Mathematical Physics, Analysis and Geometry - Volume 1

322 Pages·1998·2.55 MB·English

by  A. Boutet de Monvel, V. A. Marchenko, H. McKean (Editors)

Mathematical Physics, Analysis and Geometry 1: v, 1998.
v
Editorial
This is the first issue of Mathematical Physics, Analysis and Geometry in its
English-language, international form. A journal of the same name in the Russian
language, having its roots in the long and splendid tradition of mathematical re-
search in the former Soviet Union and, in particular, the Ukraine, was launched
in 1994 by the Kharkov mathematical community and has published papers in the
Ukrainian, Russian and English languages. The journal, in English only as of 1998,
is intended to provide an international forum for important new results not only
from the former Soviet Union but from all over the world.
Mathematical Physics, Analysis and Geometry will publish research papers and
review articles on new mathematical results with particular reference to complex
function theory; operators in function space, especially operator algebras; ordinary
and partial differential equations; differential and algebraic geometry; mathemati-
cal problems of statistical physics, fluids; etc. The Editors, supported and assisted
by an international Editorial Board, will strive to maintain the highest quality.
Papers which are too abstract will be discouraged.
It is our purpose to make Mathematical Physics, Analysis and Geometry a
leading journal of its kind attracting the best papers in the field.
VLADIMIR A. MARCHENKO
ANNE BOUTET de MONVEL
HENRY McKEAN
VTEXVR PIPS No: 168142 (mpagkap:mathfam) v.1.15
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Mathematical Physics, Analysis and Geometry 1: 1–22, 1998.
1
© 1998 Kluwer Academic Publishers. Printed in the Netherlands.
Homogenization of Harmonic Vector Fields on
Riemannian Manifolds with Complicated
Microstructure
L. BOUTET DE MONVEL
Université de Paris 6, Institut de Mathématiques de Jussieu (UMR 9994 du CNRS), 4 place Jussieu,
F-75252 Paris Cedex 05, France
E. KHRUSLOV
B. Verkin Institute for Low Temperatures, Mathematical Division, 47 Lenin Avenue,
310164 Kharkov, Ukraine
(Received: 1 October 1997)
Abstract. We study the asymptotic behaviour of harmonic vector fields with given fluxes or periods
on special manifolds consisting of one or several copies of the Euclidean space, with a large number
of small holes attached edge to edge by means of thin tubes (wormholes) when the number of
holes tends to infinity. We obtain the homogenized equations describing the leading term of the
asymptotics.
Mathematics Subject Classifications (1991): 35B27, 35B40.
Key words: electric field, homogenization, wormholes.
According to the well-known ‘Wheeler picture’, electric fields can be represented
as harmonic fields on special Riemannian manifolds Mϵ. Such a manifold can
consist of one or several copies of the Euclidean space with small holes attached
edge to edge by means of thin tubes. In geometrodynamics, these tubes are called
‘wormholes’ [1]. The flux of a vector field through a wormhole is interpreted as a
charge of the electric field. Given fluxes through all wormholes of Mϵ determine
a unique harmonic vector field on Mϵ vanishing at infinity. Such vector fields are
also determined by their periods along cycles passing through wormholes.
In this paper, we consider manifolds Mϵ depending on a small parameter ϵ > 0
such that the number of holes increases and their diameters vanish, as ϵ → 0. We
study the asymptotic behaviour of harmonic vector fields on these manifolds with
given fluxes or periods and obtain homogenized equations describing the leading
term of the asymptotics.
VTEX(EL) PIPS No.: 151041 (mpagkap:mathfam) v.1.15
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2 L. BOUTET DE MONVEL AND E. KHRUSLOV
1. Description of the Problem
n
Let � be a fixed bounded domain in the space R (n ⩾ 3) and {Bϵi, i = 1 . . . N(ϵ)}
be a family of closed pairwise disjoint balls in � depending on a small parameter
ϵ > 0. We suppose that the total number N(ϵ) of balls tends to infinity and their
diameters tend to zero, when ϵ → 0, and any open subdomain G ⊂ � contains
some balls for sufficiently small ϵ.
