On the Matrix Monge-Kantorovich Problem

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by  Yongxin Chen

1
On the Matrix Monge-Kantorovich Problem
Yongxin Chen, Wilfrid Gangbo, Tryphon T. Georgiou, and Allen Tannenbaum
Abstract
The classical Monge-Kantorovich (MK) problem as originally posed is concerned with how best to move a
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1 pileofsoilorrubbletoanexcavationorfillwiththeleastamountofworkrelativetosomecostfunction.Whenthe
0 cost is given by the square of the Euclidean distance, one can define a metric on densities called the Wasserstein
2
distance. In this note, we formulate a natural matrix counterpart of the MK problem for positive definite density

n matrices. We prove a number of results about this metric including showing that it can be formulated as a convex
a
optimization problem, strong duality, an analogue of the Poincare´-Wirtinger inequality, and a Lax-Hopf-Oleinik
J
type result.
1
1

]
A
I. INTRODUCTION
F
.
h
The mass transport problem was first formulated by Monge in 1781 and concerned finding the optimal
t
a
way, in the sense of minimal transportation cost, of moving a pile of soil from one site to another. This
m
problem of optimal mass transport (OMT) was given a modern formulation in the work of Kantorovich,
[

and so is now known as the Monge–Kantorovich (MK) problem; see [14], [16] and the many references
1
v therein. As originally formulated, the problem is static. Namely, given two probability densities, one can
6
define a metric, now known as the Wasserstein distance, that quantifies the cost of transport and enjoys a
2
8 number of remarkable properties as described in [14], [16]. Optimal mass transport is a very active area
2
of research with applications to numerous disciplines including probability, econometrics, fluid dynamics,
0
. automatic control, transportation, statistical physics, shape optimization, expert systems, and meteorology.
1
0
7 A major development in optimal mass transport theory was realized in the seminal dynamic approach
1
to optimal mass transport by Benamou and Brenier [4]. These authors base their approach to OMT on
:
v
ideas from fluid mechanics via the minimization of a kinetic energy functional subject to a continuity
i
X
constraint. In a recent paper [3] by Chen et al., a non-commutative counterpart of optimal transport was
r
a developed where density matrices ρ (i.e., Hermitian matrices that are positive-definite and have unit trace)
replace probabilitydistributions,and where “transport” corresponds to a flow on the space ofsuch matrices
that minimizes a corresponding action integral, thereby extending the fluid dynamical approach of [4].
(A similar approach to [3] was done independently by Carlen and Maas at about the same time [2].)
Here again, based on the continuity equation that imposes a “mass preservation” constraint on a quadratic
“kinetic energy,” we study a convex optimization problem that leads to a certain Riemannian structure on
densities matrices and generalizes the work of [13] to the current non-commutative setting.
Y. Chen is with the Department of Medical Physics, Memorial Sloan Kettering Cancer Center, NY; email: [email protected]
W. Gangbo is with the Department of Mathematics, UCLA, Los Angeles, CA; email: [email protected]
T. T. Georgiou is with the Department of Mechanical and Aerospace Engineering, University of California, Irvine, CA; email:
[email protected]
A.Tannenbaum iswiththeDepartmentsofComputerScienceandAppliedMathematics&Statistics,StonyBrookUniversity,NY;email:
[email protected]
2
The present paper is a continuation of [3] in which a number of the results are given rigorous
mathematical proofs. We show that indeed in line with [4], one has a convex optimization problem and
strong duality, an analogue of the Poincare´-Wirtinger inequality, and a Lax-Hopf-Oleinik type result, all
in our non-commutative Wasserstein framework.
