11affiliationtext: Max Planck Institute for Mathematics in the Sciences, 04103
Leipzig,Germany
<EMAIL_ADDRESS>
# Geometric linearisation for optimal transport with strongly $p$-convex cost

Lukas Koch

###### Abstract

We prove a geometric linearisation result for minimisers of optimal transport
problems where the cost-function is strongly $p$-convex and of $p$-growth.
Initial and target measures are allowed to be rough, but are assumed to be
close to Lebesgue measure.

## 1 Introduction

The study of the optimal transport problem:

$\displaystyle\min_{\pi\in\Pi(\lambda,\mu)}\int_{\mathbb{R}^{d}\times\mathbb{R}^{d}}c(x-y)\,{\rm
d}\pi$ (1.1)

is well established. Here $\lambda,\mu$ are two (finite) non-negative measures
on $\mathbb{R}^{d}$ satisfying $\lambda(\mathbb{R}^{d})=\mu(\mathbb{R}^{d})$.
We refer the reader to [26] and [23] for an introduction and overview of the
literature. When solutions take the form of a transport map
$\pi=(\textup{Id}\times T)_{\\#}\mu$, under mild assumptions, minimisers are
characterised by satisfying the Euler-Lagrange equation

$\displaystyle\lambda(x)\textup{det}\,(\mathrm{D}T(x))=\mu(T(x))$ (1.2)

as well as the additional structure condition

$\displaystyle T(x)=x+\nabla c^{\ast}(\mathrm{D}\phi),$ (1.3)

where $\phi$ is a $c$-concave function and $c^{\ast}$ denotes the convex
conjugate of $c$. Assuming $\mu\sim\lambda\sim 1$ and linearising the
geometric nonlinearity in (1.2), that is formally expanding
$\textup{det}(\textup{Id}+A)=1+\textup{tr}\,A+\ldots$, we find that

$\displaystyle\textup{div}\,\nabla c^{\ast}(\mathrm{D}\phi)=\mu-\lambda.$
(1.4)

Thus, at least formally, we expect solutions of (1.1) to be well approximated
by solutions of (1.4). Note that in general (1.4) is a nonlinear equation.
Thus we refer to the process of moving from (1.1) to (1.4) as geometric
linearisation. The aim of this paper is to make this connection rigorous. We
show the following:

######  Theorem 1.1.

Let $1<p<\infty$. Suppose $c\colon\mathbb{R}^{d}\to\mathbb{R}$ is a strongly
$p$-convex cost function of controlled-duality $p$-growth, that is satisfying
(1.11)-(1.14). Let $\pi$ be a minimiser of (1.1) for some non-negative
measures $\lambda$, $\mu$ satisfying
$\lambda(\mathbb{R}^{d})=\mu(\mathbb{R}^{d})$. Denote

$\displaystyle E(R)\mathrel{\mathop{\mathchar 58\relax}}=\frac{1}{\lvert
B_{R}\rvert}\int_{(B_{R}\times\mathbb{R}^{d})\cup(\mathbb{R}^{d}\times
B_{R})}c(x-y)\,{\rm d}\pi$ $\displaystyle D(R)\mathrel{\mathop{\mathchar
58\relax}}=\frac{1}{\lvert B_{R}\rvert}W_{p}^{p}(\lambda\llcorner
B_{R},\kappa_{\lambda,R}\,{\rm d}x\llcorner
B_{R})+\frac{R^{p}}{\kappa_{\lambda,R}^{p-1}}(\kappa_{\lambda,R}-1)^{p}$
$\displaystyle\qquad\qquad+\frac{1}{\lvert B_{R}\rvert}W_{p}^{p}(\mu\llcorner
B_{R},\kappa_{\mu,R}\,{\rm d}x\llcorner
B_{R})+\frac{R^{p}}{\kappa_{\mu,R}^{p-1}}(\kappa_{\mu,R}-1)^{p}.$

Here $\kappa_{\lambda,R}=\frac{\lambda(B_{R})}{\lvert B_{R}\rvert}$ and
$\kappa_{\mu,R}=\frac{\mu(B_{R})}{\lvert B_{R}\rvert}$. Then, for every
$\tau>0$, there exists $\varepsilon(\tau)>0$ such that if
${E(4)+D(4)\leq\varepsilon}$111For applications it may be useful to note that
the scale $4$ is not important and it is straightforward to carry out the
proofs under the assumption that $E(r)+D(r)\leq\varepsilon$ for some $r>3$. In
that case $\varepsilon$ will depend on $r-3$., then there exists a radius
$R\in(2,3)$, $c\in\mathbb{R}$ and $\phi$ satisfying

$\displaystyle-\mathrm{div}\,\nabla c^{\ast}(\mathrm{D}\phi)=c\text{ in
}B_{R}$

such that

$\displaystyle\int_{(B_{1}\times\mathbb{R}^{d})\cup(\mathbb{R}^{d}\times
B_{1})}c(x-y-\nabla c^{\ast}(\mathrm{D}\phi))\,{\rm d}\pi\lesssim\tau
E(4)+D(4).$ (1.5)

Moreover

$\displaystyle\sup_{B_{1}}|\mathrm{D}\phi|^{p^{\prime}}+\int_{B_{R}}\lvert\mathrm{D}\phi\rvert^{p^{\prime}}\,{\rm
d}x\lesssim E(4)+D(4).$

We remark that we explain our assumptions on the cost function in detail in
Section 1.1. Theorem 1.1 states that if at some scale the local transportation
cost $E(R)$ (and the data term) are small, then at a smaller scale, the
transportation plan is well-approximated by $\nabla c^{\ast}(\mathrm{D}\phi)$,
in the sense that estimate (1.5) holds. Note in particular that as a
consequence of (1.5) and (1.13), also

$\displaystyle\int_{(B_{1}\times\mathbb{R}^{d})\cup(\mathbb{R}^{d}\times
B_{1})}\lvert x-y-\nabla c^{\ast}(\mathrm{D}\phi)\rvert^{p}d\pi\lesssim\tau
E(4)+D(4).$

Thus Theorem 1.1 makes the intuition leading to (1.4) rigorous.

Traditionally, (1.1) has been approached via the study of (1.2) using the
theory of fully nonlinear elliptic equations developed by Caffarelli, see e.g.
[2, 1] and the references therein. Recently, an alternative approach using
variational techniques has been developed by Goldman and Otto in [9]. There,
partial $C^{1,\alpha}$-regularity for solutions to (1.1) in the case of
Hölder-regular densities $\lambda$, $\mu$ and quadratic cost function
$c(x-y)=\frac{1}{2}|x-y|^{2}$ was proven. The key tool in the proof was a
version of Theorem 1.1 in this setting. In later papers, continuous densities
[6], rougher measures [8], more general cost functions (albeit still close to
the quadratic cost functional) [21], as well as almost-minimisers of the
quadratic cost functional [21] were considered. The quadratic version of
Theorem 1.1 was also used to provide a more refined linearisation result of
(1.2) in the quadratic set-up in [8] and of a similar statement in the context
of optimal matching in [7]. Finally, quadratic versions of Theorem 1.1 played
a key role in disproving the existence of a stationary cyclically monotone
Poisson matching in $2$-d [14].

We remark that very little information is available about the regularity of
minimiser of (1.1) already in the simplest degenerate/singular case
$c(x-y)=\frac{|x-y|^{p}}{p}$. In order to attempt to extend the techniques of
[9] to this setting, an essential first step is Theorem 1.1. This result will
also play a key role in extending the results of [14] to $p$-costs with $p\neq
2$ [15].

The strategy of proof is similar to that used in [8], although with a number
of simplifications. Further, we point the reader to [16] where a detailed
account of the proof of Theorem 1.1 and the motivations behind the strategy
are given in the quadratic case. The heart of the proof is contained in
Section 7. The key insight Lemma 7.2 is a consequence of the strong
$p$-convexity, which allows to estimate (up to error terms) the left-hand side
of the main estimate in Theorem 1.1 by a sum of two terms: on the one hand the
difference between the local transportation energy of $\pi$ and a dual form of
the energy of $\phi$:

$\displaystyle\int_{(B_{R}\times\mathbb{R}^{d})\cup(\mathbb{R}^{d}\times
B_{R})}c(x-y)\,{\rm d}\pi-\int_{B_{R}}c(\nabla c^{\ast}(\mathrm{D}\phi))\,{\rm
d}x,$ (1.6)

and on the other hand

$\displaystyle\int_{\\{\exists t\colon X(t)\in
B_{R}\\}}\int_{\tau}^{\sigma}\langle y-x-\nabla
c^{\ast}(\mathrm{D}\phi(X(t)),\mathrm{D}\phi(X(t))\rangle\,{\rm d}t\,{\rm
d}\pi.$ (1.7)

Here we write $X(t)=(1-t)x+ty$ and set ${\tau=\inf\\{t\in[0,1]\colon X(t)\in
B_{R}\\}}$, as well as $\sigma=\sup\\{t\in[0,1]\colon X(t)\in B_{R}\\}$. Lemma
7.2 replaces the quasi-orthogonality property, which was employed in the
quadratic case. The quasi-orthogonality property relied on expanding the
squares, which is not an available tool if $p\neq 2$.

From formal calculations, which we make rigorous in Lemma 7.5, (1.7) will be
small, if $\phi$ solves the Neumann problem

$\displaystyle-\mathrm{div}\,\nabla c^{\ast}(\mathrm{D}\phi)=\lambda-\mu\quad$
$\displaystyle\text{ in }B_{R}$ (1.8) $\displaystyle\nabla
c^{\ast}(\mathrm{D}\phi)\cdot\nu=g_{R}-f_{R}$ $\displaystyle\text{ on
}\partial B_{R},$ (1.9)

where $f_{R},g_{R}$ are functions tracking the location of $X(\tau)$ and
$X(\sigma)$, respectively. We formally define $f_{R},g_{R}$ in (2.2). However,
as written, the problem is not well-posed and solutions do not possess
sufficient regularity to carry out the necessary estimates. Hence, we will
actually work with approximations of $f_{R}$ and $g_{R}$, which we construct
in Section 5. Controlling the error made in this approximation, will require
us to enlarge the domain on which we work and to choose a suitable radius
$R\in[2,3]$. This explains the presence of $R$ in the formulation of Theorem
1.1. The estimate of (1.7) is finally carried out in Lemma 7.5 in Section 7.

The idea in estimating (1.6) is to relate the first term with the value of a
localised optimal transportation problem. Then an appropriate competitor can
be constructed using $\phi$ in order to estimate (1.6). We carry out the first
step in Section 4, while the second is obtained in Lemma 7.4 in Section 7.

In order to carry out the estimates, both of (1.6) and (1.7), two ingredients
are essential. We require elliptic regularity estimates which follow from
strict $p^{\prime}$-convexity of $c^{\ast}$. We explain how to obtain these in
Section 1.2. In the quadratic case, the relevant equation is Laplace equation.
Hence solutions are harmonic and hence very regular- the proof in [8] requires
$C^{3}$-regularity of solutions! Already in the case
$c(x-y)=\frac{|x-y|^{p}}{p}$ with $p\neq 2$, the best regularity that is known
for solutions to (1.4) in general is $C^{1,\beta}$-regularity for some
$\beta>0$ [25, 18, 24]. Thus, at various places in the proof, more careful
estimates are needed.

The second ingredient is to obtain a $L^{\infty}$-bound for minimisers of
(1.1) in the small-energy regime, see Section 3. In the quadratic case, this
relies on the monotonicity (in the classical sense) property of solutions. In
the non-quadratic case, $c$-monotonicity needs to be used directly. Focusing
on $p$-homogeneous convex cost functions with $p>1$, $L^{\infty}$-bounds were
obtained in [10]. Note that [10] obtained $L^{\infty}$-bounds in all energy
regimes whereas the $L^{\infty}$-bounds obtained in this paper only cover the
small-energy regime. A further difference is that in this paper, we obtain the
$L^{\infty}$-bound as a consequence of the strong $p$-convexity of the energy,
whereas [10] relies on the homogeneity of the cost. Nevertheless, in the
small-energy regime and for cost functions covered by the assumptions both of
this paper and of [10], the obtained bounds agree. In particular, this is the
case for the important example of $p$-cost $c(x-y)=\frac{1}{p}\lvert
x-y\rvert^{p}$ with $p>1$.

Finally, we comment why we restrict our attention to cost functions of the
form $c(x-y)$. This is due to the fact that our proof relies on the
availability of a dynamical formulation. As (1.7) hints, we want to identify
points $(x,y)\in\mathrm{spt}\,\pi$ with the trajectory ${X(t)=tx+(1-t)y}$.
This is related to the Benamou-Brenier formulation of optimal transport, c.f.
[3], which in our case states that (1.1) can be alternatively characterised as

$\displaystyle\min_{(j,\rho)}\left\\{\int c\left(\frac{\,{\rm d}j_{t}}{\,{\rm
d}\rho_{t}}\right)\,{\rm
d}\rho\colon\partial_{t}\rho_{t}+\mathrm{div}\,j_{t}=0,\,\rho_{0}=\lambda,\,\rho_{1}=\mu\right\\}$
(1.10)

Here $\frac{\,{\rm d}j_{t}}{\,{\rm d}\rho_{t}}$ denotes the Radon-Nikodym
derivative. We refer the reader to (2.2) for an explanation on how to make
sense of (1.10) rigorously. This alternative, dynamical formulation of optimal
transport is only available for costs of the form $c(x-y)$ where $c$ is
convex.

The outline of the paper is as follows: In the remainder of this section, we
make precise our assumptions on the cost function (Section 1.1) and collect
the elliptic regularity statements we require (Section 1.2). After collecting
notation and some elementary results on optimal transportation in Section 2,
we collect the statements we require in order to prove Theorem 1.1 in Section
7: in Section 3, we prove the $L^{\infty}$-estimate. In Section 4, we prove a
localisation result on optimal transportation costs. Next, we approximate the
boundary data of (1.8) in Section 5. For technical reasons, we also need to
localise the data-term $D$, which we do in Section 6.

###  1.1 Assumptions on the cost function and its dual

In this section, we explain our assumptions on the cost function $c$. In order
to keep the statements of our results short, the conditions (1.11)–(1.14)
below will be assumed to hold throughout the entire paper. We consider cost-
functions modeled on the $p$-cost, $c(x)=\lvert x\rvert^{p}$ with $p>1$. This
is also the primary example of cost functions we have in mind. We emphasize
however that the cost functions we consider need not be homogeneous. In fact
the assumptions we impose are standard within elliptic regularity theory, see
e.g. [5, 20]. Let $p\in(1,\infty)$. We consider a $C^{1}$-cost function
$c\colon\mathbb{R}^{d}\to\mathbb{R}$ satisfying the following properties:
There is $\Lambda\geq 1$ such that

  1. (i)

$c$ is strongly $p$-convex: for any $x,y\in\mathbb{R}^{d}$ and $\tau\in[0,1]$,

$\displaystyle\Lambda^{-1}\tau(1-\tau)V_{p}(x,y)+c(\tau x+(1-\tau)y)\leq\tau
c(x)+(1-\tau)c(y).$ (1.11)

where

$V_{p}(x,y)=(|x|^{2}+|y|^{2})^{\frac{p-2}{2}}|x-y|^{2}.$

  2. (ii)

$c$ has $p$-growth: for any $x\in\mathbb{R}^{d}$,

$\displaystyle\Lambda^{-1}|x|^{p}\leq c(x)\leq\Lambda|x|^{p}.$ (1.12)

  3. (iii)

for any $x,y\in\mathbb{R}^{d}$,

$\displaystyle|c(x)-c(y)|\leq\Lambda U_{p}(x,y)$ (1.13)

where

$U_{p}(x,y)=(|x|+|y|)^{p-1}|x-y|.$

  4. (iv)

$c$ satisfies controlled $p$-growth: For any $x,y\in\mathbb{R}^{d}$,

$\displaystyle|\nabla c(x)-\nabla c(y)|\leq\Lambda(\lvert x\rvert+\lvert
y\rvert)^{p-2}\lvert x-y\rvert.$ (1.14)

If the choice of $p$ is clear from the context, we will write $V=V_{p}$ and
$U=U_{p}$. We further note the following inequality, valid for any
$z_{1},z_{2},z_{3}\in\mathbb{R}^{d}$ and with implicit constants depending
only on $p$ and $d$,

$\displaystyle\lvert
V_{p}(z_{1},z_{2})-V_{p}(z_{1},z_{3})\rvert\lesssim(\lvert z_{1}\rvert+\lvert
z_{2}\rvert+\lvert z_{3}\rvert)^{p-1}\lvert z_{2}-z_{3}\rvert.$ (1.15)

(1.15) follows from writing

$\displaystyle
V_{p}(z_{1},z_{2})-V_{p}(z_{1},z_{3})=\int_{0}^{1}\langle\nabla_{2}V_{p}(z_{1},sz_{2}+(1-s)z_{3}),z_{2}-z_{3}\rangle\,{\rm
d}t.$

Here $\nabla_{2}$ denotes a derivative with respect to the second variable of
$V_{p}$. From elementary calculations

$\displaystyle\lvert\nabla_{2}V_{p}(z_{1}+sz_{2}+(1-s)z_{3})\rvert\lesssim(\lvert
z_{1}\rvert+\lvert z_{2}\rvert+\lvert z_{3}\rvert)^{p-1},$

which gives (1.15).

(1.11) and the fact that $c$ is $C^{1}$ imply that for any
$x,y\in\mathbb{R}^{d}$,

$\displaystyle c(x)\geq c(y)+\langle\nabla c(y),x-y\rangle+\lambda V(x,y),$
(1.16) $\displaystyle\langle\nabla c(x)-\nabla c(y),x-y\rangle\geq\lambda
V(x,y).$ (1.17)

This can be seen by arguing as in the $2$-convex, $2$-growth case, which can
be found for example in [12, Chapter IV.4.1].

