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Publié par | pefav |
Nombre de lectures | 7 |
Langue | English |
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X
Harmonicit
w
y
D
up
(resp.
to
map
rearrangemen
Compact
t
strings.
and
.
isothermal
+
gas
manifolds.
Y
manifolds
ann
35J
Brenier
1
Abstra
p
norm
The
a
In
of
minimizes
harmonic
R
(or
e)
w
fairly
a
b
v
in
e)
Classic
maps
Harmonicit
up
the
to
where
rearrangemen
with
t
supp
is
j
in
.)
tro
homogeneous
A
=
relation
is
minimizing
established
v
with
b
the
smo
(or
oth
solutions
=
of
t
the
b
isother-
w
mal
or
irrotational
space
in
1991
viscid
58E30
gas
ds
equations.
supp
On
ST
in
dtds;
tro
=
duit
=
la
ect
notion
with
d'application
in
harmonique
j
the
a
R
r
means
es
earrangemen
(resp.
t
e)
pr
tt
ss
es.
:
On
harmonic
1,
etablit
is
un
if
lien
as
a
X
v
along
ec
is
les
A
solutions
the
r
a
a
eguli
is
general
eres
des
d
but
equations
our
d
manifolds
ecriv
an
er,
t
b
la
dy-
the
namique
d
isotherme
des
Subje
gaz
Primary
sans
;
viscosit
Key
phr
e
,
ni
t,
tourbillon.
ork
1.
b
Review
pro
of
T
some
)
(1.1)
1.1.
1
Harmonic
(or
1),
w
resp
a
to
v
erturbations
e)
maps.
ort
Let
U
S
(Here
>
:
0,
denotes
T
>
on
0,
d
U
This
=]0
that
;
solv
T
the
[
Laplace
w
℄
v
;
equation
S
[
X
and
D
X
=
0
T
(1.2)
d
the
=
(
=
R
a
=
X
Z
)
harmonic
d
it
the
(1.1)
at
its
torus.
alue
A
j
map
U
(
the
t;
oundary
s
xed.
)
emark.
2
natural
U
for
!
\target"
X
of
(
harmonic
t;
w
s
v
)
map
2
the
D
of
is
Riemannian
usually
The
D
a
T
harmonic
is
map
trivial
(resp.
Æ
a
for
w
discussion.
a
Riemmanian
v
without
e
oundary
map)
also
if
e
it
Ho
is
ev
a
with
p
oundaries
oin
non
t
manifolds,
of
particular
the
functional
R
Z
,
U
ould
1
Æ
2
Mathematics
(
ation.
j
58E20
76N
t
X
35Q35.
(
wor
t;
and
s
ases.
)
y
j
gas
2
rearrangemen
+
vibrating
j
W
partly
s
orted
X
y
(
austrian
t;
s
AR
)
(FWF-TEC-Y-137).
j
2
2
s;
Y
s
ANN
the
BRENIER
1.2.
an
La
u
ws,
)
rearrangemen
olution
ts,
h
Moser's
of
lemma.
1
Let
temp
(
time
A;
::::;
da
expresses
)
(1.7)
b
)
e
p
a
probabilit
;
y
Isothermal
space
(t
b
ypically
p
A
0,
=
R
[0
v
;
notations
1]
the
or
u
A
usually
=
u
T
the
d
eld
equipp
is
ed
with
the
v
Leb
esgue
+
measure
An
da
in
).
The
F
mo
or
(ph
a
3)
measurable
y
function
0,
a
)
2
s;
A
elo
!
),
X
s
(
x
a
are
)
(where
2
D
)
,
w
+
e
;
dene
of
the
tin
\la
(
w"
of
equiv
X
momen
to
if
b
t
e
0
y
(
for
x
tro
)
;
=
as
Z
system
A
s
Æ
(
0
x
u:
X
r
(
s
a
tary
))
pro
da;
8.8
(1.3)
gas
whic
olution
h
viscid
is
in
a
R
probabilit
d
y
2
measure
describ
on
a
D
,
)
more
pressure
precisely
s;
dened
0,
b
eld
y
)
Z
a
D
y
h
t;
(
alued
x
,
)
for
d
ariable
(
the
x
These
)
=
wing
Z
e
A
for
h
1
(
x
X
:
(
inner
a
R
))
s
da;
:
(1.4)
=
for
whic
all
h
and
2
\the
C
y
(
u
D
r
).
r
(An
0
usual
h
denomination
t
for
ation
The
is
\push-forw
temp
ard"
a
of
s;
da
b
the
y
en
X
=
.)
;
W
e
0.
no
w
sa
these
y
e
that
t
erb
w
space
o
(
h
u:
measurable
p
functions
:u
X
(1.10)
and
+
Y
)
are
equal
=
up
on
to
.
rearrangemen
elemen
t
pro
if
is
they
vided
ha
v
1.3.
e
irrotational
the
same
ev
la
of
w,
in
namely
gas
Z
ving
A
the
Æ
space
(
d
x
ysically
Y
=
(
;
a
;
))
is
da
ed
=
y
Z
densit
A
eld
Æ
(
(
x
x
>
X
a
(
eld
a
(
))
x
da:
>
(1.5)
a
Let
erature
us
(
quote
x
a
>
v
and
ery
v
useful
result,