HAL Id: hal-03099152
https://hal.archives-ouvertes.fr/hal-03099152
Submitted on 6 Jan 2021
HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come from
L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de
Monge-Kantorovich equation for degenerate Finsler
metrics
Van Nguyen
To cite this version:
Monge–Kantorovich equation for degenerate Finsler metrics
Van Thanh Nguyen∗
Department of Mathematics and Statistics, Quy Nhon University, Vietnam October 21, 2020
Abstract
The paper establishes that a generalization of Monge–Kantorovich equation gives rise to a necessary and sufficient optimality condition for the Kantorovich dual problem and minimal flow problem associated with a very degenerate Finsler metric without any assumption on coerciveness.
Keywords. Monge–Kantorovich equation, degenerate Finsler metric, Kantorovich dual problem, minimal flow
AMS subject classifications. 49J20, 49K20, 49Q22, 35Q93.
1
Introduction
Monge–Kantorovich equation appears as an optimality condition to the two following variational problems, called Kantorovich dual problem (see e.g. [13, 14]),
max u∈W1,∞(Ω) Z Ω u d(f1− f0) : |∇u| ≤ 1
and, called minimal flow problem (or, Beckmann’s continuous model of transportation [1], see also [12]), min Φ∈M(Ω)N Z Ω d|Φ| : −div Φ = f1− f0 in D 0 (Ω) . (1.1)
Here, Ω is a bounded Lispchitz domain of RN. Two finite Radon measures f0 and f1
on Ω satisfy the mass balance Z
Ω
df0 =
Z
Ω
df1. The space of vector-valued finite Radon
measures on Ω is denoted by M(Ω)N. The divergence constraint −divΦ = f1−f0 in D
0
(Ω)
∗
is understood in the sense of distribution, coupled with a homogeneous Neumann boundary condition, that is,
Z Ω Φ |Φ|· ∇ξ d|Φ| = Z Ω ξd(f1− f0), ∀ξ ∈ C1(Ω).
The Monge–Kantorovich equation has first been studied by Evans and Gangbo [7] for regular functions f0 and f1 with the aim of deriving the existence of optimal map
to Monge’s transportation problem via very deep PDE techniques. The equation was generalized to finite Radon measures f0 and f1 by Bouchitt´e and Buttazzo [5] as follows
−div(µDµu) = f1− f0 in D 0 (Ω) |Dµu| = 1 µ-a.e. |∇u| ≤ 1 a.e. in Ω,
where Dµu is the tangential gradient with respect to (w.r.t.) the measure µ. It turns out
to be the standard gradient ∇u if µ is absolutely continuous w.r.t. the Lebesgue measure. It is well known that φ := µDµu is an optimal flow to the minimal flow problem (1.1),
which is closely related to the Monge–Kantorovich transport problem with the Euclidean distance cost.
These relations were also generalized to Finsler metric costs in (1.1) of the form Z Ω F x, Φ |Φ| d|Φ|.
Let ω :=x ∈ Ω : F (x, .) ≡ 0 . Finsler metric F is called coercive on Ω \ ω if there exist positive constants c1, c2 such that c1|p| ≤ F (x, p) ≤ c2|p| ∀x ∈ Ω \ ω. Monge–Kantorovich
equation for such a coercive Finsler metric is studied in [3] under the assumption ω being a regular domain. Related works for non-degenerate Finsler metrics (i.e., coercive on the whole Ω and ω = ∅) are also discussed in [2, 8, 10]. The case of F (x, p) ≡ F1(p)
(independent of x) is treated in [11], where F1(.) is a continuous even convex function
satisfying F1(p) → +∞ as p → +∞.
The paper is organized as follows. We present some preliminaries in Section 2. Section 3 is devoted to the smooth approximation issue. Finally, the Monge–Kantorovich equation for degenerate Finsler metrics is presented and proven in Section 4.