II. CONTINUITY EQUATION & WASSERSTEIN DISTANCE
In thissection, weset up thecontinuityequation thatis thebasisfor ourformulationof theWasserstein
distancefordensitymatrices.Wefollowcloselytherecentpaper[3].Inthatwork,anapproachisdeveloped
based on the Lindblad equation which describes the evolution of open quantum systems. Open quantum
systems are thought of as being coupled to a larger system (heat bath), and thus cannot in general be
described by a wave function and a unitary evolution. The proper description is in terms of a density
operator ρ [8] which in turn obeys the Lindblad equation where we assume ~ = 1:
N
1 1
∗ ∗ ∗
ρ˙ = −i[H,ρ]+ (L ρL − ρL L − L L ρ). (1)
k k 2 k k 2 k k
Xk=1
Here, * as superscript denotes conjugate transpose and [H,ρ] := Hρ−ρH denotes the commutator. The
first term on the right-hand side describes the evolution of the state under the effect of the Hamiltonian H,
and it is unitary (energy preserving), while the other the terms on the right-hand side model diffusion and,
thereby, capture the dissipation of energy; these dissipative terms together represent the quantum analogue
of Laplaces operator ∆ (as it will become clear shortly). The Lindblad equation defines a non-unitary
evolution of the density and the calculus we develop next actually underscore parallels with classical
diffusion and the Fokker-Planck equation.
Denote by H and S the set of n×n Hermitian and skew-Hermitian matrices, respectively. We will
assume that all of our matrices are fixed to be n×n. Next, we denote the space of block-column vectors
consisting of N elements in S and H as SN, respectively HN. We let H and H denote the cones of
+ ++
nonnegative and positive-definite matrices, respectively, and
D := {ρ ∈ H | tr(ρ) = 1}, (2)
+
D := {ρ ∈ H | tr(ρ) = 1}. (3)
+ ++
We note that the tangent space of D , at any ρ ∈ D+ is given by
+
T = {δ ∈ H | tr(δ) = 0}, (4)
ρ
and we use the standard notion of inner product, namely
∗
hX;Yi = tr(X Y),
for both H and S. For X,Y ∈ HN (SN),
N
∗
hX;Yi = tr(X Y ).
k k
Xk=1
Given X = [X∗,··· ,X∗]∗ ∈ HN (SN), Y ∈ H (S), set
1 N
X X Y
1 1
. .
XY =  .. Y :=  .. ,
 X   X Y 
N N
   
3
and
X YX
1 1
. .
YX = Y  ..  :=  .. .
 X   YX 
N N
   
If we assume that L = L∗, i.e., L ∈ H for all k ∈ 1...,N, then we can define
k k k
L X −XL
1 1
.
∇L : H → SN, X 7→  ..  (5)
 L X −XL 
N N
 
as the gradient operator. Note that ∇ acts just like the standard gradient operator, and in particular,
L
satisfies
∇ (XY +YX) = (∇ X)Y +X(∇ Y)+(∇ Y)X +Y(∇ X), ∀X,Y ∈ H. (6)
L L L L L
The dual of ∇ , which is an analogue of the (negative) divergence operator, is defined by
L
Y
1 N
∇∗L : SN → H, Y =  ...  7→ LkYk −YkLk. (7)
 Y  Xk
N
 
The duality
∗
h∇ X;Yi = hX;∇ Yi
L L
follows by definition.
With these definitions, we define the (matricial) Laplacian as
N
∗ ∗ ∗ ∗
∆ X := −∇ ∇ X = (2L XL −XL L −L L X), X ∈ H,
L L L k k k k k k
Xk=1
whichisexactly(afterscalingby1/2)thediffusiontermintheLindbladequation(1).ThereforeLindblad’s
equation (under the assumption that L = L∗) can be rewritten as
k k
1
ρ˙ = −i[H,ρ]+ ∆ ρ,
L
2
i.e., as a continuity equation expressing flow under the influence of a suitable vector field.
In our case, we will consider a continuity equation of the form
∗
ρ˙ = ∇ m
L
for m ∈ SN a suitable “momentum field.” In particular, we are interested in the following family of
continuity equations:
∗
ρ˙ = ∇ M (v), (8)
L ρ
where the momentum field is expressed as a non-commutative product M (v) ∈ SN between a “velocity
ρ
field” v ∈ SN and the density matrix ρ.