We require some information on the convexity properties of the convex
conjugate that follow by adaptions of the $2$-convex, $2$-growth theory in
[22] and [13]. For the convenience of the reader, we provide proofs of the
statements we require. Introduce the convex conjugate $c^{\ast}$ defined on
$\mathbb{R}^{d}$ via

$c^{\ast}(\xi)=\sup_{x}\langle\xi,x\rangle-c(x).$

We remark that since $c$ is strongly convex, $C^{1}$ and superlinear at
infinity, $c^{\ast}$ is strongly convex, $C^{1}$ and superlinear at infinity.
Moreover $\nabla c$ and $\nabla c^{\ast}$ are homeomorphisms of
$\mathbb{R}^{d}$ and $\nabla c^{\ast}=(\nabla c)^{-1}$ [22, Theorem 26.5].
Note that due to (1.12), we have for any $\xi\in\mathbb{R}^{d}$,

$\displaystyle c^{\ast}(\xi)\leq\sup_{x}\langle\xi,x\rangle-\Lambda^{-1}\lvert
x\rvert^{p}\lesssim\lvert\xi\rvert^{p^{\prime}}.$

A lower bound can be obtained similarly and we deduce:

$\displaystyle|\xi|^{p^{\prime}}\lesssim
c^{\ast}(\xi)\lesssim|\xi|^{p^{\prime}}.$ (1.18)

Due to strict $p$-convexity of $c$, $c^{\ast}$ satisfies for any
$\xi_{1},\xi_{2}\in\mathbb{R}^{d}$,

$\displaystyle|\nabla c^{\ast}(\xi_{1})-\nabla
c^{\ast}(\xi_{2})|\lesssim(|\xi_{1}|+|\xi_{2}|)^{p^{\prime}-2}|\xi_{1}-\xi_{2}|.$
(1.19)

Indeed, for $\xi_{1},\xi_{2}\in\mathbb{R}^{d}$, using (1.16) with the choice
${x=\nabla c^{\ast}(\xi_{1})}$, $y=\nabla c^{\ast}(\xi_{2})$ and Cauchy-
Schwarz,

$\displaystyle V_{p}(\nabla c^{\ast}(\xi_{1}),\nabla
c^{\ast}(\xi_{2}))\lesssim$ $\displaystyle\langle\xi_{1}-\xi_{2},\nabla
c^{\ast}(\xi_{1})-\nabla c^{\ast}(\xi_{2})\rangle\leq$
$\displaystyle\lvert\nabla c^{\ast}(\xi_{1})-\nabla
c^{\ast}(\xi_{2})\rvert\lvert\xi_{1}-\xi_{2}\rvert.$

Re-arranging, we have

$\displaystyle\lvert\nabla c^{\ast}(\xi_{1})-\nabla
c^{\ast}(\xi_{2})\rvert\lesssim\lvert\xi_{1}-\xi_{2}\rvert(\lvert\nabla
c^{\ast}(\xi_{1})\rvert^{2}+\lvert\nabla
c^{\ast}(\xi_{2})\rvert^{2})^{\frac{2-p}{2}}.$ (1.20)

We claim that for any $\xi\in\mathbb{R}^{d}$,

$\displaystyle\lvert\xi\rvert^{p^{\prime}-1}\lesssim\lvert\nabla
c^{\ast}(\xi)\rvert\lesssim\lvert\xi\rvert^{p^{\prime}-1}.$ (1.21)

Then (1.20) gives (1.19). Since $\nabla c^{\ast}=(\nabla c)^{-1}$ and both
maps are homeomorphisms, to show (1.21), it suffices to show that for any
$x\in\mathbb{R}^{d}$,

$\displaystyle\lvert x\rvert^{p-1}\lesssim\lvert\nabla
c(x)\rvert\lesssim\lvert x\rvert^{p-1}.$ (1.22)

Fix $x\in\mathbb{R}^{d}$. Since difference quotients of convex functions are
non-decreasing, for any $h\in\mathbb{R}^{d}$, applying also (1.13),

$\displaystyle\langle\nabla c(x),h\rangle\leq\frac{f(x+h)-f(x)}{\lvert
h\rvert}\lesssim\frac{(\lvert x\rvert+\lvert h\rvert)^{p}}{\lvert h\rvert}.$

Applying the above with $h\to th$ and choosing $t$ such that $\lvert
th\rvert=\lvert x\rvert$, as $h$ was arbitrary, this gives the second
inequality in (1.22). Note that in particular $\nabla c(0)=0$. Thus, using
(1.11) with $y=0$ and Cauchy-Schwartz gives

$\displaystyle\lvert x\rvert^{p}\lesssim\langle\nabla
c(x),x\rangle\leq\lvert\nabla c(x)\rvert\lvert x\rvert.$

After rearranging, this proves the first inequality in (1.22). Hence (1.19) is
established.

Finally, it follows from (1.19) and the intermediate value theorem that for
$\xi_{1},\xi_{2}\in\mathbb{R}^{d}$,

$\displaystyle\lvert
c^{\ast}(\xi_{1})-c^{\ast}(\xi_{2})\rvert=\Big{\lvert}\int_{0}^{1}\langle\nabla
c^{\ast}(s\xi_{1}+(1-s)\xi_{2}),\xi_{1}-\xi_{2}\rangle\,{\rm
d}s\Big{\rvert}\lesssim U_{p^{\prime}}(\xi_{1},\xi_{2}).$ (1.23)

We also require that $c^{\ast}$ is $p^{\prime}$ convex, that is for some
$C(p,\Lambda)>0$,

$\displaystyle
c\tau(1-\tau)V_{p^{\prime}}(\xi_{1},\xi_{2})+c^{\ast}(\tau\xi_{1}+(1-\tau)\xi)2)\leq\tau
c^{\ast}(\xi_{1})+(1-\tau)c^{\ast}(\xi_{2}).$ (1.24)

Indeed, we can use Taylor’s theorem and (1.14) to obtain for any
$x,y\in\mathbb{R}^{d}$ and some $C=C(p,n)>0$,

$\displaystyle c(y)=$ $\displaystyle c(x)+\langle\nabla
c(x),y-x\rangle+\int_{0}^{1}\langle\nabla c(x+t(y-x))-\nabla
c(x),y-x\rangle\mathrm{d}t$ $\displaystyle\leq$ $\displaystyle
c(x)+\langle\nabla c(x),y-x\rangle+C(\lvert x\rvert+\lvert
x-y\rvert)^{p-2}\lvert y-x\rvert^{2}.$ (1.25)

In order to estimate the integral, we used a well-known estimate, see e.g.
[11, 4]. Recall the Fenchel-Young inequality in the form

$\displaystyle
c(\xi)+c^{\ast}(x)\geq\langle\xi,x\rangle\quad\forall\xi,x\in\mathbb{R}^{d},$
(1.26)

with equality if and only if $\xi=\nabla c^{\ast}(x)$. Hence, with the choice
$x=\nabla c^{\ast}(\xi_{1})$, (1.1) gives

$\displaystyle-c^{\ast}(\xi_{2})=$
$\displaystyle-\sup_{y}\\{\langle\xi_{2},y\rangle-c(y)\\}$ $\displaystyle\leq$
$\displaystyle-c^{\ast}(\xi_{1})-\sup_{y}\left\\{\langle\xi_{2}-\xi_{1},y\rangle-C(\lvert\nabla
c^{\ast}(\xi_{1})\rvert+\lvert\nabla c^{\ast}(\xi_{1})-y\rvert)^{p-2}\lvert
y-\nabla c^{\ast}(\xi_{1})\right\\}.$ (1.27)

Note that the supremum is nothing but $\left(CV_{p}(\nabla
c^{\ast}(\xi_{1}),\cdot-\nabla
c^{\ast}(\xi_{1})\right)^{\ast}(\xi_{2}-\xi_{1})$. Noting that for
$x,y\in\mathbb{R}^{d}$,

$\displaystyle(\lvert x\rvert+\lvert y\rvert)^{p-2}\lvert
y\rvert^{2}\lesssim\begin{cases}\lvert y\rvert^{p}\quad&\text{ if }\\{\lvert
x\rvert\leq\lvert y\rvert,p\geq 2\\}\text{ or }\\{\lvert x\rvert\geq\lvert
y\rvert,p\leq 2\\}\\\ \lvert x\rvert^{p-2}\lvert x\rvert^{2}\quad&\text{ if
}\\{\lvert x\rvert\geq\lvert y\rvert,p\geq 2\\}\text{ or }\\{\lvert
x\rvert\leq\lvert y\rvert,p\leq 2\\}\end{cases}$

and arguing case by case as for (1.18), this shows

$\displaystyle\left((\lvert
x\rvert+\lvert\cdot\rvert)^{p-2}\lvert\cdot\rvert^{2}\right)^{\ast}(\xi)\gtrsim(\lvert
x\rvert^{p-1}+\lvert\xi\rvert)^{p^{\prime}-2}\lvert\xi\rvert^{2}.$

In particular, recalling (1.21), we deduce

$\displaystyle V_{p}(\nabla
c^{\ast}(\xi_{1}),\cdot)^{\ast}(\xi_{2}-\xi_{1})\gtrsim(\lvert\nabla
c^{\ast}(\xi_{1})\rvert^{p-1}+\lvert\xi_{1}-\xi_{2}\rvert)^{p^{\prime}-2}\lvert\xi_{1}-\xi_{2}\rvert^{2}\gtrsim
V_{p^{\prime}}(\xi_{1}-\xi_{2}).$

Employing the fact that for any convex $f\colon\mathbb{R}^{d}\to\mathbb{R}$
and any $x,\xi\in\mathbb{R}^{d}$, it holds that
${f(\cdot-x)^{\ast}(\xi)=f^{\ast}(\xi)+\langle x,\xi\rangle}$, we finally
conclude,

$\displaystyle\left(CV_{p}(\nabla c^{\ast}(\xi_{1}),\cdot-\nabla
c^{\ast}(\xi_{1})\right)^{\ast}(\xi_{2}-\xi_{1})\gtrsim
V_{p^{\prime}}(\xi_{1}-\xi_{2})+\langle\nabla
c^{\ast}(\xi_{1}),\xi_{2}-\xi_{1}).$

Combining this estimate with (1.1) gives (1.24).

###  1.2 Regularity assumptions on the dual system

Let $R\in(2,3)$. In this section, we state the regularity assumptions we make
on distributional solutions $\phi\in W^{1,p^{\prime}}(B)$ of the equation

$\displaystyle-\mathrm{div}\,\nabla c^{\ast}(\mathrm{D}\phi)$
$\displaystyle=c_{g}\quad\text{ on }B_{R}$ (1.28) $\displaystyle\nabla
c^{\ast}(\mathrm{D}\phi)\cdot\nu$ $\displaystyle=g\quad\text{ on }\partial
B_{R},$ (1.29)

where $g\in L^{p}(B)$ and $c_{g}$ satisfies the compatibility condition
$\lvert B_{R}\rvert c_{g}=\int_{\partial B}g$. $\nu$ denotes the outward
pointing normal vector on $\partial B$. We will show that solutions exist and
are unique up to a constant. Hence, we usually normalise solutions by
requiring that $\int_{B}\phi=0$. Fixing $g\in L^{p}(\partial B)$, we denote by
$\phi^{r}$ the solution satisfying $\int_{B}\phi^{r}=0$ of (1.28) with data
$g^{r}$, where $g^{r}$ denotes convolution of $g$ with a smooth convolution
kernel on $\partial B$ at scale $r$.

######  Lemma 1.2.

If $c^{\ast}$ satisfies (1.11)-(1.14), then solutions $\phi$ to (1.28) exist,
are unique up to constant and the following statements hold:

  1. (i)

$\phi$ satisfies the following energy estimates:

$\displaystyle\int_{B_{R}}|\mathrm{D}\phi|^{p^{\prime}}\,{\rm
d}x\lesssim\int_{\partial B_{R}}|g|^{p},$ (1.30)
$\displaystyle\int_{B_{R}}c(\nabla c^{\ast}(\mathrm{D}\phi))\,{\rm
d}x\lesssim\int_{\partial B_{R}}|g|^{p}.$ (1.31)

  2. (ii)

$\phi$ is Lipschitz regular in the interior of $B_{R}$: For any $r<R$,

$\displaystyle\sup_{x\in
B_{r}}\lvert\mathrm{D}\phi\rvert^{p^{\prime}}\lesssim_{R-r}\int_{\partial
B_{R}}\lvert g\rvert^{p}.$ (1.32)

  3. (iii)

The difference between $\phi$ and $\phi^{r}$ is controlled: There exists
$s=s(n,p)>0$ such that

$\displaystyle\int_{B_{R}}|\mathrm{D}\phi-\mathrm{D}\phi^{r}|^{p^{\prime}}\,{\rm
d}x\lesssim r^{s}\int_{\partial B_{R}}|g|^{p}.$ (1.33)

  4. (iv)

$\mathrm{D}\phi^{r}$ is Hölder-regular up to the boundary: For any
$\beta\in(0,1)$,

$\displaystyle
r^{\beta}[\mathrm{D}\phi^{r}]_{C^{0,\beta}(B_{R})}^{p^{\prime}}+\sup_{B_{R}}|\mathrm{D}\phi^{r}|^{p^{\prime}}\lesssim\frac{1}{r^{d-1}}\int_{\partial
B_{R}}|g|^{p}.$ (1.34)

  5. (v)

Let $\beta\in(0,1)$. Whenever $\phi\in C^{0,\beta}(B)$ for some ball $B$,

$\displaystyle[c^{\ast}(\mathrm{D}\phi)+c(\nabla
c^{\ast}(\mathrm{D}\phi))]_{C^{0,\beta}(B)}\lesssim\|\mathrm{D}\phi\|_{L^{\infty}(B)}^{p^{\prime}-1}[D\phi]_{C^{0,\beta}(B)}.$

[Proof]Note that in light of the results of Section 1.1, $c^{\ast}$ is
$p^{\prime}$-convex and satisfies controlled $p^{\prime}$-growth. Hence, the
statements we need to prove are largely standard.

The existence and uniqueness up to constant of solutions follows from the
direct method. Testing the weak formulation of (1.28) with $\phi$ and applying
(1.24) in combination with Hölder’s inequality, the trace estimate and
Poincaré inequality (recall that $\int_{B}\phi=0$) gives:

$\displaystyle\|\mathrm{D}\phi\|_{L^{p^{\prime}}(B_{R})}^{p^{\prime}}\lesssim$
$\displaystyle\int_{B_{R}}\langle\nabla
c^{\ast}(\mathrm{D}\phi),\mathrm{D}\phi\rangle\,{\rm d}x=\int_{\partial
B_{R}}g\phi\leq\|\phi\|_{L^{p^{\prime}}(\partial B_{R})}\|g\|_{L^{p}(\partial
B_{R})}$ $\displaystyle\lesssim$
$\displaystyle\|\phi\|_{W^{1,p^{\prime}}(B_{R})}\|g\|_{L^{p}(\partial
B_{R})}\lesssim\|\mathrm{D}\phi\|_{L^{p^{\prime}}(B_{R})}\|g\|_{L^{p}(\partial
B_{R})}.$

Re-arranging this gives (1.30). Using (1.12) and (1.21), (1.30) implies that
also

$\displaystyle\int_{B_{R}}c(\nabla c^{\ast}(\mathrm{D}\phi))\,{\rm
d}x\lesssim\int_{B_{R}}\lvert\nabla c^{\ast}(\mathrm{D}\phi)\rvert^{p}\,{\rm
d}x\lesssim\int_{B_{R}}\lvert\mathrm{D}\phi\rvert^{p^{\prime}}\,{\rm
d}x\lesssim\int_{\partial B_{R}}|g|^{p},$

that is (1.31) holds.

(1.32) is proven in [19] and [17]. As the proofs are quite involved, we do not
comment on them here. Instead we turn to (1.33). We focus on the case
$p^{\prime}\leq n$ as the other case is easier. Testing the equations for
$\phi$ and $\phi^{r}$ with $\mathrm{D}\phi-\mathrm{D}\phi^{r}$ and applying
(1.24) and Hölder’s inequality, we find

$\displaystyle\int_{B_{R}}V_{p^{\prime}}(\mathrm{D}\phi,\mathrm{D}\phi^{r})\lesssim$
$\displaystyle\int_{B_{R}}\langle\nabla c^{\ast}(\mathrm{D}\phi)-\nabla
c^{\ast}(\mathrm{D}\phi^{r}),\mathrm{D}\phi-\mathrm{D}\phi^{r})\,{\rm
d}x=\int_{\partial B_{R}}(g-g^{r})(\phi-\phi^{r})$ $\displaystyle\lesssim$
$\displaystyle\|g-g^{r}\|_{L^{\frac{p^{\prime}(n-1)}{n(p^{\prime}-1)}}(\partial
B_{R})}\|\phi-\phi^{r}\|_{L^{\frac{p^{\prime}(n-1)}{n-p^{\prime}}}(\partial
B_{R})}$

By a standard trace estimate and Poincaré’s inequality

$\displaystyle\|\phi-\phi^{r}\|_{L^{\frac{p^{\prime}(n-1)}{n-p^{\prime}}}(\partial
B_{R})}\lesssim\|\phi-\phi^{r}\|_{W^{1,p^{\prime}}(B_{R})}\lesssim\|D\phi-\mathrm{D}\phi^{r}\|_{L^{p^{\prime}}(B_{R})}.$

Note $\frac{p^{\prime}(n-1)}{n-p^{\prime}}=\frac{(n-1)p}{n}$. By standard
properties of convolution,

$\displaystyle\|g-g^{r}\|_{L^{\frac{p(n-1)}{n}}(\partial B_{R})}\lesssim
r^{\frac{1}{n-1}}\|g\|_{L^{p}(\partial B_{R})}.$

If $p\leq 2$, since
$V_{p^{\prime}}(\mathrm{D}\phi,\mathrm{D}\phi^{r})\geq\lvert\mathrm{D}\phi-\mathrm{D}\phi^{r}\rvert^{p^{\prime}}$,
combining estimates and re-arranging concludes the proof. If $p^{\prime}\leq
2$, we apply Hölder’s inequality to see

$\displaystyle\int_{B_{R}}V_{p^{\prime}}(\mathrm{D}\phi,\mathrm{D}\phi^{r})\geq\|\mathrm{D}\phi-\mathrm{D}\phi^{r}\|_{L^{p^{\prime}}(B_{R})}^{2}(\|\lvert\mathrm{D}\phi\rvert+\lvert\mathrm{D}\phi^{r}\rvert\|_{L^{p^{\prime}}(B_{R})}^{p^{\prime}-2}.$

Since due to (1.30) and standard properties of convolution,

$\displaystyle\||\lvert\mathrm{D}\phi\rvert+\lvert\mathrm{D}\phi^{r}\rvert\|_{L^{p^{\prime}}(B_{R})}\lesssim\|D\phi\|_{L^{p^{\prime}}(B_{R})}+\|\mathrm{D}\phi^{r}\|_{L^{p^{\prime}}(B_{R})}\lesssim\|g\|_{L^{p}(\partial
B_{R})}^{p-1},$

this again gives (1.33).

(1.34) follows from [19] and [17]. Again the proof is involved and we don’t
comment on it here.

Finally, we note by direct calculation using (1.13), (1.23), (1.19) and (1.21)
that

$\displaystyle[c^{\ast}(\mathrm{D}\phi)+c(\nabla
c^{\ast}(\mathrm{D}\phi))]_{C^{0,\beta}(B)}\lesssim\|\mathrm{D}\phi\|_{L^{\infty}(B)}^{p^{\prime}-1}[D\phi]_{C^{0,\beta}(B)}.$

This concludes the proof.

## 2 Preliminaries

###  2.1 General notation

Throughout, we let $1<p<\infty$. $B_{r}(x)$ will denote a ball of radius $r>0$
centered at $x\in\mathbb{R}^{d}$. We further write $B_{r}=B_{r}(0)$ and
$B=B_{1}(0)$. $c$ denotes a generic constant that may change from line to
line. Relevant dependencies on $\Lambda$, say, will be denoted $c(\Lambda)$.
We say $a\lesssim b$ and $a\gtrsim b$, if there exists a constant $c>0$
depending only on $d$, $p$ and $\Lambda$ such that $a\leq cb$ and $a\geq cb$,
respectively.

Given $\Omega\subset\mathbb{R}^{d}$, we denote by $[\cdot]_{C^{0,\alpha}}$,
the $\alpha$-Hölder-seminorm. Given $\alpha\in(0,\infty)$, $L^{p}(\Omega)$ and
$W^{\alpha,p}(\Omega)$ denote the usual Lebesgue and (fractional) Sobolev
spaces. If $\mu$ is a measure on $\mathbb{R}^{d}$, $\mu\llcorner\Omega$
denotes its restriction to $\Omega$. We say a couple of measures $\lambda$,
$\mu$ on $\mathbb{R}^{d}$ is admissible if $\lambda,\mu$ are non-negative
finite measures satisfying $\lambda(\mathbb{R}^{d})=\mu(\mathbb{R}^{d})$.

