that, if $x\in\overline{\mathrm{D}({\boldsymbol{B}})}\cap\mathcal{Y}$, then
$\boldsymbol{J}_{\tau}x\to x$ by Theorem A.3(4) and
$\boldsymbol{J}_{\tau}x\in\mathrm{D}({\boldsymbol{B}})\cap\mathcal{Y}=\mathrm{D}({\boldsymbol{B}}_{\mathcal{Y}})$.
Assertion (v) follows again by Theorem A.3(5). ∎

######  Corollary A.9.

Let ${\boldsymbol{B}}$ be a maximal $\lambda$-dissipative operator and suppose
that $\mathcal{H}$ has finite dimension. Then the conclusions of Theorem A.7
hold.

###### Proof.

Up to a translation, we can assume that $0\in\mathrm{D}(\boldsymbol{B})$. Let
$\mathcal{Y}$ be the subspace generated by $\mathrm{D}(\boldsymbol{B})$. Since
$\mathcal{H}$ is finite dimensional, then $\mathcal{Y}$ is closed. We can thus
apply Proposition A.8 and obtain that
${\boldsymbol{B}}_{\mathcal{Y}}:={\boldsymbol{B}}\cap(\mathcal{Y}\times\mathcal{Y})$
is maximal $\lambda$-dissipative in $\mathcal{Y}$, has the same domain of
${\boldsymbol{B}}$ and its semigroup coincides with the semigroup generated by
${\boldsymbol{B}}$. Since $\mathcal{H}$ is finite dimensional, the relative
interior of $\operatorname{co}(\mathrm{D}({\boldsymbol{B}}_{\mathcal{Y}}))$ in
$\mathcal{Y}$ is nonempty and thus we conclude by Theorem A.3(3) that the
relative interior of $\mathrm{D}({\boldsymbol{B}}_{\mathcal{Y}})$ in
$\mathcal{Y}$ is nonempty, so that we can apply Theorem A.7 to
${\boldsymbol{B}}_{\mathcal{Y}}$ and obtain the conclusion of such theorem for
the semigroup generated by ${\boldsymbol{B}}$. ∎

######  Corollary A.10.

Let $\boldsymbol{B}_{1}$ and $\boldsymbol{B}_{2}$ be maximal
$\lambda$-dissipative operators with
$\overline{\mathrm{D}(\boldsymbol{B}_{1})}=\overline{\mathrm{D}(\boldsymbol{B}_{2})}$
and let $\boldsymbol{S}^{1}_{t}$ and $\boldsymbol{S}^{2}_{t}$ be the
semigroups of Lipschitz transformations associated to $\boldsymbol{B}_{1}$ and
$\boldsymbol{B}_{2}$ respectively given by Theorem A.6. If for every
$x\in\overline{\mathrm{D}(\boldsymbol{B}_{1})}=\overline{\mathrm{D}(\boldsymbol{B}_{2})}$
there exists $\delta>0$ such that
$\boldsymbol{S}^{1}_{t}x=\boldsymbol{S}^{2}_{t}x$ for every $0\leq t<\delta$,
then $\boldsymbol{B}_{1}=\boldsymbol{B}_{2}$.

###### Proof.

This can be proven as in [Bré73, Theorem 4.1]: let
$x\in\mathrm{D}(\boldsymbol{B}_{1})$ and let
$y\in\mathrm{D}(\boldsymbol{B}_{2})$; by hypotesis, we can find some
$\delta>0$ such that $\boldsymbol{S}_{t}^{1}x=\boldsymbol{S}_{t}^{2}x$ and
$\boldsymbol{S}_{t}^{1}y=\boldsymbol{S}_{t}^{2}y$ for every $0\leq t<\delta$.
Thus, for every $0\leq t<\delta$, we have

$\displaystyle\langle\frac{\boldsymbol{S}_{t}x-x}{t}-\frac{\boldsymbol{S}_{t}y-y}{t},x-y\rangle$
$\displaystyle\leq\frac{1}{t}|\boldsymbol{S}_{t}x-\boldsymbol{S}_{t}y||x-y|-\frac{1}{t}|x-y|^{2}$
$\displaystyle\leq\frac{e^{\lambda t}-1}{t}|x-y|^{2},$

where we have used that
$\boldsymbol{S}_{t}:=\boldsymbol{S}^{1}_{t}=\boldsymbol{S}^{2}_{t}$ is
$e^{\lambda t}$-Lipschitz by (A.8). Passing to the limit as $t\downarrow 0$
and using Theorem A.5(3), we get that

$\langle\boldsymbol{B}_{1}^{\circ}x-\boldsymbol{B}_{2}^{\circ}y,x-y\rangle\leq\lambda|x-y|^{2}.$

By (A.2) we get that
$\mathrm{D}(\boldsymbol{B}_{1})=\mathrm{D}(\boldsymbol{B}_{2})$ and thus that
$\boldsymbol{B}_{1}^{\circ}=\boldsymbol{B}_{2}^{\circ}$. By (A.3) we thus get
that $\boldsymbol{B}_{1}=\boldsymbol{B}_{2}$. ∎

The following proposition is a slight generalization of [Att79, Lemma 2.3] but
we report its proof for the reader’s convenience.

######  Proposition A.11.

Let $\boldsymbol{B}\subset\mathcal{H}\times\mathcal{H}$ be maximal
$\lambda$-dissipative and let $\boldsymbol{G}\subset\boldsymbol{B}$ be s.t.
$\mathrm{D}(\boldsymbol{G})$ is dense in $\mathrm{D}(\boldsymbol{B})$. Then
for every $x\in\operatorname{int}\left(\mathrm{D}(\boldsymbol{B})\right)$ it
holds

$\boldsymbol{B}x=\overline{\operatorname{co}}\left(\left\\{v\in\mathcal{H}\mid\exists(x_{n},v_{n})\in\boldsymbol{G}\text{
s.t. }x_{n}\to x,\,v_{n}\rightharpoonup v\right\\}\right).$ (A.12)

###### Proof.

Let $x\in\operatorname{int}\left(\mathrm{D}(\boldsymbol{B})\right)$ and let us
define

$\boldsymbol{M}x:=\overline{\operatorname{co}}\left(\left\\{v\in\mathcal{H}\mid\exists(x_{n},v_{n})\in\boldsymbol{G}\text{
s.t. }x_{n}\to x,\,v_{n}\rightharpoonup v\right\\}\right).$

If $(x_{n},v_{n})\in\boldsymbol{G}\subset\boldsymbol{B}$ with $x_{n}\to x$ and
$v_{n}\rightharpoonup v$, by $\lambda$-dissipativity of $\boldsymbol{B}$, we
have that

$\langle
v_{n}-w,x_{n}-y\rangle\leq\lambda|x_{n}-y|^{2}\quad\forall\,(y,w)\in\boldsymbol{B}.$

Passing to the limit we get

$\langle
v-w,x-y\rangle\leq\lambda|x-y|^{2}\quad\forall\,(y,w)\in\boldsymbol{B},$

so that $v\in\boldsymbol{B}x$ by (A.2). This, together with the closure and
convexity of $\boldsymbol{B}x$ given by Theorem A.3(2), proves that
$\boldsymbol{M}x\subset\boldsymbol{B}x$. Let us prove the other inclusion by
contradiction: suppose that there is some $v\in\boldsymbol{B}x$ s.t.
$v\notin\boldsymbol{M}x$. The sets $\\{v\\}$ and $\boldsymbol{M}x$ are
disjoint, closed, convex and $\\{v\\}$ is also compact. By Hahn-Banach theorem
we can find some $z\in\mathcal{H}$ with $|z|=1$ s.t.

