case, $\vec{\Psi}^{(sing)}_{j}=\vec{\Psi}_{j}$ for all $j=1,...,n$, so that
from (6.40) for $u_{k}^{c}\neq u_{j}^{c}$ (otherwise $c_{jk}=0$ and there is
nothing to prove)

$\vec{\Psi}_{k}(\lambda,u)\underset{\lambda\to
u_{j}}{=}\vec{\Psi}_{j}(\lambda,u)c_{jk}+\hbox{\rm reg}(\lambda-u_{j}).$
(6.41)

Using the labelling (6.1)-(6.2), from the proof of Theorem 6.1 we have the
fundamental matrix solution

$\Psi^{(\boldsymbol{p}_{1})}(\lambda,u)=\Bigl{[}\vec{\Psi}_{1}(\lambda,u)~{}\Bigr{|}~{}\cdots~{}\Bigr{|}\vec{\Psi}_{p_{1}}(\lambda,u)~{}\Bigr{|}~{}\vec{\varphi}_{p_{1}+1}^{~{}(1)}(\lambda,u)~{}\Bigr{|}~{}\cdots~{}\Bigr{|}~{}\vec{\varphi}_{n}^{~{}(1)}(\lambda,u)\Bigr{]}$

and in general at each $\lambda_{\alpha}$, $\alpha=1,...,s$ (with
$\sum_{j=1}^{\alpha-1}p_{j}=0$ for $\alpha=1$) we have

$\displaystyle\Psi^{(\boldsymbol{p}_{\alpha})}(\lambda,u)=$ $\displaystyle=$
$\displaystyle\Bigl{[}\vec{\varphi}_{1}^{~{}(\alpha)}(\lambda,u)~{}\Bigl{|}~{}\cdots~{}\Bigl{|}~{}\vec{\varphi}_{\sum_{j=1}^{\alpha-1}p_{j}}^{~{}(\alpha)}(\lambda,u)~{}\Bigl{|}\vec{\Psi}_{\sum_{j=1}^{\alpha-1}p_{j}+1}(\lambda,u)~{}\Bigl{|}~{}\vec{\Psi}_{\sum_{j=1}^{\alpha-1}p_{j}+2}(\lambda,u)~{}\Bigl{|}~{}\cdots~{}\Bigr{|}\vec{\Psi}_{\sum_{j=1}^{\alpha}p_{j}}(\lambda,u)~{}\Bigr{|}$
$\displaystyle\Bigr{|}~{}\vec{\varphi}_{\sum_{j=1}^{\alpha}p_{j}+1}^{~{}(\alpha)}(\lambda,u)~{}\Bigr{|}~{}\cdots~{}|~{}\vec{\varphi}_{n}^{~{}(\alpha)}(\lambda,u)\Bigr{]}$

where

$\vec{\Psi}_{m}(\lambda,u)=\vec{\psi}_{m}(\lambda,u)(\lambda-
u_{m})^{-\lambda^{\prime}_{m}-1},\quad
m=\sum_{j=1}^{\alpha-1}p_{j}+1,~{}\dots~{},\sum_{j=1}^{\alpha}p_{j},$

and the $\vec{\psi}_{m}(\lambda,u)$ and
$\vec{\varphi}_{r}^{~{}(\alpha)}(\lambda,u)$ are holomorphic functions in the
corresponding $\mathbb{D}_{\alpha}\times\mathbb{D}(u^{c})$. The above allows
us to explicitly rewrite (6.41), for $j$ such that
$u_{j}^{c}=\lambda_{\alpha}$, as

$\vec{\Psi}_{k}(\lambda,u)=\sum_{m=p_{1}+\cdots+p_{\alpha-1}+1}^{p_{1}+\cdots+p_{\alpha}}c_{mk}~{}\vec{\Psi}_{m}(\lambda,u)+\sum_{r\not\in\\{p_{1}+\cdots+p_{\alpha-1}+1,...,p_{1}+\cdots+p_{\alpha}\\}}h_{r}\vec{\varphi}_{r}^{~{}(\alpha)}(\lambda,u),$
(6.42)

for suitable constant coefficients $h_{r}$. Here one of the $c_{mk}$ is
$c_{jk}$ of (6.41).

Each $u_{m}$, with
$m=p_{1}+\cdots+p_{\alpha-1}+1,...,p_{1}+\cdots+p_{\alpha}$, varies in
$\mathbb{D}_{\alpha}$. Firstly, we can fix $\lambda=\lambda_{\alpha}$ in
(6.42), consider the branch cut $\mathcal{L}_{\alpha}$ from $\lambda_{\alpha}$
to infinity in direction $\eta$ (see Figure 3), and let $u$ vary in such a way
that each $u_{p_{1}+\cdots+p_{\alpha-1}+1}$, …, $u_{p_{1}+\cdots+p_{\alpha}}$
varies in $\mathbb{D}_{\alpha}\backslash\mathcal{L}_{\alpha}$, so that in the
r.h.s. of (6.42) all the $\vec{\Psi}_{m}(\lambda_{\alpha},u)$ and
$\vec{\varphi}_{r}^{~{}(\alpha)}(\lambda_{\alpha},u)$ are holomorphic with
respect to $u$, provided that $u_{m}\neq\lambda_{\alpha}$. If $u$ varies, with
the constraint that the $u_{m}$’s must remain in
$\mathbb{D}_{\alpha}\backslash\mathcal{L}_{\alpha}$, every $\tau$-cell of
$\mathbb{D}(u^{c})$ can be reached starting from an initial point in one
specific cell. This proves, by $u$-analytic continuation of (6.42) with fixed
$\lambda=\lambda_{\alpha}$, that the coefficients $c_{mk}$ are
constant161616Recall that
$\mathbb{D}(u^{c})=\bigtimes_{\beta=1}^{s}\mathbb{D}_{\beta}^{\times
p_{\beta}}$. in $(\mathbb{D}_{\alpha}\backslash\mathcal{L}_{\alpha})^{\times
p_{\alpha}}\times\left(\bigtimes_{\beta\neq\alpha}\mathbb{D}_{\beta}^{\times
p_{\beta}}\right)\subset\mathbb{D}(u^{c})$.

Now, we can slightly vary $\eta$ in $\eta_{\nu+1}<\eta<\eta_{\nu}$, so that
the cut $\mathcal{L}_{\alpha}$ is irrelevant171717The crossing locus
$X(\tau)$, $\tau=3\pi/2-\eta$, is as arbitrary as is the choice of $\tau$ in
the range $\tau_{\nu}<\tau<\tau_{\nu+1}$.. Thus, the $c_{mk}$ are constant on
$\left\\{u\in\mathbb{D}(u^{c})~{}|~{}u_{p_{1}+...+p_{\alpha-1}+1}\neq\lambda_{\alpha},\dots,u_{p_{1}+...+p_{\alpha}}\neq\lambda_{\alpha}\right\\}$.

Finally, we fix another value $\lambda=\lambda^{*}\in\mathbb{D}_{\alpha}$ in
(6.42), and repeat the above discussion with cuts $\mathcal{L}_{\alpha}$
issuing from $\lambda^{*}$, so that all the $c_{mk}$ are constant on
$\left\\{u\in\mathbb{D}(u^{c})~{}|~{}u_{p_{1}+...+p_{\alpha-1}+1}\neq\lambda^{*},\dots,u_{p_{1}+...+p_{\alpha}}\neq\lambda^{*}\right\\}$.
This proves constancy of the $c_{mk}$, $m$ associated with $\lambda_{\alpha}$,
on the whole $\mathbb{D}(u^{c})$. Then, we repeat this for all
$\alpha=1,..,s$, proving constancy of the $c_{jk}$ for all $j=1,...,n$. Hence,
Proposition 5.1 is proved in the generic case.

General case of any $A(u)$. If some of the diagonal entries
$\lambda^{\prime}_{1},...,\lambda^{\prime}_{n}$ of $A$ are integers, or some
eigenvalues are integers, there exists a sufficiently small $\gamma_{0}>0$
such that, for any $0<\gamma<\gamma_{0}$, $A-\gamma I$ has diagonal non-
integer entries $\lambda^{\prime}_{1}-\gamma,...,\lambda^{\prime}_{n}-\gamma$
and no integer eigenvalues. Take such a $\gamma_{0}$, and for any
$0<\gamma<\gamma_{0}$ consider

$(\Lambda-\lambda)\frac{d}{d\lambda}(~{}{}_{\gamma}\Psi)=\Bigl{(}(A(u)-\gamma
I)+I\Bigr{)}~{}{}_{\gamma}\Psi.$ (6.43)

namely

$\frac{d}{d\lambda}({}_{\gamma}\Psi)=\sum_{k=1}^{n}\frac{B_{k}[\gamma](u)}{\lambda-
u_{k}}{}_{\gamma}\Psi,~{}~{}~{}~{}~{}B_{k}[\gamma](u):=-E_{k}\Bigl{(}A(u)+(1-\gamma)I\Bigr{)}.$
(6.44)

######  Lemma 6.5.

The above system (6.44) is strongly isomonodromic in $\mathbb{D}(u^{0})$
contained in a $\tau$-cell, and $\lambda$-component of the integrable Pfaffian
system

$d{}_{\gamma}\Psi=P_{[\gamma]}(\lambda,u){}_{\gamma}\Psi,\quad\quad
P_{[\gamma]}(\lambda,u)=\sum_{k=1}^{n}\frac{B_{k}[\gamma](u)}{\lambda-
u_{k}}d(\lambda-u_{k})+\sum_{j=1}^{n}[F_{1}(u),E_{j}]du_{j}.$ (6.45)

where $F_{1}(u)$ is defined as in (2.8),
$(F_{1})_{ij}=\frac{A_{ij}}{u_{j}-u_{i}}$, $i\neq j$, and $[F_{1}(u),E_{j}]$
is (2.18).

###### Proof.

We do a gauge transformation

${}_{\gamma}Y(z):=z^{-\gamma}Y(z),\quad\quad\gamma\in\mathbb{C},$ (6.46)

which transforms (1.1) into

$\frac{d({}_{\gamma}Y)}{dz}=\left(\Lambda+\frac{A-\gamma
I}{z}\right)~{}{}_{\gamma}Y$ (6.47)

For $u\in\mathbb{D}(u^{0})$ contained in a $\tau$-cell, we write the unique
formal solution

${}_{\gamma}Y_{F}(z,u)=z^{-\gamma}Y_{F}(z,u),$ (6.48)

where $Y_{F}(z,u)$ is (2.4), so that

${}_{\gamma}Y_{F}(z,u)=F(z,u)z^{B-\gamma I}e^{\Lambda z},\quad\quad B-\gamma
I=\hbox{\rm diag}(A-\gamma)={\rm
diag}(\lambda^{\prime}_{1}-\gamma,~{}...~{},\lambda^{\prime}_{n}-\gamma).$

The crucial point is that $F(z,u)$ is the same as (2.5), so all the $F_{k}(u)$
are independent of $\gamma$. The fundamental matrix solutions

${}_{\gamma}Y_{\nu}(z,u):=z^{-\gamma}Y_{\nu}(z,u),$

are uniquely defined by their asymptotics ${}_{\gamma}Y_{F}(z,u)$ in
$\mathcal{S}_{\nu}(\mathbb{D}(u^{0}))$. Their Stokes matrices do not depend on
$\gamma$ because

${}_{\gamma}Y_{\nu+(k+1)\mu}(z,u)={}_{\gamma}Y_{\nu+k\mu}(z,u)\mathbb{S}_{\nu+k\mu}\quad\Longleftrightarrow\quad
Y_{\nu+(k+1)\mu}(z,u)=Y_{\nu+k\mu}(z,u)\mathbb{S}_{\nu+k\mu}.$

The system (6.47) is thus strongly isomonodromic. By Proposition 3.1 we
conclude. ∎

######  Corollary 6.2.

Let the assumptions of Theorem 5.1 hold. Then Theorem 5.1 holds also for
(6.45).

