Divergence of multivector fields on infinite-dimensional manifolds
UDC 514.763.2+515.164.17 We study the divergence of multivector fields on Banach manifolds with a Radon measure.  We propose an infinite-dimensional version of divergence consistent with the classical divergence from  finite-dimensional differential geometry.&a...
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| author | Bogdanskii, Yu. Shram, V. Богданский, Юрий Bogdanskii, Yu. Shram, V. |
| author_facet | Bogdanskii, Yu. Shram, V. Богданский, Юрий Bogdanskii, Yu. Shram, V. |
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| description | UDC 514.763.2+515.164.17
We study the divergence of multivector fields on Banach manifolds with a Radon measure.  We propose an infinite-dimensional version of divergence consistent with the classical divergence from  finite-dimensional differential geometry.  We then transfer certain natural properties of the divergence operator to the infinite-dimensional setting.  Finally, we study the relation between the divergence operator ${\rm div}_M$ on a manifold $M$ and the divergence operator ${\rm div}_S$ on a submanifold  $S \subset M.$ |
| doi_str_mv | 10.37863/umzh.v74i12.6522 |
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DOI: 10.37863/umzh.v74i12.6522
UDC 514.763.2+515.164.17
Yu. Bogdanskii (Nat. Techn. Univ. Ukraine “Igor Sikorsky Kyiv Polytechnic Institute”),
V. Shram1 (Univ. Bonn, Germany)
DIVERGENCE OF MULTIVECTOR FIELDS
ON INFINITE-DIMENSIONAL MANIFOLDS
ДИВЕРГЕНЦIЯ БАГАТОВЕКТОРНИХ ПОЛIВ
НА НЕСКIНЧЕННОВИМIРНИХ МНОГОВИДАХ
We study the divergence of multivector fields on Banach manifolds with a Radon measure. We propose an infinite-
dimensional version of divergence consistent with the classical divergence from finite-dimensional differential geometry.
We then transfer certain natural properties of the divergence operator to the infinite-dimensional setting. Finally, we study
the relation between the divergence operator \mathrm{d}\mathrm{i}\mathrm{v}M on a manifold M and the divergence operator \mathrm{d}\mathrm{i}\mathrm{v}S on a submanifold
S \subset M.
Дослiджується дивергенцiя багатовекторних полiв на банахових многовидах iз мiрою Радона. Запропоновано не-
скiнченновимiрну версiю дивергенцiї, яка узгоджується з класичним оператором дивергенцiї, що розглядається в
скiнченновимiрнiй диференцiальнiй геометрiї. Низку природних властивостей дивергенцiї перенесено на нескiнчен-
новимiрний випадок. Крiм того, дослiджено зв’язок мiж оператором дивергенцiї \mathrm{d}\mathrm{i}\mathrm{v}M на многовидi M i оператором
дивергенцiї \mathrm{d}\mathrm{i}\mathrm{v}S на пiдмноговидi S \subset M.
1. Classical divergence. Let M be an orientable differentiable real n-dimensional manifold of class
C2. A choice of a volume form \Omega on M gives rise to the divergence operator, which is defined as
follows. For a vector field \bfitX (of class C1), \mathrm{d}\mathrm{i}\mathrm{v}\bfitX is the function on M such that
\mathrm{d}\mathrm{i}\mathrm{v}\bfitX \cdot \Omega = \mathrm{d} i\bfitX \Omega , (1)
where i\bfitX denotes the interior product of a differential form by a vector field \bfitX (namely,
i\bfitX \omega (\bfitZ \bfone , . . . ,\bfitZ \bfitk - \bfone ) = \omega (\bfitX ,\bfitZ \bfone , . . . ,\bfitZ \bfitk - \bfone )).
For a decomposable m-vector field
# »
\bfitX = \bfitX \bfone \wedge . . .\wedge \bfitX \bfitm and a differential k-form \omega , the interior
product i #»
\bfitX \omega = i(
# »
\bfitX )\omega of \omega by
# »
\bfitX is given by
i #»
\bfitX \omega := i\bfitX \bfitm . . . i\bfitX \bfone \omega , if m \leq k, (2)
and
i #»
\bfitX \omega := 0, if m > k.
Throughout this paper, by an m-vector field of class Cp we mean a linear combination of
decomposable m-vector fields
\sum
i
ci\bfitZ
\bfiti
\bfone \wedge . . .\wedge \bfitZ \bfiti
\bfitm , where all \bfitZ \bfiti
\bfitj \in Cp(M). That said, one might
notice that some of the definitions and results in the article can also be transferred to multivector
fields understood in a broader sense.
In an obvious way the above definition of i #»
\bfitX extends to an arbitrary multivector field
# »
\bfitX .
This operation satisfies the following property: for any k-vector field
# »
\bfitX , m-vector field
#»
\bfitZ and
differential (k +m)-form \omega , one has the equality
1 Corresponding author, e-mail: shram.vladyslav@gmail.com.
c\bigcirc Yu. BOGDANSKII , V. SHRAM, 2022
1640 ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 12
DIVERGENCE OF MULTIVECTOR FIELDS ON INFINITE-DIMENSIONAL MANIFOLDS 1641
\langle i #»
\bfitX \omega ,
#»
\bfitZ \rangle = \langle \omega , # »
\bfitX \wedge #»
\bfitZ \rangle ,
where \langle \cdot , \cdot \rangle denotes the natural pairing between differential forms and multivector fields of the same
degree.
Then the divergence \mathrm{d}\mathrm{i}\mathrm{v}
# »
\bfitX of a k-vector field
# »
\bfitX is defined by the identity (see, for example,
[6] for an equivalent definition in terms of the Hodge operator)
i\mathrm{d}\mathrm{i}\mathrm{v} #»
\bfitX \Omega = ( - 1)k - 1 \mathrm{d} i #»
\bfitX \Omega . (3)
Remark 1. In principle, we could define the interior product by a multivector field in a different
way, namely i\prime \bfitX \bfone \wedge ...\wedge \bfitX \bfitm
= i\bfitX \bfone \circ . . . \circ i\bfitX \bfitm . In this case, Eq. (3) from the definition of divergence
becomes i\prime
\mathrm{d}\mathrm{i}\mathrm{v}
#»
\bfitX
\Omega = \mathrm{d} i\prime #»
\bfitX
\Omega . However, in this paper, we always use the definition of interior product
i #»
\bfitX given by (2).
The existence of \mathrm{d}\mathrm{i}\mathrm{v}
# »
\bfitX for a multivector field
# »
\bfitX will follow from Proposition 1, and the
uniqueness follows from general facts of multilinear algebra (see, for example, [5], Chapter III).
Let M be a manifold of class C3. Given a (k+ 1)-vector field
# »
\bfitX of class C2 and a differential
k-form \omega of class C2
0 (that is, \omega \in C2(M) and is boundedly supported) on M, Stokes’ theorem
implies
\int
M
\mathrm{d}(\omega \wedge i #»
\bfitX \Omega ) = 0, which can be written as\int
M
\mathrm{d}\omega \wedge i #»
\bfitX \Omega = ( - 1)k+1
\int
M
\omega \wedge \mathrm{d} i #»
\bfitX \Omega . (4)
Lemma 1. Let \omega and
# »
\bfitX be a differential k-form and a k-vector field on M, respectively. Then
the following equality holds:
\omega \wedge i #»
\bfitX \Omega = \langle \omega , # »
\bfitX \rangle \Omega . (5)
Proof. Without loss of generality we may assume that
# »
\bfitX is decomposable:
# »
\bfitX = \bfitX \bfone \wedge . . .\wedge \bfitX \bfitk .
We have
\omega \wedge i #»
\bfitX \Omega = \omega \wedge (i\bfitX \bfitk
. . . i\bfitX \bfone \Omega ) = ( - 1)k - 1(i\bfitX \bfitk
\omega ) \wedge (i\bfitX \bfitk - \bfone
. . . i\bfitX \bfone \Omega ) = . . .
. . . = ( - 1)
(k - 1)k
2 (i\bfitX \bfone . . . i\bfitX \bfitk
\omega ) \wedge \Omega = (i\bfitX \bfitk
. . . i\bfitX \bfone \omega ) \wedge \Omega = \langle \omega , # »
\bfitX \rangle \Omega .