$\langle v,z\rangle>\langle u,z\rangle\quad\forall\,u\in\boldsymbol{M}x.$
(A.13)

Since $x\in\operatorname{int}\left(\mathrm{D}(\boldsymbol{B})\right)$, if we
define $z_{n}:=x+z/n$, we have that
$z_{n}\in\operatorname{int}\left(\mathrm{D}(\boldsymbol{B})\right)$ for $n$
sufficiently large. We can thus find $x_{n}\in\mathrm{D}(\boldsymbol{G})$ s.t.
$|x_{n}-z_{n}|<n^{-2}$. Clearly $x_{n}\to x$ and it is easy to check that
$(x_{n}-x)/|x_{n}-x|\to z$. Since $x_{n}\in\mathrm{D}(\boldsymbol{G})$, we can
find $v_{n}\in\boldsymbol{G}x_{n}$. Since $\boldsymbol{B}$ is maximal, it is
locally bounded (cf. Theorem A.3(3)) at $x$. Being
$\boldsymbol{G}\subset\boldsymbol{B}$ and being $x_{n}\to x$, the sequence
$(v_{n})$ is bounded so that, up to an unrelabeled subsequence, it converges
weakly to some point $u\in\mathcal{H}$. By $\lambda$-dissipativity of
$\boldsymbol{B}$ we have

$\langle
v-v_{n},x-x_{n}\rangle\leq\lambda|x-x_{n}|^{2}\quad\forall\,n\in\mathbb{N},$

so that, dividing by $|x_{n}-x|$ and passing to the limit, we obtain

$\langle v-u,z\rangle\leq 0,$

a contradiction with (A.13) since, obviously, $u\in\boldsymbol{M}x$. ∎

The following proposition is an immediate consequence of [Qi83, Theorem 1] and
Remark A.1.

######  Proposition A.12.

Let $\boldsymbol{B}\subset\mathcal{H}\times\mathcal{H}$ be
$\lambda$-dissipative with open non empty convex domain. Then there exists a
unique maximal $\lambda$-disipative
$\hat{\boldsymbol{B}}\supset\boldsymbol{B}$ with
$\mathrm{D}(\hat{\boldsymbol{B}})\subset\overline{\mathrm{D}(\boldsymbol{B})}$
and it is characterized by

$\hat{\boldsymbol{B}}=\left\\{(x,v)\in\overline{\mathrm{D}(\boldsymbol{B})}\times\mathcal{H}\mid\langle
v-w,x-y\rangle\leq\lambda|x-y|^{2}\quad\forall\,(y,w)\in\boldsymbol{B}\right\\}.$

As a consequence of Propositions A.11 and A.12 we can prove the following.

######  Theorem A.13.

Let $\boldsymbol{B}\subset\mathcal{H}\times\mathcal{H}$ be
$\lambda$-dissipative with

$C:=\overline{\mathrm{D}(\boldsymbol{B})}\text{ is
convex},\quad\operatorname{int}\left(\mathrm{D}(\boldsymbol{B})\right)\neq\emptyset.$

Then there exists a unique maximal $\lambda$-dissipative
$\hat{\boldsymbol{B}}\supset\boldsymbol{B}$ with
$\mathrm{D}(\hat{\boldsymbol{B}})\subset C$ and it is characterized by

$\hat{\boldsymbol{B}}=\left\\{(x,v)\in C\times\mathcal{H}\mid\langle
v-w,x-y\rangle\leq\lambda|x-y|^{2}\quad\forall\,(y,w)\in\boldsymbol{B}\right\\}.$
(A.14)

Moreover, for every
$x\in\operatorname{int}\left(\mathrm{D}(\hat{\boldsymbol{B}})\right)$ it holds

$\hat{\boldsymbol{B}}x=\overline{\operatorname{co}}\left(\left\\{v\in\mathcal{H}\mid\exists(x_{n},v_{n})\in\boldsymbol{B}\text{
s.t. }x_{n}\to x,\,v_{n}\rightharpoonup v\right\\}\right).$ (A.15)

Finally

$\operatorname{int}\left(C\right)=\operatorname{int}\left(\mathrm{D}(\hat{\boldsymbol{B}})\right)\subset\mathrm{D}(\hat{\boldsymbol{B}})\subset\overline{\mathrm{D}(\hat{\boldsymbol{B}})}=C.$
(A.16)

###### Proof.

Let $\boldsymbol{B}^{\prime}$ be a $\lambda$-dissipative maximal extension of
$\boldsymbol{B}$ with $\mathrm{D}(\boldsymbol{B}^{\prime})\subset C$, whose
existence is granted by Theorem A.2(2); by $\lambda$-dissipativity of
$\boldsymbol{B}^{\prime}$ and since
$\boldsymbol{B}\subset\boldsymbol{B}^{\prime}$, then
$\boldsymbol{B}^{\prime}\subset\hat{\boldsymbol{B}}$, where
$\hat{\boldsymbol{B}}$ is defined as in (A.14). We need to prove the other
inclusion.
Since
$\mathrm{D}(\boldsymbol{B})\subset\mathrm{D}(\boldsymbol{B}^{\prime})\subset
C$, we have that $\overline{\mathrm{D}(\boldsymbol{B}^{\prime})}=C$. Moreover,
being $\boldsymbol{B}^{\prime}$ maximal $\lambda$-dissipative and being the
interior of its domain nonempty, we have by Theorem A.3(3) that

$\operatorname{int}\left(\mathrm{D}(\boldsymbol{B}^{\prime})\right)\text{ is
convex
},\quad\operatorname{int}\left(\mathrm{D}(\boldsymbol{B}^{\prime})\right)=\operatorname{int}\left(\overline{\mathrm{D}(\boldsymbol{B}^{\prime})}\right)=\operatorname{int}\left(C\right).$

