Francoise J.-P., Naber G.L., Tsun T.S. (editors) Encyclopedia of Mathematical Physics
Подождите немного. Документ загружается.
Theorem 11 Let X be a normed space and let
f : X ![0, þ1] be a convex and proper function;
assume that f is continuous at 0, then
(i) f
achieves its minimum on X
(ii) f (0) = f
(0) = inf f
Proof
(i) Let M be an upper bound of f on the ball {kxk
R}. Then
f
ðx
Þsup hx; x
if ðxÞ: kxkR
fg
R k x
k
X
M
Hence, for every r, the set {x
2 X
: f
(x
) r}
is bounded, thus -relatively compact, where is
the weak-star topology on X
. By assertion (i) of
Lemma 9, f
is -l.s.c. and Theorem 4 applies.
(ii) By Theorem 10, since f is convex proper and
l.s.c. at x
0
= 0, we have f (0) = f
(0) = inf f
.
&
Some Useful Consequences
Proposition 12 (Conjugate of a sum). Let f , g : X !
R [ {þ1} be convex such that
9x
0
2X : f is continuous at x
0
and gðx
0
Þ< þ1 ½2
Then
(i)
ðf þ gÞ
ðx
Þ= inf
x
1
þx
2
= x
{f
ðx
1
Þþg
ðx
2
Þ}
(the equality holds in
R).
(ii) If both sides of the equality in (i) are finite, then
the infimum in the right-hand side is achieved.
Proof Without any loss of generality, we may
assume that x
= 0 (we reduce to this case by
substituting g with g h, x
i). We let
hðpÞ¼infff ðx þ pÞþgðxÞjx 2 Xg
Noticing that ( p, x) 7!f (x þ p) þ g(x) is convex, we
infer that h(p) is convex as well. As h is majorized
by the function p 7!f (x
0
þ p) þ g(x
0
), which by [2]
continuous at 0, we deduce from Theorems 1 and 11
that h(0) = h
(0) and that h
achieves its infimum.
Now h(0) = inf(f þ g) = (f þ g)
(0) and
h
ðp
Þ¼supfhp; p
ihðpÞ: p 2 Xg
¼ supfhp; p
if ðx þpÞgðxÞ: x 2 X; p 2 Xg
¼ g
ðp
Þþf
ðp
Þ
The assertions (i), (ii) follow since h
(0) =
min h
= min {g
(p
) þf
(p
)}. &
Proposition 13 (Composition). Let X, Y be two
Banach spaces and A : X 7!Y a linear operator with
dense domain D(A). Let : Y !R [ {þ1} be a
convex l.s.c. function and let F 7!X be the convex
functional defined by
FðuÞ¼
ðAuÞ if u 2 DðAÞ
þ1 otherwise
Assume that there exists u
0
2 D(A) such that is
continuous at Au
0
. Then
(i) The Fenchel conjugate of F is given by
8f 2 X
; F
ðf Þ¼inff
ðÞ: 2 Y
; A
¼ f g
where, if both sides of the equality are finite, the
infimum on the right-hand side is achiev ed.
(ii) If, in addition, Y is reflexive and is l.s.c.
coercive, we have
FðuÞ¼F
ðuÞ¼inffðpÞjðu; pÞ2GðAÞg ½3
where G(A) denotes the graph of A.
Proof
(i) Define H, K : X Y !R [ {þ1}by
Hðu; pÞ¼
GðAÞ
ðu; pÞ; Kðu; pÞ¼ðpÞ
Then we have the identity F
(f ) = (H þ K)
(f , 0),
where the conjugate of H þ K is taken with
respect to the duality (X Y, X
Y
). From the
assumption, K is continuous at (u
0
, Au
0
) 2
dom H. By Proposition 12, we obtain
ðH þ KÞ
ðf ; 0Þ
¼ inf
ðg;Þ2X
Y
fK
ðf g;ÞþH
ðg; Þg
After a simple computation, it is easy to check
that
H
ðg; Þ¼
0ifA
¼ f
þ1 otherwise
K
ðf g;Þ¼
ðÞ if g ¼ f
þ1 otherwise
(ii) Let J(u):= inf{(p): (u, p) 2
G(A)}. As observed
for F
in the proof of (i), we have the identity
J
(f ) = (H þ K)
(f , 0). Therefore, in view of
Theorem 10,
F = F
= J
and it is enough to
prove that J is convex l.s.c. proper. Let us
consider a sequence (u
n
)inX converging to
some u 2 X. Without any loss of generality, we
may assume that lim inf J(u
n
) = lim J(u
n
) < þ1.
Then there is a sequence (p
n
)suchthat,forevery
n,(u
n
, p
n
) 2
G(A)andJ(u
n
) (u
n
) 1=n.As
is coercive, {p
n
} is bounded in the reflexive
space Y and possibly passing to a subsequence,
Convex Analysis and Duality Methods 645
we may assume that p
n
converges weakly to
some p.SinceG(A) is a (weakly) closed subspace
of X Y, we infer that (u, p) as the limit of
(u
n
, p
n
) still belongs to G(A). Thus, we conclude,
thanks to the (weak) lower-semicontinuity of
lim inf
n
Jðu
n
Þ¼lim
n
ðp
n
ÞðpÞJðuÞ
&
An immediate consequence of Pr opositions 12 and
13 is the following variant:
Proposition 14 Under the same notation as in
Proposition 13, let : X !R [ {þ1} be a convex
function and assume that there exists u
0
2 D(A)
such that F(u
0
) < þ1 and is continuous at Au
0
.