By Theorem 5.1 applied to (6.45), we receive independent vector solutions
${}_{\gamma}\vec{\Psi}_{k}(\lambda,u)\equiv{}_{\gamma}\vec{\Psi}^{(sing)}_{k}(\lambda,u)$,
$k=1,...,n$, which form a fundamental matrix

${}_{\gamma}\Psi(\lambda,u):=[{}_{\gamma}\vec{\Psi}_{1}(\lambda,u)~{}|~{}\cdots~{}|{}_{\gamma}\vec{\Psi}_{n}(\lambda,u)].$

For system (6.45) the results already proved in the generic case hold.
Therefore, the connection coefficients $c_{jk}^{(\nu)}[\gamma]$ defined by

${}_{\gamma}\vec{\Psi}_{k}(\lambda,u~{}|\nu)={}_{\gamma}\vec{\Psi}_{j}(\lambda,u~{}|\nu)~{}c_{jk}^{(\nu)}[\gamma]~{}+\hbox{\rm
reg}(\lambda-u_{j}),\quad\quad\lambda\in\mathcal{P}_{\eta},$ (6.49)

are constant on $\mathbb{D}(u^{c})$. They depend on $\gamma$, but not on
$u\in\mathbb{D}(u^{c})$.

######  Remark 6.4.

It is explained in section 8 of [23] what is the relation between
$\vec{\Psi}^{(sing)}_{k}$ and ${}_{\gamma}\vec{\Psi}_{k}$, by means of their
primitives, and that in general both $\lim_{\gamma\to
0}{}_{\gamma}\vec{\Psi}_{k}$ and $\lim_{\gamma\to 0}c_{jk}^{(\nu)}[\gamma]$
are divergent.

Now, we invoke Proposition 10 of [23], which holds with no assumptions on
eigenvalues and diagonal entries of $A(u)$.181818The proof in [23] is
laborious, because it is necessary to take into account all possible values of
the diagonal entries $\lambda^{\prime}_{k}$ of $A$, including integer values.
In [4] the proof is given only for non-integer values. This result, adapted to
our case, reads as follows.

######  Proposition 6.3.

Let $u$ be fixed in a $\tau$-cell. Let $\gamma_{0}>0$ be small enough such
that for any $0<\gamma<\gamma_{0}$ the matrix $A-\gamma I$ has no integer
eigenvalues, and its diagonal part has no integer entries.191919Recall that
eigenvalues and diagonal entries do not depend on $u$, in the isomonodromic
case. Let $c_{jk}^{(\nu)}$ be the connection coefficients of the Fuchsian
system (1.4) at the fixed $u$, as in Definition 5.1. Let
$c_{jk}^{(\nu)}[\gamma]$ be the connection coefficients in (6.49). Let

$\alpha_{k}:=\left\\{\begin{array}[]{cc}e^{-2\pi
i\lambda^{\prime}_{k}}-1,&\lambda^{\prime}_{k}\not\in\mathbb{Z}\\\ \vskip
6.0pt plus 2.0pt minus 2.0pt\cr 2\pi
i,&\lambda^{\prime}_{k}\in\mathbb{Z}\end{array}\right.;~{}~{}~{}~{}~{}~{}~{}~{}\alpha_{k}[\gamma]:=e^{-2\pi
i(\lambda^{\prime}_{k}-\gamma)}-1$

Then, the following equalities hold

$\alpha_{k}c_{jk}^{(\nu)}=e^{-2\pi
i\gamma}\alpha_{k}[\gamma]~{}c_{jk}^{(\nu)}[\gamma],\quad\hbox{ if }k\succ
j;\quad\quad\alpha_{k}c_{jk}^{(\nu)}=\alpha_{k}[\gamma]~{}c_{jk}^{(\nu)}[\gamma],\quad\hbox{
if }k\prec j;$ (6.50)

where the ordering relation $j\prec k$ means, for the fixed $u$, that
$\Re(z(u_{j}-u_{k}))<0$ for $\arg z=\tau=3\pi/2-\eta$ satisfying (5.2).

We use Proposition 6.3 to conclude the proof of Proposition 5.1 in the general
case. Indeed, the proposition is already proved in the generic case, so it
holds for the $c_{jk}^{(\nu)}[\gamma]$. Therefore, they are constant on the
whole $\mathbb{D}(u^{c})$. Equalities (6.50) hold at any fixed $u$ in
$\tau$-cell, so that each $c_{jk}^{(\nu)}$ is constant on a $\tau$-cell, and
such consta nt is the same in each $\tau$-cell. With a slight variation of
$\eta$ in $(\eta_{\nu+1},\eta_{\nu})$, equalities (6.50) hold also at the
crossing locus $X(\tau)$. They analytically extend at $\Delta$.

∎

##  7 Laplace Transform in $\mathbb{D}(u^{c})$, Theorem 7.1

By means of the Laplace transform with deformation parameters, we prove points
(I1),(I2), (I3), (II1), (II2) and (II5) of Theorem 2.2. Stokes matrices will
be expressed in terms of the isomonodromic connection coefficients satisfying
Proposition 5.1. The result is in Theorem 7.1 below, which is the last step of
our construction.

Let $\tau$ be the chosen direction in the $z$-plane admissible at $u^{c}$, and
$\eta=3\pi/2-\tau$ in the $\lambda$-plane. The Stokes rays of $\Lambda(u^{c})$
will be labelled as in (2.21), so that (5.2) holds for a certain
$\nu\in\mathbb{Z}$. We define the sectors

$\mathcal{S}_{\nu}=\\{z\in\mathcal{R}(\mathbb{C}\backslash\\{0\\})~{}\hbox{
such that }~{}\tau_{\nu}-\pi<\arg z<\tau_{\nu+1}\\}.$ (7.1)

If $u$ only varies in $\mathbb{D}(u^{0})$ contained in a ${\tau}$-cell, then
none of the Stokes rays associated with $\Lambda(u)$ crosses $\arg z={\tau}$
mod $\pi$. If $u$ varies in $\mathbb{D}(u^{c})$, some Stokes rays associated
with $\Lambda(u)$ necessarily cross $\arg z={\tau}$ mod $\pi$ (see Section
2.1.2). Consider the subset of the set of Stokes rays satisfying
$\Re(z(u_{j}-u_{k}))=0$, $z\in\mathcal{R}$, associated with pairs
$(u_{j},u_{k})$ such that $u_{j}\in\mathbb{D}_{\alpha}$ and
$u_{k}\in\mathbb{D}_{\beta}$, $\alpha\neq\beta$, namely $u_{j}^{c}\neq
u_{k}^{c}$. Following [13], we denote this subset by $\mathfrak{R}(u)$. If $u$
varies in $\mathbb{D}(u^{c})$ and $\epsilon_{0}$ satisfies (5.1), the rays in
$\mathfrak{R}(u)$ continuously rotate, but never cross the admissible rays
$\arg z={\tau}+h\pi$, where

$\tau_{\nu+h\mu}<{\tau}+h\pi<\tau_{\nu+h\mu+1},\quad\quad h\in\mathbb{Z},$
(7.2)

The above allows to define $\widehat{\mathcal{S}}_{\nu+h\mu}(u)$ to be the
unique sector containing $S\bigl{(}{\tau}+(h-1)\pi,{\tau}+h\pi\bigr{)}$ and
extending up to the nearest Stokes rays in $\mathfrak{R}(u)$. Then, let

$\widehat{\mathcal{S}}_{\nu+h\mu}:=\bigcap_{u\in\mathbb{D}(u^{c})}\widehat{\mathcal{S}}_{\nu+h\mu}(u).$
(7.3)

It has angular amplitude greater than $\pi$. The reason for the labeling is
that $\widehat{\mathcal{S}}_{\nu+h\mu}(u^{c})=\mathcal{S}_{\nu+h\mu}$ in
(7.1).

Suppose that $u$ is fixed in a $\tau$-cell. Let

$Y_{\nu+h\mu}(z,u):=\Bigl{[}\vec{Y}_{1}(z,u~{}|\nu+h\mu)~{}\Bigr{|}~{}\dots~{}\Bigr{|}~{}\vec{Y}_{n}(z,u~{}|\nu+h\mu)\Bigr{]},$

be defined by

$\displaystyle\vec{Y}_{k}(z,u~{}|\nu+h\mu):=\dfrac{1}{2\pi
i}\int_{\gamma_{k}(\eta-h\pi)}e^{z\lambda}\vec{\Psi}_{k}^{(sing)}(\lambda,u~{}|\nu+h\mu)d\lambda,$
for $\lambda^{\prime}_{k}\not\in\mathbb{Z}_{-}$, (7.4)
$\displaystyle\vec{Y}_{k}(z,u~{}|\nu+h\mu):=\int_{L_{k}(\eta-h\pi)}e^{z\lambda}\vec{\Psi}_{k}(\lambda,u~{}|\nu+h\mu)d\lambda,$
$\displaystyle\hbox{ for $\lambda^{\prime}_{k}\in\mathbb{Z}_{-}$}.$ (7.5)

In the $\lambda$-plane, the admissible directions $\eta-h\pi$ correspond to
${\tau}+h\pi$, with

$\eta_{\nu+h\mu+1}<\eta-h\pi<\eta_{\nu+h\mu}.$ (7.6)

Here, $\vec{\Psi}_{k}(\lambda,u~{}|\nu+h\mu)$,
$\vec{\Psi}_{k}^{(sing)}(\lambda,u~{}|\nu+h\mu)$ are the vector solutions of
Theorem 5.1 for $\lambda\in\mathcal{P}_{\eta-h\pi}(u)$, with $u$ fixed in a
$\tau$-cell. $L_{k}(\eta-h\pi)$ is the cut in direction $\eta-h\pi$, issuing
from $u_{k}$ and oriented from $u_{k}$ to $\infty$, and
$\gamma_{k}(\eta-h\pi)$ is the path coming from $\infty$ along the left side
of $L_{k}(\eta-h\pi)$, encircling $u_{k}$ with a small loop excluding all the
other poles, and going back to $\infty$ along the right side of
$L_{k}(\eta-h\pi)$. Here “right” and “left” refer to the orientation of
$L_{k}(\eta-h\pi)$. The label $\nu+h\mu$ keeps track of (5.2) and (7.2)-(7.6).

######  Theorem 7.1.

Let the assumptions of Theorem 5.1 hold.

  * 1)

The matrices $Y_{\nu+h\mu}(z,u)$, obtained by Laplace transform (7.4)-(7.5) at
a fixed $u\in\mathbb{D}(u^{0})$ contained in a $\tau$-cell, define holomorphic
matrix valued functions of
$(\lambda,u)\in\mathcal{R}(\mathbb{C}\backslash\\{0\\})\times\mathbb{D}(u^{c})$,
which are fundamental matrix solutions of (1.1).

