A continuous model of transportation in the Heisenberg group
-
Michele Circelli
and
Albert Clop
Abstract
We present a minimization problem with a horizontal divergence-type constraint in the Heisenberg group. Our study explores its dual formulation and examines its relationship with the congested optimal transport problem, for
1 Introduction
The optimal transport problem, originally posed by G. Monge in [37] in the late 18th century, has become a highly active research area due to its numerous applications, as highlighted in [29] and [43]. In the Monge–Kantorovich formulation, which dates back to 1942, we are given two probability measures μ and ν on a metric space
where the set
Initially, the existence of solutions to (1.1) induced by measurable maps was extensively studied within Euclidean and Riemannian settings by many researchers (see, for instance, [3, 45, 30] and references therein for comprehensive overviews). Later, the research focus shifted towards sub-Riemannian frameworks, especially Carnot groups, which arise as limits of Riemannian manifolds in a very precise sense (see [36]).
The Heisenberg group
The first significant work on the Monge–Kantorovich problem in
is [5], where L. Ambrosio and S. Rigot proved the existence and the uniqueness of the solution to (1.2) for
In this direction we continue, in this paper, the study initiated in [23] on the congested optimal transport problem in
The congested optimal transport problem was first introduced in the Euclidean setting by Carlier, Jimenez and Santambrogio in [18]. In the Heisenberg group it reads as follows: given two probability measures μ and ν over the closure of a bounded domain Ω, and a cost function
Here, the set
is a p-summable function;
where
and the minimum is taken among all p-summable horizontal vector fields
This problem is the Heisenberg version of the well-known continuous model of transportation introduced in the Euclidean setting by Beckmann in [7]. Following the approach [12] and [44], which relies on the Dacorogna–Moser flow argument, together with a standard regularization procedure, we prove that the two problems (1.4) and (1.3) are actually equivalent. Moreover, in some sense, minimizers to (1.3) and (1.4) correspond to each other, see Proposition 5.13 and Theorem 5.14. The main difference here with the Euclidean case is that, in order to apply the Dacorogna–Moser flow argument, we must pass through a Riemannian approximation of the Heisenberg group.
Using standard duality arguments, (1.4) can be formulated as a classical problem in the calculus of variations, and hence it is strongly related to the regularity theory for PDEs in the Heisenberg group. More precisely, in the spirit of [46], we show in Theorem 5.10 that (1.4) is equivalent to the following problem:
where
Here,
where
In the Euclidean setting, the equivalence between (1.3), (1.4) and (1.6) was first investigated in [13] for the autonomous case. More recently, this equivalence was extended to the non-autonomous case in [14], where the authors also assume that the right-hand side in (1.5) belongs to the dual of some Sobolev space, whose elements are not measures in general.
In the limiting case
As observed in [45, Subsection 4.2.2], this problem is not well-posed a priori: due to the non-reflexivity of
where
As byproducts of the previous equivalence, we obtain two important facts: first, if either μ or ν is absolutely continuous with respect to the Lebesgue measure (which is a Haar measure for the group structure), then (1.8) admits a solution. Second, one can conclude that the problem (1.9) also admits a dual formulation (as it does for its congested version (1.4)), which is given by the well-known Kantorovich duality theorem:
In addition, in Theorem 4.7 we prove that the Lagrangian reformulation of (1.9) is
where
The paper is organized as follows: In Section 2, we introduce the Heisenberg group and recall some well-known results on optimal transport theory in this setting. In Section 3, we prove some results about the transport set and the Kantorovich potential, which will be useful later. Section 4 deals with the aforementioned three problems in the limit case
2 Preliminaries
2.1 The Heisenberg group
ℍ
n
Let
where
The homogeneous dimension of
The exponential map
In this system of coordinates the group law will be given, through the Baker–Campbell–Hausdorff formula which defines a group structure on
the unit element
and the vector fields
According to the two step-stratification of the algebra,
See [11, Section 2] for more details about this topic.
The horizontal layer
One can equip
An absolutely continuous curve
Let us just remark that
The Rashevsky–Chow’s theorem guarantees that the sub-Riemannian distance
is well-defined and it induces the Euclidean topology, see [1, Theorem 3.31]. In particular,
If
the sub-Riemannian ball centered in x with radius r.