Let \mu be a measure on M induced by the volume form \Omega (for f \in C1(M), one has
\int
M
f d\mu =
=
\int
M
f\Omega ). Given a differential k-form \omega of class C2
0 and a (k+ 1)-vector field
# »
\bfitX of class C2, by
(4) and (5), we get\int
M
\langle \mathrm{d}\omega , # »
\bfitX \rangle d\mu =
\int
M
\mathrm{d}\omega \wedge i #»
\bfitX \Omega = ( - 1)k+1
\int
M
\omega \wedge \mathrm{d} i #»
\bfitX \Omega = -
\int
M
\omega \wedge i\mathrm{d}\mathrm{i}\mathrm{v} #»
\bfitX \Omega = -
\int
M
\langle \omega ,\mathrm{d}\mathrm{i}\mathrm{v} # »
\bfitX \rangle d\mu .
Thus, (4) is equivalent to \int
M
\langle \mathrm{d}\omega , # »
\bfitX \rangle d\mu = -
\int
M
\langle \omega ,\mathrm{d}\mathrm{i}\mathrm{v} # »
\bfitX \rangle d\mu . (6)
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 12
1642 Yu. BOGDANSKII , V. SHRAM
Using the measure \mu , one can now view the divergence of a (k + 1)-vector field
# »
\bfitX on M as a
k-vector field which satisfies (6) for any differential k-form of class C2
0 . For a manifold of class C3,
this leads to a definition of \mathrm{d}\mathrm{i}\mathrm{v}
# »
\bfitX which is equivalent to the original one.
Proposition 1. Let \bfitX and
#»
\bfitZ be a vector field and a k-vector field of class C1 on M, respec-
tively. Then one has the formula
\mathrm{d}\mathrm{i}\mathrm{v}(\bfitX \wedge #»
\bfitZ ) = \mathrm{d}\mathrm{i}\mathrm{v}\bfitX \cdot #»
\bfitZ - \bfitX \wedge \mathrm{d}\mathrm{i}\mathrm{v}
#»
\bfitZ + \scrL \bfitX
#»
\bfitZ , (7)
where \scrL \bfitX denotes the Lie derivation along the field \bfitX .
Proof. It suffices to prove formula (7) only for a decomposable multivector field
#»
\bfitZ = \bfitZ \bfone \wedge . . .
. . . \wedge \bfitZ \bfitk . We have
( - 1)k \mathrm{d} i\bfitX \wedge #»
\bfitZ \Omega = \mathrm{d} i #»
\bfitZ \wedge \bfitX \Omega = \mathrm{d} i\bfitX (i #»
\bfitZ \Omega ) = - i\bfitX \mathrm{d}(i #»
\bfitZ \Omega ) + \scrL \bfitX (i #»
\bfitZ \Omega ).
For the first term on the right-hand side we get
- i\bfitX \mathrm{d}(i #»
\bfitZ \Omega ) = - ( - 1)k - 1i\bfitX i\mathrm{d}\mathrm{i}\mathrm{v} #»
\bfitZ \Omega = - ( - 1)k - 1i\mathrm{d}\mathrm{i}\mathrm{v} #»
\bfitZ \wedge \bfitX \Omega = - i\bfitX \wedge \mathrm{d}\mathrm{i}\mathrm{v} #»
\bfitZ \Omega .
For the second term
\scrL \bfitX (i #»
\bfitZ \Omega ) = \scrL \bfitX (i\bfitZ \bfitk
. . . i\bfitZ \bfone \Omega ) = i\bfitZ \bfitk
\scrL \bfitX (i\bfitZ \bfitk - \bfone
. . . i\bfitZ \bfone \Omega ) + i\scrL \bfitX \bfitZ \bfitk
(i\bfitZ \bfitk - \bfone
. . . i\bfitZ \bfone \Omega ) = . . .
. . . = i\bfitZ \bfitk
. . . i\bfitZ \bfone \scrL \bfitX \Omega +
k\sum
r=1
i\bfitZ \bfitk
. . . i\scrL \bfitX \bfitZ \bfitr . . . i\bfitZ \bfone \Omega = i #»
\bfitZ \mathrm{d} i\bfitX \Omega +
k\sum
r=1
i\bfitZ \bfone \wedge ...\wedge \scrL \bfitX \bfitZ \bfitr \wedge ...\wedge \bfitZ \bfitk
\Omega =
= i #»
\bfitZ \mathrm{d}\mathrm{i}\mathrm{v}\bfitX \cdot \Omega + i\scrL \bfitX
#»
\bfitZ \Omega = i\mathrm{d}\mathrm{i}\mathrm{v}\bfitX \cdot #»
\bfitZ \Omega + i\scrL \bfitX
#»
\bfitZ \Omega = i\mathrm{d}\mathrm{i}\mathrm{v}\bfitX \cdot #»
\bfitZ +\scrL \bfitX
#»
\bfitZ \Omega .
Putting the two terms together, we obtain the identity (7).
Corollary 1. The divergence of a k-vector field (of class Cp) exists and is a (k - 1)-vector field
(of class Cp - 1).
Proof. The statement immediately follows from formula (7).
Given a differential k-form \omega and a decomposable m-vector field
# »
\bfitX = \bfitX \bfone \wedge . . . \wedge \bfitX \bfitm , one
defines the interior product j\omega
# »
\bfitX = j(\omega )
# »
\bfitX of
# »
\bfitX by \omega as follows:
j\omega
# »
\bfitX :=
1
k!(m - k)!
\sum
\sigma \in Sm
\mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}(\sigma )\omega (\bfitX \bfitsigma (\bfone ), . . . ,\bfitX \bfitsigma (\bfitk ))\bfitX \bfitsigma (\bfitk +\bfone ) \wedge . . . \wedge \bfitX \bfitsigma (\bfitm ), if k \leq m,
and
j\omega
# »
\bfitX := 0, if k > m.
In an obvious way this definition then extends to an arbitrary multivector field
# »
\bfitX . For a similar
definition, see, for example, [12].
The interior product of a multivector field by a differential form satisfies the following property:
for any differential k-form \omega , differential m-form \eta and (k +m)-vector field
# »
\bfitX , one has
\langle \eta , j\omega
# »
\bfitX \rangle = \langle \omega \wedge \eta , # »
\bfitX \rangle . (8)
One can prove the following generalisation of Lemma 1 (see [6]): for any differential k-form \omega
and m-vector field
# »
\bfitX , the following relation holds:
ij(\omega ) #»
\bfitX \Omega = ( - 1)k(m+1)\omega \wedge i #»
\bfitX \Omega . (9)
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 12
DIVERGENCE OF MULTIVECTOR FIELDS ON INFINITE-DIMENSIONAL MANIFOLDS 1643
Proposition 2. Let \omega and
# »
\bfitX be a differential k-form and an m-vector field (k < m), respec-
tively. Then the Leibniz rule holds
\mathrm{d}\mathrm{i}\mathrm{v}(j(\omega )
# »
\bfitX ) = ( - 1)kj(\mathrm{d}\omega )
# »
\bfitX + ( - 1)kj(\omega ) \mathrm{d}\mathrm{i}\mathrm{v}
# »
\bfitX .
Proof. Using (9), we have
( - 1)m - k - 1 \mathrm{d} ij(\omega ) #»
\bfitX \Omega = ( - 1)m - k - 1+k(m+1) \mathrm{d}\omega \wedge i #»
\bfitX \Omega + ( - 1)m - k - 1+k(m+1)+k\omega \wedge \mathrm{d} i #»
\bfitX \Omega =
= ( - 1)km+m - 1 \mathrm{d}\omega \wedge i #»
\bfitX \Omega + ( - 1)km+k\omega \wedge \mathrm{d} i\mathrm{d}\mathrm{i}\mathrm{v} #»
\bfitX \Omega =
= ( - 1)km+m - 1+(k+1)(m+1)ij(\mathrm{d}\omega )
#»
\bfitX \Omega + ( - 1)km+k+kmij(\omega ) \mathrm{d}\mathrm{i}\mathrm{v} #»
\bfitX \Omega =
= ( - 1)kij(\mathrm{d}\omega )
#»
\bfitX \Omega + ( - 1)kij(\omega ) \mathrm{d}\mathrm{i}\mathrm{v} #»
\bfitX \Omega .
2. Associated measures on Banach manifolds (see [1, 3]). Let M be a connected Hausdorff
real Banach manifold of class C2 with a model space E.