Then we have
inf
u2X
ðuÞþðAuÞ
fg
¼ sup
2Y
ðA
Þ
ðÞ
fg
where the suprem um on the right-hand side is
achieved. Furthermore, apair(
u, ) is optimal if
and only if it satisfies the relations: 2 @(A
u) and
A
2 @(
u).
Remark 15 From the assertion (ii) of Proposition
13, we may conclude that F is l.s.c. whenever the
operator A is closed. If now A is merely closable
(with closure denoted by
A), we obtain
FðuÞ¼
Gð
AuÞ if u 2 dom
A
þ1 otherwise
This is the typical situation when F is an integral
functional defined on smooth functions of the kind
FðuÞ¼
Z
f ðx; ruÞdx
where is an bounded open subset of R
n
, f :
R
n
!R is a convex integrand with quadratic growth
(i.e., cjzj
2
f (x, z) C(1 þjzj
2
for suitables C
c > 0). Then X = L
2
(), Y = L
2
(; R
n
),
GðvÞ¼
Z
f ðx; vðxÞÞdx
and A : u 2 C
1
() 7!ru 2 L
2
(; R
n
). It turns out
that A is closable and that the domain of
A
characterizes the Sobolev space W
1, 2
() on which
A coincides with the distributional gradient
operator.
The situation is more involved if we con sider
FðuÞ¼
Z
f ðx; ruÞd
is a possibly concentrated Radon measure sup-
ported on . In general, the operator A : u 2
C
1
() L
2
() 7!ru 2 L
2
(; R
n
) is not closable
and we need to come back to the general formula
[3]. The general structure of
G(A) has been given in
Bouchitte´ et al. (1997) and Bouchit te´ and Fragala`
(2002, 2003), namely
ðu;Þ2
GðAÞ()u 2 W
1;2
; 9 2 L
2
ð; R
n
Þ:
¼r
u þ ; ðxÞ2T
ðxÞ
?
where T
(x), r
(x) are suitable notions of tangent
space and tangential gradient with respect to , and
W
1, 2
denotes the domain of the ex tended tangential
gradient operator.
Remark 16 The assertion (ii) of Proposition 13
is not valid in the nonreflexive case. In
particular, for
FðuÞ¼
Z
f ðx; ruÞdx
where f (x , ) has a linear growth at infinity,
we need to take Y as the space of R
n
-values
vector measures on and the relaxed functional
F
needs to be indentified on the space BV()
of integrable functions with bounded variations.
The computation of F
is a delicate problem for
which we refer to Bouchitte´ and Dal Maso (1993)
and Bouchitte´ and Valadi er (1998).
Remark 17 By duality techniques, it is possible
also to handle variational integrals of the kind
FðuÞ¼
Z
f ðx; uðxÞ; ruðxÞÞdx
even if the dependence of f (x, u, z) with respect to u
is nonconvex. The idea consists in embedding the
space BV() in the larger space BV( R) through
the map u 7!1
u
, where 1
u
is the characteristic
function defined on R by setting
1
u
ðx; tÞ:¼
1ifuðxÞ > t
0 otherwise
Then it is possible to show, under suitable
conditions on the i ntegrand f, that there exists
a convex l.s.c., 1-homogeneous functional
G :BV( R) !R [ {þ1} such that
F(u) = G(1
u
).
This functional G is constructed as in the Example
3takingC to be a suitable convex subset of
C
0
( R). This nice new idea has been the key
tool of the calibration method developed recently
(Alberti et al. 2003).
646 Convex Analysis and Duality Methods
Convex Variational Problems in Duality
Finite-Dimensional Case
We sketch the duality scheme in two cases.
Linear programming Let c 2 R
n
, b 2 R
m
and A an
m n matrix. We denote by A
T
the transpose
matrix. We consider the linear pro gram
ðPÞ inffðcjxÞ: x 0; Ax bg
and its perturbed version (p 2 R
m
)
hðpÞ:¼ inffðcjxÞ: x 0; Ax þ p bg
An easy computation gives
8y 2 R
m
;
h
ðyÞ¼
ðbjyÞ if A
T
y þ c 0; y 0
þ1 otherwise
½4
Lemma 18 Assume that inf (P) is finite. Then:
(i) h is convex proper and l.s.c. at 0.
(ii) (P) has at least one solution.
Proof We introduce the (n þ m) (m þ 1) matrix
B defined by
B:¼
c
T
0
AI
m
(I
m
is the m-dimensional identity matrix). Denote
{b
1
, b
2
, ..., b
nþm
} R
mþ1
the columns of B and K
the convex cone K := {
P
j = nþm
j = 1
j
b
j
:
j
0}. By
Farkas lemma, this cone K is closed.
(i) Let := lim inf {h(p): p !0}. We have to prove
that h(0) = inf P. Let {p
"
} be a sequence in
R
m
such that p
"
!0andh(p
"
) !. By the
definition of h, we may choose x
"
0 such that
Ax
"
b and (c jx
"
) !. Then we see that the
column vector
~
x
"
associated with (x
"
, b Ax
"
) 2
R
nþm
satisfies: B
~
x
"
2 K and
B
~
x
"
!
b
Therefore,
b
2 K
and there exists
~
x = (x, x
0
)suchthatx 0, x
0
0,
(c jx) = and Ax þ x
0
= b. It follows that x is
admissible for (P) and then (c jx) = h(0).
(ii) We repeat the proof of (i) choosing p
"
= 0so
that = inf (P).