It is then clear that
$\boldsymbol{B}_{0}:=\boldsymbol{B}^{\prime}\cap(\operatorname{int}\left(\mathrm{D}(\boldsymbol{B}^{\prime})\right)\times\mathcal{H})$
is $\lambda$-dissipative with open and nonempty convex domain so that, by
Proposition A.12, there exists a unique maximal $\lambda$-dissipative
$\boldsymbol{B}^{\prime\prime}\supset\boldsymbol{B}_{0}$ with
$\mathrm{D}(\boldsymbol{B}^{\prime\prime})\subset\overline{\mathrm{D}(\boldsymbol{B}_{0})}=\overline{\operatorname{int}\left(\mathrm{D}(\boldsymbol{B}^{\prime})\right)}=\overline{\operatorname{int}\left(C\right)}=C$
($C$ is convex) and it is characterized by

$\boldsymbol{B}^{\prime\prime}=\left\\{(x,v)\in C\times\mathcal{H}\mid\langle
v-w,x-y\rangle\leq\lambda|x-y|^{2}\quad\forall\,(y,w)\in\boldsymbol{B}_{0}\right\\}.$
(A.17)

Since $\boldsymbol{B}^{\prime}\supset\boldsymbol{B}_{0}$,
$\boldsymbol{B}^{\prime}$ is maximal $\lambda$-dissipative and
$\mathrm{D}(\boldsymbol{B}^{\prime})\subset C$, it must be that
$\boldsymbol{B}^{\prime}=\boldsymbol{B}^{\prime\prime}$.
By (A.17), we need to prove that

$\hat{\boldsymbol{B}}\subset\left\\{(x,v)\in C\times\mathcal{H}\mid\langle
v-w,x-y\rangle\leq\lambda|x-y|^{2}\quad\forall\,(y,w)\in\boldsymbol{B}_{0}\right\\}.$
(A.18)

To this aim we apply Proposition A.11 to the maximal $\lambda$-dissipative
$\boldsymbol{B}^{\prime}$ and its subset $\boldsymbol{B}$ noticing that
$\mathrm{D}(\boldsymbol{B})$ is dense in
$\mathrm{D}(\boldsymbol{B}^{\prime})$. In this way, we obtain that

$\boldsymbol{B}_{0}y=\overline{\operatorname{co}}\left(\overline{\boldsymbol{B}}y\right),\quad
y\in\mathrm{D}(\boldsymbol{B}_{0}),$ (A.19)

where

$\overline{\boldsymbol{B}}y=\left\\{u\in\mathcal{H}\mid\exists(y_{n},u_{n})\in\boldsymbol{B}\text{
s.t. }y_{n}\to y,\,u_{n}\rightharpoonup u\right\\}.$

If $(x,v)\in\hat{\boldsymbol{B}}$ and
$(y,w)\in\mathrm{D}(\boldsymbol{B}_{0})\times\mathcal{H}$ is such that
$w\in\overline{\boldsymbol{B}}y$, we can find a sequence
$(y_{n},u_{n})\in\boldsymbol{B}$ s.t. $y_{n}\to y$ and $u_{n}\rightharpoonup
w$; then, by the very definition of $\hat{\boldsymbol{B}}$, we have

$\langle
v-u_{n},x-y_{n}\rangle\leq\lambda|x-y_{n}|^{2}\quad\forall\,n\in\mathbb{N},$

so that, passing to the limit, we get

$\langle v-w,x-y\rangle\leq\lambda|x-y|^{2}.$

This proves that, if $(x,v)\in\hat{\boldsymbol{B}}$, then

$\langle
v-w,x-y\rangle\leq\lambda|x-y|^{2}\quad\forall\,w\in\overline{\boldsymbol{B}}y,\quad\forall\,y\in\mathrm{D}(\boldsymbol{B}_{0}).$
(A.20)

Finally, if $(x,v)\in\hat{\boldsymbol{B}}$ and $(y,w)\in\boldsymbol{B}_{0}$,
we can find a sequence $(N_{n})_{n}\subset\mathbb{N}$, numbers
$(\alpha_{i}^{n})_{i=1}^{N_{n}}\subset[0,1]$ and points
$(w_{i}^{n})_{i=1}^{N_{n}}\subset\overline{\boldsymbol{B}}y$ s.t.

$\sum_{i=1}^{N_{n}}\alpha_{i}^{n}=1\,\,\forall
n\in\mathbb{N},\quad\lim_{n\to+\infty}\sum_{i=1}^{N_{n}}\alpha_{i}^{n}w_{i}^{n}=w.$

By (A.20)

$\langle v-w_{i}^{n},x-y\rangle\leq\lambda|x-y|^{2}\quad\forall
i=1,\dots,N_{n},\quad\forall n\in\mathbb{N},$

so that, multiplying by $\alpha_{i}^{n}$ and summing up w.r.t. $i$ we obtain

$\langle
v-\sum_{i=1}^{N_{n}}\alpha_{i}^{n}w_{i}^{n},x-y\rangle\leq\lambda|x-y|^{2}\quad\forall
n\in\mathbb{N}.$

Passing to the limit as $n\to+\infty$, we obtain

$\langle v-w,x-y\rangle\leq\lambda|x-y|^{2},$

so that (A.18) holds. Finally notice that (A.15) is already stated in (A.19)
since we just proved that
$\boldsymbol{B}^{\prime}=\boldsymbol{B}^{\prime\prime}=\hat{\boldsymbol{B}}$.
∎

As a consequence, we have the following corollary.

######  Corollary A.14.

Let $\boldsymbol{B}\subset\mathcal{H}\times\mathcal{H}$ be as in Theorem A.13
and let $\boldsymbol{G}:\operatorname{int}\left(C\right)\to\mathcal{H}$ be a
single-valued selection of the maximal $\lambda$-dissipative extension
$\hat{\boldsymbol{B}}$ of $\boldsymbol{B}$. Then the unique maximal
$\lambda$-dissipative extension of $\boldsymbol{G}$ with domain included in
$C$, $\hat{\boldsymbol{G}}$, coincides with $\hat{\boldsymbol{B}}$ and in
particular

$(x,v)\in\hat{\boldsymbol{B}}\Leftrightarrow x\in C,\langle
v-\boldsymbol{G}y,x-y\rangle\leq\lambda|x-y|^{2}\quad\forall
y\in\operatorname{int}\left(C\right).$ (A.21)

Let us consider a different situation when we do not assume that
$\mathrm{D}(\boldsymbol{B})$ contains interior points but there exists a
subset $D$ dense in $\mathrm{D}(\boldsymbol{B})$ which is invariant with
respect to the resolvent map ${\boldsymbol{J}_{\tau}}$, i.e.

$\overline{D}\supset\mathrm{D}(\boldsymbol{B})\ \text{and}\quad\forall\,x\in
D,\ 0<\tau<1/\lambda^{+}\quad\exists\,x_{\tau}\in
D:\,\,x_{\tau}-\tau\boldsymbol{B}x_{\tau}\ni x.$ (A.22)

Since $\boldsymbol{B}$ is $\lambda$-dissipative, the point $x_{\tau}$ solving
the inclusion in (A.22) is unique and defines a map
${\boldsymbol{J}_{\tau}}:D\to D\cap\mathrm{D}(\boldsymbol{B})$.