We say that an atlas \scrA =
\bigl\{
(U\alpha , \varphi \alpha )
\bigr\}
on M is bounded if there exists a real number K > 0
such that, for any pair of charts (U\alpha , \varphi \alpha ) and (U\beta , \varphi \beta ), the transition map F\beta \alpha = \varphi \beta \circ \varphi - 1
\alpha satisfies
the condition \bigl(
x \in \varphi \alpha (U\alpha \cap U\beta )
\bigr)
=\Rightarrow
\bigl(
\| F \prime
\beta \alpha (x)\| \leq K, \| F \prime \prime
\beta \alpha (x)\| \leq K
\bigr)
.
We then say that two bounded atlases \scrA 1 and \scrA 2 are equivalent if \scrA 1 \cup \scrA 2 is again a bounded
atlas. A bounded structure (of class C2) on M is defined as an equivalence class of bounded atlases
on M.
Let (M1,\scrA 1) and (M2,\scrA 2) be Banach manifolds M1 and M2 of class C2 modelled on E1 and
E2 together with bounded atlases \scrA 1 and \scrA 2, respectively. We say that a map f : M1 \rightarrow M2 is a
bounded morphism if there exists a real number C > 0 such that for any pair of charts (U,\varphi ) \in \scrA 1
and (V, \psi ) \in \scrA 2, the following condition is satisfied:
(p \in U, f(p) \in V ) =\Rightarrow
\Bigl( \bigm\| \bigm\| (\psi \circ f \circ \varphi - 1)(k)(\varphi (p))
\bigm\| \bigm\| \leq C for k = 1, 2
\Bigr)
.
In a natural way one then defines a bounded isomorphism between (M1,\scrA 1) and (M2,\scrA 2).
The property of being a bounded morphism does not depend on the choice of representatives of
the corresponding equivalence classes of bounded atlases on M1 and M2.
A choice of a bounded atlas on M leads to a well-defined notion of the length L(\Gamma ) of a
piecewise-smooth curve \Gamma in M. The corresponding intrinsic metric \rho is consistent with the ori-
ginal topology. A bounded morphism f : (M1,\scrA 1) \rightarrow (M2,\scrA 2) is Lipschitz with respect to the
corresponding intrinsic metrics.
A choice of a bounded atlas also allows to introduce a norm | | | \cdot p| | | on the tangent space TpM
to the manifold M, defined by | | | \xi p| | | := \mathrm{s}\mathrm{u}\mathrm{p}\alpha \| \xi \varphi \alpha \| , where
\bigl\{
(U\alpha , \varphi \alpha )
\bigr\}
is the set of charts of the
original atlas for which p \in U\alpha , and \xi \varphi \in E is the representation of a tangent vector \xi in a chart \varphi .
Furthermore, one has the property of uniform topological isomorphism of the spaces TpM and the
model space E, namely \| \xi \varphi \| \leq | | | \xi p| | | \leq K\| \xi \varphi \| , where K is the constant from the definition of a
bounded atlas, and \varphi is a chart at the point p \in M.
Remark 2. One can prove that a bounded structure on a manifold is a special case of a Finsler
structure (in this case the assignment \langle p, \xi \rangle \mapsto \rightarrow | | | \xi p| | | is a continuous function on the tangent bundle
TM ). However, in order to get the result of Theorem 2 below, it appears that further restrictions on
the Finsler structure are needed.
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 12
1644 Yu. BOGDANSKII , V. SHRAM
By a differential k-form on M of class C1 we mean a C1-section of the bundle Lk
\mathrm{a}\mathrm{l}\mathrm{t}(TM) \rightarrow M,
where Lk
\mathrm{a}\mathrm{l}\mathrm{t}(TM) is obtained by bundling together the spaces Lk
\mathrm{a}\mathrm{l}\mathrm{t}(TpM) of all bounded alternating
k-linear forms on TpM, so that the space Lk
\mathrm{a}\mathrm{l}\mathrm{t}(TpM) is the fibre at p \in M of this bundle.
On a manifold with a bounded atlas (M,\scrA ) one has a well-defined notion of a bounded vector
field \bfitX of class C1. Namely, \bfitX is said to be of class C1
b (M) if there exists a real number C > 0
such that for any chart (U,\varphi ), the local representation \bfitX \varphi of \bfitX satisfies
\bigm\| \bigm\| \bfitX \varphi (\varphi (x))
\bigm\| \bigm\| \leq C and\bigm\| \bigm\| \bfitX \prime
\varphi (\varphi (x))
\bigm\| \bigm\| \leq C for all x \in U. Boundedness of a vector field does not depend on the choice of
a bounded atlas from the corresponding equivalence class. In the same way one defines differential
forms of class C1
b (M). Finally, in a similar fashion we can also define smooth functions of class Cp
b ,
p = 0, 1, 2, Cb = C0
b . We will use this same notation also in the case when the domain of a field,
differential form or a function is a connected open subset V in M, in E or in a surface in M. A
vector field (resp., differential form) of class C1
b (V ) is said to be of class C1
0 (V ) if its support is
bounded and contained in V together with its \varepsilon -neighbourhood for some \varepsilon > 0.
We say that a bounded atlas \scrA is uniform if there exists a real number r > 0 such that for any
p \in M, there is a chart (U,\varphi ) \in \scrA such that \varphi (U) contains a ball of radius r in E centred at \varphi (p)
[1, 7, 11].
An intrinsic metric on M, induced by a uniform atlas, makes M into a complete metric space.
Furthermore, if a bounded atlas is equivalent to a uniform one, then the metric induced by this atlas is
also complete. If an equivalence class of atlases which defines a bounded structure on M contains a
uniform atlas, we call such a structure uniform. If manifolds M1 and M2 are boundedly isomorphic,
then their structures are either both uniform or nonuniform.
The flow \Phi (t, x) of a vector field \bfitX of class C1
b on a manifold M with a uniform structure is
defined on \BbbR \times M [11, p. 92].
If V is an open subset of \BbbR m, then, given a manifold with a bounded atlas (M,\scrA ), we agree
to define a bounded structure on M \times V (with a model space E \oplus \BbbR m) by the atlas \scrA \times \mathrm{i}\mathrm{d} =
=
\bigl\{
(U \times V, \varphi \times \mathrm{i}\mathrm{d}) : (U,\varphi ) \in \scrA
\bigr\}
.
An elementary surface S \subset M of codimension m is defined as follows. Let N be a manifold
with a bounded structure modelled on a subspace E1 of E of codimension m (from now on we
identify E with E1\oplus \BbbR m). Let V be an open neighbourhood of
#»
0 \in \BbbR m, and g : N\times V \rightarrow U \subset M
be a bounded (straightening) isomorphism onto an open subset U in M. Then, by definition, an
elementary surface is S = g(N \times \{ #»
0 \} ).
For \varepsilon > 0, we define
S - \varepsilon := S \cap
\bigl\{
x : \rho (x,M \setminus U) \geq \varepsilon
\bigr\}
.
Then S =
\infty \bigcup
n=1
S - 1
n
.
We say that a differential m-form \omega of class C1
b defined on U is an associated m-form of the
embedding S \subset M if for any x \in S, the tangent space TxS is an associated subspace of the exterior
form \omega (x) in TxM (i.e., TxS = \{ Y \in TxM : iY \omega (x) = 0\} , where iY is the interior product of an
exterior form by a vector Y ).
If g : N \times V \rightarrow U is a straightening isomorphism of an elementary surface S, P is a projection
of N \times V onto V, and h is a continuously differentiable function on V such that h(
#»
0 ) \not = 0, then
\omega = (g - 1)\ast P \ast (h dt1 \wedge . . .\wedge dtm) is an example of an associated m-form of the embedding S \subset M.
Note that the constructed m-form \omega is closed.
Let us now consider a Borel measure \mu on M. The associated measure \sigma = \sigma #»
\bfitY on S is
constructed as follows.
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 12
DIVERGENCE OF MULTIVECTOR FIELDS ON INFINITE-DIMENSIONAL MANIFOLDS 1645
We first consider a strictly transversal to S system
#»
\bfitY = \{ \bfitY \bfone , . . . ,\bfitY \bfitm \} of pairwise commuting
vector fields of class C1
b defined on U. Strict transversality of
#»
\bfitY is understood in the following
sense: for each \varepsilon > 0, there exists \delta > 0 such that for any x \in S - \varepsilon , one has | \omega ( #»
\bfitY )(x)| =
= | \omega (\bfitY \bfone , . . . ,\bfitY \bfitm )(x)| \geq \delta . Existence of such a system of fields was proved in [3].