2
Preliminaries
2.1 Degenerate Finsler metrics
Let Ω be a bounded Lipschitz domain of RN. A Finsler metric is a continuous function F : Ω × RN −→ [0, +∞) such that F (x, .) is convex and positively 1-homogeneous w.r.t. the second variable, i.e., F (x, tp) = tF (x, p) ∀(x, p) ∈ Ω × RN and t ≥ 0. Its polar function F∗ is defined by
F∗(x, q) = sup
{p: F (x,p)≤1}
hq, pi.
As a typical example, F (x, p) = k(x)|p| and F∗(x, q) = 1
k(x)|q| for any nonnegative continuous function k on Ω, with the convention 0
0 = 0. Another example in 1D is F (x, p) = k1(x)p− + k2(x)p+ and F∗(x, q) = 1 k1(x) q−+ 1 k2(x)
q+ for two nonnegative continuous functions k1 and k2, where p+= max(p, 0), p−= max(−p, 0).
Let ω := {x ∈ Ω : F (x, .) ≡ 0} be a closed set. As in [3], we assume that ω is a regular domain, but without any assumption about coerciveness on Ω \ ω. More precisely, in this paper, we make use of the following assumption:
(A): ω b Ω is a Lipschitz domain and F (x, p) 6= 0 for any x ∈ Ω \ ω, 0 6= p ∈ RN.
2.2 Duality issue
Given two finite Radon measures f0 and f1 on Ω satisfying
Z Ω df0= Z Ω df1. We consider
the Beckmann problem for a degenerate Finsler metric F as
(B): min Φ∈M(Ω)N Z Ω F x, Φ |Φ|(x) d|Φ|(x) : −div Φ = f1− f0 in D 0 (Ω) .
In the case of non-degenerate Finsler metric F (i.e., there exist c1 > 0, c2 > 0 such that
c1|p| ≤ F (x, p) ≤ c2|p| ∀(x, p) ∈ Ω × RN), the existence of optimal flow Φ to min (B) can
be guaranteed by the direct method. When F is degenerate, the existence can be obtained via optimal transport and the disintegration theorem.
The dual problem of (B) is given by
The existence of optimal solutions to max (DP ) can be shown by the direct method. Proposition 2.1. We have the duality relation
min Φ∈M(Ω)N Z Ω F x, Φ |Φ|(x) d|Φ|(x) : −div Φ = f1− f0 in D 0 (Ω) = max u∈W1,∞(Ω) Z Ω u d(f1− f0) : F∗(x, ∇u(x)) ≤ 1 a.e. in Ω .
Proof. As usual, it is not difficult to show that max(DP ) ≤ min(B) by taking u as a test function in the divergence constraint. It remains to prove the inverse inequality. Consider the functional H : L∞(Ω)N −→ (−∞, +∞] defined by
H(z) = inf u∈W1,∞(Ω) − Z Ω u d(f1− f0) : F∗(x, ∇u(x) + z(x)) ≤ 1 a.e. in Ω . Then we can check at once that H is convex and lower semicontinuous in L∞(Ω)N, equipped with the weak* topology. And therefore H(0) = H∗∗(0).
For any Φ ∈ L1(Ω)N, we have H∗(Φ) = sup z∈L∞(Ω)N hΦ, zi − H(z) = sup z∈L∞(Ω)N u∈W1,∞(Ω) F∗(x,∇u(x)+z(x))≤1 hΦ, zi + Z Ω ud(f1− f0).
Set v = ∇u + z, and z = v − ∇u, we get H∗(Φ) = sup v∈L∞(Ω)N F∗(x,v(x))≤1 hΦ, vi + sup u∈W1,∞(Ω) hΦ, −∇ui + Z Ω u d(f1− f0) = Z Ω F (x, Φ(x))dx + sup u∈W1,∞(Ω) hΦ, −∇ui + Z Ω u d(f1− f0).
Observe that if H∗(Φ) < +∞, we obtain hΦ, ∇ui =
Z
Ω
u d(f1− f0) ∀u ∈ W1,∞(Ω).