&
Thanks to the assertion (i) in Lemma 18,wededuce
from Theorem 10 that inf (P) = h(0) = h
(0) =
sup h
. Recalling [4], we therefore consider the dual
problem:
ðP
Þ sup b y: y 0; A
T
þ c 0
Theorem 19 The following assertions are equivalent:
(i) (P) has a solution.
(ii) (P
) has a solution.
(iii) There exists (x
0
, y
0
) 2 R
n
þ
R
m
þ
such that
Ax
0
b, A
T
y
0
þ c 0.
In this case, we have min (P) = max (P
) and
an admissible pair (
x,
y) is optimal if and
only if c
x = b
yor, equivalently, satisfies
the complem entarity relations:(A
x b)
y =
(A
T
y þ c)
x = 0.
Convex programming Let f , g
1
, ..., g
m
: X !R be
convex l.s.c. functions and the optimization problem
ðPÞ infff ðxÞ: g
j
ðxÞ0; j ¼ 1; 2 ...; mg
Here X = R
n
or any Banach space. As before, we
introduce the value function
p 2 R
m
; hðpÞ: ¼ infff ðxÞ:
g
j
ðxÞþp
j
0 j 2 1; 2; ...; mg
and compute its Fenchel conjugate:
2 R
m
; h
ðÞ¼
inf
x2X
Lðx;Þfgif 0
þ1 otherwise
where L(x, ):= f (x) þ
P
i
g
i
(x) is the so-called
Lagrangian. We notice that h is convex and that
the equality h(0) = h
(0) is equivalent to the zero-
duality gap relation
inf
x
sup
Lðx;Þ¼sup
inf
x
Lðx;Þ
This condition is fulfilled, in particular, if we make
the following qualification assumption (ensuring
that h is continuous at 0 and Theorem 11 applies):
9x
0
2 X : f continuous at x
0
; g
j
ðx
0
Þ < 0; 8j ½5
Theorem 20 Assume that [5] holds. Then
xis
optimal for (P) if and only if there ex ist Lagrangian
multipliers
1
,
2
, ...
m
in R
þ
such that
x 2 argmin
X
f þ
X
j
j
g
j
!
;
j
g
j
ð
xÞ¼0; 8j
Notice that the existence of such a solution
x
is ensured if, for example, X = R
n
and if, for some
k > 0, the function f þ k
P
j
g
j
is coercive.
Convex Analysis and Duality Methods 647
Primal–Dual Formulations in Mechanics
We present here the example of elasticity which
motivated the pioneering work by J J Moreau on
convex duality techniques. Further examples can be
found in Ekeland and Temam (1976). An elastic body is
placed in a bounded domain R
n
whose boundary
consists of two disjoint parts =
0
[
1
.The
unknown u : !R
n
(deformation) satisfies a Dirichlet
condition u = 0on
0
, where the body is clamped. The
system is subjected to a surface load g 2 L
2
(
1
; R
n
)and
to a volumic load f 2 L
2
(; R
n
). The static equilibrium
problem has the following variational formulation:
ðPÞ inf
u¼0on
0
Z
jðx; eðuÞÞdx
Z
f u dx
Z
1
g u dH
n1
where e(u):= (1=2)(u
i, j
þ u
j, i
) denotes the symmetric
strain tensor and j :(x, z) 2 R
n
2
sym
!R
þ
is a
convex integrand representing the local elastic
behavior of the material. We assume a quadratic
growth as in Remark 15 (in the case of linear
elasticity, an isotropic homogeneous material is
characterized by the quadratic form
jðx; z Þ¼
2
jtrðzÞj
2
þ jzj
2
, being the Lame´ constants).
We apply Proposition 14 with X = W
1, 2
(; R
n
),
Y = L
2
(; R
n
2
sym
), Au = e(u) and where we set
ðuÞ¼
R
f u dx
R
1
g u d H
n1
if u ¼ 0on
0
þ1 otherwise
8
>
<
>
:
ðvÞ¼
Z
jðx; vÞdx
After some computations, we may write the supre-
mum appearing in Proposition 14 as our dual
problem
ðP
Þ sup
Z
j
ðx;Þdx: 2 L
2
ð; R
n
2
sym
Þ;
div ¼ f on ; n ¼ g on
1
where j
is the Moreau–Fenchel conjugate with
respect to the second argument and n(x) denotes
the exterior unit normal on . The matrix-valued
map is called the stress tensor and j
the stress
potential. Note that the boundary conditions for n
have to be understood in the sense of traces.
Theorem 21 The problems (P) and (P
) have
solutions and we have the equality: inf(P) = sup (P
).
Futhermore, apair(
u, ) is optimal if and only if it
satisfies the following system:
div ¼ f
ðxÞ2@jðx; eð
uÞÞ
u ¼ 0
n ¼ g
on
a:e: on
a:e: on
0
on
1
ðequlibriumÞ
ðconstitutive lawÞ
Duality in Mass Transport Problems
General Cost Functions
Let X , Y be a compact metric space and c : X
Y ![0, þ1) a continuous cost function. We denote
by P(X), P(X Y) the sets of probability measures
on X and X Y, respectively. Given two elements
2P(X), 2P(Y), we denote by (, ) the subset
of probability measures in P(X Y) whose margin-
als are, respectively, and . Identified as a subset
of (C
0
(X Y))
(the space of signed Radon mea-
sures on X Y), it is convex and weakly-star
compact. The Monge–Kantorovich formulation of
the mass transport problem reads as follows:
T
c
ð;Þ:¼inf
Z
XY
cðx;yÞðdxdyÞ: 2ð;Þ
½6
This formulation, where the infimum is achieved (as
we minimize an l.s.c. functional on a compact set for
the weak star topology), is already a relaxation of
the initial Monge mass transport problem,
inf
T
Z
X
cðx; TxÞðdxÞ: T
#
ðÞ¼
where the infimum is searched among all transports
maps T : X 7!Y pushing forward on (i.e., such
that (T
1
(B) = (B ) for all Borel subset B Y).