Let us consider the infinite domain
N(ϵ)
⋃
n
�ϵ = R \ Bϵi
i=1
and the disjoint union of m copies (sheets) of �ϵ : �˜ϵ = �ϵ × {1 . . . m}.
We construct a new manifold Mϵ by attaching to �˜ϵ some n-dimension tubes,
gluing their boundaries to those of the holes of �˜ϵ. More precisely, Mϵ is defined
by the following data:
• for each ϵ > 0 the number mN(ϵ) is even and we are given a partition of
the set of all pairs {(i, k) : i = 1 . . . N(ϵ), k = 1 . . . m} into subsets of two
elements [(i, k), (j, l)]-linked pairs;
kl
• for each linked pair [(i, k), (j, l)], let T
ϵij = Sn−1 ×[0, 1] be a spherical tube
kl
of dimension n (Sn−1 is unit (n − 1)-sphere). The boundary ∂T
ϵij of this tube
consists of two components:
kl k l
∂T = Ŵ ∪ Ŵ ;
ϵij ϵi ϵj
• for each (i, k) we are given a diffeomorphism:
k k k
h : Ŵ ↔ ∂B ,
ϵi ϵi ϵi
k k k
where ∂B
ϵi is a component of the boundary ∂�ϵ of kth sheet �ϵ = �ϵ × k.
k kl
The manifold Mϵ is the union of the sheets �
ϵ (k = 1 . . . m) and tubes Tϵij identi-
k
fying boundary points pairwise according to the diffeomorphisms h . We suppose
ϵi
that Mϵ is an orientable manifold. Fragments of such a manifold are shown in
Figures 1 and 2.
We equip Mϵ with a differentiable structure inducing its canonical structure of
k kl
a differentiable manifold with a boundary on each sheet � and on each tube T .
ϵ ϵij
α
We will usually denote x as the points of Mϵ and, when needed, x (α = 1 . . . n) as
the coordinates in some local coordinate chart.
Finally, we are given a Riemannian metric on Mϵ with a positively defined
ϵ
smooth metric tensor {g (x); α, β = 1 . . . n} inducing the standard flat metric
αβ
n
of R outside some sufficiently small neighbourhoods of the tubes on Mϵ. For
simplicity we will suppose that it induces such a flat metric on each whole sheet
MPAG009.tex; 14/05/1998; 16:03; p.2
HARMONIC VECTOR FIELDS ON RIEMANNIAN MANIFOLDS 3
k
Ωϵ
i j
kk
T
ϵij
Figure 1.
l
Ωϵ j
kl
T
ϵij
i k
Ωϵ
Figure 2.
k
� . Its dependence on parameter ϵ on the tubes will be quantitatively characterized
ϵ
below (Section 2).
We will study the harmonic vector fields on Mϵ and we may identify them with
harmonic differential forms of degree 1 or n − 1. Let us recall some facts and
notations from the theory of differentiable manifolds [2].
Differential forms of degree r (r-forms) are defined in local coordinates as
follows:
∑
i1 ir
ω(x) = ωi
1...ir (x) dx ∧ · · · ∧ dx ,
16i1<···<ir6n
where functions ωi
1...ir (x) are components of a skew-symmetric tensor field of rank
r (0 ⩽ r ⩽ n) and the sign ∧ denotes the exterior product.
MPAG009.tex; 14/05/1998; 16:03; p.3
4 L. BOUTET DE MONVEL AND E. KHRUSLOV
The following three operators act on the set of differential forms. The exte-
rior differentiation operator d maps r-forms into (r + 1)-forms according to the
formulae
∑
∂f
k 2 i
d(ω ∧ µ) = dω ∧ µ + (−1) ω ∧ dµ, d = 0, df = dx ,
i
∂x
where ω and µ are any smooth forms, k is degree of ω, and f is a function (form
of degree 0).