In other words, −div Φ = f1− f0 in D
3
Approximation by smooth functions
Lemma 3.1. Let F be a degenerate Finsler metric satisfying the assumption (A) and u be a Lipschitz function such that F∗(x, ∇u(x)) ≤ 1 a.e. in Ω.
(i) Then there exists a sequence of smooth functions uε ∈ Cc∞(RN) such that uε ⇒ u
uniformly on Ω as ε → 0 and lim sup
ε→0
F∗(x, ∇uε(x)) ≤ 1 for all x ∈ Ω.
(ii) In addition, if F∗(x, q) is bounded from above on (Ω \ ω) × {|q| ≤ 1} then there exists a sequence of smooth functions uε ∈ Cc∞(RN) such that uε⇒ u uniformly on Ω as ε → 0
and F∗(x, ∇uε(x)) ≤ 1 for all x ∈ Ω without passing to the limit. In particular, this holds
whenever F is a non-degenerate Finsler metric.
Remark 3.2. This lemma generalizes [8, Lemma 3.2], [9, Lemma A.1].
Before giving the proof, let us comment that the main difficulty comes from the ap-proximation near to the boundary as well as the degeneracy of F .
Proof. Let eu be the extension of u on RN by the value zero outside Ω, i.e.,
e u(x) :=
(
u(x) if x ∈ Ω 0 if x ∈ RN\ Ω.
Step 1 (Approximating at points near the boundary of ω). Fix any z ∈ ∂ω. Since ω is a Lipschitz domain, there exist rz > 0 and a Lipschitz continuous function
γz: RN −1−→ R such that (up to rotating and relabeling if necessary)
ω ∩ B(z, rz) = {x | xN > γz(x1, ..., xN −1)} ∩ B(z, rz).
In other words, the boundary of ω is locally expressed as the graph of a Lipschitz continuous function in (N −1)-dimension. Set Uz:= ω∩B(z,
rz
2). Define a translation T
ε
z : RN −→ RN
given by Tzε(x) = x + ελzeN, where we choose a fixed sufficiently large λz and all small
ε, namely fixed λz ≥ Lip(γz) + 1, 0 < ε <
rz
2(λz+ 1)
, eN is the Nth element of canonical
basis in RN. By this choice and the Lipschitz property of γz, we see that
B(Tzε(x), ε) ⊂ ω ∩ B(z, rz) for all x ∈ Uz. (3.2) Let us define e uε(x) := Z RN ρε(y)eu (T ε z(x) − y) dy = Z B(Tε z(x),ε)
ρε(Tzε(x) − y)u(y) dy for all x ∈ Re
N, (3.3)
where ρε is the standard mollifier on RN. It is clear thatueε ∈ C
∞
c (RN). Using (3.2), (3.3)
and the continuity of u on Ω, we get e
Now, by the compactness of ∂ω and ∂ω ⊂ [
z∈∂ω
B(z,rz
2 ), there exist finite points z1, ..., zm ∈ ∂ω such that ∂ω ⊂ m [ i=1 B(zi, rzi 2 ).
By similar arguments, there exist also finite points zm+1, ..., zn∈ ∂Ω such that
∂Ω ⊂ n [ i=m+1 B(zi, rzi 2 ) and (3.2), (3.3) and (3.4) hold at z = zi, i = 1, ..., n.
Step 2 (Approximating at points far from the boundaries). Let U0 b int(ω) ∪
(Ω \ ω) be an open subset such that
Ω ⊂ n [ i=1 B(zi, rzi 2 ) [ U0.
Let {φ}ni=0be a smooth partition of unity on Ω, subordinate tonU0, B(z1,
rz1
2 ), ..., B(zn, rzn
2 ) o
(see e.g. [6, Chapter 9]), that is, φi ∈ Cc∞(RN), 0 ≤ φi≤ 1 ∀i = 0, ..., n supp(φi) b B(zi, rzi 2 ) ∀i = 1, ..., n, supp(φ0) b U0 n X i=0 φi(x) = 1 for all x ∈ Ω.