Given $R>0$, we let $\Pi_{R}(x)=R\frac{x}{|x|}$ be the projection onto
$\partial B_{R}$ and define for every measure $\rho$ on $\mathbb{R}^{d}$ the
projected measure on $\partial B_{R}$, $\hat{\rho}=\Pi_{R}\\#\rho$, i.e.

$\displaystyle\int\xi\,{\rm
d}\hat{\rho}=\int\xi\left(R\frac{x}{|x|}\right)\,{\rm d}\rho(x).$ (2.1)

A set $\Omega\subset\mathbb{R}^{d}\times\mathbb{R}^{d}$ is said to be
$c$-cyclically monotone if for any $N\in\mathbb{N}$ and any points
$(x_{1},y_{1}),\ldots,(x_{N},y_{N})\in\Omega$, there holds

$\sum_{i=1}^{N}c(x_{i}-y_{i})\leq\sum_{i=1}^{N}c(x_{i}-y_{i+1}),$

where we identify $y_{N+1}=y_{1}$.

A function $f\colon\mathbb{R}^{d}\to\mathbb{R}$ is called $c$-concave if there
exists a function $g\colon\mathbb{R}^{d}\to\mathbb{R}$ such that

$f(x)=\inf_{y\in\mathbb{R}^{d}}c(x-y)-g(y)$

for all $x\in\mathbb{R}^{d}$.

###  2.2 Optimal transportation

We recall some definitions and facts about optimal transportation, see [26]
for more details. For this subsection, the full strength of our assumptions
(1.11)-(1.14) is not needed. In fact, assuming that $c$ is lower semi-
continuous, convex and satisfies $p$-growth (1.12) would be sufficient in this
subsection.

Given a measure $\pi$ on $\mathbb{R}^{d}\times\mathbb{R}^{d}$ we denote its
marginals by $\pi_{1}$ and $\pi_{2}$ respectively. The set of measures on
$\mathbb{R}^{d}\times\mathbb{R}^{d}$ with marginals $\pi_{1}$ and $\pi_{2}$ is
denoted $\Pi(\pi_{1},\pi_{2})$. Given two positive measures with compact
support and equal mass $\lambda$ and $\mu$ we define

$W_{c}(\lambda,\mu)=\min_{\pi_{1}=\lambda,\pi_{2}=\mu}\int c(x-y)\,{\rm
d}\pi.$

While our notation is reminiscent of the Wasserstein distance, and in fact
gives the ($p$-th power of the) Wasserstein $p$-distance in the case
${c(x-y)=\lvert x-y\rvert^{p}}$, in general it is not a distance on measures.
Under our hypothesis, an optimal coupling always exists and moreover a
coupling $\pi$ is optimal, if and only if its support is $c$-cyclical
monotone.

Moreover, we note the following triangle-type inequality:

######  Lemma 2.1.

Let $\varepsilon\in(0,1)$. There is $C(\varepsilon)>0$ such that for any
admissible measures $\mu_{1},\mu_{2},\mu_{3}$ it holds that

$\displaystyle
W_{c}(\mu_{1},\mu_{3})\leq(1+\varepsilon)W_{c}(\mu_{1},\mu_{2})+C(\varepsilon)W_{c}(\mu_{2},\mu_{3}).$

[Proof]Due to the gluing lemma, see e.g. [23, Lemma 5.5.], there exists
$\sigma$, a positive measure on
$\mathbb{R}^{d}\times\mathbb{R}^{d}\times\mathbb{R}^{d}$ with marginal
$\pi_{1}$ on the first two variables and marginal $\pi_{2}$ on the last two
variables. Here $\pi_{1}$ and $\pi_{2}$ are the optimal couplings between
$\mu_{1}$ and $\mu_{2}$ and $\mu_{2}$ and $\mu_{3}$, respectively, with
respect to $W_{c}$. Set $\gamma$ to be the marginal of $\sigma$ with respect
to the first and third variable. Then $\gamma\in\Pi(\mu_{1},\mu_{3})$. It
follows using the convexity of $c$ and the triangle inequality in
$L^{p}(\gamma)$ that for any $t\in(0,1)$,

$\displaystyle W_{c}(\mu_{1},\mu_{3})\leq$ $\displaystyle\left(\int
c(x-z)\,{\rm d}\gamma\right)^{\frac{1}{p}}\leq\left(\int
tc\left(\frac{x-y}{t}\right)+(1-t)\left(\frac{y-z}{1-t}\right)\,{\rm
d}\gamma\right)^{\frac{1}{p}}$ $\displaystyle\leq$
$\displaystyle\left(\int\left(\left(tc\left(\frac{x-y}{t}\right)\right)^{\frac{1}{p}}+\left((1-t)\left(\frac{y-z}{1-t}\right)\right)^{\frac{1}{p}}\right)^{p}\,{\rm
d}\gamma\right)^{\frac{1}{p}}$ $\displaystyle\leq$ $\displaystyle\left(\int
tc\left(\frac{x-y}{t}\right)\,{\rm
d}\gamma\right)^{\frac{1}{p}}+\left(\int(1-t)c\left(\frac{y-z}{1-t}\right)\,{\rm
d}\gamma\right)^{\frac{1}{p}}.$

Using (1.12) and recalling the definition of $\gamma$, we deduce

$\displaystyle
W_{c}(\mu_{1},\mu_{3})\leq\left(\Lambda^{2}t^{1-p}\right)^{\frac{1}{p}}W_{c}(\mu_{1},\mu_{2})+\left(\Lambda^{2}(1-t)^{1-p}\right)^{\frac{1}{p}}W_{c}(\mu_{2},\mu_{3}).$

Choosing $t$ sufficiently close to $1$, this gives the desired estimate.

We require also the following consequence of Lemma 2.1.

######  Corollary 2.2.

Let $\mu_{1},\mu_{2}$ be admissible measures. Then

$\displaystyle W_{c}(\mu_{1},\mu_{2})\lesssim
W_{c}(\mu_{1}+\mu_{2},2\mu_{2}).$

[Proof]Using Lemma 2.1 and sub-additivity of $W_{c}$, we note for any
$\delta>0$,

$\displaystyle W_{c}(\mu_{1},\mu_{2})\leq$
$\displaystyle(1+\delta)W_{c}\left(\mu_{1},\frac{1}{2}(\mu_{1}+\mu_{2})\right)+C(\delta)W_{c}\left(\frac{1}{2}(\mu_{1}+\mu_{2}),\mu_{2}\right)$
$\displaystyle=$
$\displaystyle(1+\delta)W_{c}\left(\frac{1}{2}\mu_{1},\frac{1}{2}\mu_{2}\right)+c(\delta)W_{c}\left(\frac{1}{2}(\mu_{1}+\mu_{2}),\mu_{2}\right)$
$\displaystyle\leq$
$\displaystyle\frac{1+\delta}{2}W_{c}\left(\mu_{1},\mu_{2}\right)+C(\delta)W_{c}\left(\mu_{1}+\mu_{2},2\mu_{2}\right).$

Re-arranging gives the result.

We remark that the Benamou-Brenier formula (1.10) needs to be interpreted via
duality in general, that is we set

$\displaystyle\int c\left(\frac{\,{\rm d}j_{t}}{\,{\rm
d}\rho_{t}}\right)\,{\rm d}\rho_{t}=\sup_{\zeta\in
C^{0}_{c}(\mathbb{R}^{d})}\left\\{\int\zeta\,{\rm d}j_{t}-\int
c^{\ast}(\xi)\,{\rm d}\rho_{t}\right\\}.$ (2.2)

Given $O\subset\mathbb{R}^{d}$ set $\kappa_{\mu,O}$ to be the generic constant
such that $W_{c}(\mu\llcorner O,\kappa_{\mu,O}\,{\rm d}x\llcorner O)$ is well-
defined, that is $\kappa_{\mu,O}=\frac{\mu(O)}{\lvert O\rvert}$. If $O=B_{R}$,
we write $\kappa_{\mu,R}=\kappa_{\mu,B_{R}}$.

It will be convenient to denote
$\\#_{R}=(B_{R}\times\mathbb{R}^{d})\cup(\mathbb{R}^{d}\times B_{R})$. We
recall the definition of the quantities that we use to measure smallness:

$\displaystyle E(R)\mathrel{\mathop{\mathchar 58\relax}}=\frac{1}{\lvert
B_{R}\rvert}\int_{\\#_{R}}c(x-y)\,{\rm d}\pi,$ $\displaystyle
D(R)\mathrel{\mathop{\mathchar 58\relax}}=\frac{1}{\lvert
B_{R}\rvert}W_{p}^{p}(\lambda\llcorner B_{R},\kappa_{\lambda,R}\,{\rm
d}x\llcorner
B_{R})+\frac{R^{p}}{\kappa_{\lambda,R}^{p-1}}(\kappa_{\lambda,R}-1)^{p}$
$\displaystyle\qquad\qquad+\frac{1}{\lvert B_{R}\rvert}W_{p}^{p}(\mu\llcorner
B_{R},\kappa_{\mu,R}\,{\rm d}x\llcorner
B_{R})+\frac{R^{p}}{\kappa_{\mu,R}^{p-1}}(\kappa_{\mu,R}-1)^{p}.$

We will find it convenient to work with trajectories $X(t)=tx+(1-t)y$. In this
context, it is useful to work on the domain

$\displaystyle\Omega_{R}=\\{(x,y)\in\\#_{3}\colon\exists t\in[0,1]\text{ s.t.
}X(t)\in\overline{B}_{R}\\}.$

To every trajectory $X\in\Omega$, we associate entering and exiting times of
$B_{R}$:

$\displaystyle\sigma_{R}\mathrel{\mathop{\mathchar
58\relax}}=\min\\{t\in[0,1]\colon X(t)\in\overline{B}_{R}\\}$
$\displaystyle\tau_{R}\mathrel{\mathop{\mathchar
58\relax}}=\max\\{t\in[0,1]\colon X(t)\in\overline{B}_{R}\\}.$

Often, we will drop the subscripts and denote $\Omega=\Omega_{R}$,
$\sigma=\sigma_{R}$ and $\tau=\tau_{R}$. Further, we will need to track
trajectories entering and leaving $B_{R}$. This is achieved through the non-
negative measures $f_{R}$ and $g_{R}$ concentrated on $\partial B_{R}$ and
defined by the relations

$\displaystyle\int\zeta\,{\rm d}f_{R}=\int_{\Omega\cap\\{X(\sigma)\in\partial
B_{R}\\}}\zeta(X(\sigma))\,{\rm d}\pi,$ $\displaystyle\int\zeta\,{\rm
d}g_{R}=\int_{\Omega\cap\\{X(\tau)\in\partial B_{R}\\}}\zeta(X(\sigma))\,{\rm
d}\pi.$ (2.3)

Note that the set of trajectories $\Omega\cap\\{X(\sigma)\in\partial B_{R}\\}$
implicitly defines a Borel measurable subset of
$\mathbb{R}^{d}\times\mathbb{R}^{d}$, namely the pre-image under the mapping
$(x,y)\to X$, which is continuous from $\mathbb{R}^{d}\times\mathbb{R}^{d}$
into $C^{0}([0,1])$. Thus, the integrals in (2.2) are well-defined. We will
often use similar observations without further justification.

###  2.3 Estimating radial projections

We record a technical estimate concerning radial projections we will require.

######  Lemma 2.3.

For $R>0$, there exists $1\geq\varepsilon(d)>0$ such that for every $g\geq 0$
with $\mathrm{Spt}\,g\subset B_{(1+\varepsilon)R}\setminus
B_{(1-\varepsilon)R}$ we have

$\displaystyle R^{1-d}\left(\int g\right)^{p}\lesssim\int_{\partial
B_{R}}\hat{g}^{p}\lesssim\sup g^{p-1}\int|R-|x||^{p-1}g$

Here $\hat{g}$ is the radial projection of $g$ defined in (2.1).

[Proof]By scaling we may assume $R=\sup g=1$. The first inequality is then a
direct consequence of Jensen’s inequality.

For the second inequality, note that if $\varepsilon\ll 1$, $\sup_{\partial
B_{1}}|\hat{g}|\ll 1$, since we assume ${\mathrm{Spt}\,g\subset
B_{1+\varepsilon}\setminus B_{1-\varepsilon}}$. Fix $\omega\in\partial B_{1}$
and set $\psi(r)=r^{d-1}g(r\omega)$ for $r>0$. Then we have
${0\leq\psi\leq(1+\varepsilon)^{d-1}\leq 2}$ and

$\int_{0}^{\infty}\psi=\hat{g}(\omega).$

We conclude that for $\omega\in\partial B_{1}$,

$\int_{0}^{\infty}|1-r|^{p-1}r^{d-1}g(r\omega)\geq\min_{0\leq\tilde{\psi}\leq
2,\int\tilde{\psi}=\hat{g}(\omega)}\int_{0}^{\infty}|1-r|^{p-1}\tilde{\psi}(r)\gtrsim\hat{g}(\omega).$

The last inequality holds, since the minimiser of

$\min_{0\leq\tilde{\psi}\leq
2,\int\tilde{\psi}=\hat{g}(\omega)}\int_{0}^{\infty}|1-r|^{p-1}\tilde{\psi}(r)$

is given by $2I\left(|r-1|\leq\frac{1}{4}\hat{g}(\omega)\right)$.

## 3 A $L^{\infty}$-bound on the displacement

A key point in our proof will be that trajectories do not move very much.
Since we assume $E(4)\ll 1$, this is evidently true on average. However, we
will require to control the length of trajectories not just on average, but in
a pointwise sense. We establish this result in this section. In the quadratic
case, the proof in [9] relies on the fact that $2$-monotonicity is equivalent
to standard monotonicity. In our setting this is not available and we hence
provide a different proof. Our proof heavily relies on the strong
$p$-convexity of $c$.

######  Lemma 3.1.

Let $1<p<\infty$. Let $\pi$ be a coupling between two admissible measures
$\lambda$ and $\mu$. Assume that $\mathrm{Spt}\,\pi$ is cyclically monotone
with respect to $c$-cost and that ${E(4)+D(4)\ll 1}$. Then for every
$(x,y)\in\mathrm{Spt}\,\pi\cap\\#_{3}$, we have

$\displaystyle\lvert x-y\rvert\lesssim\left(E(4)+D(4)\right)^{\frac{1}{p+d}}.$
(3.1)

As a consequence, for $(x,y)\in\mathrm{Spt}\,\pi$ and $t\in[0,1]$,

$\displaystyle x\in B_{3}\text{ or }y\in B_{3}\Rightarrow(1-t)x+ty\in B_{4}.$
(3.2)

In the proof of Lemma 3.1 we require the following technical result, which we
state independently as we will require it again in the future.

######  Lemma 3.2.

Let $1<p<\infty$ and $0<\alpha<1$. For every $R>0$, $\xi\in
C^{0,\alpha}(B_{R})$ and $\mu$ supported in $B_{R}$ with $\mu(B_{R})\sim\lvert
B_{R}\rvert$,

$\displaystyle\left\lvert\int_{B_{R}}\xi(\,{\rm d}\mu-\kappa_{\mu,R}\,{\rm
d}x)\right\rvert\leq$
$\displaystyle[\xi]_{C^{0,\alpha}(B_{R})}W_{c}(\mu,\kappa_{\mu,R}\,{\rm
d}x\llcorner B_{R})^{\frac{\alpha}{p}}R^{\frac{2d(p-\alpha)}{p}}.$ (3.3)

In case $\alpha=1$, (3.3) holds with $C^{0,1}$ replaced by $C^{1}$. Further,
if in addition we have ${\xi\in
C^{\lfloor{p-1}\rfloor,p-\lfloor{p-1}\rfloor}(B_{R})}$, there is $C>0$ such
that

$\displaystyle\left\lvert\int_{B_{R}}\xi(\,{\rm d}\mu-\kappa_{\mu,R}\,{\rm
d}x)\right\rvert\leq$ $\displaystyle
C\sum_{i=1}^{\lfloor{p-1}\rfloor}\left(\kappa_{\mu,R}\int\lvert\mathrm{D}^{i}\xi\rvert^{\frac{p}{p-i}}\,{\rm
d}x\right)^{\frac{p-i}{p}}W_{c}(\mu,\kappa_{\mu,R}\,{\rm d}x\llcorner
B_{R})^{\frac{i}{p}}$
$\displaystyle\quad+[\xi]_{C^{\lfloor{p-1}\rfloor,p-\lfloor{p-1}\rfloor}}W_{c}(\mu,\kappa_{\mu,R}\
x\llcorner B_{R}).$

[Proof]Integrate the estimate

$|\xi(x)-\xi(y)|\leq[\xi]_{C^{0,\alpha}}|x-y|^{\alpha}$

against an optimal transport plan $\pi$ between $\mu$ and
$\kappa_{\mu,R}\,{\rm d}x\llcorner B_{R}$ to find,

$\displaystyle\left|\int_{B_{R}}\xi(\,{\rm d}\mu-\kappa_{\mu}\,{\rm
d}x)\right|\leq$
$\displaystyle[\xi]_{C^{0,\alpha}(B_{R})}\int_{B_{R}}|x-y|^{\alpha}\,{\rm
d}\pi.$

Applying Hölder and using (1.12) the result follows.

To obtain the second estimate, we proceed similarly, but start with the
estimate

$\displaystyle\lvert\xi(x)-\xi(y)-\sum_{\lvert\alpha\rvert=1}^{\lfloor{p-1}\rfloor}\mathrm{D}^{\alpha}\xi(y)\frac{(x-y)^{\alpha}}{\lvert\alpha\rvert!}\rvert\leq[\xi]_{C^{\lfloor{p-1}\rfloor,p-\lfloor{p-1}\rfloor}(B_{R})}\lvert
x-y\rvert^{p}.$

The result follows using (1.12) and using Hölder to estimate

$\displaystyle\int\lvert\mathrm{D}^{\alpha}\xi(y)\rvert\lvert
x-y\rvert^{\lvert\alpha\rvert}\,{\rm
d}\pi\leq\left(\kappa_{\mu,R}\int\lvert\mathrm{D}^{\lvert\alpha\rvert}\xi\rvert^{\frac{p}{p-\lvert\alpha\rvert}}\,{\rm
d}x\right)^{\frac{p-\lvert\alpha\rvert}{p}}W_{c}(\mu,\kappa_{\mu,R}\,{\rm
d}x\llcorner B_{R})^{\frac{\lvert\alpha\rvert}{p}}.$

We proceed to prove Lemma 3.1. [Proof of Lemma 3.1] Fix
$(x,y)\in\mathrm{Spt}\,\pi\cap\\#_{3}$. Without loss of generality we may
assume that $(x,y)\in B_{3}\times\mathbb{R}^{d}$.