This is equivalent to restricting the infimum in [6] to
the subclass {
T
} (, ), where
h
T
;ðx; yÞi:¼
Z
X
ðx; TxÞðdxÞ
In order to find a dual problem for [6],wefix
2P(Y) and consider the functional F : M
b
(X) !
[0, þ1) defined by
FðÞ¼
T
c
ð; Þ if 0;ðXÞ¼1
þ1 otherwise
(M
b
(X) denote the Banach space of (bounded)
signed Radon measures on X).
Lemma 22 F is convex, weakly-star l.s.c. and
proper. Its Moreau–Fenchel conjugate is given by
8’ 2 C
0
ðXÞ; F
ð’Þ¼
Z
Y
’
c
ðyÞðdyÞ
648 Convex Analysis and Duality Methods
where
’
c
ðyÞ:¼ inffcðx; yÞ’ðxÞ: x 2 Xg
Proof The convexity property is obvious and the
properness follows from the fact that
FðÞ
Z
XY
cðx; yÞ ðdxdyÞ
Let
n
be such that
n
*(weakly star). We may
assume that lim inf
n
F(
n
) = lim
n
F(
n
):= is finite.
Then
n
and the associated optimal
n
are prob-
ability measures on X and on X Y, respectively.
As X and Y are compact, possibly passin g to a
subsequence, we may assume that
n
*, and
clearly we have 2 (, ). Since c(x, y) is l.s.c.
non-negative, we conclude that
lim inf
n
Fð
n
Þ¼lim inf
n
Z
XY
cðx; yÞ
n
ðdxdyÞ
Z
XY
cðx; yÞðdxdyÞ
¼ FðÞ
Let us compute now F
(’). We have
F
ð’Þ¼inf
Z
XY
cðx; yÞðdxdyÞ
Z
X
’ d: 2PðXÞ; 2 ð; Þ
¼ inf
Z
XY
ðcðx; yÞ’ðxÞÞðdxdyÞ:
2 ð ; Þ
Z
Y
’
c
ðyÞðdyÞ
To prove that the last inequality is actually an
equality, we observe that, for every y 2 Y and ’ 2
C
0
(X), the minimum of the l.s.c. function c( , y) ’
is attained on the compact set X and there exists a
Borel selection map S(y)suchthat’
c
(y) = c(S(y), y)
’(S(y)forally 2 Y. We obtain the desired equality by
choosing defined, for every test ,by
Z
XY
ðx; yÞ ðdxdyÞ:¼
Z
Y
ðSðyÞ; yÞ ðdyÞ
&
We observe that, for every ’ 2 C
0
(X), the func-
tion ’
c
introduced in Lemma 22 is continuous (use
the uniform continuity of c) and therefore the pair
(’, ’
c
) belong to the class
F
c
:¼ð’; Þ2C
0
ðXÞC
0
ðYÞ:
’ðxÞþ ðyÞcðx; yÞg
Let us introduce the dual problem of [6]:
sup
Z
X
’ d þ
Z
Y
d : ð’; Þ2F
c
½7
We will say that (’, ) 2F
c
is a pair of c-concave
conjugate functions if = ’
c
and
c
= ’ (where
symmetrically
c
(x):= inf {c(x, y) (x): y 2 Y}).
Checking the latter condition amounts to verifying
that ’ enjoys the so-called c-concavity property
’
cc
= ’ (in general, we have only ’
cc
’, whereas
’
ccc
= ’
c
). We refer for instance to Villani (2003) for
further details about this c-duality.
Now, by exploiting Theorem 10 and Lemma 22,
we obtain a very simple proof of Kantorovich
duality theorem:
Theorem 23 The following duality formula holds:
T
c
ð; Þ¼sup
Z
X
’ d þ
Z
Y
d : ð’; Þ2F
c
Moreover, the supremum in the right-hand side
member is achieved by a pair (
’,
) of con jugate
c-concave functions such that, for any optimal
in
[6], there holds
’(x) þ
(y) = c(x, y),
-a.e.
Proof By Theorem 10 and Lemma 22, we have
T
c
ð; Þ¼F
ðÞ
¼ sup
Z
X
’ d þ
Z
Y
’
c
d: ’ 2 C
0
ðXÞ
sup
Z
X
’ d þ
Z
Y
d: ð’; Þ2F
c
T
c
ð; Þ
where the last inequality follows from the definition
of F
c
. Therefore, inf [6] = sup [7]. Furthermore, on
the right-hand side of first equality, we increase the
supremum by substituting ’ with ’
cc
(recall that
’
ccc
= ’
c
). Thus,
sup½7¼sup
Z
X
’ d
þ
Z
Y
’
c
d: ’ 2 C
0
ðXÞ;
’ c-concave
Takeamaximizingsequence(’
n
, ’
c
n
)ofc-concave
conjugate functions. It is easy to check that {f
n
}
is equicontinuous on X: this follows from the c-con-
cavity property and from the uniform continuity of
c (observe that ’
n
(x
1
) ’
n
(x
2
) = ’
cc
n
(x
1
) ’
cc
n
(x
2
)
sup
Y
{c(x
1
, ) c(x
2
, )}). Then, by Ascoli’s theorem,
possibly passing to subsequences, we may assume
that: ’
n
c
n
converges uniformly to some continuous
function
’ where {c
n
} is a suitable sequence of
reals. Then, one checks that
’ is still c-concave
and that (’
n
c
n
)
c
= ’
c
n
þ c
n
converges uniformly to
Convex Analysis and Duality Methods 649
’
c
. Thus, recalling that (X) = (Y) = 1, we
deduce that
sup½7¼lim
n
Z
X
’
n
d þ
Z
Y
’
c
n
d
¼ lim
n
Z
X
ð’
n
c
n
Þd þ
Z
Y
ð’
c
n
þ c
n
Þd
¼
Z
X
’ d þ
Z
Y
’
c
d
The last assertion is a consequence of the extrem-
ality relation:
0 ¼ inf½6sup½7
¼
Z
XY
cðx; yÞ
’ðxÞ
ðyÞ
ðdxdyÞ
&
Remark 24
(i) In their discrete version (i.e., , are atomic
measures), problems [6] and [7] can be seen as
particular linear programming problems (see the
section ‘‘Finite-dimensional case’’).