The Lebesgue measure
For any open set
its horizontal gradient, in the sense of distributions.
We say that a measurable map
where a group homomorphism
If the map L exists, it is unique and will be denoted by
If
where the map
see [33, Proposition 5.6]. If
for a.e.
For every
is a Banach space equipped with the norm
2.1.1 Geodesics in
ℍ
n
We call minimizing horizontal curve any curve
and geodesic any minimizing horizontal curve parametrized proportionally to the arc-length.
The space
Usually, in this kind of spaces, geodesics between two points may not be unique: in the case of the Heisenberg group geodesics can be computed explicitly and it is possible to detect a set in which these are unique.
We denote by
then it holds the following characterization for geodesics parametrized on
Theorem 2.1.
A non-trivial geodesics parametrized on
for some
for some
Moreover, it holds:
For all
If
From the description of geodesics given in Theorem 2.1 it follows that
We denote by
Let us just remark that there exists a map
which is γ-measurable for any positive Borel measure γ on
associates to any two points
2.1.2
H
Ω
-valued finite Radon measures
Let
The normed space
An
and its action against a horizontal vector field
We denote by
the space of all
Since the horizontal bundle
for some unique
where
Finally, if Ω is compact, then
2.1.3 Mollification in
ℍ
n
Let us consider
a mollifier for the group structure, i.e. a function
If
is smooth and it enjoys many standard properties, see for instance [32].
More generally, one may also mollify Radon measures: given a finite Radon measure
and
where
2.1.4 Riemannian approximation of
ℍ
n
The Heisenberg group
as orthonormal frame for
Moreover, for any
see [36], and the convergence is uniform on compact subsets, see [17]. In [35, Theorem 1.1] the author proved that
In the end, if
2.2 Optimal transport theory in
ℍ
n
This subsection contains some well-known results about the Monge–Kantorovich problem in
Let
Now, given
the set of transport plans between μ and ν, where
It is well known that the Monge–Kantorovich problem between μ and ν, associated with the Carnot–Carathéodory distance,
admits at least one solution. See for instance [45, Theorem 1.7]. We denote by
the set of optimal transport plans. It is a closed subset of the compact set
Let us denote by
It holds the following important theorem, see for instance [45, Proposition 3.1].
Theorem 2.2 (Kantorovich duality theorem).
There exists
Moreover, it holds that
and
Any
If at least one between μ and ν is absolutely continuous with respect to the Haar measure, then the explicit representation of geodesics implies that any optimal transport plan is concentrated on the set
Lemma 2.3.
Let us suppose that either
For the proof of this result we refer to [27, Lemma 4.1].
3 Properties of transport rays and Kantorovich potentials in
ℍ
n
In the spirit of [45, Chapter 3] and [30, Section 2] we extend to the Heisenberg setting some properties of transport rays and of transport set.
3.1 Transport rays and transport set
Let us consider
Let us just remark that if
then
where σ is any geodesic between x and y.
Definition 3.1.
A transport ray is a non-trivial geodesic
We denote by
and by
We will call transport set the set
We observe that
Moreover, as in [30, Lemma 8] the following result holds.
Theorem 3.2.
The transport set
Proof.
Thanks to Hopf–Rinow theorem it is enough to prove that
We can suppose that
Hence, A lies in the union of the L-neighborhoods of the compact sets
Let us prove that
Since
we have that
Hence there are two possibilities
3.2 Differentiability of Kantorovich potential
Following [30, Lemma 10], one can prove that any Kantorovich potential u is Pansu differentiable in the interior of transport rays.
Proposition 3.3.
Let
For the proof of this result we need the following two lemmas. The first one collects a differentiability property of the Carnot–Carathéodory distance function from a fixed point
Lemma 3.4.
Let
Moreover, if
Proof.
The core of the proof is contained in [5, Lemma 3.11].
If we denote by
Now let
where we used the fact that σ is a geodesic, then
Lemma 3.5.
Let
Proof.
Let
Therefore,
If we divide the previous inequalities by
Proof of Proposition 3.3.