Let \Phi \bfitY \bfitk
t denote the flow of \bfitY \bfitk . We then define \Phi
#»
\bfitY
#»
t
:= \Phi \bfitY \bfone
t1
. . .\Phi \bfitY \bfitm
tm . One has the property
\Phi
#»
\bfitY
#»
t + #»s
= \Phi
#»
\bfitY
#»
t
\Phi
#»
\bfitY
#»s .
For Borel sets W \in \scrB (\BbbR m) and A \in \scrB (M), the set \Phi WA = \Phi
#»
\bfitY
WA := \{ \Phi
#»
\bfitY
#»
t
(x) :
#»
t \in W,
x \in A\} is a Borel in M. Furthermore, for each \varepsilon > 0, there exists p > 0 such that (A \in \scrB (S - \varepsilon ),
W \in \scrB (Bp)) =\Rightarrow
\bigl(
\Phi
#»
\bfitY
WA \in \scrB (U)
\bigr)
, where Bp = \{ #»
t : \| #»
t \| < p\} \subset \BbbR m. For any set B \in \scrB (Bp),
we define a measure \nu B on \scrB (S - \varepsilon ) by \nu B(A) := \mu (\Phi
#»
\bfitY
BA).
Let \lambda m denote the Lebesgue measure on \BbbR m. If, for any A \in \scrB (S - \varepsilon ), the following limit exists:
\sigma (A) = \sigma #»
\bfitY (A) = \mathrm{l}\mathrm{i}\mathrm{m}
r\rightarrow 0
\nu Br(A)
\lambda m(Br)
, (10)
then Nikodym’s theorem implies that the map \scrB (S - \varepsilon ) \ni A \mapsto \rightarrow \sigma #»
\bfitY (A) \in \BbbR is a Borel measure on
S - \varepsilon . Writing A \in \scrB (S) in the form A =
\infty \bigcup
n=1
(A \cap S - 1
n
) allows to extend the measure \sigma #»
\bfitY to \scrB (S).
Sufficient conditions for existence of the limit (10) were established in [3]; the authors suggested
to call \sigma #»
\bfitY the surface measure on S of the first kind induced by the system of vector fields
#»
\bfitY .
Throughout the remainder of this paper we always assume that the surface measure exists.
Given \varepsilon > 0 and r > 0, let \sigma r denote the measure on \scrB (S - \varepsilon ) defined by
\sigma r(A) :=
1
\lambda m(Br)
\mu (\Phi BrA).
Then (10) implies that \sigma r(A) \rightarrow \sigma (A) as r \rightarrow 0 for any Borel set A \subset S - \varepsilon .
The following two lemmas were proved in [2].
Lemma 2. Suppose that \mu is a Radon measure on M. Then for any \varepsilon > 0, \sigma r and \sigma are Radon
measures on S - \varepsilon .
Lemma 3. Suppose that \mu is a (nonnegative) Radon measure on M, and u \in Cb(M). Then,
for any \varepsilon > 0 and A \in \scrB (S - \varepsilon ), the following equality holds:
\mathrm{l}\mathrm{i}\mathrm{m}
r\rightarrow 0
1
\lambda m(Br)
\int
\Phi BrA
u d\mu =
\int
A
u d\sigma .
3. Multivector fields and divergence operator. The notion of the divergence of a vector
field (as given by formula (1)) was generalized by Daletskii and Maryanin [8] to a certain class of
Banach manifolds, resulting in the so-called divergence with respect to a measure. In that work the
divergence of a vector field \bfitX with respect to a measure \mu was defined as the logarithmic derivative
of \mu along the vector field \bfitX .
In this section, we propose a definition of divergence of multivector fields on a Banach manifold,
which generalizes the finite-dimensional divergence as given by formula (3). We then establish some
of the properties which this new divergence operator satisfies.
Consider a Banach manifold M with a bounded structure and a (nonnegative) Borel measure \mu
on M. We say that a k-vector field
#»
\bfitZ on M is \mu -measurable if there exists a sequence of continuous
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 12
1646 Yu. BOGDANSKII , V. SHRAM
k-vector fields
#»
\bfitZ \bfitn such that \mathrm{l}\mathrm{i}\mathrm{m}n\rightarrow \infty
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| #»
\bfitZ \bfitn (p) -
#»
\bfitZ (p)p
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| = 0 (\mathrm{m}\mathrm{o}\mathrm{d}\mu ) (here | | | \cdot p| | | is the norm on\bigwedge k(TpM) induced by the corresponding norm | | | \cdot p| | | on TpM, see Section 2).
For a measurable multivector field
#»
\bfitZ , the function x \mapsto \rightarrow
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| #»
\bfitZ (x)x
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| is \mu -measurable on M. In
the case, when this function is integrable on M with respect to \mu , we say that
#»
\bfitZ is integrable:
#»
\bfitZ \in L1(\mu ) (see [4]). In a similar way one defines multivector fields of class Lp(\mu ) for 1 < p \leq \infty .
It is easy to check that if vector fields \bfitZ \bftwo , . . . ,\bfitZ \bfitk are measurable and bounded on M, and
\bfitZ \bfone is a vector field of class Lp(\mu ), then \bfitZ \bfone \wedge . . . \wedge \bfitZ \bfitk \in Lp(\mu ). One can also prove that if
\bfitZ \bfone \wedge . . .\wedge \bfitZ \bfitk \in Lp(\mu ), and \omega is a differential form of class Cb(M), then \langle \omega ,\bfitZ \bfone \wedge . . .\wedge \bfitZ \bfitk \rangle \in Lp(\mu ).
Let Lp
\bigwedge k(\mu ) denote the set of all linear combinations of decomposable k-vector fields of class
Lp(\mu ) (modulo the measure \mu ).
Definition 1. Let
#»
\bfitZ \in L1
\bigwedge k(\mu ). We call a (k - 1)-vector field
# »
\bfitW \in L1
\bigwedge k - 1(\mu ) a divergence
of
#»
\bfitZ
\bigl( # »
\bfitW = \mathrm{d}\mathrm{i}\mathrm{v}
#»
\bfitZ ;
#»
\bfitZ \in D(\mathrm{d}\mathrm{i}\mathrm{v})
\bigr)
if for any differential (k - 1)-form \omega \in C1
0 (M), the following
equality holds: \int
M
\langle \omega , # »
\bfitW \rangle d\mu = -
\int
M
\langle \mathrm{d}\omega , #»
\bfitZ \rangle d\mu . (11)
Uniqueness of the divergence is provided by the following theorem, which was proved in [2].
Theorem 1. Suppose that there exists a function of class C1 on E with nonempty bounded
support (it suffices to assume that E is reflexive, see [10]), and \mu is a Radon measure. Then, given a
k-vector field
#»
\bfitZ \in L1
\bigwedge k(\mu ), there cannot exist two distinct elements of L1
\bigwedge k - 1(\mu ), both of which
are divergences of
#»
\bfitZ .
Remark 3. Unlike in the finite-dimensional case, divergence need not exist in general. Thus, one
encounters the problem of describing, for a given measure, the class of (multi-)vector fields admitting
the divergence.
From now on we always assume that the assumptions of Theorem 1 are satisfied. Let us now
prove the infinite-dimensional analogues of Propositions 1 and 2.
Remark 4. Throughout this paper, by a k-vector field of class C1
b (M) we mean a linear combi-
nation of decomposable k-vector fields
\sum
i
ci\bfitZ
\bfiti
\bfone \wedge . . . \wedge \bfitZ \bfiti
\bfitk , where all \bfitZ \bfiti
\bfitj \in C1
b (M).
Proposition 3. Suppose that a vector field \bfitX and a k-vector field
#»
\bfitZ lie in C1
b (M) \cap D(\mathrm{d}\mathrm{i}\mathrm{v}).