(ii) The case X = Y R
n
and c(x, y) = (1=2)jx yj
2
is important. In this case, the notion of c-concavity
is linked to convexity and the Fenchel transform
since, for every ’ 2 C
0
(X), one has
jj
2
2
’
c
¼
jj
2
2
’
!
Then if (
’,
’
c
) is a solution of [7], we find that
’
0
ðxÞ:¼
jxj
2
2
’ðxÞ
is convex continuous and that the extremality
condition:
’(x) þ
’
c
(y) = c(x, y) is equivalent to
Fenchel equality ’
0
(x) þ ’
0
(y) = (xjy). There-
fore, any optimal
is supported in the graph
of the subdifferential map @’
0
. In the case
where is absolutely continuous with respect to
the Lebesgue measure, it is then easy to deduce
that the optimal
is unique and that
=
T
0
,
where T
0
= r’
0
is the unique gradient (a.e.
defined) of a convex function such that
r’
]
0
() = . This is a celebrated result by Y
Brenier (see, e.g., the monographs by Evans
(1997) and Villani (2003)).
The Distance Case
In the following, we assume that X = Y and that
c(x, y) is a semidistance. As an immediate
consequence of the triangular inequality, we have
the following equivalence:
’ c-concave , ’ð xÞ’ðyÞcðx; yÞ; 8ðx; yÞ
, ’
c
¼’
Let us denote Lip
1
(X):= {u 2 C
0
(X): u(x) u(y)
c(x, y)}. The first assertion of Theorem 23 becomes
the Kantorovich–Rubinte in duality formula:
T
c
ð; Þ¼max
Z
X
u dð Þ: u 2 Lip
1
ðXÞ
½8
As it appears, T
c
(, ) depends only on the differ-
ence f = , which belongs to the space M
0
(X)of
signed measure on X with zero average. Defining
N(f ):= T
c
(f
þ
, f
) provides a seminorm (Kantoro-
vich norm) on M
0
(X) (it turns out that M
0
(X)is
not complete and that in general its completion is a
strict subspace of the dual of Lip(X)).
We will now specialize to the case where X is a
compact manifold equipped with a geodesic dis-
tance. This will allow us to link the original problem
to another primal–dual formulation closer to that
considered in the section ‘‘Primal–dual formulation
in mechanics’’ and yielding to a connection with
partial differential equations. As a model example,
let us assume that K =
, where is a bounded
connected open subset of R
n
with a Lipschitz
boundary. Let
be a compact subset (on
which the transport will have zero cost) and define
cðx;yÞ:¼inf H
1
ðS nÞ:
S Lipschitz curve joining x to y; S
½9
where H
1
denotes the one-dimensional Hausdorff
measure (length). It is easy to check that
cðx; yÞ¼minf
ðx; yÞ;
ðx; Þþ
ðy; Þg
where
(x, y) is the geodesic distance on (induced
by the Euclidean norm). Furthermore, the following
characterization holds:
u 2 Lip
1
ðXÞ()u 2 W
1;1
ðÞ;
jruj1 a.e. in ; u ¼ cte on ½10
Since f := is balanced, the value of the
constant on in [10] is irrelevant and can be set
to 0. Thus we may rewrite the right hand side
member of [8] in a equivalent way as
max
Z
u df: u 2 W
1;1
ðÞ;
jruj1 a.e. on ; u ¼ 0on
½11
We will now deri ve a new dual problem for [11]
by using Proposition 14. To this aim, we consider
650 Convex Analysis and Duality Methods
X = C
1
(
) (as a closed subspace of W
1, 1
()),
Y = C
0
(
; R
n
), Y
= M
b
(
; R
n
) and the operator
A : u 2 X 7!ru 2 Y.
Theorem 25 Let , 2P(
), f = and c
defined by [9]. Then,
T
c
ð; Þ¼min
Z
jj: 2M
b
ð
; R
n
Þ;
div ¼ fon
n
½12
where the divergence condition is intended in the
sense that
Z
r’ ¼
Z
’ df
for all ’ 2 C
1
compactly supported in R
n
n.
Proof (sketch) We apply Proposition 14 with
(u) =
R
u df if u = 0on (þ1 otherwise),
A ¼r, and (v) = 0ifjvj1on
(þ1 otherwise).
We obtain that the minimum in [12] is reached
and that = , where
:¼ inf
Z
u df: u 2 C
1
ð
Þ;
jruj1on u ¼ 0on
To prove that = T
c
(, ) = sup (11), we consider a
maximizer
u in [11] and prove that it can be
approximated uniformly by a sequence {u
n
}of
functions in C
1
(
) which satisfy the same con-
straints. This technical part is done by truncation
and convolution arguments (we refer to Bouchitte´
et al. (2003) for details).