Let
The description of geodesics given in Theorem 2.1 implies that
Moreover, a and b lie on the geodesic between x and y, then
where equalities hold if
Hence, from Lemma 3.5, it follows that u is Pansu differentiable in
4 The Beckmann problem for
p
=
1
4.1 Vector transport density in
ℍ
n
In the spirit of [45, Chapter3] we define a vector version of the horizontal transport density, introduced [23].
From now on, we fix a selection of geodesics
that is γ-measurable for any
Hence, one may associate with any
for any
One may also associate with any
Definition 4.1 (Vector horizontal transport density).
For any
for any
We defined the scalar transport density, resp. the vector one, in duality with
In particular, if
as measures. Therefore, it follows from the very definition of
Proposition 4.2.
If
Proof.
Let
for every
for every
for every
for every
From (4.3) and [23, Theorem 3.3 and Theorem 3.6] the next theorem easily follows.
Theorem 4.3 (Absolute continuity and summability of vector horizontal transport densities).
If one has
4.2 The problem
Given any optimal transport plan
(4.5)
Now, given an
With this definition in mind we can rewrite (4.5) in the following way:
i.e. the horizontal divergence of the measure
This is the Heisenberg version of the well-known Beckmann problem and it will turn out to be the limit case of the wider class of problems introduced in Section 5, as explained in the introduction of the paper.
In the spirit of [45, Subsection 4.2], we show that this problem is equivalent to the Monge–Kantorovich problem (2.11).
Theorem 4.4.
The problem (BP) admits a solution. Moreover,
where
and a solution to (BP) can be built from a solution to (2.11), as in (4.2).
Proof.
First we prove the equality between (2.11) and (BP).
We start by proving that (BP)
Now we take
where we used Theorem 2.2 in the last equality. Since the previous inequality holds for any admissible
Now we will prove the converse inequality: for any
Hence
Corollary 4.5.
If either
As a consequence of Theorem 4.4, one also obtains the following dual formulation.
Corollary 4.6.
It holds that
For any
The second condition holds because
4.3 Lagrangian formulation
Following [3, Lecture 9, section 3], we now introduce a Lagrangian formulation of (BP) in
where the functional
This definition of length makes sense even for continuous curves and it extends the sub-Riemannian one, in the sense that if σ is horizontal, then
Hence, the following theorem holds.
Theorem 4.7.
If
Moreover, it holds
and Q is optimal if and only if Q is supported on the set of minimizing horizontal curves and
Proof.
Let us first observe that, for any
satisfies
by definition. This ensures that the set of admissible measures Q for (4.7) is nonempty. Moreover,
Let
(4.9)
where the first inequality follows from the definition of
To prove the converse inequality, let us consider an optimal transport plan
(4.10)
Thus, the equality of optimization problems follows. Moreover, if
This theorem is independent on the compactness of supports and on the ambient space: it holds in more general Polish metric spaces
and
5 The Beckmann problem for
1
<
p
<
+
∞
From now on we suppose Ω is a bounded domain, with regular
for any
We consider the space
viewed as subset of
5.1 Vector traffic intensities in
ℍ
n
In this subsection we will see how also the congested optimal transport problem is related with some minimization problems, over a set of vector fields with divergence constraint.
Let
Q is concentrated on the set H,
We denote by
As for the transport plans, one can also associate with any horizontal traffic plan
Example 5.1.
Let us consider the two measures
and let
and its horizontal traffic intensity is precisely
As for the vector measure, we introduce now the following definition.
Definition 5.2 (Vector horizontal traffic intensity).
Let
for any continuous horizontal vector field
The following lemma holds.
Lemma 5.3.
Let
Proof.
(1) This follows from the definition of total variation norm and of horizontal traffic plan, while (2) follows from the following inequality:
(3) By definition,
and the thesis follows. ∎
Example 5.4.
Following [14, Example 4.1], we consider
and the horizontal traffic plan
where
but
This shows that, in the previous Lemma, (2) is not an equality in general, because the curves of Q may produce some cancellations: this is due to the fact that
Lemma 5.5.
Let
then
Moreover,
Proof.