######  Lemma A.15.

Let $\boldsymbol{B}\subset\mathcal{H}\times\mathcal{H}$ be
$\lambda$-dissipative with $C:=\overline{D(\boldsymbol{B})}$ convex, let us
assume that $D\subset\mathcal{H}$ satistifies (A.22), and let us set
$\boldsymbol{B}_{0}:=\boldsymbol{B}\cap(D\times\mathcal{H})$. The following
hold:

  1. (1)

$\boldsymbol{B}$ admits a unique maximal $\lambda$-dissipative extension
$\hat{\boldsymbol{B}}$ with $\mathrm{D}(\hat{\boldsymbol{B}})\subset C$
characterized by

$\displaystyle\hat{\boldsymbol{B}}=\Big{\\{}$ $\displaystyle(x,v)\in
C\times\mathcal{H}\mid\langle v-v_{0},\rangle\leq\lambda|x-x_{0}|^{2}\text{
for every }(x_{0},v_{0})\in\boldsymbol{B}_{0}\Big{\\}}.$ (A.23)

  2. (2)

If moreover the interior of $\overline{D}$ contains $C$, we have

$\hat{\boldsymbol{B}}=\Big{\\{}(x,v)\in\mathcal{H}\times\mathcal{H}:\exists\,(x_{n},v_{n})\in\boldsymbol{B}_{0}:x_{n}\to
x,v_{n}\to v\quad\text{as }n\to\infty\Big{\\}}.$ (A.24)

###### Proof.

We first prove Claim (1). Let $\boldsymbol{B}^{\prime}$ be any maximal
$\lambda$-dissipative extension of $\boldsymbol{B}$ with domain included in
$C$ (whose existence is granted by Theorem A.2(2)) and let
$\boldsymbol{J}_{\tau}^{\prime}$ be the resolvent associated with
$\boldsymbol{B}^{\prime}$. By dissipativity of $\boldsymbol{B}^{\prime}$ and
since $\boldsymbol{B}_{0}\subset\boldsymbol{B}\subset\boldsymbol{B}^{\prime}$,
we have that $\boldsymbol{B}^{\prime}\subset\hat{\boldsymbol{B}}$ defined as
in (A.23). We need to prove the other inclusion.
Clearly, the restriction of $\boldsymbol{J}_{\tau}^{\prime}$ to $D$ coincides
with ${\boldsymbol{J}_{\tau}}$; since $\boldsymbol{J}_{\tau}^{\prime}$ is
Lipschitz and $D$ is dense in $C$, it is the unique Lipschitz extension of
${\boldsymbol{J}_{\tau}}$ to $\overline{D}\supset C$.

If $(x,v)\in\hat{\boldsymbol{B}}$, (A.23) and the fact that for every $y\in
D$,
$\frac{1}{\tau}(\boldsymbol{J}_{\tau}y-y)\in\boldsymbol{B}\boldsymbol{J}_{\tau}y$
yield by density that

$\langle
v-\tau^{-1}(\boldsymbol{J}_{\tau}^{\prime}y-y),x-\boldsymbol{J}_{\tau}^{\prime}y\rangle\leq\lambda|x-\boldsymbol{J}_{\tau}^{\prime}y|^{2}\quad\forall\,y\in\mathrm{D}(\boldsymbol{B}^{\prime}),\quad\forall\,0<\tau<1/\lambda^{+},$
(A.25)

and passing to the limit as $\tau\downarrow 0$ we obtain that

$\langle
v-\boldsymbol{B}^{\prime\circ}y,x-y\rangle\leq\lambda|x-y|^{2}\quad\forall\,y\in\mathrm{D}(\boldsymbol{B}^{\prime}),$
(A.26)

where we also used Theorem A.3(4), (5). We can then apply (A.3) and conclude
that $(x,v)\in\boldsymbol{B}^{\prime}$.

We prove Claim (2). Since
$\overline{\boldsymbol{B}_{0}}\subset\hat{\boldsymbol{B}}$, it is sufficient
to prove the opposite inclusion
$\hat{\boldsymbol{B}}\subset\overline{\boldsymbol{B}_{0}}$. Let
$(x,v)\in\hat{\boldsymbol{B}}$, let $0<\tau<1/\lambda^{+}$ and set $y:=x-\tau
v$. Clearly $\boldsymbol{J}_{\tau}^{\prime}y=x$; since $\overline{D}$ contains
a neighborhood of every element of $\mathrm{D}(\hat{\boldsymbol{B}})\subset
C$, for sufficiently small $\tau>0$ there exists a sequence
$(y_{n})_{n}\subset D$ converging to $y$ as $n\to\infty$. Setting
$x_{n}:=\boldsymbol{J}^{\prime}_{\tau}y_{n}$ and
$v_{n}:=(x_{n}-y_{n})/\tau\in\boldsymbol{B}x_{n}$, we clearly have
$\lim_{n\to\infty}x_{n}=x,$ $\lim_{n\to\infty}v_{n}=v$. ∎

######  Corollary A.16.

Let $\boldsymbol{B}\subset\mathcal{H}\times\mathcal{H}$ be maximal
$\lambda$-dissipative, let us assume that $D\subset\mathcal{H}$ satistifies
(A.22) and the interior of $\overline{D}$ contains
$C:=\overline{\mathrm{D}(\boldsymbol{B})}$. The following hold:

  1. (1)

For every $x\in\mathrm{D}(\boldsymbol{B})$ there exists a sequence $x_{n}\in
D\cap\mathrm{D}(\boldsymbol{B})$ converging to $x$ such that
$\boldsymbol{B}^{\circ}x_{n}\to\boldsymbol{B}^{\circ}x$ as $n\to\infty$.

  2. (2)

$\boldsymbol{B}$ can be determined by the restriction of the minimal section
$\boldsymbol{B}^{\circ}$ to $D$ i.e.