The star operator ∗ assigns to any r-form ω the (n − r)-form ∗ω such that
√
i1 in
µ ∧ ∗ω = (µ,ω) |gϵ| dx ∧ · · · ∧ dx
for any r-form µ. Here |gϵ| is the determinant of the metric tensor {gϵαβ(x)} on
Mϵ, and
∑
i1...ir
(µ, ω) = µi
1...ir ω
16i1<···<ir6n
is the scalar product of forms relative to the metric on Mϵ.
The operator δ maps r-forms into (r − 1)-forms according to the equality
∫ ∫
n n
(δω,µ) d xϵ = (ω, dµ) d xϵ,
Mϵ Mϵ
which is valid for any r-form ω and any smooth (r − 1)-form µ with compact
√
n i1 in
support. Here and below we denote d xϵ = |gϵ| dx · · · dx the volume element
on Mϵ.
We recall that a form ω is called exact, if there is a form φ such that ω = dφ. It
is called closed, if dω = 0, and coclosed, if δω = 0. A form ω is called harmonic,
if it is both closed and coclosed.
Let Z be a regular two-dimensional chain on Mϵ, i.e., a formal linear combina-
∑
tion Z = αkZk, where αk are real numbers and Zk are r-dimensional simplexes
α 1 r
in Mϵ, parametrized by continuously differentiable functions x
k (u . . . u ) (α =
1 r r
1 . . . n; u = {u . . . u } ∈ U ).
The integral of an r-form ω along Z is defined by the formula:
∫
∑ ∑ ∫ i1 ir
∂(x . . . x )
k k 1 r
Z ω = αk Ur ωi1...ir (x(u))∂(ui1 . . . uir ) du · · · du .
k 16i1<···<ir6n
A chain Z without boundary (∂Z = 0) is called a cycle. Two r-dimensional
cycles C1, C2 are homologous if there exists an (r + 1)-dimensional chain Z such
that the chain C1 −C2 is the boundary of Z (C1 −C2 = ∂Z). In particular, a cycle
C is homologous to zero if it is a boundary (C = ∂Z).
∫
If ω is a closed r-form and C is an r-dimensional cycle, then the integral ω
C
is called the period of ω along Z. According to Stokes’ theorem it depends only
MPAG009.tex; 14/05/1998; 16:03; p.4
HARMONIC VECTOR FIELDS ON RIEMANNIAN MANIFOLDS 5
∫
on the homology classes of ω and Z. In particular, the period ω vanishes if ω is
C
exact or Z is homologous to zero.
It follows from our construction of the manifold Mϵ, that cycles of dimension
r ≠ 1, n− 1 are homologous to zero. Besides since n ⩾ 3, one-dimensional cycles
k
Ck situated completely on one sheet �
ϵ (k = 1 . . . m) are homologous to zero.
The manifold Mϵ is not compact. We define its compactification by adjoining
k
points at infinity ∞k (k = 1 . . . m) on each of the m sheets �
ϵ and we will consider
1-forms and (n − 1)-forms on Mϵ tending to zero fast enough, i.e., satisfying the
following ‘decay condition’ when |x| → ∞:
1−n
|ω(x)| = O(|x| ), (1.1)
1/2
where |ω| = (ω,ω) is the norm of ω.
∫
For such 1-forms the integrals kl ω are still well defined along non-compact
C
ϵij
kl kl
paths C
ϵij that join ∞k and ∞l through some tube Tϵij :
k kl l
∞k → ∂B
ϵi → Tϵij → ∂Bϵj → ∞l
k
and are straight lines outside of �. Since there are only finitely many holes B and
ϵi
finitely many sheets, these contours form a basis for one-dimensional homology
∫
kl
classes. We will still call the integrals kl ω periods of the form ω (although C
C ϵij
ϵij
kl kl lk
are not cycles) and denote them P . It is clear that P = −P if [(i, k), (j, l)]
ϵij ϵij ϵji
kl
are linked pairs, and we set P = 0 otherwise (consistency conditions).