Step 1. Barrier points exist in all directions: In this step we show that in
all directions we may find points
$(x^{\prime},y^{\prime})\in\mathrm{Spt}\,\pi$ with $x^{\prime}\approx
y^{\prime}$. To be precise, consider an arbitrary unit vector
$\bm{n}\in\mathbb{R}^{d}$ and let $r>0$. We show that for any $\bm{n}$, and
all $r\ll 1$, there is $M=M(p,d,\Lambda)>0$ and
$(x^{\prime},y^{\prime})\in\mathrm{Spt}\,\pi\cap(B_{r}(x+2r\bm{n})\times\mathbb{R}^{d})$
such that

$\displaystyle c(x^{\prime}-y^{\prime})\leq\frac{ME(4)}{r^{d}}.$

Assume, for contradiction, that for any $M>0$, there is
$\bm{n}\in\mathbb{R}^{d}$ and $r>0$ such that for all
$(x^{\prime},y^{\prime})\in\mathrm{Spt}\,\pi\cap(B_{r}(x+2r\bm{n})\times\mathbb{R}^{d})$,
$|x^{\prime}-y^{\prime}|\geq\frac{ME(4)}{r^{d}}$. Let $\eta$ be a non-
negative, smooth cut-off supported in $B_{r}(x+2r\bm{n})$ satisfying

$\sum_{i=1}^{\lfloor{p-1}\rfloor}r^{i}\sup|\mathrm{D}^{i}\eta|+r^{p}[\eta]_{C^{\lfloor{p-1}\rfloor,p-\lfloor{p-1}\rfloor}}\lesssim
1.$

Then

$\displaystyle E(4)\gtrsim\int\int\eta(x)c(x-y)\,{\rm
d}\pi(x,y)\geq\int\int\frac{ME(4)}{r^{d}}\eta(x)\,{\rm
d}\pi(x,y)=\frac{ME(4)}{r^{d}}\int\eta(x)\,{\rm d}\mu(x).$

However, due to Lemma 3.2 and noting $\kappa_{\mu,4}\sim 1$,

$\displaystyle\left|\int\eta\,{\rm d}\mu(x)-\kappa_{\mu,4}\int\eta\,{\rm
d}x\right|\lesssim\sum_{i=1}^{\lfloor{p-1}\rfloor}r^{\frac{d(p-i)}{p}-i}D(4)^{\frac{i}{p}}+r^{-p}D(4).$

Normalising $\eta$ such that $\int_{B_{r}(x+2rn)}\eta\,{\rm d}x\sim r^{d}$, we
can guarantee $\kappa_{\mu,4}\int\eta\,{\rm d}x\sim\kappa_{\mu,4}r^{d}\sim
r^{d}$. Ensuring $D(4)\ll r^{p+d}$, so that
$\sum_{i=1}^{\lfloor{p-1}\rfloor}r^{\frac{d(p-i)}{p}-i}D(4)^{\frac{i}{p}}+r^{-p}D(4)\ll
r^{d}$, we may thus conclude

$\displaystyle E(4)\gtrsim\frac{ME(4)}{r^{d}}r^{d}=ME(4).$

As $M$ was arbitrary, this is a contradiction.

Step 2. Building barriers: In this step, we show that if we are given points

${(x^{\prime},y^{\prime})\in\mathrm{Spt}\,\pi\cap(B_{r}(x+2r\bm{n})\times\mathbb{R}^{d})}$

such that $|x^{\prime}-y^{\prime}|\leq\frac{ME(4)}{r^{d}}$ for some
$M=M(p,d,\Lambda)>0$, then there is a cone $C_{x,x^{\prime}}$ with vertex
$x^{\prime}+r\rho(x^{\prime}-x)$ for some $\rho=\rho(p,d,\Lambda)>0$, aperture
$\alpha=\alpha(p,d,\Lambda)$ and axis $x^{\prime}-x$ such that $y\not\in
C_{x,x^{\prime}}$.

Without loss of generality, we may assume that $x^{\prime}-x$ points in the
$e_{n}$ direction. Moreover, considering the cost $c(\cdot)-c(x)$, we may
assume that $c(x)=0$. Suppose for a contradiction that

$y\in
C_{x,x^{\prime}}=x^{\prime}+\\{a\in\mathbb{R}^{d-1}\times\mathbb{R}^{+}\colon
d(a,\Gamma)\leq\alpha(|\overline{a}-x^{\prime}|-r\rho)\\}$

for some $\alpha,\rho>0$ to be determined. Here
$\Gamma=\\{t(x^{\prime}-x)\colon t\geq 0\\}$ and $\overline{a}$ denotes the
orthogonal projection of a point $a\in\mathbb{R}^{d-1}\times\mathbb{R}^{+}$
onto $\Gamma$. We want to show that then

$\displaystyle c(x-y)\geq c(x^{\prime}-y)+c(x-y^{\prime}).$ (3.4)

(3.4) is a contradiction to the $c$-monotonicity of $\pi$ and hence proves the
stated claim.

We note that we may assume $x=0$. Indeed, setting $z=y-x$,
$z^{\prime}=y^{\prime}-x$ and $\tilde{z}=x^{\prime}-x$, (3.4) becomes

$\displaystyle c(-z)\geq c(\tilde{z}-z)+c(-z^{\prime})$

with $|\tilde{z}|\leq\frac{ME}{r^{d}}$ and $z\in C_{0,\tilde{z}}$, which we
recognise as precisely the situation we are in if $x=0$.

$C_{x,x^{\prime}}$$X$$Y$$\overline{Y}$$X^{\prime}$$Y^{\prime}$

Figure 1: Geometric situation in Step 2.

Taking $\rho\geq 4$, we then estimate using the growth assumption (1.13) and
the strict convexity assumption (1.11)

$\displaystyle c(-y)\geq$ $\displaystyle
c(-\overline{y})+c(x^{\prime}-y^{\prime})-\Lambda U(-y,-\overline{y})$
$\displaystyle\geq$
$\displaystyle\frac{|\overline{y}|}{|-\overline{y}+x^{\prime}|}c(-\overline{y}+x^{\prime})+\frac{\lambda|x^{\prime}|}{|\overline{y}|}V(-\overline{y},0)-\Lambda
U(-y,-\overline{y})$ $\displaystyle\geq$ $\displaystyle
c(-\overline{y}+x^{\prime})+c(x^{\prime})+\frac{\lambda|x^{\prime}|}{|\overline{y}|}V(-\overline{y},0)+\frac{\lambda|\overline{y}-2x^{\prime}|}{|\overline{y}-x^{\prime}|}V(-\overline{y}+x^{\prime},0)-\Lambda
U(-y,-\overline{y})$ $\displaystyle\geq$ $\displaystyle
c(-y+x^{\prime})+c(-x^{\prime})-\Lambda
U(-y+x^{\prime},-\overline{y}+x^{\prime})+\frac{\lambda|x^{\prime}|}{|\overline{y}|}V(-\overline{y},0)$
$\displaystyle\quad+\frac{\lambda|\overline{y}-2x^{\prime}|}{|\overline{y}-x^{\prime}|}V(-\overline{y}+x^{\prime},0)-\Lambda
U(-y,-\overline{y})$ $\displaystyle\geq$ $\displaystyle
c(-y+x^{\prime})+c(-y^{\prime})-\Lambda
U(-y+x^{\prime},-\overline{y}+x^{\prime})+\frac{\lambda|x^{\prime}|}{|\overline{y}|}V(-\overline{y},0)$
$\displaystyle\quad+\frac{\lambda|\overline{y}-2x^{\prime}|}{|\overline{y}-x^{\prime}|}V(-\overline{y}+x^{\prime},0)-\Lambda
U(-y,-\overline{y})-\Lambda U(-x^{\prime},-y^{\prime})$

In particular, it suffices to show

$\displaystyle
c(x^{\prime}-y^{\prime})+\frac{\lambda|x^{\prime}|}{|\overline{y}|}V(-\overline{y},0)+\frac{\lambda|\overline{y}-2x^{\prime}|}{|\overline{y}-x^{\prime}|}V(-\overline{y}+x^{\prime},0)$
$\displaystyle\geq$ $\displaystyle\Lambda U(-y,-\overline{y})+\Lambda
U(-y+x^{\prime},-\overline{y}+x^{\prime}).$ (3.5)

We note that, if $\rho\geq 8$,
$|\overline{y}-2x^{\prime}|\geq\frac{1}{2}|\overline{y}|$. Then we can
estimate

$\displaystyle\frac{\lambda|x^{\prime}|}{|\overline{y}|}V(-\overline{y},0)+\frac{\lambda|\overline{y}-2x^{\prime}|}{|\overline{y}-x^{\prime}|}V(-\overline{y}+x^{\prime},0)=$
$\displaystyle\lambda(|x^{\prime}||\overline{y}|^{p-1}+|\overline{y}-2x^{\prime}||\overline{y}-x^{\prime}|^{p-1})$
$\displaystyle\gtrsim$ $\displaystyle|\overline{y}|^{p}.$

Further, if $E(4)\leq\varepsilon r^{d+1}$,

$\displaystyle\Lambda U(-y,-\overline{y})+\Lambda
U(-y+x^{\prime},-\overline{y}+x^{\prime})+\Lambda U(-x^{\prime},-y^{\prime})$
$\displaystyle\leq$ $\displaystyle
2\Lambda(2|y|)^{p-1}|y-\overline{y}|+\Lambda\left(|x^{\prime}|+|y^{\prime}|\right)^{p-1}|x^{\prime}-y^{\prime}|$
$\displaystyle\lesssim$
$\displaystyle|y|^{p-1}\alpha|\overline{y}-x^{\prime}|+\frac{ME}{r^{d}}(2r)^{p-1}$
$\displaystyle\lesssim$
$\displaystyle\alpha|\overline{y}|^{p}+\varepsilon|\overline{y}|^{p}.$

Thus choosing $\alpha$, $\varepsilon>0$, sufficiently small, we find (3)
holds, proving our claim.

Step 3. Proving the $L^{\infty}$-bounds: Choose
$r=c(E(4)+D(4))^{\frac{1}{p+d}}$. For sufficiently large choice of $c>0$ and
selecting $c(d)$ directions $n_{i}$, by Step 1 and Step 2 we obtain points

${(x^{\prime},y^{\prime})\in\mathrm{Spt}\,\pi\cap(B_{r}(x+2rn_{i})\times\mathbb{R}^{d})}$

and cones $(C_{x,x_{i}^{\prime}})_{i\leq c(d)}$ with vertices
$x_{i}^{\prime}+r\rho(x_{i}^{\prime}-x)$, aperture $\alpha$ and axis
$x_{i}^{\prime}-x$ such that for some $c(\alpha)>0$,

${y\not\in\cup C_{y_{i}}}\text{ and }\mathbb{R}^{d}\setminus
B_{(\rho+c(\alpha))r}(x)\subset\cup C_{y_{i}}.$

In particular, we have

$\displaystyle|y-x|\leq(\rho+c(\alpha))r\lesssim(E+D)^{\frac{1}{p+d}},$

that is (3.1).

(3.2) is a direct consequence of (3.1), concluding the proof.

We record two consequences of Lemma 3.1 we will use later.

######  Corollary 3.3.

Under the assumptions of Lemma 3.1, it holds that

$\displaystyle\int_{2}^{3}\int_{\Omega\cap\\{\exists t\in[0,1]\colon
X(t)\in\partial B_{R})\\}}c(x-y)\,{\rm d}\pi\,{\rm
d}R\lesssim(E(4)+D(4))^{1+\frac{1}{p+d}},$
$\displaystyle\int_{2}^{3}\int_{\Omega}I(\\{\exists t\in[0,1]\colon
X(t)\in\partial B_{R})\\})\,{\rm d}\pi\,{\rm
d}R\lesssim(E(4)+D(4))^{\frac{1}{p+d}}.$

[Proof]We use Lemma 3.1 to deduce there is $C>0$ such that

$\displaystyle\int_{2}^{3}\int_{\Omega\cap\\{\exists t\in[0,1]\colon
X(t)\in\partial B_{R})}c(x-y)\,{\rm d}\pi\,{\rm d}R$ $\displaystyle\leq$
$\displaystyle\int_{2}^{3}\int_{(B_{7/2}\setminus
B_{3/2})\times(B_{7/2}-B_{3/2})}I(\\{\lvert\lvert x\rvert-R\rvert\leq
C(E(4)+D(4))^{\frac{1}{p+d}}\\})c(x-y)\,{\rm d}\pi\,{\rm d}R$
$\displaystyle\lesssim$
$\displaystyle(E(4)+D(4))^{\frac{1}{p+d}}\int_{\\#_{4}}c(x-y)\,{\rm d}\pi$
$\displaystyle\lesssim$ $\displaystyle(E(4)+D(4))^{1+\frac{1}{p+d}}.$

Further, again using Lemma 3.1, there is $C>0$ such that,

$\displaystyle\int_{2}^{3}\int_{\Omega}I(\\{\exists t\in[0,1]\colon
X(t)\in\partial B_{R}\\})\,{\rm d}\pi\,{\rm d}R$ $\displaystyle\leq$
$\displaystyle\int_{2}^{3}\pi(\Omega\cap\\{\lvert X(0)-R\rvert\leq
C(E(4)+D(4))^{\frac{1}{p+d}}\\})\,{\rm d}R$ $\displaystyle=$
$\displaystyle\int_{2}^{3}\mu(\\{\lvert\lvert x\rvert-R\rvert\leq
C(E(4)+D(4))^{\frac{1}{p+d}})\,{\rm d}R$ $\displaystyle\leq$
$\displaystyle\int_{B_{7/2}\setminus B_{3/2}\times B_{7/2}\setminus
B_{3/2}}\int I(\lvert\lvert x\rvert-R\rvert\leq
C(E(4)+D(4))^{\frac{1}{p+d}})\,{\rm d}R\,{\rm d}\mu$ $\displaystyle\lesssim$
$\displaystyle(E(4)+D(4))^{\frac{1}{p+d}}\mu(B_{4})$ $\displaystyle\lesssim$
$\displaystyle(E(4)+D(4))^{\frac{1}{p+d}}.$

## 4 A localisation result

In order to prove Theorem 1.1, we need to use optimality in a localised way,
as the quantity we need to estimate is a local quantity. In general, given a
minimiser $\pi$ of optimal transport with cost function $c$ between two
measures $\lambda$ and $\mu$, it is not true that the localised transport cost
of $\pi$ is approximately equal to the optimal transport cost between the
localised measures $\lambda\llcorner B_{R}$ and $\mu\llcorner B_{R}$. In other
words, it is in general not the case that

$\displaystyle\int_{\Omega}c(x-y)\,{\rm d}\pi\approx W_{c}(\lambda\llcorner
B_{R},\mu\llcorner B_{R}).$

However, if we take into account the entry points of trajectories entering
$B_{R}$ (which we denoted $f_{R}$, c.f. (2.2)) and the exit points of
trajectories exiting $B_{R}$ (which we denoted $g_{R}$, c.f. (2.2)), the
values are close as we show in the next lemma.

######  Lemma 4.1.

Let $\lambda,\mu$ be admissible measures. Suppose $\pi\in\Pi(\lambda,\mu)$
minimises (1.1). Let $R\in[2,3]$ and define $f_{R},g_{R}$ as in (2.2). Then
for any $\tau,\delta>0$, there is $\varepsilon>0$ such that if
$E(4)+D(4)\leq\varepsilon$, then

$\displaystyle\int_{\Omega}c(x-y)\,{\rm
d}\pi\leq(1+\delta)W_{c}(\lambda\llcorner B_{R}+f_{R},\mu\llcorner
B_{R}+g_{R})+\tau\left(E(4)+D(4)\right)$

[Proof]Introduce the weakly continuous family of probability measures
$\\{\lambda_{z}\\}_{z\in\partial B_{R}}$ such that

$\displaystyle\int_{\Omega\cap\\{X(\sigma)\in\partial
B_{R}\\}}\zeta(x,X(\sigma))\pi(\,{\rm d}x\,{\rm d}y)=\int_{\partial
B_{R}}\int\zeta(x,z)\lambda_{z}(\,{\rm d}x)f_{R}(\,{\rm d}z)$

for any test function $\zeta$ on $\mathbb{R}^{d}\times\mathbb{R}^{d}$.
Likewise, introduce $\\{\mu_{w}\\}_{w\in\partial B_{R}}$ via

$\displaystyle\int_{\Omega\cap\\{X(\tau)\in\partial
B_{R}\\}}\zeta(X(\tau),y)\pi(\,{\rm d}x\,{\rm d}y)=\int_{\partial
B_{R}}\int\zeta(w,y)\mu_{w}(\,{\rm d}y)g_{R}(\,{\rm d}w).$

Let $\overline{\pi}$ be an optimal plan for $W_{c}(\lambda\llcorner
B_{R}+f_{R},\mu\llcorner B_{R}+g_{R})$. Define a competitor $\tilde{\pi}$ for
$\pi$ by requiring the following formula to hold for any test function $\zeta$
on $\mathbb{R}^{d}\times\mathbb{R}^{d}$,

$\displaystyle\int\zeta(x,y)\tilde{\pi}(\,{\rm d}x\,{\rm d}y)$
$\displaystyle=$ $\displaystyle\int_{\Omega^{c}}\zeta(x,y)\,{\rm
d}\pi(x,y)+\int_{B_{R}\times B_{R}}\zeta(x,y)\overline{\pi}(\,{\rm d}x\,{\rm
d}y)+\int_{\partial B_{R}\times B_{R}}\int\zeta(x,y)\lambda_{z}(\,{\rm
d}x)\overline{\pi}(\,{\rm d}z\,{\rm d}y)$
$\displaystyle+\int_{B_{R}\times\partial B_{R}}\zeta(x,y)\mu_{w}(\,{\rm
d}y)\overline{\pi}(\,{\rm d}x\,{\rm d}w)+\int_{\partial B_{R}\times\partial
B_{R}}\int\int\zeta(x,y)\mu_{w}(\,{\rm d}y)\lambda_{z}(\,{\rm
d}x)\overline{\pi}(\,{\rm d}z\,{\rm d}w)$ $\displaystyle=$ $\displaystyle
I+II+III+IV+V.$ (4.1)

In order to see that $\tilde{\pi}\in\Pi(\lambda,\mu)$, by symmetry it suffices
to check that the first marginal is $\lambda$. Hence test (4) against
$\zeta(x)$. We begin by noting that due to the definition of $\mu_{w}$ and
using that $\overline{\pi}$ is supported in $\overline{B}_{R}$,

$\displaystyle
II+IV=\int_{B_{R}\times\mathbb{R}^{d}}\zeta(x)\overline{\pi}(\,{\rm d}x\,{\rm
d}y)=\int_{B_{R}}\zeta(x)\mu(\,{\rm d}x)=\int_{\Omega\cap\\{X(\sigma)\in
B_{R})\\}}\zeta(x)\pi(\,{\rm d}x\,{\rm d}y).$

Similarly, using also the definition of $f_{R}$,

$\displaystyle III+V=$ $\displaystyle\int_{\partial
B_{R}\times\mathbb{R}^{d}}\int\zeta(x)\lambda_{z}(\,{\rm
d}x)\overline{\pi}(\,{\rm d}z\,{\rm d}y)=\int_{\partial
B_{R}}\zeta(z)f_{R}(\,{\rm d}z)$ $\displaystyle=$
$\displaystyle\int_{\Omega\cap\\{X(\sigma)\in\partial
B_{R}\\}}\zeta(x)\pi(\,{\rm d}x\,{\rm d}y).$

In particular, we have shown

$\displaystyle\int\zeta(x)\tilde{\pi}(\,{\rm d}x\,{\rm
d}y)=\int\zeta(x)\pi(\,{\rm d}x\,{\rm d}y)=\int\zeta(x)\lambda(\,{\rm d}x)$

as desired.