$\displaystyle\boldsymbol{B}=\Big{\\{}$
$\displaystyle(x,v)\in\overline{\mathrm{D}(\boldsymbol{B})}\times\mathcal{H}\mid\langle
v-\boldsymbol{B}^{\circ}x_{0},\rangle\leq\lambda|x-x_{0}|^{2}\text{ for every
}x_{0}\in D\cap\mathrm{D}(\boldsymbol{B})\Big{\\}}.$ (A.27)

###### Proof.

We first prove Claim (1). Since $\boldsymbol{B}$ is maximal
$\lambda$-dissipative, the closure of its domain $C$ is convex (see Theorem
A.3(4)). We can thus apply the second claim of the previous Lemma A.15 (in
this case $\hat{\boldsymbol{B}}=\boldsymbol{B}$) to find a sequence
$(x_{n},v_{n})\in\boldsymbol{B}\cap(D\times\mathcal{H})$ such that $x_{n}\to
x$ and $v_{n}\to\boldsymbol{B}^{\circ}x$. Let us first prove that
$\boldsymbol{B}^{\circ}x_{n}\rightharpoonup\boldsymbol{B}^{\circ}x$ weakly in
$\mathcal{H}$ as $n\to\infty$: extracting an unrelabeled subsequence, since
$|\boldsymbol{B}^{\circ}x_{n}|\leq|v_{n}|$ is bounded, we can suppose that
there exists an increasing subsequence $k\mapsto n(k)$ and an element
$v\in\mathcal{H}$ such that $\boldsymbol{B}^{\circ}x_{n(k)}\rightharpoonup v$
as $k\to\infty$. Since the graph of $\boldsymbol{B}$ is strongly-weakly closed
(cf. Theorem A.3(1)), we deduce that $(x,v)\in\boldsymbol{B}$ so that
$|v|\geq|\boldsymbol{B}^{\circ}x|$. On the other hand, the lower
semicontinuity of the norm yields

$|\boldsymbol{B}^{\circ}x|\leq|v|\leq\liminf_{k\to\infty}|\boldsymbol{B}^{\circ}x_{n(k)}|\leq\limsup_{k\to\infty}|\boldsymbol{B}^{\circ}x_{n(k)}|\leq\limsup_{k\to\infty}|v_{n(k)}|=|\boldsymbol{B}^{\circ}x|.$

We deduce that
$\boldsymbol{B}^{\circ}x_{n(k)}\rightharpoonup\boldsymbol{B}^{\circ}x$ and
$\lim_{k\to\infty}|\boldsymbol{B}^{\circ}x_{n(k)}|=|\boldsymbol{B}^{\circ}x|$
so that the convergence is also strong. Since the starting (unrelabeled)
subsequence was arbitrary, we deduce the strong convergence of the whole
sequence.

Claim (2) now follows easily by approximation using the previous claim and
Theorem A.3(6). ∎

##  Appendix B Borel partitions and almost optimal couplings

In this appendix we summarize some of the results of [CSS23a] related to
standard Borel spaces, Borel partitions and optimal couplings between
probability measures that have been used throughout the whole paper. We refer
to [CSS23a, Section 3] for the proofs.

######  Definition B.1.

A _standard Borel space_ $(\Omega,{\mathcal{B}})$ is a measurable space that
is isomorphic (as a measurable space) to a Polish space. Equivalently, there
exists a Polish topology $\tau$ on $\Omega$ such that the Borel sigma algebra
generated by $\tau$ coincides with ${\mathcal{B}}$. We say that a probability
measure $\mathbb{P}$ on $(\Omega,{\mathcal{B}})$ is nonatomic if
$\mathbb{P}(\\{\omega\\})=0$ for every $\omega\in\Omega$ (notice that
$\\{\omega\\}\in{\mathcal{B}}$ since it is compact in any Polish topology on
$\Omega$).

If $(\Omega,{\mathcal{B}})$ is a standard Borel space endowed with a nonatomic
probability measure $\mathbb{P}$, we denote by
$\mathrm{S}(\Omega,{\mathcal{B}},\mathbb{P})$ the class of
${\mathcal{B}}$-${\mathcal{B}}$-measurable maps $g:\Omega\to\Omega$ which are
essentially injective and measure-preserving, meaning that there exists a full
$\mathbb{P}$-measure set $\Omega_{0}\in{\mathcal{B}}$ such that $g$ is
injective on $\Omega_{0}$ and $g_{\sharp}\mathbb{P}=\mathbb{P}$. If
$\mathcal{A}\subset{\mathcal{B}}$ is a sigma algebra on $\Omega$ we denote by
$\mathrm{S}(\Omega,{\mathcal{B}},\mathbb{P};\mathcal{A})$ the subset of
$\mathrm{S}(\Omega,{\mathcal{B}},\mathbb{P})$ of $\mathcal{A}-\mathcal{A}$
measurable maps.

We will often use the notation

$I_{N}:=\\{0,\dots,N-1\\},\quad N\in\mathbb{N},\,N\geq 1$

while ${\mathrm{Sym}(I_{N})}$ denotes the set of permutations of $I_{N}$ i.e.
bijective maps $\sigma:I_{N}\to I_{N}$. We will consider the partial order on
$\mathbb{N}$ given by

$m\prec n\quad\Leftrightarrow\quad m\mid n$

where $m\mid n$ means that $n/m\in\mathbb{N}$. We write
$m\mathrel{\text{\oalign{$\prec$\cr\nointerlineskip\kern
4.30554pt\cr$\smash{\raisebox{2.79857pt}{\rotatebox{90.0}{\scalebox{1.0}[-1.0]{$\nshortmid$}}}}$\cr\nointerlineskip\kern-2.15277pt\cr}}}n$
if $m\prec n$ and $m\neq n$.

This first result shows a correspondence between permutations and measure-
preserving isomorphisms.

######  Lemma B.2.

Let $(\Omega,{\mathcal{B}})$ be a standard Borel space endowed with a
nonatomic probability measure $\mathbb{P}$, and let
$\mathfrak{P}_{N}=\\{\Omega_{N,k}\\}_{k\in I_{N}}\subset{\mathcal{B}}$ be a
_$N$ -partition_ of $(\Omega,{\mathcal{B}})$ for some $N\in\mathbb{N}$, i.e.

$\bigcup_{k\in
I_{N}}\Omega_{N,k}=\Omega,\quad\Omega_{N,k}\cap\Omega_{N,h}=\emptyset\text{ if
}h,k\in I_{N},\,h\neq k;$

assume moreover that $\mathbb{P}(\Omega_{N,k})=\mathbb{P}(\Omega)/N$ for every
$k\in I_{N}$. If $\sigma\in{\mathrm{Sym}(I_{N})}$, there exists a measure-
preserving isomorphism
$g\in\mathrm{S}(\Omega,{\mathcal{B}},\mathbb{P};\sigma(\mathfrak{P}_{N}))$
such that

$(g_{k})_{\sharp}\mathbb{P}|_{\Omega_{N,k}}=\mathbb{P}|_{\Omega_{N,\sigma(k)}}\quad\forall
k\in I_{N},$

where $g_{k}$ is the restriction of $g$ to $\Omega_{N,k}$.

We introduce now the notion of _refined_ standard Borel measure space which
turns out to be useful when dealing with approximation of general measures
with discrete ones.