ϵij
kl
THEOREM 1. For a given consistent set of periods P (i, j = 1 . . . N(ϵ); k, l =
ϵij
1 . . . m) there exists a unique harmonic 1-form satisfying the decay condition (1.1).
k
As a basis for (n − 1)-homology classes we may choose (n − 1)-spheres ∂B .
ϵi
k k
We will denote � the periods of an (n − 1)-form ω along ∂B . According
ϵi ϵi
k l
to Stokes’ theorem, � = −� if [(i, k), (j, l)] are linked pairs (consistency
ϵi ϵj
conditions).
k
THEOREM 2. For a given consistent set of periods � (i = 1 . . . N(ϵ); k =
ϵi
1 . . . m) there exists a unique harmonic (n−1)-form satisfying the decay condition
(1.1).
k
Note that if ω is harmonic (n − 1)-form with periods � then 1-form v = ∗ω
ϵi
is also harmonic and corresponding vector field v = {vi(x), i = 1 . . . n} has fluxes
k k
� through (n − 1)-spheres ∂B .
ϵi ϵi
Theorems 1 and 2 are variants of Hodge’s theorem for the non-compact man-
ifold Mϵ. They may be proved by the variational method (see Section 3). The
main goal of this paper is to study the asymptotic behaviour of 1-forms defined
by Theorem 1 and the asymptotic behaviour of (n − 1)-forms defined by Theorem
2, when ϵ → 0.
MPAG009.tex; 14/05/1998; 16:03; p.5
6 L. BOUTET DE MONVEL AND E. KHRUSLOV
2. Statements of Main Results
First we describe more notations. Let xϵi be the centre of the ball Bϵi, aϵi its radius,
rϵi the distance from xϵi to the union of the other centres:
rϵi = min |xϵi − xϵj |.
i≠ j
We assume that when ϵ → 0
(i)
aϵi → 0, rϵi → 0,
so that
(ii)
n
n−2
aϵi ⩽ Cr
ϵi ,
where C doesn’t depend on ϵ.
n
Let Rϵi be the spherical annulus in R , centered at the point xϵi with inner radius
aϵi and outer radius bϵi = rϵi/2:
{ }
n rϵi
Rϵi = x ∈ R : aϵi < |x − xϵi | < .
2
It is easy to see that these annuli belong to convex non-intersecting polyhedrons
⋃
n
�ϵi (i = 1 . . . N(ϵ)) such that Rϵi ⊂ �ϵi ⊂ R , � ⊂
i �ϵi.
We will assume that
(iii)
dϵi = diam(�ϵi) ⩽ Crϵi,
where C doesn’t depend on ϵ.
kl kl k l
Let us denote D
ϵij = Tϵij ∪ Rϵi ∪ Rϵj the domain on the manifold Mϵ, where
k k
R
ϵi = Rϵi × {k} is the kth copy of the annulus Rϵi on the kth sheet �ϵ ⊂ Mϵ, and
kl kl
T is the tube corresponding to the linked pairs [(i, k), (j, l)]. Its boundary ∂D
ϵij ϵij
k k l l
consists of two components S ⊂ � and S ⊂ � . Let us consider in the domain
ϵi ϵ ϵj ϵ
kl
D the boundary value problem:
ϵij

kl
 �v(x) = 0, x ∈ Dϵij ;
k
v(x) = 0, x ∈ S ; (2.1)
ϵi

l
v(x) = 1, x ∈ S ,
ϵj
where v(x) is a function, � = �M
ϵ is the Laplace operator on the Riemannian
manifold Mϵ. In local coordinate it has the form
n ( )
1 ∑ ∂ √ ∂
αβ
� = − √
α |gϵ| gϵ β ,
|gϵ| ∂x ∂x
α,β=1
MPAG009.tex; 14/05/1998; 16:03; p.6
HARMONIC VECTOR FIELDS ON RIEMANNIAN MANIFOLDS 7
αβ
where {g
ϵ (x), α, β = 1 . . . n} is the inverse tensor to the metric tensor of Mϵ (it is
also defined on forms: � = (dδ + δ d)).