Then \bfitX \wedge #»
\bfitZ \in C1
b (M) \cap D(\mathrm{d}\mathrm{i}\mathrm{v}) and the following identity holds:
\mathrm{d}\mathrm{i}\mathrm{v}(\bfitX \wedge #»
\bfitZ ) = \mathrm{d}\mathrm{i}\mathrm{v}\bfitX \cdot #»
\bfitZ - \bfitX \wedge \mathrm{d}\mathrm{i}\mathrm{v}
#»
\bfitZ + \scrL \bfitX
#»
\bfitZ . (12)
Proof. Let \omega be a differential k-form of class C1
0 on M. One has the equality
\langle \mathrm{d}\omega ,\bfitX \wedge #»
\bfitZ \rangle = \langle i\bfitX \mathrm{d}\omega ,
#»
\bfitZ \rangle = \bfitX \langle \omega , #»
\bfitZ \rangle - \langle \mathrm{d} i\bfitX \omega ,
#»
\bfitZ \rangle - \langle \omega ,\scrL \bfitX
#»
\bfitZ \rangle . (13)
Now, by combining (11) and (13), we get\int
M
\langle \mathrm{d}\omega ,\bfitX \wedge #»
\bfitZ \rangle d\mu = -
\int
M
\langle \omega , - \mathrm{d}\mathrm{i}\mathrm{v}\bfitX \cdot #»
\bfitZ +\bfitX \wedge \mathrm{d}\mathrm{i}\mathrm{v}
#»
\bfitZ - \scrL \bfitX
#»
\bfitZ \rangle d\mu ,
which proves the proposition.
Corollary 2. If
#»
\bfitZ = \bfitZ \bfone \wedge . . .\wedge \bfitZ \bfitk , and all \bfitZ \bfiti \in C1
b (M)\cap D(\mathrm{d}\mathrm{i}\mathrm{v}), then
#»
\bfitZ \in C1
b (M)\cap D(\mathrm{d}\mathrm{i}\mathrm{v}).
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DIVERGENCE OF MULTIVECTOR FIELDS ON INFINITE-DIMENSIONAL MANIFOLDS 1647
Proposition 4. Suppose that an m-vector field
#»
\bfitZ \in D(\mathrm{d}\mathrm{i}\mathrm{v}), and let \omega be a differential k-form
(k < m) of class C1
b (M). Then j(\omega )
#»
\bfitZ \in D(\mathrm{d}\mathrm{i}\mathrm{v}), and the following Leibniz rule holds:
\mathrm{d}\mathrm{i}\mathrm{v}(j(\omega )
#»
\bfitZ ) = ( - 1)kj(\mathrm{d}\omega )
#»
\bfitZ + ( - 1)kj(\omega ) \mathrm{d}\mathrm{i}\mathrm{v}
#»
\bfitZ .
Proof. For any differential (m - k - 1)-form \eta of class C1
0 (M), using identities (8) and (11),
we have \int
M
\Bigl( \Bigl\langle
\mathrm{d} \eta , j(\omega )
#»
\bfitZ
\Bigr\rangle
+
\Bigl\langle
\eta , ( - 1)kj(\mathrm{d}\omega )
#»
\bfitZ + ( - 1)kj(\omega ) \mathrm{d}\mathrm{i}\mathrm{v}
#»
\bfitZ
\Bigr\rangle \Bigr)
d\mu =
=
\int
M
\Bigl(
\langle \omega \wedge \mathrm{d} \eta ,
#»
\bfitZ \rangle + ( - 1)k\langle \mathrm{d}\omega \wedge \eta , #»
\bfitZ \rangle + ( - 1)k\langle \omega \wedge \eta ,\mathrm{d}\mathrm{i}\mathrm{v} #»
\bfitZ \rangle
\Bigr)
d\mu =
=
\int
M
\Bigl(
( - 1)k\langle \mathrm{d}(\omega \wedge \eta ), #»
\bfitZ \rangle + ( - 1)k\langle \omega \wedge \eta ,\mathrm{d}\mathrm{i}\mathrm{v} #»
\bfitZ \rangle
\Bigr)
d\mu = 0.
4. Divergence on submanifolds. If M is a finite-dimensional (orientable) manifold endowed
with a volume form \Omega , and U is its open submanifold, then it is natural to take \Omega
\bigm| \bigm|
U
to be the volume
form on U. In this case one has the equality
\mathrm{d}\mathrm{i}\mathrm{v}U (
#»
\bfitZ
\bigm| \bigm|
U
) = (\mathrm{d}\mathrm{i}\mathrm{v}
#»
\bfitZ )
\bigm| \bigm|
U
, (14)
where \mathrm{d}\mathrm{i}\mathrm{v}U is the divergence on U induced by the volume form \Omega
\bigm| \bigm|
U
.
In the case, when U is an open submanifold of a Banach manifold M, the definition of divergence
\mathrm{d}\mathrm{i}\mathrm{v}U of a multivector field is obtained from Definition 1 by replacing (11) with\int
U
\langle \omega , # »
\bfitW \rangle d\mu = -
\int
U
\langle \mathrm{d}\omega , #»
\bfitZ \rangle d\mu ,
which now has to hold for any differential form of class C1
0 (U). In this case formula (14) also holds.
Let now M be an orientable manifold of finite dimension n; S \subset M an orientable embedded
submanifold of dimension m = n - p, which is an elementary surface in the sense of Section 2;
\alpha an associated differential p-form of the embedding S \subset M ;
#»
\bfitY = \{ \bfitY \bfone , . . . ,\bfitY \bfitp \} a commuting
strictly transversal to S system of vector fields of class C1
b (U), where U is from the definition of an
elementary surface.
For any \varepsilon > 0, there exists \gamma = \gamma (\varepsilon ) > 0 such that for each (
#»
t , x) \in B\gamma \times S - \varepsilon , one has
\Phi #»
t x \in U, and \langle \alpha , #»
\bfitY \rangle (\Phi #»
t x) \not = 0 (here B\gamma = \{ #»
t \in \BbbR p : \| #»
t \| < \gamma \} ).
Without loss of generality we may assume that \langle \alpha , #»
\bfitY \rangle (\Phi #»
t x) > 0. One has that the map q :
\Phi B\gamma S - \varepsilon \ni \Phi #»
t x \mapsto \rightarrow x \in S - \varepsilon is continuously differentiable.
Let \Omega = \Omega S be a volume form on S ; \bfitX a vector field on S ; \widetilde \bfitX the vector field on \Phi B\gamma S - \varepsilon
which is q-related to \bfitX (q\ast (\widetilde \bfitX (\Phi #»
t x)) = \bfitX (x)); \widetilde \Omega = q\ast \Omega a differential p-form on \Phi B\gamma S - \varepsilon .
Suppose that
# »
\bfitX = \bfitX \bfone \wedge . . . \wedge \bfitX \bfitm is a nowhere-vanishing multivector field on S - \varepsilon , and let
\beta = \widetilde \Omega \wedge \alpha . Then, for x \in S - \varepsilon ,
\langle \beta ,\widetilde # »
\bfitX \wedge #»
\bfitY \rangle (x) = \widetilde \Omega (\widetilde # »
\bfitX )(x) \cdot \alpha ( #»
\bfitY )(x) = (\Omega (
# »
\bfitX ) \cdot \alpha ( #»
\bfitY ))(x) > 0
(here we used (i\bfitX \bfitj
\alpha )(x) = 0). Choosing a smaller \gamma > 0 if needed, we conclude that \beta is a volume
form on \Phi B\gamma S - \varepsilon \subset M.
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1648 Yu. BOGDANSKII , V. SHRAM
Proposition 5. Let \bfitZ be a vector field of class C1
b on S, and let \mathrm{d}\mathrm{i}\mathrm{v}S \bfitZ be the divergence of
\bfitZ with respect to the volume form \Omega on S. Given \varepsilon > 0, let \widetilde \bfitZ be the vector field on \Phi B\gamma S - \varepsilon which
is q-related to \bfitZ , and let \mathrm{d}\mathrm{i}\mathrm{v} \widetilde \bfitZ be the divergence of \widetilde \bfitZ with respect to the volume form \beta . Suppose
that \alpha is closed. Then
\mathrm{d}\mathrm{i}\mathrm{v}S \bfitZ = (\mathrm{d}\mathrm{i}\mathrm{v} \widetilde \bfitZ )
\bigm| \bigm|
S
. (15)
Proof. Take x \in S - \varepsilon . The statement follows from the equalities
(\mathrm{d}\mathrm{i}\mathrm{v} \widetilde \bfitZ \cdot \beta )(x) = (\mathrm{d} i \widetilde \bfitZ (\widetilde \Omega \wedge \alpha ))(x) = (\mathrm{d} i\bfitZ \Omega )(x) \wedge \alpha (x) = (\mathrm{d}\mathrm{i}\mathrm{v}S \bfitZ \cdot \beta )(x).