&
Remark 26 By localizing the integral identity
associated with [12], it is possible to deduce
the optimality conditions which characterize optimal
pairs (
u,
) for [11], [12] (without requiring any
regularity). This is done by using a weak notion
of tangential gradient with respect to a measure
(see Bouchitte´ et al. (1997) and Bouchitte´and
Fragala` (2002)). If
= dx where 2 L
1
(; R
n
)
and if @, then we find that = ar
u, where the
pair (
u, a) solves the following system:
divðar
uÞ¼f on ðdiffusion equationÞ
jr
uj¼1 a.e. on fa > 0gðeikonal equationÞ
u ¼ 0 a.e. on
@u
@n
¼ 0on
Remark 27 Given a solution
for [6], we can
construct a solution
for [12 ] by selecting for every
(x, y) 2 spt(
) a geodesic curve S
xy
joining x and y
(possibly passing through the free-cost zone ) and
by setting, for every test :
h
; i:¼
Z
Z
S
xy
S
xy
dH
1
!
ðdxdyÞ
where
S
xy
denote the unit oriented tangent vector
(see Bouchitte´ and Buttazzo (2001)). It is also
possible to show (see Ambrosio (2003)) that any
solution
can be represented as before through a
particular solution
. As a consequence, the support
of any solution
of [12] is supported in the geodesic
envelope of the set spt() [ spt() [ . However, we
stress the fact that, in general, there is no uniqueness
at all of the optimal triple (
,
u,
) for [6], [11]
and [12].
Remark 28 An approximati on procedure for par-
ticular solutions of problems [11], [12] can be
obtained by solving a p-Laplace equation and then
by sending p to infinity. Precisely, consider the
solution u
p
2 W
1, p
()of
divðjruj
p2
ruÞ¼f on
n
u ¼ 0on
which, for p > n, exists (due to the compact
embedding W
1, p
() C
0
(
)) and is unique. In
Bouchitte´ et al. (2003) it is proved that the sequence
{(u
p
,
p
)}, where
p
= jru
p
j
p2
ru
p
, is relatively
compact in M
b
(
; R
n
) C
0
(
(weakly star with
respect to the first component) and that every cluster
point (
u,
) solves [11], [12]. It is an open pro blem
to know whether or not such a cluster point is
unique. If the answer is ‘‘yes,’’ the process described
above would select one optimal pair among all
possible solutions. As far as problem [11] is
concerned, this problem is connected with the
theory of viscosity solutions for the infinite Lapla-
cian (see Evans (1997)) although this theory does
not provide an answer as it erases the role of the
source term f. On the other hand, a new entropy
selection principle should be found for the solutions
of dual problem [12]. In fact, the following partial
result holds: let E : M
b
(
; R
n
) !R [ {þ1} be the
functional defined by
EðÞ:¼
R
jjlogðj jÞ dx if dx and ¼
d
djj
þ1 otherwise
8
<
:
Assume that [12] admits at least one solution
0
such that E(
0
) < þ1. Then it can be shown that
Convex Analysis and Duality Methods 651
the sequence {
p
} does converge weakly-star to
,
the unique minimizer of the problem
inf EðÞ: solution of ½12
fg
The general case, in particular when all optimal
measures are singular, is open.
Remark 29 Variational problems [11], [12] have
important counterparts in the theory of elasticity
and in optimal design problems (see Bouchitte´ and
Buttazo (2001)). They read, respectively, as
max
Z
u df: u 2\
p>1
W
1;p
ð; R
n
Þ;
ruðxÞ2K a:e: on ; u ¼ 0on
min
Z
0
K
ðÞ: 2M
b
ð
; R
n
2
sym
Þ;
div ¼ f on
n
where K R
n
2
sym
) is a convex compact subset of
symmetric second-order tensors associated with the
elastic material,
0
K
() = sup { z: z 2 K} is convex
positively 1-homogeneous and the functional on
measures
R
0
K
() is intended in the sense given in
[1]. A celebrated example is given by Michell’s
problem (Michell 1904) where n = 2 and K := {z 2
R
n
2
sym
, j(z)j1}, (z) being the largest singular value
of z. The potential
0
K
is given by the nondifferenti-
able convex function
0
K
() =
1
() þ
2
(), where the
i
()’s are the singular values of .
Unfortunately, it is not known if the vector
variational problem above can be linked to an
optimal transportation problem of the type [6],
even if the analogous of equivalence [10] does exist
in the Michell’s case, namely (for co nvex):
ðeðuÞÞ 1on
() j ð uðxÞuðyÞjx yÞj jx yj
2
; 8ðx; yÞ
Further Reading
Alberti G, Bouchitte´ G, and Dal Maso G (2003) The calibration
method for the Mumford–Shah functional and free-disconti-
nuity problems. Calculus of Variations and Partial Differential
Equations 16(3): 299–333.
Ambrosio L (2003) Lecture notes on optimal transport problems.
In: Mathematical Aspects of Evolving Interfaces (Funchal
2000), Lecture Notes in Mathematics, vol. 1812, pp. 1–52.
Berlin: Springer.
Borwein M and Lewis SA (2000) Convex Analysis and Nonlinear
Optimization. Theory and Examples, CMS Series. Berlin:
Springer.
Bouchitte´ G and Buttazzo G (2001) Characterization of optimal
shapes and masses through Monge–Kantorovich equations.