kl kl
There is a unique solution to problem (2.1). We denote it v = v (x) and set
ϵij ϵij
∫
kl kl kl
V = dv ∧ ∗ dv . (2.2)
ϵij ϵij ϵij
kl
D
ϵij
kl lk
These quantities are positive and possess the symmetry V = V . They charac-
ϵij ϵji
kl
terize the metric on the tubes T . We assume that this metric satisfies an additional
ϵij
condition which provides the inequality
(iv)
kl n−2
V ⩾ Caˆ ,
ϵij ϵij
where aˆϵij = max{aϵi, aϵj }, C > 0 does not depend on ϵ. Here [(i, k), (j, l)] are
linked pairs.
kl
Note, that solution v (x) of problem (2.1) minimizes the functional (2.2) over
ϵij
1 kl k l
class H (D ) of functions satisfying the boundary conditions on S and S .
ϵij ϵi ϵj
Taking this into account, it is easy to obtain the opposite inequality
kl n−2
V ⩽ Caˇ , (2.3)
ϵij ϵij
where aˇϵij = min{aϵi, aϵj }. Therefore inequality (iv) means that radii of linked
balls have the same order of magnitude when ϵ → 0.
n n
Now we introduce the distributions on R × R
∑
kl
Vϵkl(x, y) = V
ϵij δ(x − xϵi)δ(y − xϵj ); k, l = 1 . . . m,
ij
n kl
where δ(x) is the delta-function on R , V = 0, if pairs [(i, k), (j, l)] are not
ϵij
linked. These distributions are non-negative and by virtue of (2.3) and (i)–(ii)
∫ ∫
Vϵkl(x, y) dx dy < C|�|; k, l = 1 . . . m,
n
where |�| is the volume of the domain � ⊂ R , C does not depend on ϵ. Thus, the
sets of distributions {Vϵkl(x, y), ϵ > 0} (k, l = 1 . . . m) are compact in the sense of
n n
weak topology of distributions on R × R . We will assume that there exist weak
limits
(v)
w − lim Vϵkl(x, y) = Vkl(x, y), k, l = 1 . . . m,
ϵ→0
where Vkl(x, y) are non-negative distributions (densities of measures) with sup-
ports in � × �. Symmetry Vkl(x, y) = Vlk(y, x) holds true and using (i), (ii), (v)
and (2.3), one can show that integrals
∫
Vkl(x, y) dy = Ikl(x)
are bounded functions of x: Ikl(x) ∈ L∞(�).
MPAG009.tex; 14/05/1998; 16:03; p.7
8 L. BOUTET DE MONVEL AND E. KHRUSLOV
n
We further introduce the distributions on R :
∑
kl kl
Pϵk(x) = V
ϵijPϵij δ(x − xϵi) (k = 1 . . . m),
ij
and
∑
k
�ϵk(x) = �
ϵiδ(x − xϵi) (k = 1 . . . m),
i
kl k
where P are periods of harmonic 1-form and � are periods of harmonic (n −
ϵij ϵi
1)-form, which were described in Theorems 1 and 2, respectively.
′ n
We will also assume that these distributions weakly converge (in D (R ))
(j)
w − lim Pϵk(x) = Pk(x) (k = 1 . . . m),
ϵ→0
(jj)
w − lim �ϵk(x) = �k(x) (k = 1 . . . m)
ϵ→0
and, in addition, that the following inequalities hold
(jjj)
∑
n−2 kl 2
a
ϵi |Pϵij | < C1;
i,j,k,l
(jv)
∑
−n+2 k 2
a
ϵi |�ϵi| < C2
i,k
with constants C1, C2 independent of ϵ. It follows from (jjj), (jv) and (i), (ii) that
limit distributions Pk(x), �k(x)(k = 1 . . . m) are functions of L2(�).