  * 2)

They have structure

$Y_{\nu+h\mu}(z,u)=\widehat{Y}_{\nu+h\mu}(z,u)z^{B}e^{z\Lambda},\quad\quad
B=\hbox{\rm diag}(\lambda^{\prime}_{1},...,\lambda^{\prime}_{n}),$

with asymptotic behaviour, uniform in $u\in\mathbb{D}(u^{c})$,

$\widehat{Y}_{\nu+h\mu}(z,u)\sim
F(z,u)=I+\sum_{l=1}^{\infty}\frac{F_{l}(u)}{z^{l}},\quad\quad z\to\infty\hbox{
in }\widehat{S}_{\nu+h\mu},$

given by the formal solution $Y_{F}(z,u)=F(z,u)z^{B}e^{z\Lambda}$. The
coefficients $F_{l}(u)$ are holomorphic in $\mathbb{D}(u^{c})$. The explicit
expression of their columns is (7.12), (7.13), (7.15) (or (7.16)) and (7.17).

  * 3)

Stokes matrices defined by

$Y_{\nu+(h+1)\mu}(z,u)=Y_{\nu+h\mu}(z,u)\mathbb{S}_{\nu+h\mu},\quad\quad
z\in\widehat{\mathcal{S}}_{\nu+h\mu}\cap\widehat{\mathcal{S}}_{\nu+(h+1)\mu},$
(7.7)

are constant in the whole $\mathbb{D}(u^{c})$ and satisfy

$(\mathbb{S}_{\nu+h\mu})_{ab}=(\mathbb{S}_{\nu+h\mu})_{ba}=0\quad\hbox{ for
$a\neq b$ such that $u_{a}^{c}=u_{b}^{c}$}.$ (7.8)

  * 4)

The following representation in terms of the constant connection coefficients
$c_{jk}^{(\nu)}$ of Proposition 5.1 holds on $\mathbb{D}(u^{c})$:

$\begin{array}[]{c}(\mathbb{S}_{\nu})_{jk}=\\\ \\\
\left\\{\begin{array}[]{cc}e^{2\pi
i\lambda^{\prime}_{k}}\alpha_{k}~{}c_{jk}^{(\nu)},&\hbox{ $j\prec k$,
$u_{j}^{c}\neq u_{k}^{c}$},\\\ \vskip 6.0pt plus 2.0pt minus 2.0pt\cr
1&j=k,\\\ \vskip 6.0pt plus 2.0pt minus 2.0pt\cr 0&\hbox{ $j\succ k$,
$u_{j}^{c}\neq u_{k}^{c}$},\\\ \vskip 6.0pt plus 2.0pt minus 2.0pt\cr 0&\hbox{
$j\neq k$ ,
$u_{j}^{c}=u_{k}^{c}$},\end{array}\right.;\end{array}\quad\quad\begin{array}[]{c}(\mathbb{S}_{\nu+\mu}^{-1})_{jk}=\\\
\\\ \left\\{\begin{array}[]{cc}0&\hbox{ $j\neq k$, $u_{j}^{c}=u_{k}^{c}$},\\\
\vskip 6.0pt plus 2.0pt minus 2.0pt\cr 0&\hbox{ $j\prec k$, $u_{j}^{c}\neq
u_{k}^{c}$},\\\ \vskip 6.0pt plus 2.0pt minus 2.0pt\cr 1&j=k,\\\ \vskip 6.0pt
plus 2.0pt minus 2.0pt\cr-e^{2\pi
i(\lambda^{\prime}_{k}-\lambda^{\prime}_{j})}\alpha_{k}~{}c_{jk}^{(\nu)}&\hbox{
$j\succ k$, $u_{j}^{c}\neq u_{k}^{c}$},\end{array}\right.\end{array}$ (7.9)

where the relation $j\prec k$ is defined for $j\neq k$ such that
$u_{j}^{c}\neq u_{k}^{c}$ and means that $\Re(z(u_{j}^{c}-u_{k}^{c}))<0$ when
$\arg z=\tau$.

######  Remark 7.1.

The above (7.9) generalises Theorem 2.3 in presence of isomonodromic
deformation parameters, including coalescences. Notice that the ordering
relation $\prec$ here is referred to $u^{c}$, while in Theorem 2.3 it refers
to $u^{0}$.

###### Proof.

We use the labelling (6.1)-(6.2) for $u^{c}$. a) Case
$\lambda^{\prime}_{k}\not\in\mathbb{Z}$.

$\bullet$ Construction of $\vec{Y}_{k}(z,u~{}|\nu)$. We have
$\vec{\Psi}_{k}^{(sing)}(\lambda,u|~{}\nu)=\vec{\Psi}_{k}(\lambda,u|~{}\nu)$
and (7.4) is

$\vec{Y}_{k}(z,u~{}|\nu):=\dfrac{1}{2\pi
i}\int_{\gamma_{k}(\eta)}e^{z\lambda}\vec{\Psi}_{k}(\lambda,u~{}|\nu)d\lambda$
(7.10)

Since $\vec{\Psi}_{k}(\lambda,u~{}|\nu)$ grows at infinity no faster than some
power of $\lambda$, the integral converges in a sector of amplitude at most
$\pi$. Now, $\vec{\Psi}_{k}(\lambda,u~{}|\nu)$ satisfies Theorem 5.1, hence if
$u$ varies in $\mathbb{D}(u^{c})$ the following facts hold.

  * 1.

$\vec{\Psi}_{k}(\lambda,u~{}|\nu)$ is branched at $\lambda=u_{k}$ and possibly
at other poles $u_{l}$ such that $u_{l}^{c}\neq u_{k}^{c}$.

  * 2.

$\vec{\Psi}_{k}(\lambda,u~{}|\nu)$ is holomorphic at all $\lambda=u_{j}$ such
that $u_{j}^{c}=u_{k}^{c}$, $j\neq k$.

It follows from 1. and 2. that the path of integration can be modified: for
$\alpha$ such that $u_{k}^{c}=\lambda_{\alpha}$, we have

$\vec{Y}_{k}(z,u~{}|\nu)=\dfrac{1}{2\pi
i}\int_{\Gamma_{\alpha}(\eta)}e^{z\lambda}\vec{\Psi}_{k}(\lambda,u~{}|\nu)d\lambda,$
(7.11)

where $\Gamma_{\alpha}(\eta)$ is the path which comes from $\infty$ in
direction $\eta-\pi$, encircles $\lambda_{\alpha}$ along
$\partial\mathbb{D}_{\alpha}$ anti-clockwise and goes to $\infty$ in direction
$\eta$. This path encloses all the $u_{j}$ such that
$u_{j}^{c}=\lambda_{\alpha}$, end excludes the others. See figure 4. We
conclude that $u$ can vary in $\mathbb{D}(u^{c})$ and the integral (7.11)
converges for $z$ in the sector

$\mathcal{S}(\eta):=\Bigl{\\{}z\in\mathcal{R}(\mathbb{C}\backslash\\{0\\})~{}\hbox{
such that }\frac{\pi}{2}-\eta<\arg z<\frac{3\pi}{2}-\eta\Bigr{\\}},$

defining $\vec{Y}_{k}(z,u~{}|\nu)$ as a holomorphic function of
$(z,u)\in\mathcal{S}(\eta)\times\mathbb{D}(u^{c})$. Now, if $u$ varies in
$\mathbb{D}(u^{c})$ and $\epsilon_{0}$ satisfies (5.1) none of the vectors
$u_{i}-u_{j}$ such that $u_{i}^{c}=\lambda_{\alpha}$ and
$u_{j}^{c}=\lambda_{\beta}$, $1\leq\alpha\neq\beta\leq s$, cross a direction
$\eta$ mod $\pi$, for every $\eta_{\nu+1}<\eta<\eta_{\nu}$. Due to 1. and 2.
above, a vector function $\vec{\Psi}_{k}(\lambda,u~{}|\nu)$ is well defined in
$\mathcal{P}_{\eta}$ and $\mathcal{P}_{\tilde{\eta}}$ for any
$\eta_{\nu+1}<\eta<\tilde{\eta}<\eta_{\nu}$, and so on
$\mathcal{P}_{\eta}\cup\mathcal{P}_{\tilde{\eta}}$. Therefore, the integral in
(7.11) satisfies

$\dfrac{1}{2\pi
i}\int_{\Gamma_{\alpha}(\eta)}e^{z\lambda}\vec{\Psi}_{k}(\lambda,u~{}|\nu)d\lambda=\dfrac{1}{2\pi
i}\int_{\Gamma_{\alpha}(\tilde{\eta})}e^{z\lambda}\vec{\Psi}_{k}(\lambda,u~{}|\nu)d\lambda,\quad\quad
z\in\mathcal{S}(\eta)\cap\mathcal{S}(\tilde{\eta}),$

namely one is the analytic continuation of the other, so defining the function
$\vec{Y}_{k}(z,u~{}|\nu)$ as analytic on
$\widehat{\mathcal{S}}_{\nu}\times\mathbb{D}(u^{c})$, where

$\widehat{\mathcal{S}}_{\nu}:=\bigcup_{\eta_{\nu+1}<\eta<\eta_{\nu}}\mathcal{S}(\eta)$

coincides with (7.3) (with $h=0$).

Figure 4: The paths of integration $\Gamma_{\alpha}$, $\Gamma_{\beta}$, etc
$\alpha,\beta,...\in\\{1,...,s\\}$.

Finally, notice that $e^{\lambda
z}(\lambda-\Lambda)\vec{\Psi}_{k}(\lambda,u~{}|\nu)\Bigl{|}_{\Gamma(\alpha)}=0$,
due to the exponential factor. By (2.25), the vector solutions
$\vec{Y}_{k}(z,u~{}|\nu)$ satisfy system (1.1).