Corollary 3. In the assumptions of Proposition 5, suppose that
#»
\bfitZ is a multivector field of class
C1
b on S; \widetilde #»
\bfitZ is the multivector field on V = \Phi B\gamma S - \varepsilon which is q-related to
#»
\bfitZ ; \mathrm{d}\mathrm{i}\mathrm{v}S and \mathrm{d}\mathrm{i}\mathrm{v} are the
divergence operators on (S,\Omega ) and (V, \beta ), respectively. Then
\mathrm{d}\mathrm{i}\mathrm{v}S
#»
\bfitZ = (\mathrm{d}\mathrm{i}\mathrm{v}
\widetilde #»
\bfitZ )
\bigm| \bigm|
S
. (16)
Proof. Formula (16) follows by induction from formula (15); recurrent formula (7), applied to
\mathrm{d}\mathrm{i}\mathrm{v}S(\bfitX \wedge #»
\bfitZ ) and \mathrm{d}\mathrm{i}\mathrm{v}(\widetilde \bfitX \wedge \widetilde #»
\bfitZ ); equalities \widetilde \bfitX \wedge #»
\bfitZ = \widetilde \bfitX \wedge \widetilde #»
\bfitZ and \widetilde \scrL \bfitX
#»
\bfitZ = \scrL \widetilde \bfitX \widetilde #»
\bfitZ .
Throughout the remainder of this article, M is a Banach manifold with a uniform atlas, modelled
on a space E, where E satisfies the assumptions of Theorem 1. Suppose that S is an elementary
surface in M of codimension m; \mu is a (nonnegative) Radon measure on M, and the corresponding
measure \sigma = \sigma #»
\bfitY on the surface S - \varepsilon \subset S is constructed as described in Section 2.
It follows from general theory of differential equations in Banach spaces that there exists
\gamma = \gamma (\varepsilon ) > 0 for which one has a well-defined map q : \Phi B\gamma S - \varepsilon \ni \Phi #»
t x \mapsto \rightarrow x \in S - \varepsilon of class
C1
b . Let \bfitZ be a vector field of class C1
b on S. Then the q-related vector field \widetilde \bfitZ is defined on
V = \Phi B\gamma S - \varepsilon and is also of class C1
b .
Theorem 2. Suppose that \widetilde \bfitZ admits the divergence \mathrm{d}\mathrm{i}\mathrm{v} \widetilde \bfitZ \in L\infty (V, \mu ). Then \bfitZ admits the
divergence \mathrm{d}\mathrm{i}\mathrm{v}S \bfitZ \in L\infty (S, \sigma ), and for any \varepsilon > 0 and a bounded Borel function u : S - \varepsilon \rightarrow \BbbR , we
have the identity \int
S - \varepsilon
u\mathrm{d}\mathrm{i}\mathrm{v}S \bfitZ d\sigma = \mathrm{l}\mathrm{i}\mathrm{m}
r\rightarrow 0
1
\lambda m(Br)
\int
\Phi BrS - \varepsilon
\widehat u\mathrm{d}\mathrm{i}\mathrm{v} \widetilde \bfitZ d\mu (17)
(here and henceforth \widehat u(\Phi #»
t x) = u(x) for (
#»
t , x) \in B\gamma \times S - \varepsilon ).
Proof. Step 1. Let u \in C1
0 (S). Then u \in C1
0 (S - \varepsilon ) for some \varepsilon > 0. We shall prove that, for
any r \in (0, \gamma ), the following holds:\int
\Phi BrS - \varepsilon
\widehat u\mathrm{d}\mathrm{i}\mathrm{v} \widetilde \bfitZ d\mu = -
\int
\Phi BrS - \varepsilon
\widetilde \bfitZ \widehat u d\mu . (18)
The function \widehat u is not of class C1
0 (V ). We will use the fact that \widetilde \bfitZ is tangent to the surface
\Phi #»
t S - \varepsilon for each
#»
t \in B\gamma .
Let us define a sequence of functions \varphi n \in C[0, r] for n > 3 as follows:
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DIVERGENCE OF MULTIVECTOR FIELDS ON INFINITE-DIMENSIONAL MANIFOLDS 1649
\varphi n(s) =
\left\{
0, if s \in
\biggl[
0,
n - 3
n
r
\biggr]
\cup
\biggl[
n - 1
n
r, r
\biggr]
,
- n
2
r2
s+
n(n - 3)
r
, if s \in
\biggl[
n - 3
n
r,
n - 2
n
r
\biggr]
,
n2
r2
s - n(n - 1)
r
, if s \in
\biggl[
n - 2
n
r,
n - 1
n
r
\biggr]
.
Then for the sequence of functions hn(s) = 1+
\int s
0
\varphi n(s) ds, one has that the functions un(\Phi #»
t x) =
= hn
\bigl(
\| #»
t \|
\bigr)
\cdot u(x) coincide with \widehat u\bigl( \Phi #»
t x
\bigr)
for \| #»
t \| \leq n - 3
n
r, and un \in C1
0 (\Phi BrS\varepsilon ).
Hence, we have \int
\Phi BrS - \varepsilon
un \mathrm{d}\mathrm{i}\mathrm{v} \widetilde \bfitZ d\mu = -
\int
\Phi BrS - \varepsilon
\widetilde \bfitZ un d\mu (19)
and
( \widetilde \bfitZ un)(\Phi #»
t x) = hn(\|
#»
t \| ) \cdot ( \widetilde \bfitZ \widehat u)(\Phi #»
t x) for x \in S - \varepsilon .
Passing in (19) to the limit as n\rightarrow \infty , we obtain (18).
Since the function \widetilde \bfitZ \widehat u \in Cb(\Phi B\gamma S - \varepsilon ), Lemma 3 implies the existence of the limit
\mathrm{l}\mathrm{i}\mathrm{m}
r\rightarrow 0
1
\lambda m(Br)
\int
\Phi BrS - \varepsilon
\widetilde \bfitZ \widehat u d\mu =
\int
S - \varepsilon
\bfitZ u d\sigma .
Therefore, using (18), we obtain the equality
\mathrm{l}\mathrm{i}\mathrm{m}
r\rightarrow 0
1
\lambda m(Br)
\int
\Phi BrS - \varepsilon
\widehat u\mathrm{d}\mathrm{i}\mathrm{v} \widetilde \bfitZ d\mu = -
\int
S - \varepsilon
\bfitZ u d\sigma , (20)
that holds for any function u \in C1
0 (S - \varepsilon ).
Step 2. The model space E1 of the manifold S has a finite codimension in E and therefore also
admits a function of class C1(E1) with bounded nonempty support. The argument used in the proof
of Theorem 1 also proves that there exists a family of functions \{ u\alpha \} of class C1
0 (S - \varepsilon ) such that the
sets U\alpha = \{ x : u\alpha (x) > 0\} constitute a base of the topology of S - \varepsilon .
For any choice of u \in \{ u\alpha \} , let U = \{ x : u(x) > 0\} be the corresponding set of this base.
Taking a sequence of smooth functions hn \in C1(\BbbR ) that approximate the Heaviside step function \chi ,
we construct a sequence of functions vn = hn \circ u for which \{ x : vn(x) > 0\} = U ; vn \nearrow 1U = \chi \circ u
and Vn = \{ x : vn(x) = 1\} \nearrow U (where 1U denotes the indicator function of U and the notation
Vn \nearrow U means that for any n \in \BbbN , Vn \subset Vn+1 and
\bigcup
n\in \BbbN Vn = U).
Nikodym’s theorem implies the uniform in r \in (0, \gamma ) convergence
\sigma r(U \setminus Vn) =
1
\lambda m(Br)
\mu (\Phi Br(U \setminus Vn)) \rightarrow 0, n\rightarrow \infty .
Since \mathrm{d}\mathrm{i}\mathrm{v} \widetilde \bfitZ \in L\infty (\mu ), one also has the uniform in r \in (0, \gamma ) convergence
1
\lambda m(Br)
\int
\Phi BrS - \varepsilon
\bigm| \bigm| \bigm| (\widehat vn - \widehat 1U ) \mathrm{d}\mathrm{i}\mathrm{v} \widetilde \bfitZ \bigm| \bigm| \bigm| d\mu \rightarrow 0, n\rightarrow \infty .