Several such “non-commutative products” have been considered (see [3]), however, in the present
work, we consider the following case:
1
M (v) := (vρ+ρv), (9a)
ρ
2
4
which gives
1
∗
ρ˙ = ∇ (vρ+ρv) (9b)
2 L
and v = [v∗,...,v∗ ]∗ ∈ SN. Clearly vρ+ρv ∈ SN, which is consistent with the definition of ∇∗. In [3],
1 N L
we call this the anti-commutator case, since
vρ+ρv =: {v,ρ}
is the anti-commutator when applied to elements of an associative algebra. In [3], another possibility is
considered for the multiplication operator M (v).
ρ
Given two density matrices ρ ,ρ ∈ D we formulate the optimization problem (following [3])
0 1 +
1
W (ρ ,ρ )2 := inf tr(ρv∗v)dt, (10a)
2 0 1
ρ∈D+,v∈SN Z0
1
∗
ρ˙ = ∇ (vρ+ρv), (10b)
2 L
ρ(0) = ρ , ρ(1) = ρ , (10c)
0 1
and define the Wasserstein distance W (ρ ,ρ ) between ρ and ρ to be the square root of the infimum
2 0 1 0 1
of the cost (10a). Other choices for M (v) in (9a) give alternative Wasserstein metrics, as noted in [3]. In
ρ
order for the metric W to be well-defined for all ρ ,ρ ∈ D , we need to assume that ker(∇ ) is spanned
2 0 1 + L
by the identity matrix. The results in the present paper, however, carry through without this assumption.
III. QUADRATIC FORMS AND POINCARE´-WIRTINGER INEQUALITY
In this section, we prove some initial convexity results as well as a Poincare´-Wirtinger type inequality
that we will need in the sequel. We begin with some notation. If m ,··· ,m ∈ Cn×n, we define the
1 N
column vector m ∈ CnN×n with matrix entries the m ’s by
i
∗ ∗ ∗
m = (m ,··· ,m ) ,
1 N
the column vector m ∈ CnN×n with entries m∗’s, by
∗ i
∗
m = (m ,··· ,m ) .
∗ 1 N
For m,b ∈ CnN×n, i.e., with b = (b∗,··· ,b∗ )∗ for b ,··· ,b ∈ Cn×n, we define the inner products
1 N 1 N
N
∗ ∗
hm ;b i = tr(m b ), hm;bi = tr(m b) = hm ;b i
i i i i i i
Xi=1
and introduce
1
m·b = hm;bi+hb;mi ∈ R.
2
(cid:0) (cid:1)
Then, for v ∈ CnN×n and ρ ∈ H , we define the quadratic form
+
∗
Q (v) := tr(ρv v) = hvρ;vi. (11)
ρ
The following lemma is an easy consequence of the definition and can be readily verified.
Lemma 1: Let ρ ∈ H and v,w ∈ CnN×n. The following hold:
+
5
(i) Q (v) ≥ 0 and Q (v) = 0 if v = 0; when ρ ∈ H , it becomes if and only if,
ρ ρ ++
(ii) Q (v +w) = Q (v)+Q (w)+hvρ;wi+hwρ;vi,
ρ ρ ρ
(iii) Q (1−t)v +tw = (1−t)Q (v)+tQ (w)−t(1−t)Q (v −w),
ρ ρ ρ ρ
(iv) If w(cid:0)e further assu(cid:1)me that v,w ∈ SN, then hwρ;vi = hρv;wi.
Since ||·|| (the standard norm on H) is uniformly convex and ker(∇ ) is a finite dimensional space,
L
there exists a unique proj(X) ∈ ker(∇ ), the orthogonal projection of X onto ker(∇ ), such that
L L
min ||X −Z|| = ||X −proj(X)||.