Using optimality of $\pi$ in the form

$\displaystyle\int_{\Omega}c(x-y)\,{\rm d}\pi+\int_{\Omega^{c}}c(x-y)\,{\rm
d}\pi\leq\int c(x-y)\,{\rm d}\tilde{\pi}$

and testing (4) against $\zeta(x,y)=c(x-y)$, we learn

$\displaystyle\int_{\Omega}c(x-y)\,{\rm d}\pi$ $\displaystyle\leq$
$\displaystyle\int_{B_{R}\times B_{R}}c(x-y)\overline{\pi}(\,{\rm d}x\,{\rm
d}y)+\int_{\partial B_{R}\times B_{R}}\int c(x-y)\lambda_{z}(\,{\rm
d}x)\overline{\pi}(\,{\rm d}z\,{\rm d}y)$
$\displaystyle\quad+\int_{B_{R}\times\partial B_{R}}c(x-y)\mu_{w}(\,{\rm
d}y)\overline{\pi}(\,{\rm d}x\,{\rm d}w)$ $\displaystyle\qquad+\int_{\partial
B_{R}\times\partial B_{R}}\int\int c(x-y)\mu_{w}(\,{\rm d}y)\lambda_{z}(\,{\rm
d}x)\overline{\pi}(\,{\rm d}z\,{\rm d}w)$ $\displaystyle=$
$\displaystyle\int_{B_{R}\times B_{R}}f_{1}\overline{\pi}(\,{\rm d}x\,{\rm
d}y)+\int_{\partial B_{R}\times B_{R}}f_{2}\overline{\pi}(\,{\rm d}z\,{\rm
d}y)+\int_{B_{R}\times\partial B_{R}}f_{3}\overline{\pi}(\,{\rm d}x\,{\rm
d}w)$ $\displaystyle\quad+\int_{\partial B_{R}\times\partial
B_{R}}f_{4}\overline{\pi}(\,{\rm d}z\,{\rm d}w)$

As in the proof of Lemma 2.1, for any $\delta>0$, there is $C_{\delta}>0$ such
that for any $x,y,z$,

$\displaystyle c(x-z)\leq(1+\delta)c(x-y)+C_{\delta}c(y-z).$

Using this in combination with the fact that $\lambda_{z}$, $\mu_{w}$ are
probability measures we deduce

$\displaystyle f_{2}\leq$
$\displaystyle(1+\delta)c(z-y)+C(\delta)\tilde{f}_{2},\quad
f_{3}\leq(1+\delta)c(x-w)+C(\delta)\tilde{f}_{3}$ $\displaystyle f_{4}\leq$
$\displaystyle(1+\delta)c(z-w)+C(\delta)\tilde{f}_{4},$

where

$\displaystyle\tilde{f}_{2}(z,y)=\int c(x-z)\lambda_{z}(\,{\rm
d}x),\quad\tilde{f}_{3}(x,w)=\int c(w-y)\mu_{w}(\,{\rm d}y),$
$\displaystyle\tilde{f}_{4}(z,w)=\tilde{f}_{2}(z,y)+f_{3}(x,w).$

In particular, we deduce

$\displaystyle\int_{\Omega}c(x-y)\,{\rm d}\pi$ $\displaystyle\leq$
$\displaystyle(1+\delta)\int_{\overline{B}_{R}\times\overline{B}_{R}}c(x-y)\overline{\pi}(\,{\rm
d}x\,{\rm d}y)+C(\delta)\int_{\partial B_{R}\times
B_{R}}\tilde{f}_{2}\overline{\pi}(\,{\rm d}z\,{\rm d}y)$
$\displaystyle\quad+C(\delta)\int_{B_{R}\times\partial
B_{R}}\tilde{f}_{3}\overline{\pi}(\,{\rm d}x\,{\rm
d}w)+C(\delta)\int_{\partial B_{R}\times\partial
B_{R}}\tilde{f}_{4}\overline{\pi}(\,{\rm d}z\,{\rm d}w)$ $\displaystyle=$
$\displaystyle(1+\delta)\int_{\overline{B}_{R}\times\overline{B}_{R}}c(x-y)\overline{\pi}(\,{\rm
d}x\,{\rm d}y)+2C(\delta)\int_{\partial B_{R}\times\mathbb{R}^{d}}\int
c(x-z)\lambda_{z}(\,{\rm d}x)\overline{\pi}(\,{\rm d}z\,{\rm d}y)$
$\displaystyle\quad+2C(\delta)\int_{\mathbb{R}^{d}\times\partial
B_{R}}c(w-y)\mu_{y}(\,{\rm d}y)\overline{\pi}(\,{\rm d}x\,{\rm d}w)$
$\displaystyle=$
$\displaystyle(1+\delta)\int_{\overline{B}_{R}\times\overline{B}_{R}}c(x-y)\overline{\pi}(\,{\rm
d}x\,{\rm d}y)+2C(\delta)\int_{\partial B_{R}\times\mathbb{R}^{d}}\int
c(x-z)\lambda_{z}(\,{\rm d}x)f_{R}(\,{\rm d}z)$
$\displaystyle\quad+2C(\delta)\int_{\mathbb{R}^{d}\times\partial
B_{R}}c(w-y)\mu_{y}(\,{\rm d}y)g_{R}(\,{\rm d}w)$ $\displaystyle=$
$\displaystyle(1+\delta)\int_{\overline{B}_{R}\times\overline{B}_{R}}c(x-y)\overline{\pi}(\,{\rm
d}x\,{\rm d}y)+2C(\delta)\int_{\Omega\cap\\{X(\sigma)\in\partial
B_{R}\\}}c(x-X(\sigma))\,{\rm d}\pi$
$\displaystyle\quad+2C(\delta)\int_{\Omega\cap\\{X(\tau)\in\partial
B_{R}\\}}c(X(\tau)-y)\,{\rm d}\pi$

In order to obtain the second to last line we used the admissibility of
$\overline{\pi}$. Now note on the one hand, that due to optimality of
$\overline{\pi}$,

$\displaystyle\int_{\overline{B}_{R}\times\overline{B}_{R}}c(x-y)\,{\rm
d}\overline{\pi}=W_{c}(\lambda\llcorner B_{R}+f_{R},\mu\llcorner
B_{R}+g_{R}).$

On the other hand, on $\Omega\cap\\{X(\sigma)\in\partial
B_{R})\\}\cap\\{X(\tau)\in\partial B_{R}\\}$, for some $\rho_{1},\rho_{2}\geq
0$ with $\rho_{1}+\rho_{2}\leq 1$, due to convexity of $c$ and $c(0)=0$,

$\displaystyle
c(x-X(\sigma))+c(X(\tau)-y)=c(\rho_{1}(x-y))+c(\rho_{2}(x-y))\leq c(x-y).$

Thus, we have shown

$\displaystyle\int_{\Omega}c(x-y)\,{\rm d}\pi$ $\displaystyle\leq$
$\displaystyle(1+\delta)W_{c}(\lambda\llcorner B_{R}+f_{R},\mu\llcorner
B_{R}+g_{R})+C(\delta)\int_{\Omega\cap(\\{X(\sigma\in\partial
B_{R}\\}\cup\\{X(\tau)\in\partial B_{R}\\})}c(x-y)\,{\rm d}\pi$
$\displaystyle=$ $\displaystyle(1+\delta)W_{c}(\lambda\llcorner
B_{R}+f_{R},\mu\llcorner B_{R}+g_{R})+C(\delta)\int_{\Omega\cap\\{\exists
t\in[0,1]\colon X(t)\in\partial B_{R}\\}}c(x-y)\,{\rm d}\pi$
$\displaystyle\leq$ $\displaystyle(1+\delta)W_{c}(\lambda\llcorner
B_{R}+f_{R},\mu\llcorner B_{R}+g_{R})+C(\delta)(E(4)+D(4))^{1+\frac{1}{p+d}}.$

To obtain the last line, we used Corollary 3.3. Choosing $\varepsilon$
sufficiently small the result follows.

## 5 Approximating the boundary data

Before we can implement the $c^{\ast}$-harmonic approximation, we face another
problem. Lemma 4.1 suggests that the $c^{\ast}$-harmonic function $\phi$ we
should use in Theorem 1.1 is given as a solution of the following Neumann-
problem

$\displaystyle\begin{cases}-\mathrm{div}\,\nabla
c^{\ast}(\mathrm{D}\phi)=\lambda-\mu\quad&\text{ in }B_{R}\\\ \nabla
c^{\ast}(\mathrm{D}\phi)\cdot\nu=g_{R}-f_{R}&\text{ on }\partial
B_{R}.\end{cases}$

However, $f_{R}$, $g_{R}$ as well as $\lambda,\mu$ are not sufficiently smooth
for $\phi$ to make sense as a weak solution and we will not be able to apply
the regularity results of Lemma 1.2 as it stands. Hence, we will approximate
$f_{R}$, $g_{R}$ by suitable $L^{p}(\partial B_{R})$-functions $\bar{f}_{R}$
and $\bar{g}_{R}$ and will replace $\lambda-\mu$ with $c=\int_{\partial
B_{R}}g_{R}-f_{R}$. After choosing a suitable radius $R\in[2,3]$, this
approximation is given by the following result:

######  Lemma 5.1.

Let $\tau>0$. Let $\lambda,\mu$ be admissible measures on $\mathbb{R}^{d}$.
There is $\varepsilon>0$ such that if $E(4)+D(4)\leq\varepsilon$, then for
every $R\in[2,3]$ there exist non-negative functions $\overline{f}_{R}$,
$\overline{g}_{R}$ such that

$\displaystyle
W_{c}(f_{R},\overline{f}_{R})+W_{c}(g_{R},\overline{g}_{R})\lesssim\tau
E(4)+D(4)$ $\displaystyle\int_{2}^{3}\int_{\partial
B_{R}}\overline{g}_{R}^{p}+\overline{f}_{R}^{p}\,{\rm d}R\lesssim E(4)+D(4)$

Here $f_{R},g_{R}$ are the functions defined in (2.2).

[Proof]By symmetry it suffices to focus on the terms involving $g$.

We begin by constructing $\overline{g}_{R}$. Let $\overline{\pi}$ be optimal
for $W_{c}(\mu\llcorner B_{4},\kappa_{\mu,4}\,{\rm d}z\llcorner B_{4})$.
Extend $\overline{\pi}$, which is supported on $B_{4}\times B_{4}$ by
$\frac{\mu\otimes\mu}{\mu(\mathbb{R}^{d})}$ to
$\mathbb{R}^{d}\times\mathbb{R}^{d}$. Note that this extension, still denoted
$\overline{\pi}$ has marginals $\mu$ and $\kappa_{\mu,4}\,{\rm d}z\llcorner
B_{4}+\mu\llcorner B_{4}^{c}$. Further due to the definition of $D$,
$W_{c}(\mu\llcorner B_{4},\kappa_{\mu,4}\,{\rm d}z\llcorner B_{4})\lesssim
D(4)$. Introduce the family of probability measures
$\\{\overline{\pi}(\cdot|y)\\}_{y\in\mathbb{R}^{d}}$ via asking that for every
test function $\zeta$

$\displaystyle\int\zeta(y,z)\overline{\pi}(\,{\rm d}z|y)\mu(\,{\rm
d}y)=\int\zeta(y,z)\overline{\pi}(\,{\rm d}y\,{\rm d}z).$

Let $\pi$ be the minimiser of $W_{c}(\lambda,\mu)$. Then define $\tilde{\pi}$
on $\mathbb{R}^{d}\times\mathbb{R}^{d}\times\mathbb{R}^{d}$ by the formula

$\displaystyle\int\zeta(x,y,z)\tilde{\pi}(\,{\rm d}x\,{\rm d}y\,{\rm
d}z)=\int\int\zeta(x,y,z)\overline{\pi}(\,{\rm d}z|y)\pi(\,{\rm d}x\,{\rm
d}y)$

valid for any test function $\zeta$. We note that with respect to the $(x,y)$
variables $\tilde{\pi}$ has marginal $\pi$, while with respect to the $(y,z)$
variables $\tilde{\pi}$ has marginal $\overline{\pi}$.

Fix $R\in[2,3]$. Extend a trajectory $X\in\Omega$ in a piecewise affine
fashion by setting for $t\in[1,2]$,

$\displaystyle X(t)=(t-1)z+(2-t)y.$

Note that the distribution $g^{\prime}$ of the endpoint of those trajectories
that exit $\overline{B}_{R}$ during the time interval $[0,1]$ is given by

$\displaystyle\int\zeta\,{\rm
d}g^{\prime}=\int_{\Omega\cap\\{X(\tau)\in\partial
B_{R}\\}}\zeta(z)\tilde{\pi}(\,{\rm d}x\,{\rm d}y\,{\rm d}z).$ (5.1)

Note that due to Lemma 3.1, $y=X(1)\in B_{4}$ for any trajectory $X$ that
contributes to (5.1). Since $\overline{\pi}(B_{4},B_{4}^{c})=0$, we deduce
that also $z=X(2)\in B_{4}$ and hence that $g^{\prime}$ is supported in
$B_{4}$. In particular, we may estimate for any $\zeta\geq 0$, using that the
second marginal of $\overline{\pi}$ is $\kappa_{\mu,4}\,{\rm d}z\llcorner
B_{4}+\mu\llcorner B_{4}^{c}$,

$\displaystyle\int\zeta\,{\rm d}g^{\prime}\leq\int_{\\{z\in
B_{4}\\}}\zeta(z)\overline{\pi}(\,{\rm d}y\,{\rm
d}z)=\kappa_{\mu,4}\int_{B_{4}}\zeta.$

This shows that $g^{\prime}$ has a density, still denoted $g^{\prime}$,
satisfying $g^{\prime}\leq\kappa_{\mu,4}$ and allows us to conclude the
construction of $\overline{g}_{R}$ by defining

$\displaystyle\int\zeta\,{\rm
d}\overline{g}_{R}=\int\zeta\left(R\frac{z}{\lvert
z\rvert}\right)g^{\prime}(\,{\rm d}z).$

We now turn to establishing the claimed estimates for $\overline{g}_{R}$. Note
that, directly from the definitions of $\tilde{\pi}$, $g^{\prime}$ and
$\overline{g}_{R}$, an admissible plan for $W_{c}(g_{R},\overline{g}_{R})$ is

$\displaystyle\int_{\Omega\cap\\{X(\tau)\in\partial
B_{R}\\}}\zeta\left(X(\tau),R\frac{z}{\lvert z\rvert}\right)\,{\rm
d}\tilde{\pi}.$

Indeed, due to the definition of $\tilde{\pi}$ and the definition of $g_{R}$
in (2.2), for any test function $\zeta$,

$\displaystyle\int_{\Omega\cap\\{X(\tau)\in\partial
B_{R}\\}}\zeta(X(\tau))\,{\rm
d}\tilde{\pi}=\int_{\Omega\cap\\{X(\tau)\in\partial
B_{R}\\}}\zeta(X(\tau))\,{\rm d}\pi=\int\xi\,{\rm d}g_{R}.$

On the other hand, using (5.1) and the definition of $\overline{g}_{R}$,

$\displaystyle\int_{\Omega\cap\\{X(\tau)\in\partial
B_{R}\\}}\zeta\left(R\frac{z}{\lvert z\rvert}\right)\,{\rm
d}\tilde{\pi}=\int\zeta\left(R\frac{z}{\lvert z\rvert}\right)\,{\rm
d}g^{\prime}=\int\zeta\,{\rm d}\overline{g}_{R}.$

In particular, using the $p$-growth of $c$ (1.12),

$\displaystyle
W_{c}(g_{R},\overline{g}_{R})\lesssim\int_{\Omega\cap\\{X(\tau)\in\partial
B_{R}\\}}c\left(X(\tau)-R\frac{z}{\lvert z\rvert}\right)\,{\rm
d}\tilde{\pi}\lesssim\int_{\Omega\cap\\{X(\tau)\in\partial
B_{R}\\}}\left\lvert X(\tau)-R\frac{z}{\lvert z\rvert}\right\rvert^{p}\,{\rm
d}\tilde{\pi}.$

Noting that $\lvert X(\tau)-R\frac{z}{\lvert z\rvert}\rvert\leq 2\lvert
X(\tau)-z\rvert$, we deduce

$\displaystyle\left\lvert X(\tau)-R\frac{z}{\lvert
z\rvert}\right\rvert^{p}\lesssim\lvert x-y\rvert^{p}+\lvert y-z\rvert^{p}.$

Thus, we deduce using once again the $p$-growth of $c$ (1.12) and Corollary
3.3,

$\displaystyle W_{c}(g_{R},\overline{g}_{R})\lesssim\int_{\Omega\cap\\{\exists
t\in[0,1]\colon X(t)\in\partial B_{R}\\}}c(x-y)\,{\rm d}\pi+D(4)\lesssim
E(4)^{1+\frac{1}{p+d}}+D(4).$

Choosing $\varepsilon$ sufficiently small, the first estimate holds.

Noting $\sup g^{\prime}\leq\kappa_{\mu,4}\lesssim 1$, in order to prove the
second inequality, it suffices to prove

$\displaystyle\int_{2}^{3}\int\lvert R-\lvert x\rvert\rvert^{p-1}\,{\rm
d}g^{\prime}\lesssim E(4)+D(4)$

and to apply Lemma 2.3. The condition on the support of $g$ in Lemma 2.3
applies due to Lemma 3.1. Note that by definition of $g^{\prime}$,

$\displaystyle\int\lvert R-\lvert x\rvert\rvert^{p-1}\,{\rm d}g^{\prime}=$
$\displaystyle\int_{\Omega\cap\\{X(\tau)\in\partial B_{R}\\}}\lvert\lvert
z\rvert-R\rvert^{p-1}\tilde{\pi}(\,{\rm d}x\,{\rm d}y\,{\rm d}z)$
$\displaystyle\lesssim$ $\displaystyle\int_{\\{y\in
B_{4}\\}\cap\\{\min_{[0,1]}\lvert X\rvert\leq R\leq\max_{[0,1]}\lvert
X\rvert\\}}\lvert x-y\rvert^{p-1}+\lvert y-z\rvert^{p-1}\tilde{\pi}(\,{\rm
d}x\,{\rm d}y\,{\rm d}z).$

In order to obtain the second line, we observed that since $\lvert
X(\tau)\rvert=R$, it holds that ${\lvert\lvert z\rvert-R\rvert\leq\lvert
x-y\rvert+\lvert y-z\rvert}$. In addition we noted that, since
$X(\tau)\in\partial B_{R}$, we have ${\min_{[0,1]}X\leq R\leq\max_{[0,1]}X}$
and $X(1)\in B_{4}$ due to Lemma 3.1. Integrating over $R$, this gives

$\displaystyle\int_{2}^{3}\int\lvert\lvert z\rvert-R\rvert^{p-1}\,{\rm
d}g^{\prime}\,{\rm d}R$ $\displaystyle\lesssim$ $\displaystyle\int_{\\{y\in
B_{4}\\}}(\max_{[0,1]}X-\min_{[0,1]}X)\left(\lvert x-y\rvert^{p-1}+\lvert
y-z\rvert^{p-1}\right)\tilde{\pi}(\,{\rm d}x\,{\rm d}y\,{\rm d}z)$
$\displaystyle\leq$ $\displaystyle\int_{\\{y\in B_{4}\\}}\lvert
x-y\rvert\left(\lvert x-y\rvert^{p-1}+\lvert
y-z\rvert^{p-1}\right)\tilde{\pi}(\,{\rm d}x\,{\rm d}y\,{\rm d}z)$
$\displaystyle\lesssim$ $\displaystyle\int_{\\{y\in B_{4}\\}}\lvert
x-y\rvert^{p}\pi(\,{\rm d}x\,{\rm d}y)+\int\lvert
y-z\rvert^{p}\overline{\pi}(\,{\rm d}y\,{\rm d}z)$ $\displaystyle\leq$
$\displaystyle E(4)+D(4).$

The second-to last line was obtained applying Young’s inequality. This
concludes the proof.