Let
As a consequence, we see that if
5.2 Well-posedness of the problem
In the beginning of this subsection, we want to determine the image of the horizontal divergence operator
Lemma 5.6.
If
Proof.
Any given
Moreover, since
This allows us to extend the action of
The operator
while there are many other elements in
We endow
Proposition 5.7.
It holds that
Moreover, there is a constant
and
Proof.
Let us consider a horizontal vector field
Conversely, let us consider the following minimization problem
where
This problem admits at least one solution. Indeed, we can restrict the minimization to the subspace
which is a convex subset and closed with respect to weak topology of
Hence, the existence of minimizers follows again from (5.1).
Computing Euler–Lagrange equations, a solution φ of the problem above turns out to satisfy
This means that there exists
as desired. ∎
Let now consider a function
Let us suppose in addition that
and it satisfies
for some
Following [46], we are interested in the Beckmann-type problem
Since
Theorem 5.8.
If
Proof.
Let
Since
This proves that
5.3 Dual formulation
The aim of this subsection is to show that the Beckmann-type problem ((${\mathcal{B}}$)) admits the dual formulation
where
By definition the function
and
The condition (5.4) implies the following q-growth condition on
where
We follow the proof developed in [13], which is based on the following classical theorem of convex analysis.
Theorem 5.9 (Convex duality, [28, Proposition 5]).
Let
where
Moreover, if the supremum in (5.7) is attained at some
where
We are now ready to show that ((${\mathcal{B}}$))
Theorem 5.10.
If
Moreover, if
Proof.
First we observe that the existence of a minimizer for the problem ((${\mathcal{B}}$)) follows from Theorem 5.8. Thanks to the Direct method in calculus of variations also ((${\mathcal{D}}$)) admits at least a solution belonging the space
is lower semicontinuous with respect to the weak topology of
where
Let now denote by
This operator is linear; moreover, it is bounded since
We denote by
By the convexity and the lower semicontinuity of
Moreover, it is a straightforward consequence of the definition of the continuous linear operator
that
The rest of the thesis follows from the primal-dual optimality condition (5.8). ∎
Remark 5.11.
In [46] the author provides a different proof of the equality
which works also in the Heisenberg group. For the sake of completeness we just highlight the main steps.
Let
Lemma 5.7 and (5.4) imply that the functional
By definition,
One can prove that
Let us remark that if the function
where
Example 5.12.
If
Based on the results in [16] and [48, Theorem 3.1], it can be shown that if
Consequently, following [13], the solution
where
On the other hand, the optimal regularity for solutions is the Hölder continuity of the horizontal derivatives: for
See also [24] for Lipschitz regularity for solutions to (5.10), with a different choice of
5.4 Lagrangian formulation
The congested optimal transport problem, adapted to the Heisenberg group in [23], consists of the following minimization problem:
where the function
and
Additionally, we assume:
and it exhibits p-growth in the second variable:
for some
In this subsection, we investigate the equivalence between ((${\mathcal{W}}$)) and ((${\mathcal{B}}$)), where
Moreover, we will demonstrate how to pass from a solution of one problem to a solution of the other, and vice versa.
First, we state the following result.
Proposition 5.13.
If
Hence, the equivalence between the two problems is reduced to the converse inequality.
To prove this, we follow [12, Theorem 2.1] and [44], which are based on the Dacorogna–Moser scheme combined with a regularization procedure. In the Euclidean setting, this regularization procedure is unnecessary if μ and ν are sufficiently smooth, because every solution
Theorem 5.14.
If
Proof.
Let
where
and we are supposing that both μ and ν are extended by 0 outside Ω. Let us introduce the non-autonomous horizontal vector field
where
Let us notice that
where
for any
Since
is the flow of the vector field
Let us remark that, since
Hence
If we denote by
the trajectory map, i.e.
satisfies
in particular,
By construction,
because
Hence, one can define the horizontal traffic intensity
for any
(5.20)
for any
Analogously we define and compute the vector traffic intensity
hence
For any
therefore
and we consider
with
From the fact that
In the end,
by the lower semicontinuity of the map
hence
By using the monotonicity of G,
An alternative proof of the previous theorem can be found in the literature, specifically in [13, Theorem 3.2], and it is also applicable in
take a minimizer
We know that
for any
and we can apply [8, Theorem 5.8]: it follows that there exists a probability measure
and Q-a.e. curve σ is an integral curve of
Following the same computation as (5.20), one can prove that
for every
Proposition 5.13 and Theorem 5.14, combined with the existence of solutions for ((${\mathcal{B}}$)) as established by Theorem 5.8, lead to the following two straightforward consequences.