Z∈ker(∇L)
If we denote by ker(∇ )⊥ the orthogonal complement of ker(∇ ) in H, then
L L
⊥
H = ker(∇ )⊕ker(∇ ) .
L L
Lemma 2: For any ρ ∈ H , the map X → Q (∇ X) is convex on H. If in addition ρ > 0, then the
+ ρ L
map is strictly convex on ker(∇ )⊥.
L
Proof: From Lemma 1, we obtain that, for t ∈ (0,1) and X,Y ∈ H,
Q (1−t)∇ X +t∇ Y = (1−t)Q (∇ X)+tQ (∇ Y)−t(1−t)Q (∇ X −∇ Y).
ρ L L ρ L ρ L ρ L L
(cid:0) (cid:1)
The convexity follows from the fact that Q (·) ≥ 0. Furthermore, if ρ > 0, then Q (∇ X −∇ Y) > 0
ρ ρ L L
unless ∇ X −∇ Y = 0. Hence, X → Q (∇ X) strictly convex on ker(∇ )⊥.
L L ρ L L
Theorem 3 (Poincare´–Wirtinger inequality): Let K ⊂ D be a compact set. Then there exists a
+
constant cK > 0 such that for all X ∈ H and ρ ∈ K,
2
Qρ ∇L(X −proj(X)) ≥ cK X −proj(X) .
(cid:0) (cid:1) (cid:13) (cid:13)
(cid:13) (cid:13)
Proof: Define
cK := inf tr ρ(∇LX)∗∇LX | ρ ∈ K,X ∈ ker(∇L)⊥,||X|| = 1 (12)
ρ,Xn o
(cid:0) (cid:1)
and let (ρ ,X ) be a minimizing sequence in (12). This infimum of a continuous function over a compact
k k k
setisaminimum,attainedatacertain(ρ,X).SinceX ∈ ker(∇ )⊥,wecannothaveX ∈ ker(∇ )because,
L L
we would otherwise have X = 0 which will contradict the fact that ||X|| = 1. Since ρ > 0 we conclude
that
∗
tr ρ(∇ X) ∇ X > 0 (13)
L L
(cid:16) (cid:17)
In conclusion
∗
tr ρ(∇LY) ∇LY ≥ cK > 0
(cid:0) (cid:1)
for any ρ ∈ K and any Y ∈ ker(∇ )⊥ such that ||Y|| = 1. By homogeneity, this completes the proof of
L
the theorem.
Lemma 4: Given ρ ∈ H and v ∈ SN, there is a unique element ∇ X,X ∈ H, such that
++ L
Q (v −∇ X) ≤ Q (v −∇ Y)
ρ L ρ L
for all Y ∈ H. Furthermore, the minimizer is characterized by the Euler–Lagrange equations
∗
(v −∇ X)ρ+ρ(v −∇ X) ∈ ker(∇ ). (14)
L L L
6
Proof: Let (X ) ⊂ H be a sequence such that
ℓ ℓ
lim Q (v −∇ X ) = inf Q (v −∇ Y).
ρ L ℓ ρ L
ℓ→∞ Y∈H
Note that Q (∇ X ) is bounded by definition and the convexity of Q (·). Replacing X by X −
ρ L ℓ ℓ ρ ℓ ℓ
proj(X ) i(cid:0)f necessary,(cid:1)we use the Poincare´–Wirtinger inequality to conclude that (X ) is a bounded
ℓ ℓ ℓ
sequence. Passing to a subsequence if necessary, we may assume that (X ) converges to some X which
ℓ ℓ
minimizes Q (v−∇ X) over H. The uniqueness follows from the strict convexity of Q (·), and condition
ρ L ρ
(14) expresses stationarity.