## 6 Restricting the data

As we see from Lemma 5.1, we will not be able to work on $B_{4}$ directly, but
will have to pass to a smaller ball $B_{R}$ with some suitably chosen
$R\in[2,3]$. Hence, we need to control– for a well-chosen $R$– $D(R)$, while
at the moment we only control $D(4)$. Unfortunately, this does not follow
immediately from the definition but requires a technical proof utilising ideas
of the previous sections. The outcome of these considerations is the following
lemma:

######  Lemma 6.1.

For any non-negative measure $\mu$ there is $\varepsilon>0$ such that if
$D(4)\leq\varepsilon$, then

$\displaystyle\int_{2}^{3}\left(W_{c}(\mu\llcorner B_{R},\kappa_{\mu,R}\,{\rm
d}x\llcorner
B_{R})+\frac{1}{\kappa_{\mu,R}}(\kappa_{\mu,R}-1)^{p}\right)\,{\rm d}R\lesssim
D(4).$

[Proof]Fix $R\in[2,3]$. In this proof $\pi$ will denote the optimal
transference plan for the problem ${W_{c}(\mu\llcorner
B_{4},\kappa_{\mu,4}\,{\rm d}x\llcorner B_{4})}$. Note that due to the
$L^{\infty}$-bound Lemma 3.1, if $X(0)\in B_{R}$, then $X(1)\in B_{4}$ and if
$X(1)\in B_{R}$, then $X(0)\in B_{4}$. Define the measures $0\leq
f^{\prime}\leq\kappa_{\mu,4}$ on $\overline{B}_{R}$ and $0\leq
g^{\prime}\leq\kappa_{\mu,4}$ on $\overline{B}_{4}\setminus B_{R}$, which
record where exiting and entering trajectories end up by asking that for all
test functions $\zeta$,

$\displaystyle\int\zeta\,{\rm
d}f^{\prime}\coloneqq\int_{\Omega\cap\\{X(0)\not\in B_{R}\\}\cap\\{X(1)\in
B_{R}\\}}\zeta(X(1))\,{\rm d}\pi$ $\displaystyle\int\zeta\,{\rm
d}g^{\prime}\coloneqq\int_{\Omega\cap\\{X(0)\in B_{R}\\}\cap\\{X(1)\not\in
B_{R}\\}}\zeta(X(1))\,{\rm d}\pi.$

Introduce the mass densities

$\displaystyle\kappa_{f}=\frac{f^{\prime}(\mathbb{R}^{d})}{\lvert
B_{R}\rvert}\leq\kappa_{\mu,R},\qquad\kappa_{g}=\frac{g^{\prime}(\mathbb{R}^{d})}{\lvert
B_{R}\rvert}.$

We use Lemma 2.1 to deduce

$\displaystyle W_{c}(\mu\llcorner B_{R},\kappa_{\mu,R}\,{\rm d}x\llcorner
B_{R})$ $\displaystyle\lesssim$ $\displaystyle W_{c}(\mu\llcorner
B_{R},\kappa_{\mu,4}\,{\rm d}x\llcorner
B_{R}-f^{\prime}+g^{\prime})+W_{c}(\kappa_{\mu,4}\,{\rm d}x\llcorner
B_{R}-f^{\prime}+g^{\prime},(\kappa_{\mu,4}-\kappa_{f})\,{\rm d}x\llcorner
B_{R}+g^{\prime})$ $\displaystyle\quad+W_{c}((\kappa_{\mu,4}-\kappa_{f})\,{\rm
d}x\llcorner B_{R}+g^{\prime},\kappa_{\mu,R}\,{\rm d}x\llcorner B_{R})$
$\displaystyle=$ $\displaystyle I+II+III.$

Restricting $\pi$ to trajectories that start in $B_{R}$ gives an admissible
plan for $I$. Consequently,

$\displaystyle I=W_{c}(\mu\llcorner B_{R},\kappa_{\mu,4}\,{\rm d}x\llcorner
B_{R}-f^{\prime}+g^{\prime})\leq W_{c}(\mu\llcorner B_{4},\kappa_{\mu,4}\,{\rm
d}x\llcorner B_{4})\leq D(4).$

Since $II$ will be estimated in the same way as $III$, but is slightly more
tricky, we first estimate $III$. In order to estimate $III$, introduce the
projection $\hat{g}$ of $g^{\prime}$ onto $\partial B_{R}$ via (2.1). Using
Lemma 2.1, we deduce

$\displaystyle III\lesssim$ $\displaystyle
W_{c}((\kappa_{\mu,4}-\kappa_{f})\,{\rm d}x\llcorner
B_{R}+\hat{g},\kappa_{\mu,R}\,{\rm d}x\llcorner
B_{R})+W_{c}(\hat{g},g^{\prime}).$ (6.1)

Regarding the first term, we claim that

$\displaystyle W_{c}((\kappa_{\mu,4}-\kappa_{f})\,{\rm d}x\llcorner
B_{R}+\hat{g},\kappa_{\mu,R}\,{\rm d}x\llcorner
B_{R})\lesssim\int(\hat{g})^{p}.$

Indeed, an admissible density-flux pair $(\rho,j)$ for the Benamou-Brenier
formulation (1.10) is given by

$\displaystyle\begin{cases}\rho_{t}=(\kappa_{\mu,4}\,{\rm d}x\llcorner
B_{4}-\kappa_{f}+t\kappa_{g})+(1-t)\hat{g}g\\\ j_{t}=\nabla
c^{\ast}(\mathrm{D}\phi)\,{\rm d}x\llcorner B_{R},\end{cases}$

where $\phi$ solves

$\displaystyle\begin{cases}-\mathrm{div}\,\nabla
c^{\ast}(\mathrm{D}\phi)=\kappa_{g}\quad&\text{ in }B_{R}\\\ \nu\cdot\nabla
c^{\ast}(\mathrm{D}\phi)=\hat{g}&\text{ on }\partial B_{R}.\end{cases}$

We find, writing $s=(\kappa_{\mu,4}-\kappa_{f}+t\kappa_{g})$, for any $\zeta$
supported in $B_{R}$,

$\displaystyle\int\zeta\,{\rm d}j_{t}-\int c^{\ast}(\zeta)\,{\rm d}\rho_{t}=$
$\displaystyle\int_{B_{R}}\zeta\cdot\nabla
c^{\ast}(\mathrm{D}\phi)-c^{\ast}(\zeta)s\,{\rm d}x$ $\displaystyle\leq$
$\displaystyle\int_{B_{R}}\int_{0}^{1}sc\left(\frac{1}{s}\nabla
c^{\ast}(\mathrm{D}\phi)\right)\,{\rm d}t\,{\rm d}x$ (6.2)

To obtain the second line, we used the Fenchel-Young inequality. Assuming
$\lvert s-1\rvert\ll 1$ for now, using the $p$-growth of $c$ (1.12) it is
straightforward to see that

$\displaystyle\int_{B_{R}}\int_{0}^{1}sc\left(\frac{1}{s}\nabla
c^{\ast}(\mathrm{D}\phi)\right)\,{\rm d}t\,{\rm
d}x\lesssim\int_{B_{R}}c\left(\nabla c^{\ast}(\mathrm{D}\phi)\right)\,{\rm
d}x\lesssim\int_{\partial B_{R}}(\hat{g})^{p}.$

To obtain the last inequality, we used the energy inequality (1.31). We now
estimate $\int(\hat{g})^{p}$ and $W_{c}(\hat{g},g^{\prime})$ arguing exactly
as in Lemma 6.1 in order to conclude

$\displaystyle\int\hat{g}^{p}+W_{c}(\hat{g},g^{\prime})\lesssim D(4).$

Since $\kappa_{\mu,R}=\kappa_{4,R}-\kappa_{f}+\kappa_{g}$ and
$\lvert\kappa_{4,R}-1\rvert\lesssim D(4)$, to deduce that $\lvert s-1\rvert\ll
1$ and to conclude the estimate of $III$, it suffices to show
$\kappa_{f}^{p}+\kappa_{g}^{p}\lesssim D(4)$. By symmetry it suffices to
consider $\kappa_{g}$. Since $\hat{g}$ is supported on $\partial B_{R}$, using
Young’s inequality, we find

$\displaystyle\kappa_{g}^{p}=\frac{\hat{g}(\mathbb{R}^{d})^{p}}{\lvert
B_{R}\rvert^{p}}\leq\frac{\lvert\partial B_{R}\rvert^{p-1}}{\lvert
B_{R}\rvert^{p}}\int_{\partial B_{R}}(\hat{g})^{p}\lesssim D(4).$ (6.3)

This concludes the estimate for $III$.

It remains to estimate $II$. Using the subadditivity of $W_{c}$, we have

$\displaystyle W_{c}(\kappa_{\mu,4}\,{\rm d}x\llcorner
B_{R}-f^{\prime}+g^{\prime},(\kappa_{\mu,4}-\kappa_{f})\,{\rm d}x\llcorner
B_{R}+g^{\prime})$ $\displaystyle\leq$ $\displaystyle
W_{c}(\kappa_{\mu,4}\,{\rm d}x\llcorner
B_{R}-f^{\prime},(\kappa_{\mu,4}-\kappa_{f})\,{\rm d}x\llcorner
B_{R})+W_{c}(g^{\prime},g^{\prime})$ $\displaystyle=$ $\displaystyle
W_{c}(\kappa_{\mu,4}\,{\rm d}x\llcorner
B_{R}-f^{\prime},(\kappa_{\mu,4}-\kappa_{f})\,{\rm d}x\llcorner B_{R}).$

We want to proceed exactly as we did in the estimate for $III$, the only
delicate issue being that we do not have $\kappa_{\mu,4}\,{\rm d}x\llcorner
B_{R}-f^{\prime}\geq c>0$. This is necessary in (6). However, this can be
remedied by using Corollary 2.2 to deduce

$\displaystyle W_{c}(\kappa_{\mu,4}\,{\rm d}x\llcorner
B_{R}-f^{\prime},(\kappa_{\mu,4}-\kappa_{f})\,{\rm d}x\llcorner B_{R})$
$\displaystyle\lesssim$ $\displaystyle
W_{c}((2\kappa_{\mu,4}-\kappa_{f})\,{\rm d}x\llcorner
B_{R}-f^{\prime},2(\kappa_{\mu,4}-\kappa_{f})\,{\rm d}x\llcorner B_{R}).$

Note that since $\kappa_{\mu,4}\,{\rm d}x\llcorner B_{R}-f^{\prime}\geq 0$ and
using (6.3), after choosing $\varepsilon$ sufficiently small,

$\displaystyle(2\kappa_{\mu,4}-\kappa_{f})\,{\rm d}x\llcorner
B_{R}-f^{\prime}\geq\kappa_{\mu,4}-\kappa_{f}\geq\frac{1}{2}\kappa_{\mu,4}\geq
c>0.$

In order to obtain the last inequality, we used the definition of $D$ and
chose $\varepsilon$ sufficiently small. This allows to proceed using the same
argument as for $III$ to conclude

$\displaystyle W_{c}(\kappa_{\mu,4}\,{\rm d}x\llcorner
B_{R}-f^{\prime},(\kappa_{\mu,4}-\kappa_{f})\,{\rm d}x\llcorner B_{R})\lesssim
D(4).$

This completes the proof.

## 7 The $c^{\ast}$-harmonic approximation result

The goal of this section is to prove the $c^{\ast}$-harmonic approximation
result. With the results of the previous sections in hand, we can give a more
precise version of Theorem 1.1. In particular, we can make explicit the
problem that $\phi$ solves.

To this end, in light of Lemma 5.1 and Lemma 6.1, given $\tau>0$, we fix
$R\in(2,3)$ such that there exist non-negative $\overline{f}_{R}$,
$\overline{g}_{R}$ such that

$\displaystyle
W_{c}(f_{R},\overline{f}_{R})+W_{c}(g_{R},\overline{g}_{R})\lesssim\tau
E(4)+D(4)$ $\displaystyle D(R)+\int_{\partial
B_{R}}\overline{f}_{R}^{p}+\overline{g}_{R}^{p}\lesssim E(4)+D(4).$

Then let $\phi$ be a solution with $\int_{B_{R}}\phi\,{\rm d}x=0$ of

$\displaystyle\begin{cases}-\mathrm{div}\,\nabla
c^{\ast}(\mathrm{D}\phi)=c_{R}\quad&\text{ in }B_{R}\\\ \nabla
c^{\ast}(\mathrm{D}\phi)\cdot\nu=\overline{g}_{R}-\overline{f}_{R}&\text{ on
}\partial B_{R}.\end{cases}$ (7.1)

where $c_{R}=\lvert B_{R}\rvert^{-1}\left(\int_{\partial
B_{R}}\overline{g}_{R}-\overline{f}_{R}\right)$ is the constant so that (7.1)
is well-posed. We emphasize that while we do not make the dependence explicit
in our notation, $\phi$ depends on the choice of radius $R$.

With this notation in place, we state a precise version of our main result.

######  Theorem 7.1.

For every $\tau>0$, there exist positive constants
$\varepsilon(\tau),C(\tau)>0$ such that if $E(4)+D(4)\leq\varepsilon(\tau)$,
then there exists $R\in(2,3)$ such that

$\displaystyle\int_{\\#_{1}}c(x-y-\nabla c^{\ast}(\mathrm{D}\phi))\leq\tau
E(4)+C(\tau)D(4),$

where $\phi$ solves (7.1).

The proof will be a direct consequence of the lemmata we prove in the
following subsections. We begin in Section 7.1 by using strong $p$-convexity
in order to bound a quantity related to the left-hand side of the estimate in
Theorem 7.1 by a difference of energies, as well as two error terms. The first
error term arises from the approximation of the boundary data, while the
second error term comes from passing to the perspective of trajectories. We
construct a competitor to estimate the difference of energies and estimate the
two error terms in Section 7.2. Collecting estimates, we conclude the proof of
Theorem 7.1 in Section 7.3.

###  7.1 Quasi-orthogonality

The key observation in order to prove Theorem 7.1 is contained in the
following elementary lemma, which relies on the strong $p$-convexity of $c$.

######  Lemma 7.2.

For any $\pi\in\Pi(\lambda,\mu)$ and $\phi$ continuously differentiable in
$\overline{B}_{R}$, there is $c(p,\Lambda)$ such that we have

$\displaystyle
c(p,\Lambda)\int_{\Omega}\int_{\sigma}^{\tau}V\left(\dot{X}(t),\nabla
c^{\ast}(\nabla\phi(X(t))\right)\,{\rm d}t\,{\rm d}\pi$ $\displaystyle\leq$
$\displaystyle\int_{\Omega}c(x-y)\,{\rm
d}\pi-\int_{B_{R}}c(\nabla^{\ast}(\mathrm{D}\phi))\,{\rm
d}x-\int_{\Omega}\int_{\sigma}^{\tau}\langle\dot{X}(t)-\nabla
c^{\ast}(\mathrm{D}\phi(X(t)),\mathrm{D}\phi(X(t))\rangle\,{\rm d}t\,{\rm
d}\pi$ $\displaystyle\quad+\int_{B_{R}}c(\nabla
c^{\ast}(\mathrm{D}\phi))\,{\rm d}x-\int_{\Omega}\int_{\sigma}^{\tau}c(\nabla
c^{\ast}(\mathrm{D}\phi(X(t)))\,{\rm d}t\,{\rm d}\pi.$

[Proof]We apply (1.16) with $x=\dot{X}(t)$ and $y=\nabla
c^{\ast}(\mathrm{D}\phi(X(t)))$. Noting that we have ${\nabla c(\nabla
c^{\ast}(\mathrm{D}\phi))=\mathrm{D}\phi}$ and
$\int_{\Omega}\int_{\sigma}^{\tau}c(\dot{X}(t))\,{\rm d}t\,{\rm
d}\pi\leq\int_{\Omega}c(x-y)\,{\rm d}\pi$ this gives the desired result.

###  7.2 Error estimates

We would like to apply Lemma 7.2 with $\phi$ solving (7.1). In order to do so,
we require $\phi\in C^{1}(\overline{B}_{R})$. However note that
$\overline{g}_{R}$ and $\overline{f}_{R}$ will in general not be sufficiently
smooth to ensure that $\phi\in C^{1}(\overline{B}_{R})$. Thus, we approximate
them using mollification. To be precise, let $0<r\ll 1$ and denote by
$\overline{f}_{R}^{r}$ and $\overline{g}_{R}^{r}$, respectively, the
convolution with a smooth convolution kernel (on $\partial B_{R})$ at scale
$r$ of $\overline{f}_{R}$ and $\overline{g}_{R}$. Set $\phi^{r}$ to be the
solution with $\int_{B_{R}}\phi^{r}\,{\rm d}x=0$ of

$\displaystyle\begin{cases}-\mathrm{div}\,\nabla
c^{\ast}(\mathrm{D}\phi^{r})=c^{r}\quad&\text{ in }B_{R}\\\ \nabla
c^{\ast}(\mathrm{D}\phi^{r})\cdot\nu=\overline{g}_{R}^{r}-\overline{f}_{R}^{r}&\text{
on }\partial B_{R}.\end{cases}$ (7.2)

Here $c^{r}=\lvert B_{R}\rvert^{-1}\left(\int_{\partial
B_{R}}\overline{g}_{R}^{r}-\overline{f}_{R}^{r}\right)$ is the constant such
that (7.2) is well-posed.

We begin by showing that replacing $\overline{f}_{R}$ and $\overline{g}_{R}$
with $\overline{f}_{R}^{r}$ and $\overline{g}_{R}^{r},$ respectively is not
detrimental on the left-hand side of the estimate in Lemma 7.2.