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1650 Yu. BOGDANSKII , V. SHRAM
This uniform convergence and the convergence (20), together with the inequality\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
1
\lambda m(Br)
\int
\Phi BrU
\mathrm{d}\mathrm{i}\mathrm{v} \widetilde \bfitZ d\mu - 1
\lambda m(Bs)
\int
\Phi BsU
\mathrm{d}\mathrm{i}\mathrm{v} \widetilde \bfitZ d\mu
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq
\leq 1
\lambda m(Br)
\int
\Phi BrS - \varepsilon
\bigm| \bigm| \bigm| (\widehat vn - \widehat 1U ) \mathrm{d}\mathrm{i}\mathrm{v} \widetilde \bfitZ \bigm| \bigm| \bigm| d\mu +
+
1
\lambda m(Bs)
\int
\Phi BsS - \varepsilon
\bigm| \bigm| \bigm| (\widehat vn - \widehat 1U ) \mathrm{d}\mathrm{i}\mathrm{v} \widetilde \bfitZ \bigm| \bigm| \bigm| d\mu +
+
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
1
\lambda m(Br)
\int
\Phi BrS - \varepsilon
\widehat vn \cdot \mathrm{d}\mathrm{i}\mathrm{v} \widetilde \bfitZ d\mu - 1
\lambda m(Bs)
\int
\Phi BsS - \varepsilon
\widehat vn \cdot \mathrm{d}\mathrm{i}\mathrm{v} \widetilde \bfitZ d\mu
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
allow us to conclude that the following limit exists:
\mathrm{l}\mathrm{i}\mathrm{m}
r\rightarrow 0
1
\lambda m(Br)
\int
\Phi BrU
\mathrm{d}\mathrm{i}\mathrm{v} \widetilde \bfitZ d\mu . (21)
Step 3. Let K be a compact subset of S - \varepsilon . Then there is a sequence of sets Un \in \{ U\alpha \} such
that Un \searrow K (i.e., for any n \in \BbbN , Un \supset Un+1 and
\bigcap
n\in \BbbN Un = K ).
Again, using Nikodym’s theorem and the fact that \mathrm{d}\mathrm{i}\mathrm{v} \widetilde \bfitZ \in L\infty (\mu ), we obtain uniform in r \in
\in (0, \gamma ) convergence
\mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
1
\lambda m(Br)
\int
\Phi Br (Un\setminus K)
\bigm| \bigm| \bigm| \mathrm{d}\mathrm{i}\mathrm{v} \widetilde \bfitZ \bigm| \bigm| \bigm| d\mu = 0.
From this uniform convergence and the convergence (21), together with the next inequality (here
r, s \in (0, \gamma )) \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
1
\lambda m(Br)
\int
\Phi BrK
\mathrm{d}\mathrm{i}\mathrm{v} \widetilde \bfitZ d\mu - 1
\lambda m(Bs)
\int
\Phi BsK
\mathrm{d}\mathrm{i}\mathrm{v} \widetilde \bfitZ d\mu
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq
\leq 1
\lambda m(Br)
\int
\Phi Br (Un\setminus K)
\bigm| \bigm| \bigm| \mathrm{d}\mathrm{i}\mathrm{v} \widetilde \bfitZ \bigm| \bigm| \bigm| d\mu +
1
\lambda m(Bs)
\int
\Phi Bs (Un\setminus K)
\bigm| \bigm| \bigm| \mathrm{d}\mathrm{i}\mathrm{v} \widetilde \bfitZ \bigm| \bigm| \bigm| d\mu +
+
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
1
\lambda m(Br)
\int
\Phi BrUn
\mathrm{d}\mathrm{i}\mathrm{v} \widetilde \bfitZ d\mu - 1
\lambda m(Bs)
\int
\Phi BsUn
\mathrm{d}\mathrm{i}\mathrm{v} \widetilde \bfitZ d\mu
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| ,
we conclude that the following limit exists:
\mathrm{l}\mathrm{i}\mathrm{m}
r\rightarrow 0
1
\lambda m(Br)
\int
\Phi BrK
\mathrm{d}\mathrm{i}\mathrm{v} \widetilde \bfitZ d\mu . (22)
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 12
DIVERGENCE OF MULTIVECTOR FIELDS ON INFINITE-DIMENSIONAL MANIFOLDS 1651
Step 4. Let A be an arbitrary Borel subset of S - \varepsilon . Let Kn be a non decreasing sequence of
compact subsets of A satisfying \sigma (A \setminus Kn) <
1
n
. Then, for C =
\infty \bigcap
n=1
(A \setminus Kn), one has \sigma (C) = 0,
and therefore
\mathrm{l}\mathrm{i}\mathrm{m}
r\rightarrow 0
1
\lambda m(Br)
\int
\Phi BrC
\bigm| \bigm| \bigm| \mathrm{d}\mathrm{i}\mathrm{v} \widetilde \bfitZ \bigm| \bigm| \bigm| d\mu = 0. (23)
Analogously to Step 3, we first obtain a uniform in r \in (0, \gamma ) convergence
\mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
1
\lambda m(Br)
\int
\Phi Br ((A\setminus C)\setminus Kn)
\bigm| \bigm| \bigm| \mathrm{d}\mathrm{i}\mathrm{v} \widetilde \bfitZ \bigm| \bigm| \bigm| d\mu = 0,
and then use (23) and the existence of the limit (22) in order to conclude that the following limit
exists:
\mathrm{l}\mathrm{i}\mathrm{m}
r\rightarrow 0
1
\lambda m(Br)
\int
\Phi BrA
\mathrm{d}\mathrm{i}\mathrm{v} \widetilde \bfitZ d\mu . (24)
Let now \tau r denote the measure on \scrB (S - \varepsilon ) defined by
\tau r(A) :=
1
\lambda m(Br)
\int
\Phi BrA
\mathrm{d}\mathrm{i}\mathrm{v} \widetilde \bfitZ d\mu .
Existence of the limit (24) means that for any Borel set A \in \scrB (S - \varepsilon ), there exists a limit
\mathrm{l}\mathrm{i}\mathrm{m}r\rightarrow 0 \tau r(A) =: \tau (A). Since \mathrm{d}\mathrm{i}\mathrm{v} \widetilde \bfitZ \in L\infty (\mu ), the measure \tau is absolutely continuous with re-
spect to \sigma , and, additionally, g\varepsilon =
d\tau
d\sigma
\in L\infty (S - \varepsilon , \sigma ), and
\| g\varepsilon \| L\infty (\sigma ) \leq \| \mathrm{d}\mathrm{i}\mathrm{v} \widetilde \bfitZ \| L\infty (\mu ). (25)
For any bounded Borel function u on S - \varepsilon , one has
\mathrm{l}\mathrm{i}\mathrm{m}
r\rightarrow 0
1
\lambda m(Br)
\int
\Phi BrS - \varepsilon
\widehat u\mathrm{d}\mathrm{i}\mathrm{v} \widetilde \bfitZ d\mu = \mathrm{l}\mathrm{i}\mathrm{m}
r\rightarrow 0
\int
S - \varepsilon
ud\tau r =
\int
S - \varepsilon
u \cdot g\varepsilon d\sigma . (26)
Since (26) holds for any bounded Borel function on S - \varepsilon , it follows that g\varepsilon 1 = g\varepsilon 2
\bigm| \bigm|
S - \varepsilon 1
for
\varepsilon 2 \in (0, \varepsilon 1) and, hence, there exists a Borel function g, defined on the whole of S, such that
g\varepsilon = g
\bigm| \bigm|
S - \varepsilon
for any \varepsilon > 0; moreover, by (25), g \in L\infty (S, \sigma ).
In particular, by (20), for any function u \in C1
0 (S), one has
-
\int
S
\bfitZ ud\sigma =
\int
S
u \cdot gd\sigma .
Therefore, there exists \mathrm{d}\mathrm{i}\mathrm{v}S \bfitZ = g on S ; \mathrm{d}\mathrm{i}\mathrm{v}S \bfitZ \in L\infty (\sigma ), and for any bounded Borel function u,
defined on S - \varepsilon for some \varepsilon > 0, equality (17) holds.
Theorem 2 is proved.
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 12
1652 Yu. BOGDANSKII , V. SHRAM
Remark 5. Analogously to Lemma 3, one can prove that\int
S - \varepsilon
u\mathrm{d}\mathrm{i}\mathrm{v}S \bfitZ d\sigma = \mathrm{l}\mathrm{i}\mathrm{m}
r\rightarrow 0
1
\lambda m(Br)
\int
\Phi BrS - \varepsilon
u\mathrm{d}\mathrm{i}\mathrm{v} \widetilde \bfitZ d\mu
for any function u \in Cb(M).