IV. FLOW RATES IN THE SPACE OF DENSITIES
We now return to the continuity equation
ρ˙ = f
with the flow rate f being the divergence of a momentum field p, i.e.,
∗
f = ∇ p, (15)
L
with p ∈ SN, so that f ∈ H as well as tr(f) = 0. In particular, we are interested in the case where
the momentum is a linear function of ρ of the form p = M (v) (see (9a)); then p = 1(m − m ) with
ρ 2 ∗
m = vρ ∈ CNn×n, ρ ∈ H , and v ∈ SN, and
++
1
∗
f = ∇ (vρ+ρv).
2 L
Since the range of ∇∗ coincides with ker(∇ )⊥, any f belongs to ker(∇ )⊥. The next theorem states
L L L
that in this case not only the converse holds, namely, that given ρ ∈ H , any f ∈ ker(∇ )⊥ can be
++ L
written as above with m = vρ, but that v ∈ SN can be selected in the range of ∇ and that this choice
L
is unique.
Theorem 5: For any ρ ∈ H and f ∈ ker(∇ )⊥, there exists a unique X ∈ ker(∇ )⊥ such that
++ L L
1
∗
f = ∇ ∇ Xρ+ρ∇ X . (16)
2 L L L
(cid:0) (cid:1)
Furthermore, if K is a compact subset of D+ and ρ/tr(ρ) ∈ K then there exists cK > 0 such that
||f|| ≥ cK tr(ρ)||X||. (17)
Proof: Define the functional
1
I(Y) = Q (∇ Y)−hf;Yi, ∀ Y ∈ H.
ρ L
2
To avoid trivialities, we assume that f 6= 0. Observe that I(Y) ≡ 0 on ker(∇ ) and for 0 < ǫ << 1 we
L
have that
ǫ2
I(ǫf) = Q (∇ f)−ǫ||f||2 < 0
ρ L
2
Thus, λ , the infimum of I over H is negative. Let (Y ) be a minimizing sequence. Since λ < 0, for ℓ
0 ℓ ℓ 0
large enough, I(Y ) < 0 and so, Y ∈ H\ker(∇ ). Replacing Y by Y −proj(Y ) if necessary, we may
ℓ ℓ L ℓ ℓ ℓ
assume that Y ∈ ker(∇ )⊥. By the Poincare´–Wirtinger inequality
ℓ L
0 > I(Yℓ) ≥ cKtr(ρ)||Yℓ||2 −||f||||Yℓ||.
7
Consequently, (Y ) is a bounded sequence and so, passing to a subsequence if necessary, we may assume
ℓ ℓ
that (Yℓ)ℓ converges to some X ∈ ker(∇L)⊥. We have 0 ≥ cKtr(ρ)||X||2−||f||||X|| and so, (17) holds.
If Y ∈ H is arbitrary, then for any real number ǫ, we use Lemma 1 to conclude that
1
∗
I(X +ǫY) = I(X)+ǫ ∇ ∇ Xρ+ρ∇ X −f;Y +o(ǫ).
2 L L L
(cid:10) (cid:0) (cid:1) (cid:11)
By Lemma 2, I is convex on H and so, X is a critical point of I if and only if X minimizes I. Thus (16)
holds if and only if X minimizes I. Since, the same lemma gives that I is strictly convex on ker(∇ )⊥,
L
I admits a unique minimizer on ker(∇ )⊥ which means that there exists a unique X ∈ ker(∇ )⊥ such
L L
that (16) holds.
Remark 6: The uniqueness and existence of the representation may also be proven as follows. We
first note that provided ρ ∈ H , the non-commutative multiplication in (9a), defines a positive definite
++
Hermitian operator
1
M : SN → SN : v 7→ (vρ+ρv).
ρ
2
It follows that ∇∗M ∇ , when restricted to ker(∇ )⊥ = range(∇∗), is positive and therefore invertible.