$\bullet$ Asymptotic behaviour. From (5.4)-(5.5), we write (7.11) as

$\vec{Y}_{k}(z,u~{}|\nu)=\dfrac{1}{2\pi
i}\int_{\Gamma_{\alpha}(\eta)}e^{z\lambda}\Bigl{(}\Gamma(\lambda^{\prime}_{j}+1)\vec{e}_{j}+\sum_{l\geq
1}\vec{b}_{l}^{(k)}(u)(\lambda-u_{k})^{l}\Bigr{)}(\lambda-
u_{k})^{-\lambda^{\prime}_{k}-1},d\lambda.$

with holomorphic $\vec{b}_{l}^{(k)}(u)$ on $\mathbb{D}(u^{c})$. We split the
series as $\sum_{l\geq 1}=\sum_{l=1}^{\mathcal{N}}+\sum_{l\geq\mathcal{N}+1}$,
and recall the standard formula (see [18])

$\int_{\Gamma_{\alpha}(\eta)}(\lambda-\lambda_{k})^{a}e^{z\lambda}d\lambda=\int_{\gamma_{k}(\eta)}(\lambda-\lambda_{k})^{a}e^{z\lambda}d\lambda=\frac{z^{-a-1}e^{\lambda_{k}z}}{\Gamma(-a)}$

so that

$\vec{Y}_{k}(z,u~{}|\nu)=\left(\vec{e}_{k}+\sum_{l=1}^{\mathcal{N}}\frac{\vec{b}_{l}^{(k)}(u)}{\Gamma(\lambda^{\prime}_{k}+1-l)}z^{-l}+R_{\mathcal{N}}(z)\right)~{}z^{\lambda^{\prime}_{k}}e^{\lambda_{k}z},$

with remainder

$R_{\mathcal{N}}(z)=\oint_{\Gamma_{0}(\eta)}\sum_{l\geq\mathcal{N}}\frac{\vec{b}_{l}^{(k)}(u)}{z^{l}}~{}e^{x}x^{l-\lambda^{\prime}_{k}-1}~{}dx~{}=O(z^{-\mathcal{N}+1}).$

The integral is along a path $\Gamma_{0}(\eta)$, coming from $\infty$ along
the left part of the half line oriented from $0$ to $\infty$ in direction
$\eta+\arg z$, going around $0$, and back to $\infty$ along the right part.
The estimate $O(z^{-\mathcal{N}+1})$ is standard. We conclude that

$\vec{Y}_{k}(z,u~{}|\nu)\left(z^{\lambda^{\prime}_{k}}e^{u_{k}z}\right)^{-1}\sim\vec{e}_{k}+\sum_{l=1}^{\infty}\frac{\vec{b}_{l}^{(k)}(u)}{\Gamma(\lambda^{\prime}_{k}+1-l)}z^{-l}\equiv\vec{e}_{k}+\sum_{l=1}^{\infty}\vec{f}_{l}^{~{}(k)}(u)z^{-l},\quad\quad
z\to\infty\hbox{ in }\widehat{\mathcal{S}}_{\nu}$

with

$\vec{f}_{l}^{~{}(k)}(u):=\frac{\vec{b}_{l}^{(k)}(u)}{\Gamma(\lambda^{\prime}_{k}+1-l)}.$
(7.12)

b) Case $\lambda^{\prime}_{k}\in\mathbb{N}=\\{0,1,2,...\\}$.

$\bullet$ Construction of $\vec{Y}_{k}(z,u~{}|\nu)$. Definition (7.4) is

$\displaystyle\vec{Y}_{k}(z,u~{}|\nu)$ $\displaystyle:=\frac{1}{2\pi
i}\int_{\gamma_{k}(\eta)}e^{z\lambda}\vec{\Psi}_{k}^{(sing)}(\lambda,u~{}|\nu)d\lambda$
$\displaystyle\underset{(\ref{17ottobre2020-1})}{=}\frac{1}{2\pi
i}\int_{\gamma_{k}(\eta)}e^{z\lambda}\left(\frac{\vec{\psi}_{k}(\lambda,u~{}|\nu)}{(\lambda-
u_{k})^{\lambda^{\prime}_{k}+1}}+\vec{\Psi}_{k}(\lambda,u~{}|\nu)\ln(\lambda-
u_{k})\right)d\lambda.$

The same facts 1. and 2. of the previous case apply to
$\vec{\Psi}_{k}(\lambda,u~{}|\nu)$ and $\vec{\psi}_{k}(\lambda,u~{}|\nu)$ and
allow to rewrite

$\displaystyle\vec{Y}_{k}(z,u~{}|\nu)$ $\displaystyle=\frac{1}{2\pi
i}\int_{\Gamma_{\alpha}(\eta)}e^{z\lambda}\left(\frac{\vec{\psi}_{k}(\lambda,u~{}|\nu)}{(\lambda-
u_{k})^{\lambda^{\prime}_{k}+1}}+\vec{\Psi}_{k}(\lambda,u~{}|\nu)\ln(\lambda-
u_{k})\right)d\lambda$ $\displaystyle=\frac{1}{2\pi
i}\int_{\Gamma_{\alpha}(\eta)}e^{z\lambda}\vec{\Psi}_{k}^{(sing)}(\lambda,u~{}|\nu)d\lambda.$

We conclude that $\vec{Y}_{k}(z,u~{}|\nu)$ is analytic on
$\widehat{\mathcal{S}}_{\nu}\times\mathbb{D}(u^{c})$. Moreover, $e^{\lambda
z}(\lambda-\Lambda)\vec{\Psi}_{k}^{(sing)}(\lambda,u~{}|\nu)\Bigl{|}_{\Gamma(\alpha)}=0$,
due to the exponential factor. By (2.25), the vector solution
$\vec{Y}_{k}(z,u~{}|\nu)$ satisfies the system (1.1).

$\bullet$ Asymptotic behaviour. By (5.7) and (5.11), and the fact that
$\vec{\psi}_{k}$ has no singularities at $u_{j}\in\mathbb{D}_{\alpha}$, $j\neq
k$, so that the terms $\sum_{l\geq
1+\lambda^{\prime}_{k}}\vec{b}_{l}^{~{}(k)}(u)(\lambda-u_{k})^{l}$ in
$\vec{\psi}_{k}(\lambda,u~{}|\nu)$ do not contribute to the integration, we
can write

$\vec{Y}_{k}(z,u~{}|\nu)=\frac{1}{2\pi
i}\int_{\Gamma_{\alpha}(\eta)}\left(\frac{\lambda^{\prime}_{k}!\vec{e}_{k}+\sum_{l=1}^{\lambda^{\prime}_{k}}\vec{b}_{l}^{~{}(k)}(u)(\lambda-
u_{k})^{l}}{(\lambda-
u_{k})^{\lambda^{\prime}_{k}+1}}+\sum_{l=0}^{\infty}\vec{d}_{l}^{~{}(k)}(u)(\lambda-
u_{k})^{l}~{}\ln(\lambda-u_{k})\right)e^{z\lambda}~{}d\lambda.$

By Cauchy formula

$\displaystyle\frac{1}{2\pi
i}\int_{\Gamma_{\alpha}(\eta)}\left(\frac{\lambda^{\prime}_{k}!\vec{e}_{k}+\sum_{l=1}^{\lambda^{\prime}_{k}}\vec{b}_{l}^{~{}(k)}(u)(\lambda-
u_{k})^{l}}{(\lambda-
u_{k})^{\lambda^{\prime}_{k}+1}}\right)e^{z\lambda}~{}d\lambda$
$\displaystyle=\frac{1}{\lambda^{\prime}_{k}!}\frac{d^{\lambda^{\prime}_{k}}}{d\lambda^{\lambda^{\prime}_{k}}}\left.\left[\left(\lambda^{\prime}_{k}!\vec{e}_{k}+\sum_{l=1}^{\lambda^{\prime}_{k}}\vec{b}_{l}^{~{}(k)}(u)(\lambda-
u_{k})^{l}\right)e^{z\lambda}\right]\right|_{\lambda=u_{k}}$
$\displaystyle=z^{\lambda^{\prime}_{k}}e^{u_{k}z}\left(\vec{e}_{k}+\sum_{l=1}^{\lambda^{\prime}_{k}}\vec{f}_{l}^{~{}(k)}(u)\frac{1}{z^{l}}\right),$

where

$\vec{f}_{l}^{~{}(k)}(u):=\frac{\vec{b}_{l}^{(k)}(u)}{(\lambda^{\prime}_{k}-l)!},\quad
l=1,...,\lambda^{\prime}_{k}.$ (7.13)

In order to evaluate the terms with logarithm, we observe that for any
function $g(\lambda)$ holomorphic along $L_{k}(\eta)$, including
$\lambda=u_{k}$, we have

$\int_{\gamma_{k}(\eta)}g(\lambda)\ln(\lambda-
u_{k})d\lambda=\int_{L_{k}(\eta)^{-}}g(\lambda)\ln(\lambda-
u_{k})_{-}d\lambda-\int_{L_{k}(\eta)^{+}}g(\lambda)\ln(\lambda-
u_{k})_{+}d\lambda,$

where $L_{k}(\eta)^{+}$ and $L_{k}(\eta)^{-}$ respectively are the left and
right parts of $L_{k}(\eta)$, oriented from $0$ to $\infty$. Since
$\ln(\lambda-u_{k})_{+}=\ln(\lambda-u_{k})_{+}-2\pi i$, we conclude that

$\int_{\gamma_{k}(\eta)}g(\lambda)\ln(\lambda-u_{k})d\lambda=2\pi
i\int_{L_{k}(\eta)}g(\lambda)d\lambda.$ (7.14)

Keeping into account that the integral along $\Gamma_{\alpha}$ can be
interchanged with that along $\gamma_{k}$, it follows that

$\displaystyle\frac{1}{2\pi
i}\int_{\Gamma_{\alpha}(\eta)}\vec{\Psi}_{k}(\lambda,u~{}|\nu)\ln(\lambda-
u_{k})e^{z\lambda}~{}d\lambda$
$\displaystyle=\int_{L_{k}(\eta)}\vec{\Psi}_{k}(\lambda,u~{}|\nu)e^{z\lambda}~{}d\lambda$
$\displaystyle=\int_{L_{k}(\eta)}\sum_{l=0}^{\infty}\vec{d}_{l}^{~{}(k)}(u)(\lambda-
u_{k})^{l}~{}e^{z\lambda}~{}d\lambda.$

We conclude, by the standard evaluation of the remainder analogous to
$R_{\mathcal{N}}(z)$ considered before, and the variation of $\eta$ in the
range $(\eta_{\nu+1},\eta_{\nu})$, that202020Notice that, by abuse of
notation, if $f(\lambda)e^{-u_{k}\lambda}\sim\sum_{0}^{\infty}c_{l}z^{-l}$ we
write $f(\lambda)\sim e^{u_{k}\lambda}\sum_{0}^{\infty}c_{l}z^{-l}$.

$\displaystyle\int_{L_{k}(\eta)}\vec{\Psi}_{k}(\lambda,u~{}|\nu)e^{z\lambda}~{}d\lambda$
$\displaystyle\sim
e^{u_{k}z}\left(\sum_{l=0}^{\infty}(-1)^{l+1}l!~{}\vec{d}_{l}^{(k)}(u)~{}z^{-l-1}\right),\quad\quad
z\to\infty\hbox{ in }\widehat{\mathcal{S}}_{\nu}.$
$\displaystyle=z^{\lambda^{\prime}_{k}}e^{u_{k}z}\left(\sum_{l=\lambda^{\prime}_{k}+1}^{\infty}\vec{f}_{l}^{~{}(k)}(u)~{}z^{-l}\right),$

where

$\vec{f}_{l}^{~{}(k)}(u):=(-1)^{l-\lambda^{\prime}_{k}}(l-\lambda^{\prime}_{k}-1)!~{}\vec{d}_{l-\lambda^{\prime}_{k}-1}^{~{}(k)}(u),\quad\quad
l\geq\lambda^{\prime}_{k}+1.$ (7.15)

In conclusion, we have the expansion

$\vec{Y}_{k}(z,u~{}|\nu)\sim
z^{\lambda^{\prime}_{k}}e^{u_{k}z}\left(\vec{e}_{k}+\sum_{l=1}^{\infty}\vec{f}_{l}^{~{}(k)}(u)~{}z^{-l}\right),\quad\quad
z\to\infty\hbox{ in }\widehat{\mathcal{S}}_{\nu},$

with coefficients $\vec{f}_{l}^{~{}(k)}(u)$ holomorphic in $\mathbb{D}(u^{c})$
defined in (7.13)-(7.15). Notice that, in exceptional cases, $\vec{\Psi}_{k}$
may be identically zero, so that

$\hbox{$\vec{f}_{l}^{~{}(k)}=0$ for $l\geq\lambda^{\prime}_{k}+1$}.$ (7.16)

c) Case $\lambda^{\prime}_{k}\in\mathbb{Z}_{-}=\\{-1,-2,...\\}$ $\bullet$
Construction of $\vec{Y}_{k}(z,u~{}|\nu)$. Definition (7.5) is

$\vec{Y}_{k}(z,u~{}|\nu):=\int_{L_{k}(\eta)}e^{\lambda
z}\vec{\Psi}_{k}(\lambda,u~{}|\nu)d\lambda\equiv\int_{\mathcal{L}_{\alpha}(\eta)}e^{\lambda
z}\vec{\Psi}_{k}(\lambda,u~{}|\nu)d\lambda.$

In the last equality, we have used the fact that
$\vec{\Psi}_{k}(\lambda,u~{}|\nu)$ is analytic in
$\mathbb{D}_{\alpha}\times\mathbb{D}(u^{c})$, where
$\lambda_{\alpha}=u_{k}^{c}$.