######  Lemma 7.3.

For every $0<\tau$ there exists $\varepsilon(\tau)$ and
$C(\tau),r_{0}(\tau)>0$, such that if it holds that
${E(4)+D(4)\leq\varepsilon(\tau)}$ and $0<r\leq r_{0}$, then there exists
$R\in[2,3]$ such that if $\phi$ solves (7.1) and $\phi^{r}$ solves (7.2), then

$\displaystyle\int_{\Omega_{3/2}}\int_{\sigma_{3/2}}^{\tau_{3/2}}V\left(\dot{X}(t),\nabla
c^{\ast}(\mathrm{D}\phi(X(t))\right)\,{\rm d}t\,{\rm
d}\pi-\int_{\Omega_{3/2}}\int_{\sigma_{3/2}}^{\tau_{3/2}}V\left(\dot{X}(t),\nabla
c^{\ast}(\mathrm{D}\phi^{r}(X(t))\right)\,{\rm d}t\,{\rm d}\pi$
$\displaystyle\lesssim$ $\displaystyle\tau(E(4)+D(4)).$

[Proof]Write $\xi(x)=\nabla c^{\ast}(\mathrm{D}\phi(x))$, $\xi^{r}(x)=\nabla
c^{\ast}(\mathrm{D}\phi^{r}(x))$. We focus on the case $p\leq 2$. The case
$p>2$ follows by similar arguments, but is easier. In light of (1.15), the
controlled $p^{\prime}$-growth of $c^{\ast}$ (1.19) and using Hölder, we find

$\displaystyle\left\lvert\int_{\Omega_{3/2}}\int_{\sigma_{3/2}}^{\tau_{3/2}}V\left(\dot{X}(t),\xi(X(t))\right)\,{\rm
d}t\,{\rm
d}\pi-\int_{\Omega}\int_{\sigma}^{\tau}V\left(\dot{X}(t),\xi^{r}(X(t))\right)\,{\rm
d}t\,{\rm d}\pi\right\rvert$ $\displaystyle\lesssim$
$\displaystyle\int_{\Omega_{3/2}}\int_{\sigma_{3/2}}^{\tau_{3/2}}\lvert\xi(X(t))-\xi^{r}(X(t))\rvert\left(\lvert\dot{X}(t)+\lvert\xi(X(t))\rvert+\lvert\xi^{r}(X(t))\rvert\right)^{p-1}\,{\rm
d}t\,{\rm d}\pi$ $\displaystyle\lesssim$
$\displaystyle\int_{\Omega_{3/2}}\int_{\sigma_{3/2}}^{\tau_{3/2}}\lvert\mathrm{D}\phi(X(t))-\mathrm{D}\phi^{r}(X(t))\rvert\left(\lvert\mathrm{D}\phi(X(t))\rvert+\lvert\mathrm{D}\phi^{r}(X(t))\rvert\right)^{p^{\prime}-1}\,{\rm
d}t\,{\rm d}\pi$ $\displaystyle\leq$
$\displaystyle\left(\int_{\Omega_{3/2}}\int_{\sigma_{3/2}}^{\tau_{3/2}}\lvert\mathrm{D}\phi(X(t))-\mathrm{D}\phi^{r}(X(t))\rvert^{p^{\prime}}\,{\rm
d}t\,{\rm d}\pi\right)^{\frac{1}{p^{\prime}}}$
$\displaystyle\quad\times\left(\int_{\Omega_{3/2}}\int_{\sigma_{3/2}}^{\tau_{3/2}}\lvert\mathrm{D}\phi(X(t))\rvert^{p^{\prime}}+\lvert\mathrm{D}\phi^{r}(X(t))\rvert^{p^{\prime}}\,{\rm
d}t\,{\rm d}\pi\right)^{\frac{1}{p}}.$

Note that due to Lemma 3.1, if $X\in\Omega_{3/2}$, then $X(0)\in B_{7/8}$.
Thus using elliptic regularity in the form of (1.32) and (1.34),

$\displaystyle\Big{\lvert}\int_{\Omega_{3/2}}\int_{\sigma_{3/2}}^{\tau_{3/2}}\lvert\mathrm{D}\phi(X(t)-\mathrm{D}\phi^{r}(X(t))\rvert^{p^{\prime}}\,{\rm
d}t\,{\rm
d}\pi-\int_{\Omega_{3/2}}\int_{\sigma_{3/2}}^{\tau_{3/2}}\lvert\mathrm{D}\phi(X(0)-\mathrm{D}\phi^{r}(X(0))\rvert^{p^{\prime}}\,{\rm
d}t\,{\rm d}\pi\Big{\rvert}$ $\displaystyle\lesssim$
$\displaystyle\left([\mathrm{D}\phi]_{C^{0,\beta}(B_{7/8})}+[\mathrm{D}\phi^{r}]_{C^{0,\beta}(B_{7/8})}\right)\left(\sup_{B_{7/8}}\lvert\mathrm{D}\phi\rvert+\lvert\mathrm{D}\phi^{r}\rvert\right)^{p^{\prime}-1}\int_{\Omega_{3/2}}\int_{\sigma_{3/2}}^{\tau_{3/2}}\lvert
X(t)-X(0)\rvert^{\beta}\,{\rm d}t\,{\rm d}\pi$ $\displaystyle\lesssim$
$\displaystyle
c(r)(E(4)+D(4))\pi(\Omega_{3/2})(E(4)+D(4))^{\frac{\beta}{p+d}}\lesssim
c(r)(E(4)+D(4))^{1+\frac{\beta}{p+d}}.$

In order to obtain the last line, we used the $L^{\infty}$-bound Lemma 3.1.
Moreover, using Lemma 3.2 and elliptic regularity in the form of (1.33),

$\displaystyle\int_{\Omega_{3/2}}\int_{\sigma_{3/2}}^{\tau_{3/2}}\lvert\mathrm{D}\phi(X(0))-\mathrm{D}\phi^{r}(X(0))\rvert^{p^{\prime}}\,{\rm
d}t\,{\rm
d}\pi\leq\int_{B_{7/8}}\lvert\mathrm{D}\phi(x)-\mathrm{D}\phi^{r}(x)\rvert^{p^{\prime}}\,{\rm
d}\mu$ $\displaystyle\leq$
$\displaystyle\Big{\lvert}\int_{B_{7/8}}\lvert\mathrm{D}\phi(x)-\mathrm{D}\phi^{r}(x)\rvert^{p^{\prime}}(\,{\rm
d}\mu-\kappa_{\mu,R}\,{\rm
d}x)\Big{\rvert}+\kappa_{\mu,R}\int_{B_{R}}\lvert\mathrm{D}\phi(x)-\mathrm{D}\phi^{r}(x)\rvert^{p^{\prime}}\,{\rm
d}x$ $\displaystyle\lesssim$
$\displaystyle\left(\sup_{B_{7/8}}\lvert\mathrm{D}\phi\rvert^{p^{\prime}-1}+\lvert\mathrm{D}\phi^{r}\rvert^{p^{\prime}-1}\right)\left([\mathrm{D}\phi]_{C^{0,\beta}(B_{(3/2+R)/2})}+[\mathrm{D}^{r}\phi]_{C^{0,\beta}(B_{(3/2+R)/2})}\right)$
$\displaystyle\quad\times W_{c}(\mu\llcorner B_{R},\kappa_{\mu,R}\,{\rm
d}x\llcorner B_{R})^{\frac{\beta}{p}}+r^{s}(E(4)+D(4))$
$\displaystyle\lesssim$ $\displaystyle
c(r)(E(4)+D(4))^{1+\frac{\beta}{p}}+r^{s}(E(4)+D(4)).$

Arguing similarly, that is first replacing $X(t)$ with $X(0)$, at the cost of
making an error of size $c(r)\left(E(4)+D(4)\right)^{1+\frac{\beta}{p+d}}$,
and then replacing $\,{\rm d}\mu$ with $\,{\rm d}x$, making an error
$c(r)(E(4)+D(4))^{1+\frac{\beta}{p}}+r(E(4)+D(4))$, we find

$\displaystyle\int_{\Omega_{3/2}}\int_{\sigma_{3/2}}^{\tau_{3/2}}\lvert\nabla\phi(X(t))\rvert^{p^{\prime}}+\lvert\nabla\phi^{r}(X(t))\rvert^{p^{\prime}}\,{\rm
d}t\,{\rm d}\pi$ $\displaystyle\lesssim$ $\displaystyle
r^{s}(E(4)+D(4))+c(r)(E(4)+D(4))^{1+\frac{\beta}{p+d}}.$

Collecting estimates and choosing $r$ as well as $\varepsilon(\tau)$
sufficiently small, the desired result follows.

We now turn to estimating each of the three terms on the right-hand side of
the estimate in Lemma 7.2 in turn. We will see that the second and third term
are errors that arise from the approximation of the boundary data and from
passing to the perspective of trajectories, respectively. Accordingly,
estimating them will be essentially routine. In contrast, estimating the first
term requires us to contrast an appropriate competitor to $\pi$.

######  Lemma 7.4.

For every $0<\tau$ there exists $\varepsilon(\tau),C(\tau),r_{0}(\tau)>0$ such
that if it holds that ${E(4)+D(4)\leq\varepsilon(\tau)}$ and $0<r\leq r_{0}$,
then there exists $R\in[2,3]$ such that if $\phi^{r}$ solves (7.2), then

$\displaystyle\int_{\Omega}c(x-y)\,{\rm d}\pi-\int_{B_{R}}c(\nabla
c^{\ast}(\mathrm{D}\phi^{r}))\,{\rm d}x\lesssim\tau E(4)+D(4).$

[Proof]We note, in the case $p\leq 2$, using the $p$-growth of $c$ (1.13), the
$p^{\prime}-1$-growth of $\nabla c^{\ast}$ (1.19) and Hölder,

$\displaystyle\int_{B_{R}}c(\nabla c^{\ast}(\mathrm{D}\phi^{r}))-c(\nabla
c^{\ast}(\mathrm{D}\phi))\,{\rm d}x$ $\displaystyle\lesssim$
$\displaystyle\int_{B_{R}}\lvert\nabla c^{\ast}(\mathrm{D}\phi^{r})-\nabla
c^{\ast}(\mathrm{D}\phi)\rvert(\lvert\nabla
c^{\ast}(\mathrm{D}\phi^{r})\rvert+\lvert\nabla
c^{\ast}(\mathrm{D}\phi)\rvert)^{p-1}\,{\rm d}x$ $\displaystyle\lesssim$
$\displaystyle\int_{B_{R}}\lvert\mathrm{D}\phi-\mathrm{D}\phi^{r}\rvert\left(\lvert
D\phi\rvert+\lvert\mathrm{D}\phi^{\prime}\rvert\right)^{p^{\prime}-1}$
$\displaystyle\lesssim$
$\displaystyle\|\mathrm{D}\phi-\mathrm{D}\phi^{r}\|_{L^{p^{\prime}}(B_{R})}\left(\|\mathrm{D}\phi\|_{L^{p^{\prime}}(B_{R})}+\|\mathrm{D}\phi^{r}\|_{L^{p^{\prime}}(B_{R})}\right)^{p^{\prime}-1}$
$\displaystyle\lesssim$ $\displaystyle r^{\frac{s}{p^{\prime}}}(E(4)+D(4)).$

To obtain the last line, we used the elliptic estimates (1.30) and (1.33). In
case $p\geq 2$ a similar estimate holds by the same argument. Due to the
localisation result of Lemma 4.1 and the $L^{\infty}$-bound in the form of
Corollary 3.3,

$\displaystyle\int_{\Omega}c(x-y))\,{\rm d}\pi\leq W_{c}(\lambda\llcorner
B_{R}+f_{R},\mu\llcorner B_{R}+g_{R})+2\int_{\Omega\cap\\{\exists
t\in[0,1]\colon X(t)\in\partial B_{R}\\}}c(x-y)\,{\rm d}\pi$
$\displaystyle\leq W_{c}(\lambda\llcorner B_{R}+f_{R},\mu\llcorner
B_{R}+g_{R})+c\left(\tau E(4)+D(4)\right).$

In particular, combining the previous two estimates and choosing $r$
sufficiently small, it suffices to prove

$\displaystyle W_{c}(\lambda\llcorner B_{R}+f_{R},\mu\llcorner
B_{R}+g_{R})-\int_{B_{R}}c(\nabla c^{\ast}(\mathrm{D}\phi))\,{\rm
d}x\lesssim\tau E(4)+D(4).$

Using Lemma 2.1, we obtain for $\delta\in(0,1)$ to be fixed

$\displaystyle W_{c}(\lambda\llcorner B_{R}+f_{R},\mu\llcorner B_{R}+g_{R})$
$\displaystyle\leq$ $\displaystyle(1+\delta)W_{c}(\kappa_{\lambda,R}\,{\rm
d}x\llcorner B_{R}+\overline{f}_{R},\kappa_{\mu,R}\,{\rm d}x\llcorner
B_{R}+\overline{g}_{R})$
$\displaystyle\quad+c(\delta)\left(W_{c}(\lambda\llcorner
B_{R},\kappa_{\lambda,R}\,{\rm d}x\llcorner B_{R})+W_{c}(\mu\llcorner
B_{R},\kappa_{\mu,R}\,{\rm d}x\llcorner B_{R})\right)$
$\displaystyle\qquad+c(\delta)\left(W_{c}(f_{R},\overline{f}_{R})+W_{c}(g_{R},\overline{g}_{R})\right).$

Noting that due to the definition of $D$ and our choice of $R$,

$\displaystyle W_{c}(\lambda\llcorner B_{R},\kappa_{\lambda,R}\,{\rm
d}x\llcorner B_{R})+W_{c}(\mu\llcorner B_{R},\kappa_{\mu,R}\,{\rm d}x\llcorner
B_{R})+W_{c}(f_{R},\overline{f}_{R})+W_{c}(g_{R},\overline{g}_{R})$
$\displaystyle\lesssim$ $\displaystyle\tau E(4)+D(4),$

we claim that for some $C>0$,

$\displaystyle W_{c}(\kappa_{\lambda,R}\,{\rm d}x\llcorner
B_{R}+\overline{f}_{R},\kappa_{\mu,R}\,{\rm d}x\llcorner
B_{R}+\overline{g}_{R})\leq(1+CD(4)^{\frac{1}{p}})\int_{B_{R}}c(\nabla
c^{\ast}(\mathrm{D}\phi))\,{\rm d}x.$ (7.3)

Collecting estimates, choosing first $\delta$ and $r$ small, then
$\varepsilon$ small, once (7.3) is established, the proof is complete.

Establishing (7.3) is easy to do using the Benamou-Brenier formulation (1.10).
For $t\in[0,1]$ introduce the non-singular, non-negative measure

$\displaystyle\rho_{t}=t(\kappa_{\mu,R}\,{\rm d}x\llcorner
B_{R}+\overline{f}_{R})+(1-t)(\kappa_{\lambda,R}\,{\rm d}x\llcorner
B_{R}+\overline{g}_{R}),$

and the vector-valued measure

$\displaystyle j_{t}=\nabla c^{\ast}(\mathrm{D}\phi)\,{\rm d}x\llcorner
B_{R}.$

Note that (7.2) can be rewritten as

$\displaystyle\frac{\,{\rm d}}{\,{\rm d}t}\int\zeta\,{\rm
d}\rho_{t}=\int\nabla\zeta\cdot\,{\rm d}j_{t}$

for all test functions $\zeta$, that is $(j_{t},\rho_{t})$ satisfy the
continuity equation in the sense of distributions. The Benamou-Brenier formula
(1.10) gives

$\displaystyle W_{c}(\kappa_{\lambda,R}\,{\rm d}x\llcorner
B_{R}+\overline{f}_{R},\kappa_{\mu,R}\,{\rm d}x\llcorner
B_{R}+\overline{g}_{R})\leq\int_{0}^{1}\int c\left(\frac{\,{\rm
d}j_{t}}{\,{\rm d}\rho_{t}}\right)\,{\rm d}\rho_{t}\,{\rm d}t.$

Here the right-hand side needs to be interpreted in the sense of (2.2). Since
$j_{t}$ is supported in $B_{R}$ it suffices to consider $\zeta$ supported in
$B_{R}$. Then by definition of $(j_{t},\rho_{t})$ and the Fenchel-Young
inequality for any $s>0$,

$\displaystyle\int\zeta\,{\rm d}j_{t}-\int c^{\ast}(\zeta)\,{\rm d}\rho_{t}=$
$\displaystyle\int_{B_{R}}\zeta\cdot\nabla
c^{\ast}(\mathrm{D}\phi)-c^{\ast}(\zeta)(t\kappa_{\mu,R}+(1-t)\kappa_{\lambda,R})\,{\rm
d}x$ $\displaystyle\leq$
$\displaystyle\int_{B_{R}}sc^{\ast}(\zeta)+sc\left(\frac{1}{s}\nabla
c^{\ast}(\mathrm{D}\phi)\right)-c^{\ast}(\zeta)(t\kappa_{\mu,R}+(1-t)\kappa_{\lambda,R})\,{\rm
d}x.$

Choosing $s=t\kappa_{\mu,R}+(1-t)\kappa_{\lambda,R}$ and integrating in $t$,
we deduce

$\displaystyle W_{c}(\kappa_{\mu,R}\,{\rm d}x\llcorner
B_{R}+\overline{f}_{R},\kappa_{\lambda,R}\,{\rm d}x\llcorner
B_{R}+\overline{g}_{R})\leq\int_{B_{R}}\int_{0}^{1}sc\left(\frac{\nabla
c^{\ast}(\mathrm{D}\phi))}{s}\right)\,{\rm d}t\,{\rm d}x.$

Now

$\displaystyle\int_{B_{R}}\int_{0}^{1}sc\left(\frac{\nabla
c^{\ast}(\mathrm{D}\phi))}{s}\right)\,{\rm d}t\,{\rm d}x\leq$
$\displaystyle\int_{B_{R}}c(\nabla c^{\ast}(\mathrm{D}\phi))\,{\rm
d}x+\int_{B_{R}}\int_{0}^{1}(s-1)c(\nabla c^{\ast}(\mathrm{D}\phi))\ dx\,{\rm
d}t$ $\displaystyle\quad+\int_{B_{R}}\int_{0}^{1}s\left(c\left(\frac{\nabla
c^{\ast}(\mathrm{D}\phi)}{s}\right)-c(\nabla
c^{\ast}(\mathrm{D}\phi))\right)\,{\rm d}t\,{\rm d}x.$

Note that

$\displaystyle\lvert s-1\rvert\lesssim D(4)^{\frac{1}{p}}.$

Further using (1.13) and (1.12),

$\displaystyle\int_{B_{R}}\int_{0}^{1}s\left(c\left(\frac{\nabla
c^{\ast}(\mathrm{D}\phi)}{s}\right)-c(\nabla
c^{\ast}(\mathrm{D}\phi))\right)\,{\rm d}t\,{\rm d}x$ $\displaystyle\lesssim$
$\displaystyle\int_{B_{R}}\int_{0}^{1}\left\lvert
1-\frac{1}{s}\right\rvert\left(1+\frac{1}{s}\right)^{p-1}\lvert\nabla
c^{\ast}(\mathrm{D}\phi)\rvert^{p}\,{\rm d}t\,{\rm d}x\lesssim
D(4)^{\frac{1}{p}}\int_{B_{R}}c(\nabla c^{\ast}(\mathrm{D}\phi))\,{\rm d}x.$

Thus the proof of (7.3) is complete.

We turn to the second term on the right-hand side of Lemma 7.2. This term will
be small due to the definition of $\phi^{r}$.