L ρ L L L
Thus, for all f ∈ ker(∇ )⊥, (16) has a unique solution X ∈ ker(∇ )⊥. One also gets a lower bound on
L L
the norm of f as follows. If λ > 0 denotes the smallest eigenvalue of
min
∗ ⊥ ⊥
∇ M ∇ | : ker(∇ ) → ker(∇ ) ,
L ρ L ker(∇L)⊥ L L
then
kfk ≥ λ kXk.
min
V. FLOWS IN THE SPACE OF DENSITIES
We begin with establishing a canonical representation of flow rates that minimize a certain analogue
of kinetic energy of our matrix-valued flows.
Proposition 1: Suppose ρ ∈ H , f ∈ ker(∇ )⊥, X ∈ H satisfy (16), and that v ∈ SN is such that
+ L
1
∗
f = ∇ vρ+ρv .
2 L
(cid:0) (cid:1)
The following hold:
(i) For all Y ∈ H,
1 1
∗
tr(ρv v) ≥ hf;Yi− Q (∇ Y)
ρ L
2 2
and equality holds if and only if v = ∇ X.
L
(ii) Further assume that ρ > 0 (which by Theorem 5 is a sufficient condition for (16) to hold). Then
1 1 1
min hm;mρ−1i | f = ∇∗(m−m ) = max hf;Yi− Q (∇ Y) .
m∈CnN×nn2 2 L ∗ o Y∈H (cid:26) 2 ρ L (cid:27)
Besides, the maximum is uniquely attained by the X which satisfies (16) and the minimum is
uniquely attained by m = ∇ Xρ for the same X.
L
8
Proof: (i) We have
1 1 1 1 1
∗ 1 1 ∗ 1 1
2 tr(ρv v) = 2hvρ2;vρ2i+hf − 2∇L vρ+ρv ;Yi = 2hvρ2;vρ2i+hf;Yi− 2hvρ+ρv;∇LYi.
(cid:0) (cid:1)
Since both v as well as ∇ Y belong to SN (cf. Lemma 1 (iv)),
L
1 1 1 1
hvρ+ρv;∇LYi = hvρ2;∇LYρ2i+h∇LYρ2;vρ2i.
We conclude that
1 1 1 1
tr(ρv∗v) = kvρ12 −∇LYρ21k2 +hf;Yi− k∇LYρ21k2 ≥ hf;Yi− Qρ(∇LY).
2 2 2 2
(ii) Computations similar to the ones in (i) reveal that
1 1 1
hm;mρ−1i = kmρ−12 −∇LYρ12k2 +hf;Yi− k∇LYρ21k2 (18)
2 2 2
and so, for Y ∈ H,
1 1
hm;mρ−1i ≥ hf;Yi− Q (∇ Y).
ρ L
2 2
We proceed to consider paths ρ(t) ∈ H for t ∈ [0,1] along with corresponding flow rates and action
+
integrals. A corollary of the above proposition ascertains the measurability of the canonical representation
of the velocity field v.
Corollary 7: Let L ⊂ H and denote by A : ker(∇ )⊥×L → H the map which to (f,ρ) associates
++ L
X ∈ ker(∇ )⊥ such that (16) holds.
L
(i) If L is a compact subset of H , then A is continuous.
++
(ii) If ρ : [0,1] → H and f : [0,1] → ker(∇ )⊥ are continuous at t ∈ [0,1] then A(f,ρ) is
++ L 0
continuous at t .
0
(iii) If ρ ∈ L1(0,1;H ) and f ∈ L1(0,1;ker(∇ )⊥) are measurable, then A(f,ρ) is measurable.
++ L
Proof: (i) Let K be the set of ρ/tr(ρ) such that ρ ∈ L. Let (f ,ρ ) be a sequence in ker(∇ )⊥×L
ℓ ℓ ℓ L
converging to (f,ρ). By Theorem 5, (X ) := A(f ,ρ ) is a bounded sequence in ker(∇ )⊥ and so,
ℓ ℓ ℓ ℓ l L
has all its points of accumulation in ker(∇ )⊥.(cid:0)If X is an(cid:1)y such point of accumulation, then clearly
L
1
∗
f = ∇ (∇ Xρ+ρ∇ X).