We conclude analogously to previous cases that $\vec{Y}_{k}(z,u~{}|\nu)$ is
analytic in $\widehat{\mathcal{S}}_{\nu}\times\mathbb{D}(u^{c})$. It is a
solution of (1.1), by (2.25), because $\vec{\Psi}_{k}(\lambda,u~{}|\nu)$ is
analytic at $\lambda=u_{k}$ and behaves as in (5.4)-(5.5), so that

$e^{\lambda z}(\lambda
I-\Lambda(u))\vec{\Psi}_{k}(\lambda,u~{}|\nu)\Bigr{|}_{\mathcal{L}_{\alpha}}=e^{\lambda
z}(\lambda
I-\Lambda(u))\vec{\Psi}_{k}(\lambda,u~{}|\nu)\Bigr{|}_{L_{k}}=0-(u_{k}I-\Lambda(u))\vec{\Psi}_{k}(\lambda,u_{k}~{}|\nu)=0.$

$\bullet$ Asymptotic behaviour. We have, from (5.4)-(5.5),

$\vec{Y}_{k}(z,u~{}|\nu)=\int_{\mathcal{L}_{\alpha}(\eta)}e^{\lambda
z}\left(\frac{(-1)^{\lambda^{\prime}_{k}}~{}\vec{e}_{k}}{(-\lambda^{\prime}_{k}-1)!}(\lambda-
u_{k})^{-\lambda^{\prime}_{k}-1}+\sum_{l\geq 1}\vec{b}_{l}^{(k)}(u)(\lambda-
u_{k})^{l-\lambda^{\prime}_{k}-1}\right)d\lambda$

We integrate term by term in order to obtain the asymptotic expansion (the
remainder for the truncated series is evaluate in standard way, as
$R_{\mathcal{N}}(z)$ above). For the integration, we use

$\int_{L_{k}(\eta)}(\lambda-u_{k})^{m}e^{\lambda
z}d\lambda=\frac{e^{u_{k}z}}{z^{m+1}}\int_{+\infty
e^{i\phi}}^{0}x^{m}e^{x}dx=\frac{e^{u_{k}z}}{z^{m+1}}m!~{}(-1)^{m+1},\quad\quad\quad\frac{\pi}{2}<\phi<\frac{3\pi}{2}.$

We obtain, analogously to previous cases,

$\vec{Y}_{k}(z,u~{}|\nu)\sim
z^{\lambda^{\prime}_{k}}e^{u_{k}z}\left(\vec{e}_{k}+\sum_{l=1}^{\infty}\vec{f}_{l}^{~{}(k)}(u)z^{-l}\right),\quad\quad
z\to\infty\hbox{ in }\widehat{\mathcal{S}}_{\nu},$

where the holomorphic in $\mathbb{D}(u^{c})$ coefficients are

$\vec{f}_{l}^{~{}(k)}(u):=(-1)^{l-\lambda^{\prime}_{k}}(l-\lambda^{\prime}_{k}-1)!~{}\vec{b}_{l}^{(k)}(u).$
(7.17)

######  Remark 7.2.

We cannot use $\vec{\Psi}_{k}^{(sing)}(\lambda,u~{}|\nu)$ in (5.8) to define
$\vec{Y}_{k}(z,u~{}|\nu)$ if $u$ varies in the whole $\mathbb{D}(u^{c})$. On
the other hand, if $u$ is restricted to a $\tau$-cell, so that the eigenvalues
$u_{j}$ are all distinct, by (7.14) we can write

$\vec{Y}_{k}(z,u~{}|\nu)=\int_{L_{k}(\eta)}e^{\lambda
z}\vec{\Psi}_{k}(\lambda,u~{}|\nu)d\lambda\underset{(\ref{18ottobre2020-4})}{=}\frac{1}{2\pi
i}\int_{\gamma_{k}(u)}e^{\lambda
z}\vec{\Psi}_{k}(\lambda,u~{}|\nu)\ln(\lambda-u_{k})d\lambda.$

Then, we can use the local expansion (5.9) and the fact that
$\int_{\gamma_{k}(u)}\hbox{\rm reg}(\lambda-u_{k})d\lambda=0$, receiving

$\vec{Y}_{k}(z,u~{}|\nu)=\frac{1}{2\pi i}\int_{\gamma_{k}(u)}e^{\lambda
z}\vec{\Psi}_{k}^{(sing)}(\lambda,u~{}|\nu)d\lambda$

#### Fundamental matrix solutions

The vector solutions $\vec{Y}_{k}(z,u~{}|\nu)$ constructed above can be
arranged as columns of the matrix

$Y_{\nu}(z,u):=\Bigl{[}\vec{Y}_{k}(z,u~{}|\nu)~{}\Bigl{|}~{}\cdots~{}\Bigr{|}~{}\vec{Y}_{n}(z,u~{}|\nu)\Bigr{]},$

which thus solves system (1.1). From the general theory of differential
systems, it admits analytic continuation as analytic matrix valued function on
$\mathcal{R}(\mathbb{C}\backslash\\{0\\})\times\mathbb{D}(u^{c})$. Letting
$B=\hbox{\rm diag}A=\hbox{\rm
diag}(\lambda^{\prime}_{1},...,\lambda^{\prime}_{n})$, the asymptotic
expansions obtained above are summarized as

$Y_{\nu}(z,u~{}|\nu)~{}z^{-B}e^{-\Lambda(u)z}\sim
F(z,u)=I+\sum_{l=1}^{\infty}F_{l}(u)z^{-l},\quad\quad z\to\infty\hbox{ in
}\widehat{\mathcal{S}}_{\nu},$
$F_{l}(u)=\Bigl{[}\vec{f}_{l}^{~{}(1)}(u)~{}|~{}\cdots~{}|~{}\vec{f}_{l}^{~{}(n)}(u)\Bigr{]}.$

Therefore, the coefficients $F_{l}(u)$ of the formal solution
$Y_{F}(z,u)=F(z,u)z^{B}e^{\Lambda(u)z}$ are holomorphic in
$\mathbb{D}(u^{c})$. Moreover, the leading term is the identity $I$, which
implies that $Y_{\nu}(z,u)$ is a fundamental matrix solution.

Consider now another direction $\eta$, satisfying
$\eta_{\nu+\mu+1}<\eta<\eta_{\nu+\mu}$. The above discussion can be repeated.
We obtain a fundamental matrix solution $Y_{\nu+\mu}(z,u)$ with canonical
asymptotics $Y_{F}(z,u)$ in $\widehat{S}_{\nu+\mu}$. Again, for $\eta$
satisfying $\eta_{\nu+2\mu+1}<\eta<\eta_{\nu+2\mu}$ we obtain the analogous
result for $Y_{\nu+2\mu}(z,u)$ with canonical asymptotics in
$\widehat{S}_{\nu+2\mu}$. This can be repeated for every $\nu+h\mu$,
$h\in\mathbb{Z}$, obtaining the fundamental matrix solutions
$Y_{\nu+h\mu}(z,u)$ with canonical asymptotics $Y_{F}(z,u)$ in
$\widehat{S}_{\nu+h\mu}$. So, Points 1) and 2) of Theorem 7.1 are proved.

Stokes matrices are defined by (7.7). Thus,
$\mathbb{S}_{\nu+h\mu}(u)=Y_{\nu+h\mu}(z,u)^{-1}Y_{\nu+(h+1)\mu}(z,u)$ is
holomorphic in $\mathbb{D}(u^{c})$. Let us consider the relations for $h=0,1$:

$Y_{\nu+\mu}(z,u)=Y_{\nu}(z,u)\mathbb{S}_{\nu}(u),\quad\quad
Y_{\nu+2\mu}(z,u)=Y_{\nu+\mu}(z,u)\mathbb{S}_{\nu+\mu}(u).$ (7.18)

Let $u$ be fixed in a $\tau$-cell, so that $\Lambda$ has distinct eigenvalues.
From Theorem 2.3 at the fixed $u$ we receive

$\bigl{(}\mathbb{S}_{\nu}(u)\bigr{)}_{jk}=\left\\{\begin{array}[]{cc}e^{2\pi
i\lambda^{\prime}_{k}}\alpha_{k}~{}c_{jk}^{(\nu)}&~{}~{}~{}\hbox{ for }j\prec
k,\\\ \\\ 1&~{}~{}~{}\hbox{ for }j=k,\\\ \\\ 0&~{}~{}~{}\hbox{ for }j\succ
k,\end{array}\right.~{}~{}~{}~{}~{}~{}~{}~{}\bigl{(}\mathbb{S}_{\nu+\mu}^{-1}(u)\bigr{)}_{jk}=\left\\{\begin{array}[]{cc}0&~{}~{}~{}\hbox{
for }j\prec k,\\\ \\\ 1&~{}~{}~{}\hbox{ for }j=k,\\\ \\\ -e^{2\pi
i(\lambda^{\prime}_{k}-\lambda^{\prime}_{j})}\alpha_{k}~{}c_{jk}^{(\nu)}&~{}~{}~{}\hbox{
for }j\succ k.\end{array}\right.$

Here, for $j\neq k$ the ordering relation $j\prec k$ $\Longleftrightarrow$
$\Re(z(u_{j}-u_{k}))|_{\arg z=\tau}<0$ is well defined for every $u$ in the
$\tau$-cell, because no Stokes rays $\Re(z(u_{j}-u_{k}))=0$ cross $\arg
z=\tau$ as $u$ varies in the $\tau$-cell.

The relation $j\prec k$ may change to $j\succ k$ when passing from one
$\tau$-cell to another only for a pair $u_{j}$, $u_{k}$ such that
$u_{j}^{c}=u_{k}^{c}$. This is due to the choice of $\epsilon_{0}$ as in
(5.1). On the other hand, $c_{jk}^{(\nu)}=0$ whenever $u_{j}^{c}=u_{k}^{c}$.
This means that (7.9) is true at every fixed $u$ in every $\tau$-cell, with
ordering relation $j\prec k$ defined for $j\neq k$ such that $u_{j}^{c}\neq
u_{k}^{c}$, namely $\Re(z(u_{j}^{c}-u_{k}^{c}))<0$ when $\arg z=\tau$.

Since the $\mathbb{S}_{\nu+h\mu}$ are holomorphic in $\mathbb{D}(u^{c})$ and
the $c_{jk}^{(\nu)}$ are constant in $\mathbb{D}(u^{c})$, we conclude that
Stokes matrices are constant in $\mathbb{D}(u^{c})$ and hence (7.9) holds in
$\mathbb{D}(u^{c})$. The vanishing conditions (7.8) follow from the vanishing
conditions (5.14) for the connection coefficients, plus the fact that we can
generate all the $\mathbb{S}_{\nu+h\mu}$ from the formula
$\mathbb{S}_{\nu+2\mu}=e^{-2\pi iB}\mathbb{S}_{\nu}e^{2\pi iB}$.