######  Definition B.3.

Let $(\Omega,{\mathcal{B}})$ be a standard Borel space endowed with a
nonatomic probability measure $\mathbb{P}$, and let
${\mathfrak{N}}\subset\mathbb{N}$ be an unbounded directed set w.r.t. $\prec$.
We say that a collection of partitions
$(\mathfrak{P}_{N})_{N\in{\mathfrak{N}}}$ of $\Omega$, with corresponding
sigma algebras ${\mathcal{B}}_{N}:=\sigma(\mathfrak{P}_{N})$, is a
_${\mathfrak{N}}$ -segmentation of $(\Omega,{\mathcal{B}},\mathbb{P})$_ if

  1. (1)

$\mathfrak{P}_{N}=\\{\Omega_{N,k}\\}_{k\in I_{N}}$ is a $N$-partition of
$(\Omega,{\mathcal{B}})$ for every $N\in{\mathfrak{N}}$,

  2. (2)

$\mathbb{P}(\Omega_{N,k})=\mathbb{P}(\Omega)/N$ for every $k\in I_{N}$ and
every $N\in{\mathfrak{N}}$,

  3. (3)

if $M\mid N=KM$ then $\bigcup_{k=0}^{K-1}\Omega_{N,mK+k}=\Omega_{M,m}$, $m\in
I_{M}$,

  4. (4)

$\sigma\left(\left\\{{\mathcal{B}}_{N}\mid
N\in{\mathfrak{N}}\right\\}\right)={\mathcal{B}}$.

In this case we call
$(\Omega,{\mathcal{B}},\mathbb{P},(\mathfrak{P}_{N})_{N\in{\mathfrak{N}}})$ a
_${\mathfrak{N}}$ -refined standard Borel probability space_.

######  Proposition B.4.

For any standard Borel space $(\Omega,{\mathcal{B}})$ endowed with a nonatomic
probability measure $\mathbb{P}$ and any unbounded directed set
${\mathfrak{N}}\subset\mathbb{N}$ w.r.t. $\prec$, there exists a
${\mathfrak{N}}$-segmentation of $(\Omega,{\mathcal{B}},\mathbb{P})$. If
${\mathfrak{N}}\subset\mathbb{N}$ is an unbounded directed subset w.r.t.
$\prec$, then there exists a totally ordered cofinal sequence
$(b_{n})_{n}\subset{\mathfrak{N}}$ satisfying

  * •

$b_{n}\mathrel{\text{\oalign{$\prec$\cr\nointerlineskip\kern
4.30554pt\cr$\smash{\raisebox{2.79857pt}{\rotatebox{90.0}{\scalebox{1.0}[-1.0]{$\nshortmid$}}}}$\cr\nointerlineskip\kern-2.15277pt\cr}}}b_{n+1}$
for every $n\in\mathbb{N}$,

  * •

for every $N\in\mathbb{N}$ there exists $n\in\mathbb{N}$ such that $N\mid
b_{n}.$

In particular, for every ${\mathfrak{N}}$-refined standard Borel measure space
$(\Omega,{\mathcal{B}},\mathfrak{m},(\mathfrak{P}_{N})_{N\in{\mathfrak{N}}})$
it holds that $({\mathcal{B}}_{b_{n}})_{n\in\mathbb{N}}$ is a filtration on
$(\Omega,{\mathcal{B}})$,

$\text{for every $N\in{\mathfrak{N}}$ there exists $n\in\mathbb{N}$ such that
}{\mathcal{B}}_{N}\subset{\mathcal{B}}_{b_{n}},$ (B.1)

and $\sigma\left(\left\\{{\mathcal{B}}_{b_{n}}\mid
n\in\mathbb{N}\right\\}\right)={\mathcal{B}}$.

For every every separable Hilbert space $\mathsf{X}$, we thus have that

$\bigcup_{N\in{\mathfrak{N}}}L^{2}(\Omega,{\mathcal{B}}_{N},\mathfrak{m};\mathsf{X})\text{
is dense in }L^{2}(\Omega,{\mathcal{B}},\mathfrak{m};\mathsf{X}).$ (B.2)

The next theorem contains approximation results for couplings by means of maps
in different situations.

######  Theorem B.5.

Let
$(\Omega,{\mathcal{B}},\mathbb{P},(\mathfrak{P}_{N})_{N\in{\mathfrak{N}}})$ be
a ${\mathfrak{N}}$-refined standard Borel probability space. Then:

  1. (1)

For every $\boldsymbol{\gamma}\in\Gamma(\mathbb{P},\mathbb{P})$ there exist a
totally ordered strictly increasing sequence
$(N_{n})_{n}\subset{\mathfrak{N}}$ and maps
$g_{n}\in\mathrm{S}(\Omega,{\mathcal{B}},\mathbb{P};{\mathcal{B}}_{N_{n}})$
such that, for every separable Hilbert space $\mathsf{X}$ and every $X,Y\in
L^{2}(\Omega,{\mathcal{B}},\mathbb{P};\mathsf{X})$ it holds

$(X,Y)_{\sharp}(\boldsymbol{i}_{\Omega},g_{n})_{\sharp}\mathbb{P}\to(X\otimes
Y)_{\sharp}\boldsymbol{\gamma}\text{ in }\mathcal{P}_{2}(\mathsf{X}^{2}).$
(B.3)

  2. (2)

If $\mathsf{X}$ is a separable Hilbert space and $X,X^{\prime}\in
L^{2}(\Omega,{\mathcal{B}},\mathbb{P};\mathsf{X})$, then for every
$\boldsymbol{\mu}\in\Gamma(X_{\sharp}\mathbb{P},X^{\prime}_{\sharp}\mathbb{P})$
there exist a totally ordered strictly increasing sequence
$(N_{n})_{n}\subset{\mathfrak{N}}$ and maps
$g_{n}\in\mathrm{S}(\Omega,{\mathcal{B}},\mathbb{P};{\mathcal{B}}_{N_{n}})$
such that

$(X,X^{\prime}\circ g_{n})_{\sharp}\mathbb{P}\to\boldsymbol{\mu}\text{ in
}\mathcal{P}_{2}(\mathsf{X}^{2}).$ (B.4)

In particular, if $X_{\sharp}\mathbb{P}=X^{\prime}_{\sharp}\mathbb{P}$, there
exist a totally ordered strictly increasing sequence
$(N_{n})_{n}\subset{\mathfrak{N}}$ and maps
$g_{n}\in\mathrm{S}(\Omega,{\mathcal{B}},\mathbb{P};{\mathcal{B}}_{N_{n}})$
such that $X^{\prime}\circ g_{n}\to X$ in
$L^{2}(\Omega,{\mathcal{B}},\mathbb{P};\mathsf{X})$ as $n\to\infty$.