2 L L L
Since ρ is invertible, X is unique and so, A(f,ρ) = X. This establishes (i).
(ii) Condition (ii) is a direct consequence of (i).
(iii) Approximate ρ in the L1–norm by a sequence (ρ ) ⊂ C([0,1];H ) which converges pointwise
ℓ ℓ ++
almosteverywheretoρ.Similarly,approximatef intheL1–normbyasequence(f ) ⊂ C([0,1];ker(∇ )⊥)
ℓ ℓ L
which converges pointwise almost everywhere to f. By (i), A(f ,ρ ) converges pointwise almost
ℓ ℓ ℓ
everywhere to A(f,ρ) and so, A(f,ρ) is measurable. This estab(cid:0)lishes (iii(cid:1)).
Lemma 8: If ρ ,ρ ∈ D , then the following hold:
0 1 +
9
(i) If ρ − ρ ∈ ker(∇ )⊥, then there exists a Borel map t → X(t) ∈ ker(∇ )⊥ and a Borel map
1 0 L L
t → ρ(t) starting at ρ(0) = ρ and ending at ρ(1) = ρ such that
0 1
1
∗
ρ˙(t) = ∇ ∇ X(t)ρ(t)+ρ(t)∇ X(t) (19)
2 L L L
(cid:0) (cid:1)
in the sense of distributions, and
1
Q (∇ X(t))dt < ∞. (20)
ρ(t) L
Z
0
(ii) Conversely, assume that there exist a Borel map t → v(t) ∈ SN and a Borel map t → ρ(t) starting
at ρ and ending at ρ such that
0 1
1
∗
ρ˙(t) = ∇ v(t)ρ(t)+ρ(t)v(t) (21)
2 L
(cid:0) (cid:1)
in the sense of distributions, and
1
∗
tr(ρ(t)v(t) v(t))dt < ∞. (22)
Z
0
Then ρ −ρ ∈ ker(∇ )⊥.
1 0 L
Proof: (i) Assume ρ −ρ ∈ ker(∇ )⊥, Set ρ(t) = (1−t)ρ +tρ . Then K := {ρ(t) | t ∈ [0,1]} is
1 0 L 0 1
a compact subset of D . For each t ∈ [0,1], we use Theorem 5 to find a unique X(t) ∈ ker(∇ )⊥ such
+ L
that
1
∗
ρ −ρ = ∇ ∇ X(t)ρ(t)+ρ(t)∇ X(t)
1 0 2 L L L
(cid:0) (cid:1)
and
||X(t)k ≤ cK.
By Corollary 7, t → X(t) ∈ ker(∇ )⊥ is continuous. Hence, (19) and (20) hold.
L
(ii) Conversely, assume (21) and (22) hold. Let Y ∈ ker(∇ ), then
L
1 1 1 1
∗
hρ −ρ ;Yi = h∇ v(t)ρ(t)+ρ(t)v(t) ;Yidt = hv(t)ρ(t)+ρ(t)v(t);∇ Yidt = 0.
1 0 2 Z L 2 Z L
0 (cid:0) (cid:1) 0
Since Y ∈ ker(∇ ) is arbitrary, we conclude that ρ −ρ ∈ ker(∇ )⊥.
L 1 0 L
Remark 9: ObservethatinLemma8 (ii), ifwerelax theassumptionsonρ and ρ by merelyimposing
0 1
that ρ ,ρ ∈ H , then (21) and (22) still imply ρ −ρ ∈ ker(∇ )⊥.
0 1 + 1 0 L
For ρ ∈ H and m ∈ CnN×n we set
++
1
F(ρ,m) := hm,mρ−1i.