∎

##  8 Non-Uniqueness at $u=u^{c}$ of the Formal Solution

By Laplace transform we prove Corollary 2.1 in Background 1, asserting that
system (2.19) has unique formal solution if and only if the constant diagonal
entries of $A(u)$ satisfy the partial non-resonance

$\lambda^{\prime}_{i}-\lambda^{\prime}_{j}\not\in\mathbb{Z}\backslash\\{0\\}\quad\hbox{
for every $i\neq j$ such that $u_{i}^{c}=u_{j}^{c}$}.$

Otherwise, the Laplace transform will be proved to generate a family of formal
solutions

$\mathring{Y}_{F}(z)=\Bigl{(}I+\sum_{l=1}^{\infty}\mathring{F}_{l}z^{-l}\Bigr{)}z^{B}e^{\Lambda(u^{c})z},$

whose coefficients $\mathring{F}_{l}$ depend on a finite number of arbitrary
parameters.

Due to the strategy of Section 6.6, it will suffice to consider the generic
case when all
$\lambda^{\prime}_{1},...,\lambda^{\prime}_{n}~{}\not\in\mathbb{Z}$ and $A$
has no integer eigenvalues. Indeed, if this is not the case, the gauge
transformation (6.46) relates a formal solution ${}_{\gamma}Y_{F}$ to $Y_{F}$
at any point $u$, through (6.48), so that the coefficients $F_{l}$ of a formal
expansion do not depend on $\gamma$. We are interested in these coefficients.

Consider system (1.4) under the assumptions that it is (strongly)
isomonodormic in $\mathbb{D}(u^{c})$, so that $(A)_{ij}(u^{c})=0$ for
$u_{i}^{c}=u_{j}^{c}$. For simplicity, we order the eigenvalues as in
(6.1)-(6.2). Since $B_{1}(u)$, …, $B_{n}(u)$ are holomorphic at $u^{c}$,
system (1.4) at $u=u^{c}$ is

$\frac{d\Psi}{d\lambda}=\left(\frac{\sum_{j=1}^{p_{1}}B_{j}(u^{c})}{\lambda-\lambda_{1}}+\frac{\sum_{j=p_{1}+1}^{p_{1}+p_{2}}B_{j}(u^{c})}{\lambda-\lambda_{2}}+\cdots+\frac{\sum_{j=p_{1}+...+p_{s-1}+1}^{n}B_{j}(u^{c})}{\lambda-\lambda_{s}}\right)\Psi$
(8.1)

Let $G^{(\boldsymbol{p}_{1})}$ be as in (6.24). The gauge transformation
$\Psi(\lambda)=G^{(\boldsymbol{p}_{1})}\widetilde{\Psi}(\lambda)$ yields

$\frac{d\widetilde{\Psi}}{d\lambda}=\left(\frac{T^{(\boldsymbol{p}_{1})}}{\lambda-\lambda_{1}}+\sum_{\alpha=2}^{s}\frac{D_{\alpha}^{(\boldsymbol{p}_{1})}}{\lambda-\lambda_{\alpha}}\right)\widetilde{\Psi},$
(8.2)

where

$T^{(\boldsymbol{p}_{1})}:=T^{(1)}+...+T^{(p_{1})}=\hbox{diag}(-\lambda^{\prime}_{1}-1,~{}...,~{}-\lambda^{\prime}_{p_{1}}-1,~{}\underbrace{0,~{}...~{},~{}0}_{n-p_{1}}).$

and
$D_{\alpha}^{(\boldsymbol{p}_{1})}:={G^{(\boldsymbol{p}_{1})}}^{-1}\cdot\sum_{j=p_{1}+...+p_{\alpha-1}+1}^{p_{1}+...+p_{\alpha}}B_{j}(u^{c})\cdot
G^{(\boldsymbol{p}_{1})}$. The matrix coefficient in system (8.2) has
convergent Taylor series at $\lambda=\lambda_{1}$

$\frac{d\widetilde{\Psi}}{d\lambda}=\frac{1}{\lambda-\lambda_{1}}\left(T^{(\boldsymbol{p}_{1})}+\sum_{m=1}^{\infty}\mathfrak{D}_{m}(\lambda-\lambda_{1})^{m}\right)\widetilde{\Psi},\quad\quad\mathfrak{D}_{m}=\sum_{\alpha=2}^{s}\frac{(-1)^{m+1}}{(\lambda_{1}-\lambda_{\alpha})^{m}}D_{\alpha}^{(\boldsymbol{p}_{1})}.$

We consider $\eta_{\nu+1}<\eta<\eta_{\nu}$ and $\lambda$ in the plane with
branch cuts $\mathcal{L}_{\alpha}=\mathcal{L}_{\alpha}(\eta)$ issuing from
$\lambda_{1},...,\lambda_{s}$ to infinity in direction $\eta$, as in (5.2).
Close to the Fuchsian singularity $\lambda=\lambda_{1}$ a fundamental matrix
solution to (8.1) has Levelt form

$\mathring{\Psi}^{(\boldsymbol{p}_{1})}(\lambda)=G^{(\boldsymbol{p}_{1})}\Bigl{(}I+\sum_{l=1}^{\infty}\mathfrak{G}_{l}(\lambda-\lambda_{1})^{l}\Bigr{)}(\lambda-\lambda_{1})^{T^{(\boldsymbol{p}_{1})}},$
(8.3)

where the matrix entries $(\mathfrak{G}_{l})_{ij}$, $1\leq i\leq j\leq n$, are
recursively computed by the following formulae (see Appendix C for an
explanation of (8.3), or [27, 62]).

  * •

If $T^{(\boldsymbol{p}_{1})}_{ii}-T^{(\boldsymbol{p}_{1})}_{jj}=l$ positive
integer, $(\mathfrak{G}_{l})_{ij}$ is arbitrary.

  * •

If $T^{(\boldsymbol{p}_{1})}_{ii}-T^{(\boldsymbol{p}_{1})}_{jj}\neq l$
(positive integer)

$(\mathfrak{G}_{l})_{ij}=\frac{1}{T^{(\boldsymbol{p}_{1})}_{jj}-T^{(\boldsymbol{p}_{1})}_{ii}+l}\left(\sum_{p=1}^{l-1}\mathfrak{D}_{l-p}\mathfrak{G}_{l}+\mathfrak{D}_{l}\right)_{ij}\quad\hbox{(sum
is zero for $l=1$)}.$

Since we have assumed that all the $\lambda^{\prime}_{k}$ are not integers,
the only possibility to have
$T^{(\boldsymbol{p}_{1})}_{ii}-T^{(\boldsymbol{p}_{1})}_{jj}=l$ occurs for
$1\leq i,j\leq p_{1}$, precisely

$T^{(\boldsymbol{p}_{1})}_{ii}-T^{(\boldsymbol{p}_{1})}_{jj}=\lambda^{\prime}_{j}-\lambda^{\prime}_{i}=l.$
(8.4)

In this case, (8.3) is a family depending on a finite number of parameters due
to the arbitrary $(\mathfrak{G}_{l})_{ij}$. Thus, in the first $p_{1}$ columns
of a solution of type (8.3)

$\vec{\mathring{\Psi}}_{j}(\lambda~{}|\nu)=\Bigl{(}\Gamma(\lambda^{\prime}_{k}+1)\vec{e}_{k}+\sum_{l=1}^{\infty}\mathring{b}_{l}^{(j)}(\lambda-\lambda_{1})\Bigr{)}(\lambda-\lambda_{1})^{-\lambda^{\prime}_{j}-1},\quad\quad
j=1,...,p_{1}.$

the vectors $\mathring{b}_{l}^{(j)}$ contain a finite number of parameters.
The Laplace transform

$\vec{\mathring{Y}}_{j}(z~{}|\nu)=\int_{\Gamma_{1}(\eta)}e^{z\lambda}\vec{\mathring{\Psi}}_{j}(\lambda~{}|\nu)d\lambda,\quad\quad
j=1,...,p_{1},$

yields the first $p_{1}$ columns of a fundamental matrix solution of (2.19).
Repeating the same computations of Section 7, we obtain, for $j=1,...,p_{1}$,

$\vec{\mathring{Y}}_{j}(z~{}|\nu)~{}z^{-\lambda^{\prime}_{j}}e^{-\lambda_{1}z}\sim\vec{e}_{j}+\sum_{l=1}^{\infty}\frac{\mathring{b}_{l}^{(j)}}{\Gamma(\lambda^{\prime}_{j}+1-l)}\frac{1}{z^{l}},\quad
z\to\infty\hbox{ in }\mathcal{S}_{\nu},$

where $\mathcal{S}_{\nu}$ is given in (7.1). We repeat the same construction
at all $\lambda_{1}$, …, $\lambda_{s}$. This yields a family of fundamental
matrix solutions of (2.19)

$\mathring{Y}_{\nu}(z)=\Bigl{[}\vec{\mathring{Y}}_{1}(z~{}|\nu)~{}|~{}\cdots~{}|~{}\vec{\mathring{Y}}_{n}(z~{}|\nu)\Bigr{]},$

depending on a finite number of parameters, with the behaviour for
$z\to\infty$ in ${S}_{\nu}$

$\mathring{Y}_{\nu}(z)\sim\mathring{Y}_{F}(z)=\Bigl{(}I+\sum_{l=1}^{\infty}\mathring{F}_{l}z^{-l}\Bigr{)}z^{B}e^{\Lambda(u^{c})z};\quad\quad\mathring{F}_{l}=\Bigl{[}\vec{\mathring{f}}_{1}^{~{}(l)}~{}|~{}\cdots~{}|~{}\vec{\mathring{f}}_{n}^{~{}(l)}\Bigr{]},\quad\quad\vec{\mathring{f}}_{j}^{~{}(l)}=\frac{\vec{\mathring{b}}_{j}^{~{}(l)}}{\Gamma(\lambda^{\prime}_{j}+1-l)}.$

We conclude that the formal solution is not unique whenever a condition (8.4)
occurs. Only one element in the family satisfies
$\mathring{Y}_{F}(z)=Y_{F}(z,u^{c})$.

######  Remark 8.1.

If we choose one formal solution $\mathring{Y}_{F}(z)$, then the corresponding
$\mathring{Y}_{\nu}(z)$ with asymptotic expansion $\mathring{Y}_{F}(z)$ in
$\mathcal{S}_{\nu}$ is unique. For more details on the Stokes phenomenon at
$u=u^{c}$ see [13].

##  9 Appendix A. Non-normalized Schlesinger System

######  Lemma 9.1.

The integrability condition $dP=P\wedge P$ of the Pfaffian system (3.2)
defined on a polydisc $\mathbb{D}(u^{0})$ contained in a $\tau$-cell is the
non-normalized Schlesinger system (3.3)-(3.5).