Journal of the European Mathematical Society 3: 139–168.
Bouchitte´ G, Buttazzo G, and De Pascale L (2003) A p-Laplacian
approximation for some mass optimization problems. Journal
of Optimization Theory and Applications 118: 1–25.
Bouchitte´ G, Buttazzo G, and Seppecher P (1997) Energies with
respect to a measure and applications to low dimensional
structures. Calculus of Variations and Partial Differential
Equations 5: 37–54.
Bouchitte´ G and Dal Maso G (1993) Integral representation and
relaxation of convex local functionals on BVðÞ. Annali della
Scuola Superiore di Pisa 20(4): 483–533.
Bouchitte´ G and Fragala` I (2002) Variational theory of weak
geometric structures: the measure method and its applications.
Variational Methods for Discontinuous Structures, Ser.
PNLDE, vol. 51, pp. 19–40. Basel: Birkha
¨
user.
Bouchitte´ G and Fragala` I (2003) Second order energies on thin
structures: variational theory and non-local effects. Journal of
Functional Analysis 204(1): 228–267.
Bouchitte´ G and Valadier M (1988) Integral representation of
convex functionals on a space of measures. Journal of
Functional Analysis 80: 398–420.
Ekeland I and Temam R (1976) Analyse convexe et proble`mes
variationnels. Paris: Dunod-Gauthier Villars.
Evans LC (1997) Partial differential equations and Monge–
Kantorovich mass transfer. In: Bott R, Jaffe A, Jerison D,
Lutsztig G, Singer I, and Yau JT (eds.) Current Developments
in Mathematics, pp. 65–126. Cambridge.
Michell AGM (1904) The limits of economy of material in frame
structures. Philosophical Magazine and Journal of Science
6: 589–597.
Rockafellar RT (1970) Convex Analysis. Princeton: Princeton
University Press.
Villani C (2003) Topics in Optimal transportation, Graduate
studies in Mathematics, vol. 58. Providence, RI: AMS.
Cosmic Censorship see Spacetime Topology, Causal Structure and Singularities
652 Convex Analysis and Duality Methods
Cosmology: Mathematical Aspects
G F R Ellis, University of Cape Town,
Cape Town, South Africa
ª 2006 Elsevier Ltd. All rights reserved.
Introduction
Mathematical cosmology focuses on the geometrical
and mathematical aspects of the study of the
universe as a whole. Because the structure of
spacetime (with metric tensor g
ab
(x
j
)) is governed
by gravity, with matter and energy causing space-
time curvature according to the nonlinear gravita-
tional field equations of the theory of general
relativity, it has its roots in differential geometry. It
is to be distinguished from the three other major
aspects of modern cosmology, namely astrophysical
cosmology, high-energy physics cosmology, and
observational cosmology; see Peacock (1999) for
these aspects.
The Einstein field equations (EFEs) are
R
ab
1
2
Rg
ab
þ g
ab
¼ T
ab
½1
where R
ab
is the Ricci tensor, R the Ricci scalar, T
ab
the matter tensor, the cosmological constant, and
the gravitational constant. Cosmological models
differ from generic solutions of these equations in
that they have preferred world lines in spacetime
associated with the motion of matter and distribu-
tion of radiation (Ellis 1971). This is a classic case of
a broken symmetry: the underlying equations [1] are
locally Lorentz invariant but their solutions are not.
These preferred world lines, characterized by a unit
4-velocity vector u
a
, are associated at late times with
‘‘fundamental observers,’’ and a key aspect of
cosmological modeling is determining the observa-
tional relations such observers would determine
through astronomical observations.
The dynamics of cosmological models is deter-
mined by their matter content. This is usually
represented in simplified form, often using the
‘‘perfect-fluid’’ approximation to represent the effect
of matter or radiation; that is,
T
ab
¼ð þ pÞu
a
u
b
þ pg
ab
½2
where is the energy density and p the pressure, and
the matter 4-velocity u
b
is the preferred cosmo-
logical 4-velocity. This description can include a
scalar field with dynamics governed by the
Klein–Gordon equation, provided u
a
is normal to
spacelike surfaces { = const}. Suitable equations of
state describe the nature of the matter envisaged
(e.g., p = 0 for baryons, whereas p = =3 for
radiation); in the case of a scalar field with potential
V() and spacelike surfaces { = const:}, on choosing
u
a
orthogonal to these surfaces, the stress tensor has
a perfect-fluid form with = (1/2)
˙
2
þ V(),
p = (1/2)
˙
2
V(). A cosmological constant can
be represented as a perfect fluid with þ p = 0,
=p. More general matter may involve a momen-
tum flux density q
a
and anisotropic pressures
ab
(Ehlers 1961). Whatever the nature of the matter, it
will usually be required to satisfy energy conditions
(Hawking and Ellis 1973). All realistic matter has a
positive inertial mass density:
þ p > 0 ½3
(note that realistic cosmological models are non-
empty), whereas all ordinary matter has a positive
gravitational mass density:
þ 3p > 0 ½4
but this is not necessarily true for a scalar field or
effective cosmological constant.
Mathematical cosmology (Ellis and van Elst 1999)
studies (1) generic properties of solutions with a
preferred 4-velocity field and matter content as
indicated above, (2) the standard FLRW models,
(3) approximate FLRW solutions, and (4) other
exact and approximate cosmological solutions. The
ultimate underlying issue is (5) the origin of the
universe. We look at these in turn. We aim to use
covariant methods as far as possible, to avoid being
misled by coordinate effects, and to obtain exact
solutions and exact results as far as possible, because
approximate methods can be misleading in the case
of these nonlinear field equations.