2
Given ρ ,ρ ∈ D , denote by C(ρ ,ρ ) the set of paths (ρ,v) such that ρ ∈ C1([0,1],D ) start at ρ and
0 1 + 0 1 + 0
end at ρ , v : (0,1) → SN is Borel, Q (v) ∈ L1(0,1) and
1 ρ
1
∗
ρ˙ = ∇ (vρ+ρv)
2 L
in the sense of distributions on (0,1). Similarly, we define C˜(ρ ,ρ ) to be the set of paths (ρ,m) such
0 1
that ρ ∈ C1([0,1],D ) start at ρ and end at ρ , m : (0,1) → CnN×n is Borel, F(ρ,m) ∈ L1(0,1) and
+ 0 1
1
∗
ρ˙ = ∇ (m−m )
2 L ∗
10
in the sense of distributions on (0,1).
Observe that if v ∈ SN and we set m = vρ then
1
∗
F(ρ,m) = tr(ρv v).
2
and so, the embedding (ρ,v) → (ρ,vρ) of C(ρ ,ρ ) into C˜(ρ ,ρ ), extends 1 tr(ρv∗v) to F(ρ,m).
0 1 0 1 2
Consequently,
1 1 1
inf tr(ρv∗v)dt | (ρ,v) ∈ C(ρ ,ρ ) ≥ inf F(ρ,m)dt | (ρ,m) ∈ C˜(ρ ,ρ ) .
0 1 0 1
(ρ,v)nZ0 2 o (ρ,m)nZ0 o
We next show that the inequality can be turned into an equality.
Lemma 10: If ρ ,ρ ∈ D then
0 1 +
1 1 1
inf tr(ρv∗v)dt | (ρ,v) ∈ C(ρ ,ρ ) = inf F(ρ,m)dt | (ρ,m) ∈ C˜(ρ ,ρ )
0 1 0 1
(ρ,v)nZ0 2 o (ρ,m)nZ0 o
Proof: It suffices to show that for any (ρ,m) ∈ C˜(ρ ,ρ ), there exists (ρ,v) ∈ C(ρ ,ρ ) such that
0 1 0 1
1 1 1
∗
tr(ρv v)dt ≤ F(ρ,m)dt. (23)
Z 2 Z
0 0
Observe that for almost every t ∈ (0,1) we have
1
∗
ρ˙(t) = ∇ (m(t)−m (t)). (24)
2 L ∗
Since both t → ρ(t) and t → ρ˙(t) are continuous, by Corollary 7, there exists a continuous map X :
[0,1] → H such that
1
∗
ρ˙(t) = ∇ ∇ X(t)ρ(t)+ρ(t)∇ X(t) . (25)
2 L L L
(cid:16) (cid:17)
Thus, for almost every t ∈ (0,1), both (24) and (25) hold and so, by Proposition 1
1
∗
tr ρ(t)v (t)v(t) ≤ F ρ(t),m(t)
2
(cid:0) (cid:1) (cid:0) (cid:1)
for almost every t ∈ (0,1) with v(t) = ∇ X(t). Thus, (23) holds, which concludes the proof.
L
VI. RELAXATION OF VELOCITY-MOMENTUM FIELDS
Given ρ ,ρ ∈ D we are interested in characterizing the paths (ρ,v) in H ×SN that minimize the
0 1 + +
“action integral,” i.e., paths that possibly attain
1 1
∗ ∗
inf tr(ρv v)dt ρ˙ = ∇ (vρ+ρv), ρ(0) = ρ , ρ(1) = ρ . (26)
ρ∈H+,v∈SNnZ0 (cid:12) 2 L 0 1o
(cid:12)
(cid:12)
When ρ > 0, Lemma 10 replaced tr(ρv∗v) by a new expression F(ρ,vρ), introducing a new problem
which, under appropriate conditions, is a relaxation of (26). It then becomes neccesary to extend F to
the whole set H × CnN×n and study the convexity properties of the extended functional. We start by
introducing the open sets
O := H ×CnN×n, O := {ρ ∈ H\H }×CnN×n.
0 ++ ∞ +