######  Lemma 7.5.

For every $0<\tau$ there exists $\varepsilon(\tau),C(\tau),r_{0}(\tau)>0$ such
that if it holds that ${E(4)+D(4)\leq\varepsilon(\tau)}$ and $0<r\leq r_{0}$,
then there exists $R\in[2,3]$ such that if $\phi^{r}$ solves (7.2), then

$\displaystyle\int_{\Omega}\int_{\sigma}^{\tau}\langle\dot{X}(t)-\nabla
c^{\ast}(\mathrm{D}\phi^{r}(X(t)),\mathrm{D}\phi^{r}(X(t))\rangle\,{\rm
d}t\,{\rm d}\pi\lesssim\tau E(4)+D(4).$

[Proof]Note that $\frac{\,{\rm d}}{\,{\rm
d}t}\phi^{r}(X(t))=\langle\dot{X}(t),\nabla\phi^{r}(X(t))\rangle$. Thus, since
$\pi\in\Pi(\lambda,\mu)$ as well as using the definition of $f_{R}$ and
$g_{R}$,

$\displaystyle\int_{\Omega}\int_{\sigma}^{\tau}\langle\dot{X}(t),\mathrm{D}\phi^{r}(X(t))\rangle\,{\rm
d}t\,{\rm d}\pi=$
$\displaystyle\int_{\Omega}\phi^{r}(X(\tau)-\phi^{r}(X(\sigma))\,{\rm d}\pi$
$\displaystyle=$ $\displaystyle\int_{\Omega\cap\\{X(\tau)\in\partial
B_{R}\\}}\phi^{r}(X(\tau))\,{\rm d}\pi+\int_{\\{x\in
B_{R}\\}}\phi^{r}(x)\,{\rm d}\pi$
$\displaystyle\quad-\int_{\Omega\cap\\{X(\sigma)\in\partial
B_{R}\\}}\phi^{r}(X(\sigma))\,{\rm d}\pi-\int_{\\{y\in
B_{R}\\}}\phi^{r}(y)\,{\rm d}\pi$ $\displaystyle=$
$\displaystyle\int_{B_{R}}\phi^{r}\,{\rm d}(\mu-\lambda)+\int_{\partial
B_{R}}\phi^{r}\,{\rm d}(g_{R}-f_{R}).$

On the other hand, as in Lemma 7.3, at cost of an error
$c(r)(E(4)+D(4))^{1+\frac{\beta}{p+d}}$, we may replace
$\mathrm{D}\phi^{r}(X(t))$ with $\mathrm{D}\phi^{r}(x)$ in the expression

$\displaystyle\int_{\Omega}\int_{\sigma}^{\tau}\langle\nabla
c^{\ast}(\mathrm{D}\phi^{r}(X(t)),\mathrm{D}\phi^{r}(X(t))\rangle\,{\rm
d}t\,{\rm d}\pi$

and $\int_{\Omega}\int_{\sigma}^{\tau}\,{\rm d}t\,{\rm d}\pi$ with
$\int_{B_{R}}\,{\rm d}x$ at the cost of a further error
$c(r)(E(4)+D(4))^{1+\frac{\beta}{p}}$. Thus it suffices to consider

$\displaystyle\int_{B_{R}}\langle\nabla
c^{\ast}(\mathrm{D}\phi),\mathrm{D}\phi\rangle\,{\rm
d}x=c^{r}\int_{B_{R}}\phi^{r}\,{\rm d}x+\int_{\partial B_{R}}\phi^{r}\,{\rm
d}\left(\overline{g}^{r}_{R}-\overline{f}^{r}_{R}\right).$

Collecting estimates, we have shown

$\displaystyle\int_{\Omega}\int_{\sigma}^{\tau}\langle\dot{X}(t)-\nabla
c^{\ast}(\mathrm{D}\phi^{r}(X(t)),\mathrm{D}\phi^{r}(X(t))\rangle\,{\rm
d}t\,{\rm d}\pi$ $\displaystyle\lesssim$
$\displaystyle\int_{B_{R}}\phi^{r}\,{\rm d}(\mu-\lambda-c^{r})+\int_{\partial
B_{R}}\phi^{r}\,{\rm
d}(g_{R}-\overline{g}_{R}^{r}+f_{R}-\overline{f}_{R}^{r})+c(r)\left(E(4)+D(4)\right)^{1+\frac{\beta}{p+d}}$
$\displaystyle=$ $\displaystyle I+II+III.$

We find using Lemma 3.2, Young’s inequality and elliptic regularity in the
form of (1.34) as well as (1.30),

$\displaystyle I\leq$ $\displaystyle\lvert\int_{B_{R}}\phi^{r}\,{\rm
d}(\mu-\kappa_{\mu,R}-\lambda+\kappa_{\lambda,R})\rvert+\lvert\int_{B_{R}}\phi^{r}\,{\rm
d}(\kappa_{\mu,R}-\kappa_{\lambda,R}-c^{r})\rvert$ $\displaystyle\lesssim$
$\displaystyle\sup_{B_{R}}\lvert\mathrm{D}\phi^{r}\rvert\left(W_{c}(\lambda\llcorner
B_{R},\lambda_{\mu,R}\,{\rm d}x\llcorner
B_{R})^{\frac{1}{p}}+W_{c}(\mu\llcorner B_{R},\kappa_{\mu,R}\,{\rm
d}x\llcorner B_{R})^{\frac{1}{p}}\right)$
$\displaystyle\quad+\rvert\kappa_{\mu,R}-\kappa_{\lambda}-c^{r}\rvert\|\phi^{r}\|_{\mathrm{L}^{p}(B_{R})}$
$\displaystyle\lesssim$ $\displaystyle\tau
E(4)+c(\tau,r)D(4)+c(\tau)\lvert\kappa_{\mu,R}-\kappa_{\lambda,R}-c^{r}\rvert^{p^{\prime}}.$

Noting that as $r\to 0$,

$c^{r}=\lvert B_{R}\rvert^{-1}\left(g^{r}(\partial B_{R})-f^{r}(\partial
B_{R})\right)\to\lvert B_{R}\rvert^{-1}\left(g(\partial B_{r})-f(\partial
B_{R})\right)=\kappa_{\mu,R}-\kappa_{\lambda,R},$

we deduce, choosing $r_{0}$ sufficiently small, that

$\displaystyle I=\int_{B_{R}}\phi^{r}\,{\rm d}(\mu-\lambda-c^{r})\lesssim\tau
E(4)+D(4).$

Finally consider $II$. By symmetry it suffices to consider the terms involving
$g$. We estimate, denoting by $(\phi^{r})^{r}$ convolution of $\phi^{r}$ with
the convolution kernel used to construct $g_{R}^{r}$,

$\displaystyle\int_{\partial B_{R}}\phi^{r}\,{\rm
d}(g_{R}-\overline{g}_{R}^{r})=\int_{\partial
B_{R}}(\phi^{r})^{r}-\phi^{r}\,{\rm d}\overline{g}_{R}+\int_{\partial
B_{R}}\phi^{r}\,{\rm d}(\overline{g}_{R}-g_{R}).$

Now note that a standard mollification argument shows

$\displaystyle\int_{\partial B_{R}}\phi^{r}\,{\rm
d}(g_{R}-\overline{g}_{R}^{r})\lesssim
r^{\frac{1}{p}}\|\mathrm{D}\phi^{r}\|_{\mathrm{L}^{p^{\prime}}(B_{R})}\|\overline{g}_{R}\|_{\mathrm{L}^{p}(\partial
B_{R})}\lesssim r^{\frac{1}{p}}(E(4)+D(4)),$

and that moreover using Lemma 3.2 and Young’s inequality,

$\displaystyle\int_{\partial B_{R}}\phi^{r}(\,{\rm
d}\overline{g}_{R}-g)\lesssim[\mathrm{D}_{\mathrm{tan}}\phi^{r}]_{C^{0,\beta}(\partial
B_{R})}W{c}(\overline{g}_{R},\overline{g})^{\frac{1}{p}}\lesssim c(r)(\tau
E(4)+D(4)).$

Thus, collecting estimates and first choosing $r_{0}$ sufficiently small, then
$\varepsilon$ small, we conclude the proof.

We next estimate the third term on the right-hand side of the estimate in
Lemma 7.2.

######  Lemma 7.6.

For every $0<\tau$ there exists $\varepsilon(\tau),C(\tau),r_{0}(\tau)>0$ such
that if it holds that ${E(4)+D(4)\leq\varepsilon(\tau)}$ and $0<r\leq r_{0}$,
then there exists $R\in[2,3]$ such that if $\phi^{r}$ solves (7.2), then

$\displaystyle\int_{B_{R}}c(\nabla c^{\ast}(\mathrm{D}\phi^{r}))\,{\rm
d}x-\int_{\Omega}\int_{\sigma}^{\tau}c(\nabla
c^{\ast}(\mathrm{D}\phi^{r}(X(t)))\,{\rm d}t\,{\rm d}\pi\lesssim\tau
E(4)+D(4).$

[Proof]Set $\xi=c(\nabla c^{\ast}(\mathrm{D}\phi^{r}))$. Then

$\displaystyle\int_{B_{R}}c(\nabla c^{\ast}(\mathrm{D}\phi^{r}))\,{\rm
d}x-\int_{\Omega}\int_{\sigma}^{\tau}c(\nabla
c^{\ast}(\mathrm{D}\phi^{r}(X(t))\,{\rm d}t\,{\rm d}\pi$ $\displaystyle=$
$\displaystyle(1-\kappa_{\mu,R})\int_{B_{R}}\xi\,{\rm
d}x+\left(\kappa_{\mu,R}\int_{B_{R}}\xi\,{\rm
d}x-\int_{B_{R}\times\mathbb{R}^{d}}\xi\,{\rm d}\pi\right)$
$\displaystyle\quad+\int_{B_{R}\times\mathbb{R}^{d}}\xi\,{\rm
d}\pi-\int_{\Omega}\int_{\sigma}^{\tau}\xi(X(t))\,{\rm d}t\,{\rm d}\pi$
$\displaystyle=$ $\displaystyle I+II+III.$

Using (1.31) and Young’s inequality, we find

$\displaystyle I\leq D(4)^{\frac{1}{p}}(E(4)+D(4))\lesssim\tau E(4)+D(4).$

Employing Lemma 3.2 we deduce

$\displaystyle
II\lesssim\|\xi\|_{C^{0,\beta}(\overline{B}_{R})}W_{c}(\mu\llcorner
B_{R},\kappa_{\mu,R}\llcorner B_{R})^{\frac{\beta}{p}}.$

It is straightforward to check that
$\|\xi\|_{C^{0,\beta}(\overline{B}_{R})}\lesssim\|\mathrm{D}\phi^{r}\|_{C^{0,\beta}(\overline{B}_{R})}\left(\sup_{\overline{B}_{R}}\mathrm{D}\phi^{r}\right)^{p^{\prime}-1}$,
so that using (1.34) and Young’s inequality,

$\displaystyle II\lesssim c(r)(E(4)+D(4))D(4)^{\frac{\beta}{p}}\lesssim\tau
E(4)+D(4).$

In order to estimate $III$, we first find

$\displaystyle I(X(0)\in B_{R})\xi(X(0))-I(X(t)\in B_{R})\xi(X(t))$
$\displaystyle\leq$ $\displaystyle I(\exists s\in[0,1]\colon X(s)\in\partial
B_{R},X(0)\in\overline{B}_{R})\xi(X(0))$ $\displaystyle\quad+I(\forall
s\in[0,1]\;X(s)\in B_{R})\left(\xi(X(0))-\xi(X(t)\right).$

Thus, using also (1.34) and Jensen’s inequality,

$\displaystyle III\leq$ $\displaystyle\int_{0}^{1}\int I(X(0)\in
B_{R})\xi(X(0))-I(X(t)\in B_{R})\xi(X(t))\,{\rm d}\pi\,{\rm d}t$
$\displaystyle\leq$
$\displaystyle\sup_{\overline{B}_{R}}\lvert\xi\rvert\int_{0}^{1}\int I(\exists
s\in[0,1]\colon X(s)\in\partial B_{R},X(0)\in\partial B_{R})\,{\rm d}\pi\,{\rm
d}t$
$\displaystyle\quad+\|D\xi\|_{C^{0,\beta}(\overline{B}_{R})}\int_{0}^{1}\int
I(\forall s\in[0,1]\;X(s)\in B_{R})\lvert X(t)-X(0)\rvert^{\beta}\,{\rm
d}t\,{\rm d}\pi$ $\displaystyle\lesssim$ $\displaystyle
c(r)(E(4)+D(4))\pi(\exists t\in[0,1]\colon X(t)\in\partial
B_{R})+c(r)(E+D)\int_{\Omega}\lvert x-y\rvert^{\beta}\,{\rm d}\pi$
$\displaystyle\lesssim$ $\displaystyle
c(r)(E(4)+D(4))^{1+\frac{1}{p+d}}+c(r)(E(4)+D(4))^{1+\frac{\beta}{p}}.$

In order to obtain the last line we used Corollary 3.3. Collecting estimates
and choosing first $r_{0}$ small, then $\varepsilon$ small, we conclude the
proof.

###  7.3 Proof of Theorem 7.1

We are now ready to prove Theorem 7.1. [Proof of Theorem 7.1] Applying Lemma
7.2 to $\phi^{r}$ and collecting the output of Lemma 7.3, Lemma 7.4, Lemma 7.5
and Lemma 7.6, we have shown that for any $0<\tau$, there is $\varepsilon,C>0$
such that if $E(4)+D(4)\leq\varepsilon$, then

$\displaystyle\int_{\Omega_{3/2}}\int_{\sigma_{3/2}}^{\tau_{3/2}}V\left(\dot{X}(t),\nabla
c^{\ast}(\mathrm{D}\phi(X(t))\right)\,{\rm d}t\,{\rm d}\pi\leq\tau
E(4)+CD(4).$

Arguing as in Lemma 7.3, we may replace $\mathrm{D}(\phi(X(t)))$ by
$\mathrm{D}(\phi(X(0)))$ at the cost of an error of size
$\left(E(4)+D(4)\right)^{1+\frac{\beta}{p+d}}$. Noting that due to Lemma 3.1,
$\\#_{1}\subset\Omega_{3/2}$, we find

$\displaystyle\int_{\\#_{1}}V\left(x-y-\nabla
c^{\ast}(\mathrm{D}\phi(x))\right)\,{\rm
d}\pi\leq\int_{\Omega_{3/2}}\int_{0}^{1}V\left(\dot{X}(t)-\nabla
c^{\ast}(\mathrm{D}\phi(X(0))\right)\,{\rm d}t\,{\rm d}\pi.$

Now using (1.23), elliptic regularity (1.32), as well as the
$L^{\infty}$-bound in the form of Corollary 3.3,

$\displaystyle\int_{\Omega_{3/2}\cap(\\{\sigma_{3/2}>0\\}\cup\\{\tau_{3/2}<1\\})}\int_{\sigma_{3/2}}^{\tau_{3/2}}V\left(\dot{X}(t)-\nabla
c^{\ast}(\mathrm{D}\phi(X(0))\right)\,{\rm d}t\,{\rm d}\pi$
$\displaystyle\lesssim$
$\displaystyle\int_{\Omega_{3/2}\cap(\\{\sigma_{3/2}>0\\}\cup\\{\tau_{3/2}<1\\})}\lvert\dot{X}(t)\rvert^{p}+\lvert\mathrm{D}\phi(X(0)))\rvert^{p^{\prime}}$
$\displaystyle\lesssim$
$\displaystyle(E(4)+D(4))\pi(\Omega_{3/2}\cap(\\{\sigma_{3/2}>0\\}\cup\\{\tau_{3/2}<1\\}))$
$\displaystyle\lesssim$ $\displaystyle(E(4)+D(4))^{1+\frac{1}{p+d}}.$

Thus, we conclude

$\displaystyle\int_{\\#_{1}}V(x-y-\nabla c^{\ast}(\mathrm{D}\phi(x)))\,{\rm
d}\pi\lesssim\tau E(4)+D(4).$

This concludes the proof in the case $p\geq 2$. In the case $p\leq 2$, an
application of Hölder’s inequality combined with (1.12) concludes the proof.

## References

  * [1] De Philippis, G., and Figalli, A. Partial regularity for optimal transport maps.  Publ. Math. Inst. Hautes Etudes Sci. 121, 1 (June 2015), 81–112.
  * [2] Figalli, A., and Kim, Y.-H. Partial regularity of brenier solutions of the Monge-Ampère equation.  Discrete Contin. Dyn. Syst. Ser. A 28, 2 (2010), 559–565.
  * [3] Gangbo, W., and McCann, R. The geometry of optimal transportation.  C. R. Acad. Sci. Paris Ser. Mat. 321, 12 (1995), 1653–1658.
  * [4] Giaquinta, M., and Modica, G. Remarks on the regularity of the minimizers of certain degenerate functionals.  Manuscripta Math. 57, 1 (Mar. 1986), 55–99.
  * [5] Giusti, E. Direct Methods in the Calculus of Variations.  World Scientific, 2003.
  * [6] Goldman, M. An $\varepsilon$-regularity result for optimal transport maps between continuous densities.  arXiv preprint arXiv:2005.06785 (2020).
  * [7] Goldman, M., and Huesmann, M. A fluctuation result for the displacement in the optimal matching problem.  Ann. Probab. 50, 4 (2022), 144601477.
  * [8] Goldman, M., Huesmann, M., and Otto, F. Quantitative linearization results for the monge‐ampère equation.  Commun. Pure Appl. Math. 74, 12 (Dec. 2021), 2483–2560.
  * [9] Goldman, M., and Otto, F. A variational proof of partial regularity for optimal transportation maps.  Ann. Sci. Ec. Norm. Super. 53, 5 (2020), 1209–1233.
  * [10] Gutiérrez, C. E., and Montanari, A. $L^{\infty}$-estimates in optimal transport for non quadratic costs.  arXiv preprint arXiv:2106.02899 (2021).
  * [11] Hamburger, C. Regularity of differential forms.  J. Reine Angew. Math. 431 (1992), 7–64.
  * [12] Hiriart-Urruty, J.-B., and Lemar, C. Convex Analysis and Minimization Algorithms II.  Springer-Verlag Berlin Heidelberg, 1993.
  * [13] Hiriart-Urruty, J.-B., and Lemaréchal, C. Convex Analysis and Minimization Algorithms I.  Springer Berlin Heidelberg, 1993.
  * [14] Huesmann, M., Mattesini, F., and Otto, F. There is no stationary cyclically monotone poisson matching in 2d.  arXiv preprint arXiv:2109.13590 (2021).
  * [15] Huesmann, M., Mattesini, F., and Otto, F. There is no stationary p-cyclically monotone poisson matching in 2d.  in preparation (2023).
  * [16] Koch, L., and Otto, F. Lectures notes on the harmonic approximation to quadratic optimal transport.  arXiv preprint arXiv:2303.14462 (2023).
  * [17] Ladyzhenskaya, O., and Ural’tseva, N. Linear and quasilinear elliptic equations.  Academic Press Inc., New York, 1968.
  * [18] Lewis, J. L. Regularity of the derivatives of solutions to certain degenerate elliptic equations.  Indiana Univ. Math. J. (1983).
  * [19] Lieberman, G. M. Boundary regularity for solutions of degenerate elliptic equations.  Nonlinear Anal. Theory Methods Appl. 12, 11 (1988), 1203–1219.
  * [20] Mingione, G. Regularity of minima: An invitation to the dark side of the calculus of variations.  Appl. Math. 51, 4 (2006), 355–426.
  * [21] Otto, F., Prod’homme, M., and Ried, T. Variational approach to regularity of optimal transport maps: General cost functions.  Ann. PDE 7, 2 (Dec. 2021).
  * [22] Rockafellar, R. T. Convex analysis.  Princeton University Press, Princeton, N.J., 1970.
  * [23] Santambrogio, F. Optimal Transport for Applied Mathematicians.  Birkhäuser Press, 2015.
  * [24] Tolksdorf, P. Everywhere regularity for some quasi-linear systems with a lack of ellipticity.  Ann. Mat. Pura Appl. 134 (1983), 241–266.
  * [25] Ural’tseva, N. Degenerate quasilinear elliptic systems.  Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 7 (1968), 184–222.
  * [26] Villani, C. Optimal Transport. Old and new.  Springer-Verlag Berlin Heidelberg, 2009.