Exact Properties
We can split the equations into spacelike and
timelike parts relative to the 4-velocity u
a
, obtain-
ing the (1 þ 3) covariant dynamical equations and
identities in terms of the fluid shear
ab
,vorticity
!
ab
,expansion=u
a
;a
,andaccelerationa
b
=
u
a;b
u
b
(Ehlers 1961, Ellis 1971, Ellis and van Elst
1999). The energy density of a perfect fluid obeys
the conservation equation
_
¼3ð þ pÞ
_
S
S
½5
with extra terms occurring in the case of more
complex matter. From the momentum equations,
pressure-free solutions are geodesic (a
b
= 0). The
crucial Raychaudhuri–Ehlers equation for the
Cosmology: Mathematical Aspects 653
time derivative of the expansion (Ehlers 1961)
can be written as
3
€
S
S
¼ 2ð!
2
2
Þþa
b
;b
2
ð þ 3pÞþ ½6
where the representative length scale S is defined by
=3
˙
S=S. This is the basis of the ‘‘fundamental
singularity theorem’’: if in an expanding universe
! = 0 = a
b
and the combined matter present satisfies
[4],with 0, then there was a singularity where
S ! 0 a finite time t
0
< 1=H
0
ago, H
0
= (
˙
S=S)
0
being
the present value of the Hubble constant. The energy
density will diverge there, so this is a spacetime
singularity: an origin of physics, matter, and space-
time itself. However, the deduction does not follow if
there is rotation or acceleration, which could
conceivably avoid the singularity, so this result is by
itself inconclusive for realistic cosmologies.
The vorticity obeys conservation laws analogous
to those in Newtonian theory (Ehlers 1961).
Vorticity-free solutions (! = 0) occur whenever the
fluid flow lines are hypersurface-orthogonal in
spacetime, that is, there exists a cosmic time
function for the comoving observers, which will
measure proper time along the flow lines if
additionally the fluid flow is geodesic. The rate of
change of shear is related to the conformal curvature
(Weyl) tensor, which represents the free gravita-
tional field, and which splits into an electric part E
ab
and a magnetic part H
ab
in close analogy with
electromagnetic theory. Shear-free solutions ( = 0)
are very special because they strongly constrain the
Weyl tensor; indeed if the flow is shear free and
geodesic, then it either does not expand (=0), or
does not rotate (! = 0) (Ellis 1967). The set of
cosmological observations associated with generic
cosmological models has been characterized in
power series form by Kristian and Sachs (1966),
and that result has been extended to general models
by Ellis et al. (1985).
The local regularity of the theory is expressed in
existence and uniqueness theorems for the EFEs,
provided the matter behavior is well defined through
prescription of suitable equations of state (Hawking
and Ellis 1973). However, in general the theory
breaks down in the large, and this feature is
specified by the Hawking–Penrose singularity theo-
rems, predicting the existence of a geodesic incom-
pleteness of spacetime under conditions applicable
to realistic cosmological models satisfying the energy
conditions given by eqns [3] and [4] (Hawking and
Ellis 1973, Tipler et al. 1980). However, the
conclusion does not follow if the energy conditions
are not satisfied. Furthermore, the deduction follows
only if the gravitational field equations remain valid
to arbitrarily early times; but we would in fact
expect that, at high enough energy densities,
quantum gravity would take over from classical
gravity, so whether or not there was indeed a
singularity would depend on the nature of the as
yet unknown theory of quantum gravity. The cash
value of the singularity theorems then is the
implication that, when the energy conditions are
satisfied, one would indeed be involved in such a
quantum gravity realm in the very early universe.
The Standard Friedmann–Lemaıˆtre
Models
The standard models of cosmology are the Fried-
mann–Lemaıˆtre (FL) models with Robertson–Walker
(RW) geometry: that is, they are exactly spatially
homogeneous and locally isotropic, invariant under a
G
6
of isometries (Robertson 1933, Ehlers 1961).
They have a unique cosmic time function t,with
space sections {t = const:} of constant spatial curva-
ture orthogonal to the uniquely preferred 4-velocity
u
a
. The fluid acceleration, vorticity, and shear all
vanish, and all physical quantities depend only on the
time coordinate t. They can be represented by a
metric with scale factor S(t):
ds
2
g
ab
dx
a
dx
b
¼dt
2
þ S
2
ðtÞfdr
2
þ f
2
ðrÞðd
2
þ sin
2
d
2
Þg
½7
in comoving coordinates (x
a
) = (t, r, , ), where f (r) =
{sinr, r,sinhr}if{k = þ1, 0, 1}, and the matter is a
perfect fluid with 4-velocity vector u
a
= dx
a
=ds =
a
0
.
The curvature of the space sections {t = const:}is
K = k=S
2
; these 3-spaces are necessarily closed (com-
pact) if they are positively curved (k = þ1), but may be
open or closed in the flat (k = 0) and negatively curved
(k = 1) cases, depending on their topology
(Lachieze-Rey and Luminet 1995).
Matter obeys the conservation equation [5],whose
outcome depends on the equation of state; for
baryons = M=S
3
, whereas for radiation = M=S
4
,
where M is a constant. The dynamics of the models is
governed by the Raychaudhuri equation
3
€
S
S
¼
2
ð þ 3pÞþ ½8
which has the Friedmann equation
3
_
S
2
S
2
¼ þ
3k
S
2
½9
as a first integral whenever
˙
S 6¼ 0. Depending on the
matter components present, one can qualitatively
654 Cosmology: Mathematical Aspects