###### Proof.

For every $i\in\\{1,...,n\\}$, the Pfaffian system (3.2) can be rewritten as

$P=\left(\frac{B_{i}}{\lambda-u_{i}}+\sum_{j\neq i}\frac{B_{j}}{\lambda-
u_{j}}\right)d(\lambda-u_{i})+\sum_{j\neq
i}\left(\gamma_{j}-\frac{B_{j}}{\lambda-
u_{j}}\right)d(u_{j}-u_{i})+\sum_{j=1}^{n}\gamma_{j}(u)d\lambda.$

We study $\lambda-u_{i}\to 0$, while $u_{j}-u_{i}\neq 0$ for $j\neq i$ in
$\mathbb{D}(u^{0})$. In new variables

$\lambda=\lambda,\quad\quad y_{i}=\lambda-u_{i},\quad y_{j}=u_{j}-u_{i},\quad
j\neq i,$

$P$ is rewritten in the following way (which defines the matrices
$\mathcal{A}_{j}(y)$)

$\displaystyle P=$ $\displaystyle\left(\frac{B_{i}}{y_{i}}+\sum_{j\neq
i}\frac{B_{j}}{y_{i}-y_{j}}\right)dy_{i}+\sum_{j\neq
i}\left(\gamma_{j}-\frac{B_{j}}{y_{i}-y_{j}}\right)dy_{j}+\sum_{j=1}^{n}\gamma_{j}(y)d\lambda$
$\displaystyle=:\mathcal{A}_{i}(y)dy_{i}+\sum_{j\neq
i}\mathcal{A}_{j}(y)dy_{j}+\sum_{j=1}^{n}\gamma_{j}(y)d\lambda.$

The only singular term at $y_{i}=0$ is ${B_{i}/y_{i}}$ in
$\mathcal{A}_{i}(y)$. The components relative to $dy_{1},...,dy_{n}$ of
$dP=P\wedge P$ are

$\frac{\partial{\mathcal{A}}_{l}}{\partial
y_{k}}+{\mathcal{A}}_{l}{\mathcal{A}}_{k}=\frac{\partial{\mathcal{A}}_{k}}{\partial
y_{l}}+{\mathcal{A}}_{k}{\mathcal{A}}_{l},\quad k\neq l,$ (9.1)

For $k\neq i$ and $l=i$, from (9.1) we receive

$\frac{\partial}{\partial y_{k}}\left(\frac{B_{i}}{y_{i}}+\hbox{\rm
reg}(y_{i})\right)+\left(\frac{B_{i}}{y_{i}}+\hbox{\rm
reg}(y_{i})\right){\mathcal{A}}_{k}=\frac{\partial{\mathcal{A}}_{k}}{\partial
y_{i}}+{\mathcal{A}}_{k}\left(\frac{B_{i}}{y_{i}}+\hbox{\rm
reg}(y_{i})\right),$

where $\hbox{\rm reg}(y_{i})$ stands for an analytic term at $y_{i}=0$. We
expand the above in Taylor series at $y_{i}=0$. The singular term (the residue
at $y_{i}=0$) is

$\frac{\partial B_{i}}{\partial
y_{k}}=\bigl{[}\mathcal{A}_{k}|_{y_{i}=0},B_{i}\bigr{]}=\frac{[B_{k},B_{i}]}{u_{k}-u_{i}}+[\gamma_{k},B_{i}],\quad
k\neq i.$ (9.2)

The above gives the non-normalized Schlesinger equations (3.4)-(3.5), because

$\displaystyle\frac{\partial B_{i}}{\partial y_{k}}=\frac{\partial
B_{i}}{\partial(u_{k}-u_{i})}=\frac{\partial
u_{k}}{\partial(u_{k}-u_{i})}\frac{\partial B_{i}}{\partial
u_{k}}=\frac{\partial B_{i}}{\partial u_{k}},$ (9.3)
$\displaystyle\frac{\partial B_{i}}{\partial u_{i}}=\sum_{k\neq
i}\frac{\partial(u_{k}-u_{i})}{\partial u_{i}}\frac{\partial
B_{i}}{\partial(u_{k}-u_{i})}=-\sum_{k\neq i}\frac{\partial B_{i}}{\partial
u_{k}}\quad\Longrightarrow\quad\sum_{k=1}^{n}\frac{\partial B_{i}}{\partial
u_{k}}=0.$ (9.4)

If we write the components of $dP=P\wedge P$ referring to $dy_{l}$ ad
$d\lambda$, and we substitute into them (9.3)-(9.4), we receive (3.3), namely
$\partial_{l}\gamma_{k}-\partial_{k}\gamma_{l}=\gamma_{l}\gamma_{k}-\gamma_{k}\gamma_{l}$.
∎

######  Corollary 9.1.

A solution $B_{i}(u)$, $i=1,...,n$, of (3.3)-(3.5) is holomorphically similar
to a constant Jordan form on $\mathbb{D}(u^{0})$. The similarity is realized
by a fundamental matrix solution $G^{(i)}(u)$ of the Pfaffian system (9.6)
below.

###### Proof.

We must show that there exists a holomorphically invertible $G^{(i)}(u)$ on
$\mathbb{D}(u^{0})$ such that $(G^{(i)})^{-1}B_{i}G^{(i)}$ is a constant
Jordan form. The conditions (9.1) for $k,l\neq i$ can be evaluated at
$y_{i}=0$, and become

$\frac{\partial{\mathcal{A}}_{l}|_{y_{i}=0}}{\partial
y_{k}}+{\mathcal{A}}_{l}|_{y_{i}=0}{\mathcal{A}}_{k}|_{y_{i}=0}=\frac{\partial{\mathcal{A}}_{k}|_{y_{i}=0}}{\partial
y_{l}}+{\mathcal{A}}_{k}|_{y_{i}=0}{\mathcal{A}}_{l}|_{y_{i}=0},\quad\quad
k\neq i,~{}l\neq i,~{}k\neq l.$

Hence, the following Pfaffian system is Frobenius integrable

$\frac{\partial G}{\partial
y_{k}}=\mathcal{A}_{k}|_{y_{i}=0}~{}G~{}\equiv\left(\frac{B_{k}}{u_{k}-u_{i}}+\gamma_{k}\right)G,\quad
k\neq i.$ (9.5)

Using the chain rule as in (9.3), we receive (6.8)

$\frac{\partial G}{\partial
u_{k}}=\left(\frac{B_{k}}{u_{k}-u_{i}}+\gamma_{k}\right)G,\quad k\neq
i,\quad\quad\quad\frac{\partial G}{\partial u_{i}}=-\sum_{k\neq
i}\left(\frac{B_{k}}{u_{k}-u_{i}}+\gamma_{k}\right)G$ (9.6)

Notice that for both $\varphi(u)=B_{i}(u)$ and $\varphi(u)=G(u)$ we have

$\sum_{k=1}^{n}\frac{\partial\varphi}{\partial
u_{k}}=0\quad\Longrightarrow\quad\varphi(u)=\varphi(u_{1}-u_{i},~{}\dots~{},u_{n}-u_{i}).$
(9.7)

We can take a solution $G(u)$ which holomorphically reduces $B_{i}$ to Jordan
form. Indeed

$\displaystyle\hbox{ for $k\neq i$, }\quad\frac{\partial}{\partial
y_{k}}(G^{-1}B_{i}G)$ $\displaystyle=-G^{-1}\frac{\partial G}{\partial
y_{k}}G^{-1}B_{i}G+G^{-1}\frac{\partial B_{i}}{\partial
y_{k}}G+G^{-1}B_{i}\frac{\partial G}{\partial y_{k}}$
$\displaystyle\underset{(\ref{23agosto2020-4}),(\ref{23agosto2020-7})}{=}-G^{-1}\mathcal{A}_{k}|_{y_{i}=0}B_{i}G+G^{-1}\bigl{[}\mathcal{A}_{k}|_{y_{i}=0},~{}B_{i}\bigr{]}G+G^{-1}B_{i}\mathcal{A}_{k}|_{y_{i}=0}G$
$\displaystyle\quad\quad=0.$

Therefore, keeping into account (9.7), we see that
$\mathcal{B}_{i}:=G^{-1}(u)B_{i}(u)G(u))$ is independent of $u$. Thus, there
exists a constant matrix $\mathcal{G}$ such that
$\mathcal{G}^{-1}\mathcal{B}_{i}\mathcal{G}$ is a constant Jordan form, and
$G^{(i)}(u):=G(u)\mathcal{G}$ realises the holomorphic "Jordanization" . The
above arguments are standard, see for example [28].

∎

If the $B_{i}(u)$ are holomorphic on $\mathbb{D}(u^{c})$ and the vanishing
conditions (4.1) hold, the coefficients of the Pfaffian system (6.39) are
holomorphic on $\mathbb{D}(u^{c})$, so that $G^{(i)}(u)$ extends
holomorphically there, and Corollary 9.1 holds on $\mathbb{D}(u^{c})$.

##  10 Appendix B. Proof of Proposition 3.1

According to Theorem 2.1, system (1.1) is strongly isomonodromic in
$\mathbb{D}(u^{0})$ contained in a $\tau$-cell of $\mathbb{D}(u^{c})$ if and
only if (3.1) holds. In this case $G^{(0)}$ in (2.12) holomorphically reduces
$A(u)$ to constant Jordan form and satisfies

$dG^{(0)}=\sum_{j=1}^{n}\omega_{j}(u)du_{j}~{}G^{(0)}.$ (10.1)

###### Proof of Proposition 3.1.

Suppose that (1.1) is strongly isomonodromic, so that (3.1) holds. Let
$\mathcal{A}:=-A-I$, so that $E_{k}\mathcal{A}=B_{k}$, and (3.1) are rewritten
as $\partial_{i}\mathcal{A}=[\omega_{i}(u),\mathcal{A}]$. We multiply these
equations to the left by $E_{k}$, with $k\neq i$. We receive

$E_{k}\partial_{i}\mathcal{A}=E_{k}[\omega_{i}(u),\mathcal{A}].$

The l.h.s. is $E_{k}\partial_{i}\mathcal{A}=\partial_{i}B_{k}$. The r.h.s. is

$E_{k}[\omega_{i},\mathcal{A}]=E_{k}\omega_{i}\mathcal{A}-E_{k}\mathcal{A}\omega_{i}=E_{k}\omega_{i}\mathcal{A}-B_{k}\omega_{i}=\bigl{(}E_{k}\omega_{i}\mathcal{A}-\omega_{i}B_{k}\bigr{)}+[\omega_{i},B_{k}].$

In conclusion

$\partial_{i}B_{k}=\bigl{(}E_{k}\omega_{i}\mathcal{A}-\omega_{i}B_{k}\bigr{)}+[\omega_{i},B_{k}],\quad
i\neq k.$

The only terms we need to evaluate are

$E_{k}\omega_{i}\mathcal{A}-\omega_{i}B_{k}=E_{k}[F_{1},E_{i}]\mathcal{A}-[F_{1},E_{i}]B_{k}=$
$=E_{k}F_{1}E_{i}\mathcal{A}+E_{i}F_{1}B_{k}=E_{k}F_{1}E_{i}B_{i}+E_{i}F_{1}E_{k}B_{k}.$

In the second line we have used $E_{i}E_{k}=E_{k}E_{i}=E_{i}B_{k}=0$, for
$i\neq k$, and $E_{i}^{2}=E_{i}$. Now, observe that $E_{k}F_{1}E_{i}$ has zero
entries, except for the entry $(k,i)$, which is
$(F_{1})_{ki}=(A)_{ki}/(u_{i}-u_{k})$. This implies that

$E_{k}F_{1}E_{i}B_{i}+E_{i}F_{1}E_{k}B_{k}=\frac{[B_{i},B_{k}]}{u_{i}-u_{k}}.$

In conclusion, we have proved that (3.1) implies (3.4). On the other hand
(3.4)-(3.5) are equivalent to the system given by (3.4) and the equations

$\partial_{i}\sum_{k}B_{k}=[\omega_{i},\sum_{k}B_{k}],\quad i=1,...,n.$

which are exactly (3.1) if $B_{k}=E_{k}\mathcal{A}$. Finally, notice that
(3.3), here with $\gamma_{j}=\omega_{j}$, is the integrability condition on
$\mathbb{D}(u^{0})$ of $dG=\sum_{j=1}^{n}\omega_{j}(u)du_{j}~{}G$. On the
other hand, it is a computation to see that (3.1) implies the the same
conditions.