Finally, if $(\Omega,{\mathcal{B}})$ is a standard Borel space endowed with a
nonatomic probability measure $\mathbb{P}$, $\mathsf{X}$ is a separable
Hilbert space, $\mu,\nu\in\mathcal{P}_{2}(\mathsf{X})$ and $X\in
L^{2}(\Omega,{\mathcal{B}},\mathbb{P};\mathsf{X})$ is s.t.
$X_{\sharp}\mathbb{P}=\mu$, then, for every $\varepsilon>0$, there exists
$Y\in L^{2}(\Omega,{\mathcal{B}},\mathbb{P};\mathsf{X})$ s.t.
$Y_{\sharp}\mathbb{P}=\nu$ and

$|X-Y|_{L^{2}(\Omega,{\mathcal{B}},\mathbb{P};\mathsf{X})}\leq
W_{2}(\mu,\nu)+\varepsilon.$

## References

  * [AGS08] Luigi Ambrosio, Nicola Gigli and Giuseppe Savaré  “Gradient Flows In Metric Spaces and in the Space of Probability Measures”, Lectures in Mathematics. ETH Zürich  Birkhäuser Basel, 2008
  * [AK22] Yurii Averboukh and Dmitry Khlopin  “Pontryagin maximum principle for the deterministic mean field type optimal control problem via the Lagrangian approach”  In _arXiv:2207.01892_ , 2022  DOI: 10.48550/ARXIV.2207.01892
  * [Amb+21] Luigi Ambrosio, Massimo Fornasier, Marco Morandotti and Giuseppe Savaré  “Spatially Inhomogeneous Evolutionary Games”  In _Communications on Pure and Applied Mathematics_ 74.7, 2021, pp. 1353–1402  DOI: https://doi.org/10.1002/cpa.21995
  * [AMQ21] Yurii Averboukh, Antonio Marigonda and Marc Quincampoix  “Extremal shift rule and viability property for mean field-type control systems”  In _J. Optim. Theory Appl._ 189.1, 2021, pp. 244–270  DOI: 10.1007/s10957-021-01832-z
  * [Att79] Hedy Attouch  “Familles d’opérateurs maximaux monotones et mesurabilite”  In _Ann. Mat. Pura Appl. (4)_ 120 Springer, Berlin/Heidelberg; Fondazione Annali di Matematica Pura ed Applicata c/o Dipartimento di Matematica “U. Dini”, Firenze, 1979, pp. 35–111  DOI: 10.1007/BF02411939
  * [Ave22] Yu.. Averboukh  “A mean field type differential inclusion with upper semicontinuous right-hand side”  In _Vestn. Udmurt. Univ. Mat. Mekh. Komp’yut. Nauki_ 32.4, 2022, pp. 489–501
  * [Bén72] Philippe Bénilan  “Solutions intégrales d’équations d’évolution dans un espace de Banach”  In _C. R. Acad. Sci. Paris Sér. A-B_ 274, 1972, pp. A47–A50
  * [BF21] Benoît Bonnet and Hélène Frankowska  “Necessary optimality conditions for optimal control problems in Wasserstein spaces”  In _Appl. Math. Optim._ 84.suppl. 2, 2021, pp. S1281–S1330  DOI: 10.1007/s00245-021-09772-w
  * [BF21a] Benoı̂t Bonnet and Hélène Frankowska  “Differential inclusions in Wasserstein spaces: the Cauchy-Lipschitz framework”  In _J. Differential Equations_ 271, 2021, pp. 594–637  DOI: 10.1016/j.jde.2020.08.031
  * [BF22] Zeinab Badreddine and Hélène Frankowska  “Solutions to Hamilton-Jacobi equation on a Wasserstein space”  In _Calc. Var. Partial Differential Equations_ 61.1, 2022, pp. Paper No. 941  DOI: 10.1007/s00526-021-02113-3
  * [BF22a] Zeinab Badreddine and Hélène Frankowska  “Viability and invariance of systems on metric spaces”  In _Nonlinear Analysis_ 225, 2022, pp. 113–133  DOI: https://doi.org/10.1016/j.na.2022.113133
  * [BF22b] Benoît Bonnet and Hélène Frankowska  “Viability and Exponentially Stable Trajectories for Differential Inclusions in Wasserstein Spaces”  In _2022 IEEE 61st Conference on Decision and Control (CDC)_ , 2022, pp. 5086–5091  DOI: 10.1109/CDC51059.2022.9992888
  * [BF23] Benoı̂t Bonnet and Hélène Frankowska  “Carathéodory theory and a priory estimates for continuity inclusions in the space of probability measures”  In _arXiv:2302.00963_ , 2023
  * [Bre10] H. Brezis  “Functional Analysis, Sobolev Spaces and Partial Differential Equations”, Universitext  Springer New York, 2010  URL: https://books.google.at/books?id=GAA2XqOIIGoC
  * [Bré73] Haim Brézis  “Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert” North-Holland Mathematics Studies, No. 5. Notas de Matemática (50)  North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York, 1973, pp. vi+183
  * [BW09] Heinz H. Bauschke and Xianfu Wang  “The kernel average for two convex functions and its application to the extension and representation of monotone operators”  In _Trans. Amer. Math. Soc._ 361.11, 2009, pp. 5947–5965  DOI: 10.1090/S0002-9947-09-04698-4
  * [BW10] Heinz H. Bauschke and Xianfu Wang  “Firmly nonexpansive and Kirszbraun-Valentine extensions: a constructive approach via monotone operator theory”  In _Nonlinear analysis and optimization I. Nonlinear analysis_ 513, Contemp. Math.  Amer. Math. Soc., Providence, RI, 2010, pp. 55–64  DOI: 10.1090/conm/513/10075
  * [Cam+21] Fabio Camilli, Giulia Cavagnari, Raul De Maio and Benedetto Piccoli  “Superposition principle and schemes for measure differential equations”  In _Kinet. Relat. Models_ 14.1, 2021, pp. 89–113  DOI: 10.3934/krm.2020050
  * [Car13] Pierre Cardaliaguet  “Notes on Mean Field Games” From P.-L. Lions’ lectures at College de France, link to the notes., 2013
  * [Cav+22] Giulia Cavagnari, Stefano Lisini, Carlo Orrieri and Giuseppe Savaré  “Lagrangian, Eulerian and Kantorovich formulations of multi-agent optimal control problems: equivalence and gamma-convergence”  In _J. Differential Equations_ 322, 2022, pp. 268–364  DOI: 10.1016/j.jde.2022.03.019
  * [CB18] Lenaic Chizat and Francis Bach  “On the Global Convergence of Gradient Descent for Over-parameterized Models using Optimal Transport”  In _Advances in neural information processing systems_ , 2018, pp. 3036–3056
  * [CCR11] J.. Cañizo, J.. Carrillo and J. Rosado  “A well-posedness theory in measures for some kinetic models of collective motion”  In _Math. Models Methods Appl. Sci._ 21.3, 2011, pp. 515–539  DOI: 10.1142/S0218202511005131