For short, we will write Tiε, Ui from Step 1 instead of Tzεi, Uzi and x
ε
i = xεzi = T
ε
i(x), i =
1, ..., n.
Due to Step 1, there existue
1 ε, ...,ue n ε ∈ C ∞ c (RN) such that e uiε ⇒ u on Ui, i = 1, ..., n.
For i = 0, since U0 b int(ω) ∪ (Ω \ ω), we can define T0ε(x) ≡ x and the standard
convo-lutionue 0 ε:= ρε?eu. Theneu 0 ε ∈ C ∞ c (RN) and eu 0 ε ⇒ u on U0.
Step 3 (proof of (i)). Set
uε(x) := n X i=0 φi(x)eu i ε(x) = n X i=0 φi(x) Z B(Tε i(x),ε)
ρε(Tiε(x) − y)eu(y) dy for all x ∈ R
Let us show that the sequence {uε} satisfies the desired properties. We check at once that uε ∈ Cc∞(RN) and uε⇒ n X i=0 φiu = u on Ω.
It remains to prove that lim sup
ε→0
F∗(x, ∇uε(x)) ≤ 1 for all x ∈ Ω. By the construction,
B(xεi, ε) ⊂ Ω for sufficiently small ε and xεi → x as ε → 0. Observe that
∇uε(x) = n X i=0 ∇φi(x)euiε(x) + n X i=0 φi(x)∇eu i ε(x) = n X i=0 ∇φi(x) Z B(xε i,ε) ρε(xεi − y)u(y)dy + n X i=0 φi(x) Z B(xε i,ε) ρε(xεi − y)∇u(y)dy.
• If x ∈ ω, then B(xεi, ε) ⊂ ω and ∇u(y) = 0 a.e. y in ω (since u is a constant on ω), we have ∇uε(x) = 0 and therefore F∗(x, ∇uε(x)) = 0 ≤ 1.
• If x ∈ Ω \ ω, there exists a sequence of small ε (depending on x) such that xεi ∈ Ω \ ω for all i = 0, ..., n. We have that
F∗(x, ∇uε(x)) ≤ F∗ x, n X i=0 ∇φi(x) Z B(xε i,ε) ρε(xεi − y)u(y)dy + n X i=0 φi(x) Z B(xε i,ε) F∗(x, ∇u(y)) ρε(xεi − y)dy ≤ F∗ x, n X i=0 ∇φi(x) Z B(xε i,ε) ρε(xεi − y)u(y)dy + 1 + n X i=0 φi(x) Z B(xε i,ε)
(F∗(x, ∇u(y)) − F∗(y, ∇u(y))) ρε(xεi − y)dy.
(3.5) Since F∗(x, q) is finite and continuous on (Ω \ ω) × RN, we have F∗(x, ∇u(y)) − F∗(y, ∇u(y)) → 0 as y → x whenever ∇u(y) exists.
Letting ε → 0, we obtain lim sup
ε→0
F∗(x, ∇uε(x)) ≤ 1.
where the constant C is chosen later and w(ε) := sup
x,y
{|F∗(x, q) − F∗(y, q)| : x, y ∈ Ω \ ω, |x − y| ≤ M ε, |q| ≤ k∇ukL∞},
with a fixed constant M ≥ max
1≤i≤n{λzi+ 1}, λzi is given in Step 1. We show that uεsatisfies
all the desired properties. By the assumption of (ii), w(ε) → 0 as ε → 0. It is clear that uε ∈ Cc∞(RN) and uε⇒ n X i=0 φiu = u on Ω.