Corollary 5.15.
We have
Corollary 5.16.
If
( (${\mathcal{B}}$) )
for every Q solving ( (${\mathcal{W}}$) ),
for every
Remark 5.17.
If the function
Indeed, Corollary 5.16, Lemma 5.3 and the monotonicity of G imply
Therefore the result follows.
On the other hand, the convexity assumption on
Funding statement: Michele Circelli is supported by the project PRIN 2022 F4F2LH – CUP J53D23003760006, Regularity problems in sub-Riemannian structures. Albert Clop acknowledges partial support from grant PID2020-112881GB-I00, PID2021-125021NAI00, PID2024-156429NB-I00 (Spanish Government) and 2021-SGR-00071 (Catalan Government). Albert Clop is also supported by the Spanish State Research Agency, through the Severo Ochoa and María de Maeztu Program for Centers and Units of Excellence in R&D (CEX2020-001084-M), as well as the CERCA Programme/Generalitat de Catalunya.
Acknowledgements
The authors would like to thank the anonymous referee for the careful reading of the draft and for the valuable suggestions, which significantly contributed to the improvement of this paper.
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Articles in the same Issue
- Frontmatter
- On the asymptotic behavior of a diffraction problem with a thin layer
- Degenerate parabolic p-Laplacian equations: Existence, uniqueness, and asymptotic behavior of solutions
- Iterative blow-ups for maps with bounded 𝒜-variation: A refinement, with application to BD and BV
- A Sobolev gradient flow for the area-normalised Dirichlet energy of H 1 maps
- An elliptic approximation for phase separation in a fractured material
- The L 1-relaxed area of the graph of the vortex map: Optimal upper bound
- Compactness of Palais–Smale sequences with controlled Morse index for a Liouville-type functional
- Characterization of sets of finite local and non-local perimeter via non-local heat equation
- On nonexpansiveness of metric projection operators on Wasserstein spaces
- A continuous model of transportation in the Heisenberg group
- Universality of renormalisable mappings in two dimensions: the case of polar convex integrands
- Convergence of a heterogeneous Allen–Cahn equation to weighted mean curvature flow
- Harnack inequality for degenerate elliptic equations with matrix weights
- Two-scale density of almost smooth functions in sphere-valued Sobolev spaces: A high-contrast extension of the Bethuel–Zheng theory
- The Capacitary John–Nirenberg Inequality Revisited
Articles in the same Issue
- Frontmatter
- On the asymptotic behavior of a diffraction problem with a thin layer
- Degenerate parabolic p-Laplacian equations: Existence, uniqueness, and asymptotic behavior of solutions
- Iterative blow-ups for maps with bounded 𝒜-variation: A refinement, with application to BD and BV
- A Sobolev gradient flow for the area-normalised Dirichlet energy of H 1 maps
- An elliptic approximation for phase separation in a fractured material
- The L 1-relaxed area of the graph of the vortex map: Optimal upper bound
- Compactness of Palais–Smale sequences with controlled Morse index for a Liouville-type functional
- Characterization of sets of finite local and non-local perimeter via non-local heat equation
- On nonexpansiveness of metric projection operators on Wasserstein spaces
- A continuous model of transportation in the Heisenberg group
- Universality of renormalisable mappings in two dimensions: the case of polar convex integrands
- Convergence of a heterogeneous Allen–Cahn equation to weighted mean curvature flow
- Harnack inequality for degenerate elliptic equations with matrix weights
- Two-scale density of almost smooth functions in sphere-valued Sobolev spaces: A high-contrast extension of the Bethuel–Zheng theory
- The Capacitary John–Nirenberg Inequality Revisited