  * [CD18] René Carmona and François Delarue  “Probabilistic theory of mean field games with applications. I” Mean field FBSDEs, control, and games 83, Probability Theory and Stochastic Modelling  Springer, Cham, 2018, pp. xxv+713
  * [CM22] Rossana Capuani and Antonio Marigonda  “Constrained Mean Field Games Equilibria as Fixed Point of Random Lifting of Set-Valued Maps” 25th International Symposium on Mathematical Theory of Networks and Systems MTNS 2022  In _IFAC-PapersOnLine_ 55.30, 2022, pp. 180–185  DOI: https://doi.org/10.1016/j.ifacol.2022.11.049
  * [CMP20] Giulia Cavagnari, Antonio Marigonda and Benedetto Piccoli  “Generalized dynamic programming principle and sparse mean-field control problems”  In _J. Math. Anal. Appl._ 481.1, 2020, pp. 123–13745  DOI: 10.1016/j.jmaa.2019.123437
  * [CMQ21] Giulia Cavagnari, Antonio Marigonda and Marc Quincampoix  “Compatibility of state constraints and dynamics for multiagent control systems”  In _J. Evol. Equ._ 21.4, 2021, pp. 4491–4537  DOI: 10.1007/s00028-021-00724-z
  * [CSS23] Giulia Cavagnari, Giuseppe Savaré and Giacomo Enrico Sodini  “Dissipative probability vector fields and generation of evolution semigroups in Wasserstein spaces”  In _Probab. Theory Related Fields_ 185.3-4, 2023, pp. 1087–1182  DOI: 10.1007/s00440-022-01148-7
  * [CSS23a] Giulia Cavagnari, Giuseppe Savaré and Giacomo Enrico Sodini  “Extension of monotone operators and Lipschitz maps invariant for a group of isometries”  In _arXiv:2305.04678_ , 2023
  * [DOr+06] M.. D’Orsogna, Y.-L. Chuang, A.. Bertozzi and L. Chayes  “Self-propelled particles with soft-core interactions: Patterns, stability, and collapse”  In _Phys. Rev. Lett._ 96, 2006, pp. 104302–1/4
  * [For+19] M. Fornasier, S. Lisini, C. Orrieri and G. Savaré  “Mean-field optimal control as gamma-limit of finite agent controls”  In _European J. Appl. Math._ 30.6, 2019, pp. 1153–1186  DOI: 10.1017/s0956792519000044
  * [FSS22] Massimo Fornasier, Giacomo E. Sodini and Giuseppe Savaré  “Density of subalgebras of Lipschitz functions in metric Sobolev spaces and applications to Sobolev-Wasserstein spaces”  In _arXiv:2209.00974_ , 2022
  * [GT19] Wilfrid Gangbo and Adrian Tudorascu  “On differentiability in the Wasserstein space and well-posedness for Hamilton-Jacobi equations”  In _J. Math. Pures Appl. (9)_ 125, 2019, pp. 119–174  DOI: 10.1016/j.matpur.2018.09.003
  * [JKO98] Richard Jordan, David Kinderlehrer and Felix Otto  “The variational formulation of the Fokker-Planck equation”  In _SIAM J. Math. Anal._ 29.1, 1998, pp. 1–17  DOI: 10.1137/S0036141096303359
  * [JKO98a] Richard Jordan, David Kinderlehrer and Felix Otto  “The variational formulation of the Fokker-Planck equation”  In _SIAM J. Math. Anal._ 29.1, 1998, pp. 1–17  DOI: 10.1137/S0036141096303359
  * [JMQ20] Chloé Jimenez, Antonio Marigonda and Marc Quincampoix  “Optimal control of multiagent systems in the Wasserstein space”  In _Calc. Var. Partial Differential Equations_ 59.2, 2020, pp. Paper No. 5845  DOI: 10.1007/s00526-020-1718-6
  * [Lio07] Pierre-Louis Lions  “Théorie des jeux à champs moyen et applications”, Lectures at the Collège de France, 2007  URL: http://www.college-de-france.fr/default/EN/all/equ_der/cours_et_seminaires.htm
  * [McC97] Robert J. McCann  “A convexity principle for interacting gases”  In _Adv. Math._ 128.1, 1997, pp. 153–179  DOI: 10.1006/aima.1997.1634
  * [NS06] Ricardo H. Nochetto and Giuseppe Savaré  “Nonlinear evolution governed by accretive operators in Banach spaces: error control and applications”  In _Math. Models Methods Appl. Sci._ 16.3, 2006, pp. 439–477  DOI: 10.1142/S0218202506001224
  * [NS09] Luca Natile and Giuseppe Savaré  “A Wasserstein approach to the one-dimensional sticky particle system”  In _SIAM J. Math. Anal._ 41.4, 2009, pp. 1340–1365  DOI: 10.1137/090750809
  * [NS21] Emanuele Naldi and Giuseppe Savaré  “Weak topology and Opial property in Wasserstein spaces, with applications to gradient flows and proximal point algorithms of geodesically convex functionals”  In _Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl._ 32.4, 2021, pp. 725–750  DOI: 10.4171/rlm/955
  * [NSV00] Ricardo H. Nochetto, Giuseppe Savaré and Claudio Verdi  “A posteriori error estimates for variable time-step discretizations of nonlinear evolution equations”  In _Comm. Pure Appl. Math._ 53.5, 2000, pp. 525–589  DOI: 10.1002/(SICI)1097-0312(200005)53:5¡525::AID-CPA1¿3.0.CO;2-M
  * [Ott01] Felix Otto  “The geometry of dissipative evolution equations: the porous medium equation”  In _Comm. Partial Differential Equations_ 26.1-2, 2001, pp. 101–174  DOI: 10.1081/PDE-100002243
  * [Pic18] Benedetto Piccoli  “Measure differential inclusions”  In _2018 IEEE Conference on Decision and Control (CDC)_ , 2018, pp. 1323–1328
  * [Pic19] Benedetto Piccoli  “Measure differential equations”  In _Arch. Ration. Mech. Anal._ 233.3, 2019, pp. 1289–1317  DOI: 10.1007/s00205-019-01379-4
  * [Pic23] Benedetto Piccoli  “Control of multi-agent systems: results, open problems, and applications”  In _arXiv:2302.12308_ , 2023  DOI: 10.48550/ARXIV.2302.12308
  * [Pog16] Nikolay Pogodaev  “Optimal control of continuity equations”  In _NoDEA Nonlinear Differential Equations Appl._ 23.2, 2016, pp. Art. 2124  DOI: 10.1007/s00030-016-0357-2
  * [Qi83] Liqun Qi  “Uniqueness of the maximal extension of a monotone operator”  In _Nonlinear Anal., Theory Methods Appl._ 7 Elsevier Science Ltd. (Pergamon), Oxford, 1983, pp. 325–332  DOI: 10.1016/0362-546X(83)90086-X
  * [Szn91] Alain-Sol Sznitman  “Topics in propagation of chaos”  In _École d’Été de Probabilités de Saint-Flour XIX—1989_ 1464, Lecture Notes in Math.  Springer, Berlin, 1991, pp. 165–251  DOI: 10.1007/BFb0085169