At last, we show that F∗(x, ∇uε(x)) ≤ 1 ∀x ∈ Ω. Similar to Step 3, we have F∗(x, ∇uε(x)) =
0 ≤ 1 for all x ∈ ω. Let us now prove for the case x ∈ Ω \ ω. Using the fact that
n
X
i=0
∇φi(x)u(x) = 0 for all x ∈ Ω,
we have n X i=0 ∇φi(x) Z B(xε i,ε) ρε(xεi − y)u(y)dy = n X i=0 ∇φi(x) Z B(xε i,ε) ρε(xεi − y)u(y)dy − u(x) . (3.6) Moreover, Z B(xε i,ε) ρε(xεi − y)u(u) dy − u(x) ≤ Z B(xε i,ε) ρε(xεi − y) (u(y) − u(xεi)) dy + |u(xεi) − u(x)| ≤ C1ε ∀i = 0, ..., n,
where the constant C1 depends only on Lip(γzi) and the Lipschitz constant of u on Ω.
Thus, by combining this with (3.6), n X i=0 ∇φi(x) Z B(xε i,ε) ρε(xεi − y)u(y)dy ≤ C2ε ∀x ∈ Ω,
where C2 depends only on C1 and k∇φikL∞.
Since F∗(x, q) is bounded from above on (Ω \ ω) × {|q| ≤ 1}, there exists K > 0 such that F∗(x, q) ≤ K|q| for all x ∈ Ω \ ω, q ∈ RN and therefore
Now, if y ∈ B(xεi, ε) then |x − y| ≤ |x − xεi| + |xεi− y| ≤ M ε. So we obtain, in view of (3.5), F∗(x, ∇uε(x)) ≤ 1 1 + Cε + w(ε)
[
C3ε + 1 + n X i=0 φi(x) Z B(xεi,ε) ρε(xεi − y) (F ∗(x, ∇u(y)) − F∗(y, ∇u(y))) dy
]
≤ C3ε + 1 + w(ε)1 + Cε + w(ε)
≤ 1 (choose a constant C ≥ C3).
Here, we have used ∇u(y) = 0 in ω and the boundary ∂ω is negligble. Thus, Z
B(xε i,ε)
ρε(xεi − y) (F∗(x, ∇u(y)) − F∗(y, ∇u(y))) dy
= Z
B(xε
i,ε)∩(Ω\ω)
ρε(xεi − y) (F∗(x, ∇u(y)) − F∗(y, ∇u(y))) dy ≤ w(ε).
4
Monge–Kantorovich equation
In this section, we present a generalization of Monge–Kantorovich equation for degenerate Finsler metrics: (MKE) −div Φ = f1− f0 in D0(Ω) Φ |Φ|(x) · D|Φ|u(x) = F x, Φ |Φ|(x) |Φ|-a.e. F∗(x, ∇u(x)) ≤ 1 a.e. in Ω,
where D|Φ|u is the tangential gradient w.r.t. the total variation measure |Φ| (see [4, 5, 11]).
This equation turns out to be a standard Monge–Kantorovich equation if F (x, p) = |p| is the Euclidean norm (independent of x). Also, it is worth noting that we do not use the constraint F∗(x, D|Φ|u(x)) ≤ 1 as in [8].
Theorem 4.3. (i) For any u such that F∗(x, ∇u(x)) ≤ 1 a.e. in Ω and any Φ ∈ M(Ω)N such that −div Φ = f1− f0 in D
0 (Ω), then Φ |Φ|(x) · D|Φ|u(x) ≤ F x, Φ |Φ|(x) |Φ| − a.e. x ∈ Ω.
Proof. (i) Let B be any Borel subset of Ω. By Lemma 3.1, there exists a sequence of smooth functions uε such that uε⇒ u on Ω and lim sup
ε→0
F∗(x, ∇uε(x)) ≤ 1 for all x ∈ Ω.