For a differential k-form \alpha of class C1
b on S, we define \widehat \alpha := q\ast \alpha . For each \varepsilon > 0, the form \widehat \alpha
is defined on \Phi B\gamma (\varepsilon )
S - \varepsilon .
Corollary 4. Let
#»
\bfitZ = \bfitZ \bfone \wedge . . . \wedge \bfitZ \bfitk +\bfone be a decomposable multivector field of class C1
b on
S. Given \varepsilon > 0, let \widetilde #»
\bfitZ = \widetilde \bfitZ \bfone \wedge . . . \wedge \widetilde \bfitZ \bfitk +\bfone be the q-related multivector field on \Phi B\gamma S - \varepsilon , and
suppose that, for each i \in \{ 1, . . . , k + 1\} , there exists \mathrm{d}\mathrm{i}\mathrm{v}\widetilde \bfitZ \bfiti \in L\infty (\mu ). Then
#»
\bfitZ \in D(\mathrm{d}\mathrm{i}\mathrm{v}S) and
\mathrm{d}\mathrm{i}\mathrm{v}S \bfitZ \bfiti \in L\infty (\sigma ) for each i \in \{ 1, . . . , k + 1\} . Moreover, for any \varepsilon > 0 and differential k-form \alpha
of class C1
0 (S), the following equality holds:\int
S - \varepsilon
\langle \alpha ,\mathrm{d}\mathrm{i}\mathrm{v}S
#»
\bfitZ \rangle d\sigma = \mathrm{l}\mathrm{i}\mathrm{m}
r\rightarrow 0
1
\lambda m(Br)
\int
\Phi BrS - \varepsilon
\langle \widehat \alpha ,\mathrm{d}\mathrm{i}\mathrm{v} \widetilde #»
\bfitZ \rangle d\mu .
Proof. Induction on k. Theorem 2 constitutes the basis of the induction. The induction step is
based on formula (12).
Let
#»
\bfitZ = \bfitX \wedge #»
\bfitY , where
#»
\bfitY is a k-vector fiel‘ \widetilde #»
\bfitZ = \widetilde \bfitX \wedge \widetilde #»
\bfitY and \langle \widehat \alpha ,\mathrm{d}\mathrm{i}\mathrm{v} \widetilde #»
\bfitZ \rangle =
= \mathrm{d}\mathrm{i}\mathrm{v} \widetilde \bfitX \cdot \langle \widehat \alpha , \widetilde #»
\bfitY \rangle - \langle i\widetilde \bfitX \widehat \alpha ,\mathrm{d}\mathrm{i}\mathrm{v} \widetilde #»
\bfitY \rangle + \langle \widehat \alpha ,\scrL \widetilde \bfitX \widetilde #»
\bfitY \rangle .
Since \langle \widehat \alpha , \widetilde #»
\bfitY \rangle = \widehat \langle \alpha , #»
\bfitY \rangle , Theorem 2 implies that\int
S - \varepsilon
\mathrm{d}\mathrm{i}\mathrm{v}S \bfitX \cdot \langle \alpha , #»
\bfitY \rangle d\sigma = \mathrm{l}\mathrm{i}\mathrm{m}
r\rightarrow 0
1
\lambda m(Br)
\int
\Phi BrS - \varepsilon
\mathrm{d}\mathrm{i}\mathrm{v} \widetilde \bfitX \cdot \langle \widehat \alpha , \widetilde #»
\bfitY \rangle d\mu .
Since one has i\widetilde \bfitX \widehat \alpha = \widehat i\bfitX \alpha , the equality\int
S - \varepsilon
\langle i\bfitX \alpha ,\mathrm{d}\mathrm{i}\mathrm{v}S
#»
\bfitY \rangle d\sigma = \mathrm{l}\mathrm{i}\mathrm{m}
r\rightarrow 0
1
\lambda m(Br)
\int
\Phi BrS - \varepsilon
\langle i\widetilde \bfitX \widehat \alpha ,\mathrm{d}\mathrm{i}\mathrm{v} \widetilde #»
\bfitY \rangle d\mu
follows from the induction hypothesis.
We have \langle \widehat \alpha ,\scrL \widetilde \bfitX \widetilde #»
\bfitY \rangle = \widehat u, where u = \langle \alpha ,\scrL \bfitX
#»
\bfitY \rangle is a function of class Cb(S - \varepsilon ), and therefore
the identity \int
S - \varepsilon
\langle \alpha ,\scrL \bfitX
#»
\bfitY \rangle d\sigma = \mathrm{l}\mathrm{i}\mathrm{m}
r\rightarrow 0
1
\lambda m(Br)
\int
\Phi BrS - \varepsilon
\langle \widehat \alpha ,\scrL \widetilde \bfitX \widetilde #»
\bfitY \rangle d\mu
is a direct consequence of Lemma 3.
Applying now formula (12) to \mathrm{d}\mathrm{i}\mathrm{v}S(\bfitX \wedge #»
\bfitY ), we obtain the statement of the corollary.
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DIVERGENCE OF MULTIVECTOR FIELDS ON INFINITE-DIMENSIONAL MANIFOLDS 1653
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Received 12.01.21
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 12
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| id | umjimathkievua-article-6522 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T03:28:36Z |
| publishDate | 2023 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/3a/8e7e4cf80ea31cd3dea9fee794dbe03a.pdf |
| spelling | umjimathkievua-article-65222023-01-23T14:02:42Z Divergence of multivector fields on infinite-dimensional manifolds Дивергенция поливекторных полей на бесконечномерных многообразиях Divergence of multivector fields on infinite-dimensional manifolds Bogdanskii, Yu. Shram, V. Богданский, Юрий Bogdanskii, Yu. Shram, V. Banach manifold Radon measure multivector field divergence surface measure UDC 514.763.2+515.164.17 We study the divergence of multivector fields on Banach manifolds with a Radon measure.&nbsp;&nbsp;We propose an infinite-dimensional version of divergence consistent with the classical divergence from&nbsp; finite-dimensional differential geometry.&nbsp;&nbsp;We then transfer certain natural properties of the divergence operator to the infinite-dimensional setting.&nbsp;&nbsp;Finally, we study the relation between the divergence operator ${\rm div}_M$ on a manifold $M$ and the divergence operator ${\rm div}_S$ on a submanifold&nbsp;&nbsp;$S \subset M.$ УДК 514.763.2+515.164.17 Дивергенція багатовекторних полів на нескінченновимірних многовидах&nbsp; Досліджується дивергенція багатовекторних полів на банахових многовидах із мірою Радона.&nbsp;&nbsp;Запропоновано нескінченновимірну версію дивергенції, яка узгоджується з класичним оператором дивергенції, що розглядається в скінченновимірній диференціальній геометрії.&nbsp;&nbsp;Низку природних властивостей дивергенції перенесено на нескінченновимірний випадок.&nbsp;&nbsp;Крім того, досліджено зв’язок між оператором дивергенції ${\rm div}_M$ на многовиді $M$ і оператором дивергенції ${\rm div}_S$ на підмноговиді $S \subset M.$&nbsp; Institute of Mathematics, NAS of Ukraine 2023-01-17 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/6522 10.37863/umzh.v74i12.6522 Ukrains’kyi Matematychnyi Zhurnal; Vol. 74 No. 12 (2022); 1640 - 1653 Український математичний журнал; Том 74 № 12 (2022); 1640 - 1653 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/6522/9341 Copyright (c) 2023 Vladyslav Shram |
| spellingShingle | Bogdanskii, Yu. Shram, V. Богданский, Юрий Bogdanskii, Yu. Shram, V. Divergence of multivector fields on infinite-dimensional manifolds |
| title | Divergence of multivector fields on infinite-dimensional manifolds |
| title_alt | Дивергенция поливекторных полей на бесконечномерных многообразиях Divergence of multivector fields on infinite-dimensional manifolds |
| title_full | Divergence of multivector fields on infinite-dimensional manifolds |
| title_fullStr | Divergence of multivector fields on infinite-dimensional manifolds |
| title_full_unstemmed | Divergence of multivector fields on infinite-dimensional manifolds |
| title_short | Divergence of multivector fields on infinite-dimensional manifolds |
| title_sort | divergence of multivector fields on infinite-dimensional manifolds |
| topic_facet | Banach manifold Radon measure multivector field divergence surface measure |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/6522 |
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