(r) n (r)
We denote L2(Mϵ) and L2(R ) the Hilbert spaces of r-forms (r = 1, n−1)
n
on Mϵ and R , respectively, with relation to the metrics scalar products [2].
(r)
Let us introduce the operators Qϵk (k = 1 . . . m) mapping L2(Mϵ) into
n (r)
L2(R ) , defined by the formula:
[ ] {
k
Qϵkvϵ (x) = vϵ(x), x × {k
n} ∈ �ϵ, ⋃
0, x ∈ R \ �ϵ =
i Bϵi,
(r)
for any vϵ ∈ L2(Mϵ) .
The main results of this paper are contained in the following theorems describ-
ing the asymptotic behaviour of harmonic 1-forms and (n − 1)-forms on Mϵ, as
ϵ → 0.
THEOREM 3. Let vϵ be the harmonic 1-form defined by Theorem 1 and let con-
ditions (i)–(v) and (j), (jjj) be fulfilled, when ϵ → 0. Then, for any k
n (1)
Qϵkvϵ → vk weakly in L2(R ) ,
MPAG009.tex; 14/05/1998; 16:03; p.8
HARMONIC VECTOR FIELDS ON RIEMANNIAN MANIFOLDS 9
n
where vk (k = 1 . . . m) are exact 1-forms on R such that vk = dUk and the col-
lection of functions {Uk(x), k = 1 . . . m} is the generalized solution of the problem
m ∫
∑
n
�Uk − Vkl(x, y)[Uk(x) − Ul(y)] dy = Pk(x), x ∈ R ;
l=1
2−n
Uk(x) = O(|x| ), x → ∞ (k = 1 . . . m).
THEOREM 4. Let vϵ be the harmonic (n − 1)-form defined by Theorem 2 and let
conditions (i)–(iv) and (jj), (jv) be fulfilled, when ϵ → 0. Then, for any k
n (n−1)
Qϵkvϵ → vk weakly in L2(R ) ,
n
where vk (k = 1 . . . m) are (n−1)-forms on R such that vk = ∗ dUk and functions
Uk(x) are the solutions of the problems
n
�Uk = �k(x), x ∈ R ;
(2.4)
2−n
Uk(x) = O(|x| ), x → ∞ (k = 1 . . . m).
We will prove these theorems in Sections 3, 4 and 5. In Section 3 we will con-
struct suitable representations for harmonic 1-forms and (n−1)-forms on Mϵ. Then
using these representations, in Sections 4 and 5 we study the asymptotic behaviour
of 1-form and (n − 1)-forms, respectively.
3. Representation of Harmonic Forms on Mϵ
kl
Let us make cuts on Mϵ along the middle section on each tube T
ϵij , as indicated
k l
in Figure 3. We will denote by T and T the corresponding halves of the tube
ϵi ϵj
kl kl k l
T (T = T ∪ T ).
ϵij ϵij ϵi ϵj
k
Thus, we cut Mϵ in m components M
ϵ (k = 1 . . . m) whose boundaries con-
kl k n
sist of N(ϵ) spheres S . Each M is homeomorphic to the space R with N(ϵ)
ϵij ϵ
k
removed balls Bϵi. Therefore since n > 2, M
ϵ has no 1-homology.
k
Hence, any closed 1-form vϵ on Mϵ is exact on each M
ϵ , i.e.,
k
vϵ = duϵk, x ∈ M
ϵ (k = 1 . . . m). (3.1)
k
Here uϵk is a smooth function on M
ϵ such that
2−n
uϵk = O(|x| ), |x| → ∞ (3.2)
and vϵ satisfying the vanishing condition (1.1).
kl
According to the definition of the periods P
ϵij of vϵ and (3.1), (3.2),
kl
uϵk(x)|
Skl − uϵl(x)|Slk = Pϵij . (3.3)
ϵij ϵji
MPAG009.tex; 14/05/1998; 16:03; p.9