We get Z B Φ |Φ|(x) · D|Φ|u(x)d|Φ|(x) = limε→0 Z B Φ |Φ|(x) · ∇uε(x)d|Φ|(x) ≤ lim sup ε→0 Z B F∗(x, ∇uε(x))F x, Φ |Φ|(x) d|Φ|(x) ≤ Z B F x, Φ |Φ|(x) d|Φ|(x), where the last inequality follows from Fatou’s lemma and lim sup
ε→0
F∗(x, ∇uε(x)) ≤ 1 for
all x in Ω. By the arbitrariness of Borel set B, it follows that Φ |Φ|(x) · D|Φ|u(x) ≤ F x, Φ |Φ|(x)
for all |Φ|-a.e. x in Ω.
(ii) Let u and Φ be optimal solutions to the Kantorovich dual problem and minimal flow problem, that is, F∗(x, ∇u(x)) ≤ 1 a.e. x ∈ Ω and Φ ∈ M(Ω)N, −div Φ = f1− f0 in
D0(Ω), and by the duality in Subsection 2.2, Z Ω u d(f1− f0) = Z Ω F x, Φ |Φ|(x) d|Φ|(x).
On the other hand, taking u as a test function in the divergence constraint, we get Z Ω ud(f1− f0) = Z Ω Φ |Φ|(x) · D|Φ|u(x)d|Φ|(x) and therefore Z Ω Φ |Φ|(x) · D|Φ|u(x)d|Φ|(x) = Z Ω F x, Φ |Φ|(x) d|Φ|(x). By (i) above, we deduce that
Φ |Φ|(x) · D|Φ|u(x) = F x, Φ |Φ|(x) |Φ|-a.e. x in Ω. Conversely, if (u, Φ) satisfies the system (MKE), then
Z Ω ud(f1− f0) = Z Ω Φ |Φ|(x) · D|Φ|u(x)d|Φ|(x) = Z Ω F x, Φ |Φ|(x) d|Φ|(x), which implies the optimalities of u and Φ, in view of the duality.
References
[1] M. Beckmann. A continuous model of transportation. Econometrica: Journal of the Econometric Society, 20(4):643–660, 1952.
[2] J.-D. Benamou, G. Carlier, and R. Hatchi. A numerical solution to Monge’s problem with a Finsler distance as cost. ESAIM: Mathematical Modelling and Numerical Analysis, 52(6):2133–2148, 2018.
[3] G. Bouchitt´e and G. Buttazzo. Characterization of optimal shapes and masses through Monge–Kantorovich equation. Journal of the European Mathematical So-ciety, 3(2):139–168, 2001.
[4] G. Bouchitte, G. Buttazzo, and P. Seppecher. Energies with respect to a measure and applications to low dimensional structures. Calculus of Variations and Partial Differential Equations, 5(1):37–54, 1997.
[5] G. Bouchitt´e, G. Buttazzo, and P. Seppecher. Shape optimization solutions via Monge–Kantorovich equation. Comptes Rendus de l’Acad´emie des Sciences-Series I-Mathematics, 324(10):1185–1191, 1997.
[6] H. Brezis. Functional analysis, Sobolev spaces and partial differential equations. Springer Science & Business Media, 2010.
[7] L. C. Evans and W. Gangbo. Differential equations methods for the Monge– Kantorovich mass transfer problem. Memoirs of the American Mathematical Society, 137:1–66, 1999.
[8] N. Igbida, J. M. Maz´on, J. D. Rossi, and J. Toledo. Optimal mass transportation for costs given by Finsler distances via p-Laplacian approximations. Advances in Calculus of Variations, 11(1):1–28, 2018.
[9] N. Igbida and V. T. Nguyen. Augmented Lagrangian method for optimal partial transportation. IMA Journal of Numerical Analysis, 38(1):156–183, 2018.
[10] N. Igbida and V. T. Nguyen. Optimal partial mass transportation and obstacle Monge–Kantorovich equation. Journal of Differential Equations, 264(10):6380–6417, 2018.
[11] C. Jimenez. Dynamic formulation of optimal transport problems. Journal of Convex Analysis, 15(3):593–622, 2008.
[13] C. Villani. Topics in optimal transportation, volume 58. American Mathematical Society, 2003.
