENCES
[1] C. Amrouche, P.G. Ciarlet and P. Ciarlet, Jr., Vector and scalar potentials, Poincaré
theorem and Korn's inequality. C.R. Math. Acad. Sci. Paris 345 (2007), 603608.
[2] C. Amrouche and V. Girault, Decomposition of vector spaces and application to the
Stokes problem in arbitrary dimension. Czechoslovak Math. J. 44 (1994), 109140.
[3] H. Brezis, Analyse fonctionnelle. Théorie et applications. Masson, Paris, 1983.
[4] J. Bourgain, H. Brezis and P. Mironescu, Lifting in Sobolev spaces. J. Anal. Math. 80
(2000), 3786.
[5] D. Bleecker, Gauge Theory and Variational Principles. Addison-Wesley Publishing Co.,
1981.
19
On Poincaré and de Rham's theorems
541
[6] W. Borchers, H. Sohr, On the equations rot v = g and div u = f with zero boundary
conditions. Hokaido Math. J. 19 (1990), 6787.
[7] P.G. Ciarlet et P. Ciarlet, Jr., Another approach to linearized elasticity and a new proof
of Korn's inequality. Math. Models Methods Appl. Sci. 15 (2005) 259271.
[8] G. Duvaut et J.L. Lions, Les inéquations en Mécanique et en Physique. Dunod, Paris,
1972; English translation: Inequalities in Mechanics and Physics. Springer-Verlag,
BerlinNew York, 1976.
[9] V. Girault and P.A. Raviart, Finite Element Methods for Navier-Stokes Equations.
Springer-Verlag, BerlinHeidelberg, 1986.
[10] S. Kesavan, On Poincaré's and J.L. Lions' lemmas. C.R. Math. Acad. Sci. Paris 340
(2005) 2730.
[11] V.G. Maz'ja, Sobolev Spaces. Springer-Verlag, Berlin, 1985.
[12] S. Mardare, The fundamental theorem of surface theory with little regularity. J. Elasticity
73 (2003), 251290.
[13] J. Ne£as, Equations aux dérivées partielles. Presses de l'Univ. de Montréal, 1966.
[14] J. Peetre, Espaces d'interpolation et théorème de Sobolev. Ann. Institut Fourrier (Grenoble) 16 (1966), 279317.
[15] G. de Rham, Dierentiable manifolds. Springer-Verlag, BerlinHeidelberg, 1984.
[16] L. Schwartz, Théorie des distributions. Hermann, Paris, 1966.
[17] L. Tartar, Nonlinear partial dierential equations using compactness methods. Report
1584. Mathematics Research Center, Univ. of Wisconsin, Madison, 1975.
Received 5 May 2008
Universität Zürich
Institüt für Mathematik
Winterthurerstrasse 190
CH-8057 Zürich, Switzerland