Conversely, let system (1.4) be strongly isomonodromic, so that the
integrability conditions (3.3)-(3.5) hold. Firstly, we show that (3.4)-(3.5)
imply a Pfaffian system for $A$ of type (3.1). To this end, we sum (3.4) and
(3.5):

$\sum_{k=1}^{n}\partial_{i}B_{k}=\sum_{k\neq
i}\frac{[B_{i},B_{k}]}{u_{i}-u_{k}}-\sum_{k\neq
i}\frac{[B_{i},B_{k}]}{u_{i}-u_{k}}+[\gamma_{i},\sum_{k=1}^{n}B_{k}]=[\gamma_{i},\sum_{k=1}^{n}B_{k}].$

Using $B_{k}=-E_{k}(A+I)$ and $\sum_{k}E_{k}=I$, the above becomes

$\partial_{i}A=[\gamma_{i},A],\quad i=1,...,n.$ (10.2)

Since $\gamma_{1},...,\gamma_{n}$ satisfy (3.3), it is directly verified that
(10.2) is Frobenius integrable. Secondly, we must show that we can choose

$\gamma_{j}:=\omega_{j}=\left(\frac{A_{ab}(\delta_{aj}-\delta_{bj})}{u_{a}-u_{b}}\right)_{a,b=1}^{n}\quad\hbox{
as in (\ref{22agosto2020-6}).}$

Substituting this choice into (3.3), we see that if (10.2) holds, in the form
$\partial_{i}A=[\omega_{i},A]$, then (3.3) are satisfied.212121This is exactly
what has been said before: (3.1) implies by computation the integrability
conditions of (10.1), namely exactly equations (3.3) with
$\gamma_{j}=\omega_{j}$. Now, since (10.2) follows from (3.4)-(3.5) with
matrices $B_{k}=-E_{k}(A+I)$, we conclude that (3.4)-(3.5) and the choice
$\gamma_{j}=\omega_{j}$ guarantee that both (3.3) and (3.1) are satisfied.

∎

The Schlesinger system can be used to show that there is a fundamental matrix
solution $G^{(0)}$ of (10.1) that holomorphically reduces $A$ to constant
Jordan form on $\mathbb{D}(u^{0})$. Since (3.3) is the integrability condition
on $\mathbb{D}(u^{0})$ of the linear Pfaffian system

$dG=\sum_{j=1}^{n}\omega_{j}(u)du_{j}~{}G.$ (10.3)

the latter admits holomorphic fundamental matrix solutions in
$\mathbb{D}(u^{0})$. Let $G(u)$ be one of them and define

$\widehat{B}_{k}:=G(u)^{-1}B_{k}G(u).$ (10.4)

By direct computation, using (10.3) and its integrability (3.3), it is
verified that (3.4)-(3.5) (with $\gamma_{j}=\omega_{j}$) are equivalent to the
normalized Schlesinger equations for the matrices $\widehat{B}_{k}$,

$\partial_{i}\widehat{B}_{k}=\frac{[\widehat{B}_{i},\widehat{B}_{k}]}{u_{i}-u_{k}},\quad
i\neq k;\quad\quad\partial_{i}\widehat{B}_{i}=-\sum_{k\neq
i}\frac{[\widehat{B}_{i},\widehat{B}_{k}]}{u_{i}-u_{k}}.$

The above equations imply that

$\forall~{}i=1,...,n,\quad\quad\partial_{i}\widehat{B}_{\infty}=0,\quad\quad\hbox{
where $\widehat{B}_{\infty}:=-\sum_{k=1}^{n}\widehat{B}_{k}$}$

Being $\widehat{B}_{\infty}$ constant, it can be put in constant Jordan form
by a constant invertible matrix $P$, say
$-\mathcal{J}=P^{-1}\widehat{B}_{\infty}P$. Since also $G(u)P$ solves (10.3),
we can choose from the beginning $G(u)$ such that

$G^{-1}(u)\left(\sum_{k=1}^{n}B_{k}(u)\right)G(u)=\mathcal{J}\quad\hbox{
constant Jordan form}.$ (10.5)

Now, observe that $\sum_{k=1}^{n}E_{k}=I$, so that

$\sum_{k=1}^{n}B_{k}=-\sum_{k=1}^{n}E_{k}(A+I)=-A-I.$

Thus, $G(u)$ also puts $A$ in constant Jordan form, so that222222Up to the
freedom $G\mapsto GG_{*}$ where $G_{*}$ commutes with the Jordan form.

$G(u)=G^{(0)}(u),\quad\quad\hbox{ where $G^{(0)}$ is in (\ref{25nov2018-7})}.$

In particular, $G^{(0)}$ satisfies (10.1).

The second part of the statement of Proposition 2.3 (Prop. 19.2 of [13]) is
now easily proved. Indeed, if $A(u)=G^{(0)}(u)J(G^{(0)})^{-1}$ holomorphically
on $\mathbb{D}(u^{c})$, where $J$ is Jordan, then $G^{(0)}$ satisfies (10.1)
on $\mathbb{D}(u^{0})$ (and $J$ is constant). Since $G^{(0)}(u)$ is
holomorphic on $\mathbb{D}(u^{c})$, the $\omega_{j}$ must be as well, so that
the vanishing conditions (2.22) must hold. Conversely, if $A$ is holomorphic
on on $\mathbb{D}(u^{0})$ and satisfies the vanishing conditions (2.22) (or,
more weakly, if $dA=\sum_{j}[\omega_{j},A]du_{j}$ on
$\mathbb{D}\backslash\Delta$, which automatically implies (2.22) – see Remark
2.1), then $dG=\sum_{j}\omega_{j}du_{j}~{}G$ is integrable with holomorphic
coefficients on $\mathbb{D}(u^{0})$, and admits a fundamental matrix solution
that can be chosen so that $(G^{(0)})^{-1}AG^{(0)}(u)=J$ (the proof is as done
before on $\mathbb{D}(u^{0})$).

##  11 Appendix C. The normal form (8.3)

We prove the expression (8.3) of Section 8, where it was sufficient to only
consider the generic case of all
$\lambda^{\prime}_{1},...,\lambda^{\prime}_{n}~{}\not\in\mathbb{Z}$ and no
integer eigenvalues of $A$. A fundamental matrix solution in Levelt form at
$\lambda=\lambda_{1}$ for system (8.1) has structure

$\mathring{\Psi}(\lambda)=G^{(\boldsymbol{p}_{1})}\Bigl{(}I+\sum_{l=1}^{\infty}\mathfrak{G}_{l}(\lambda-\lambda_{1})^{l}\Bigr{)}(\lambda-\lambda_{1})^{T^{(\boldsymbol{p}_{1})}}(\lambda-\lambda_{1})^{R},$
(11.1)

with

$R=R_{1}+R_{2}+\dots
R_{\kappa},\quad\quad\kappa:=\max\\{T^{(\boldsymbol{p}_{1})}_{ii}-T^{(\boldsymbol{p}_{1})}_{jj}\hbox{
integer}\\}.$

where $R$ is a nilpotent matrix with $R_{ij}\neq 0$ only if
$T^{(\boldsymbol{p}_{1})}_{ii}-T^{(\boldsymbol{p}_{1})}_{jj}$ is a positive
integer. We prove that $R=0$ in our case. The formulae for
$(\mathfrak{G}_{l})_{ij}$ and $(R_{l})_{ij}$ are obtained recursively by
substituting the series into the differential system, and are as follows.

  * •

If $T^{(\boldsymbol{p}_{1})}_{ii}-T^{(\boldsymbol{p}_{1})}_{jj}=l$ (positive
integer), $(\mathfrak{G}_{l})_{ij}$ is arbitrary, and

$(R_{l})_{ij}=\left(\sum_{p=1}^{l-1}(\mathfrak{D}_{l-p}\mathfrak{G}_{l}-\mathfrak{G}_{l}{R}_{l-p})+\mathfrak{D}_{l}\right)_{ij},$

  * •

If $T^{(\boldsymbol{p}_{1})}_{ii}-T^{(\boldsymbol{p}_{1})}_{jj}\neq l$
(positive integer)

$(\mathfrak{G}_{l})_{ij}=\frac{1}{T^{(\boldsymbol{p}_{1})}_{jj}-T^{(\boldsymbol{p}_{1})}_{ii}+l}\left(\sum_{p=1}^{l-1}(\mathfrak{D}_{l-p}\mathfrak{G}_{l}-\mathfrak{G}_{l}{R}_{l-p})+\mathfrak{D}_{l}\right)_{ij}$

The claim that $R=0$ follows from two facts. First, if we evaluate at
$u=u^{c}$ the isomonodromic fundamental matrix solution (6.24), we receive a
fundamental matrix solution of (1.4) at $u=u^{c}$,

$\Psi^{(\boldsymbol{p}_{1})}(\lambda,u^{c})=G^{(\boldsymbol{p}_{1})}\cdot
U^{(\boldsymbol{p}_{1})}(\lambda,u^{c})\cdot(\lambda-\lambda_{1})^{T^{(\boldsymbol{p}_{1})}},$
(11.2)

which has $R=0$, because in the generic case here considered all $R^{(j)}=0$
in (6.24). The expression (11.2) belongs to the class of solutions (11.1).

The second fact is that other solutions in the class (11.1) may have different
matrix-exponents (see [27] and [13]; see also [20, 14] for the case of
Frobenius manifolds), but if $R$ corresponds to one solution, all the other
solutions in the class can only have exponent

$\widetilde{R}=\mathcal{D}^{-1}R\mathcal{D},$ (11.3)

where $\mathcal{D}$ is an invertible matrix explained below. Now, since $R=0$
in (11.2), then by (11.3) all the other $\widetilde{R}=0$. This proves that
(8.3) is the correct form.

Finally, we explain (11.3). System (1.4) at $u=u^{c}$ is holomorphically
equivalent to "Birkhoff-normal forms"

$\frac{d\Psi}{d\lambda}=\left(\frac{T^{(\boldsymbol{p}_{1})}}{\lambda-\lambda_{1}}+\sum_{l=1}^{\kappa}R_{l}(\lambda-\lambda_{1})^{l}\right)\Psi\quad\hbox{
and
}\quad\frac{d\widetilde{\Psi}}{d\lambda}=\left(\frac{T^{(\boldsymbol{p}_{1})}}{\lambda-\lambda_{1}}+\sum_{l=1}^{\kappa}\widetilde{R}_{l}(\lambda-\lambda_{1})^{l}\right)\widetilde{\Psi},$

which are related to each other by a gauge transformations
$\Psi=\mathcal{D}(\lambda)\widetilde{\Psi}$, with
$\mathcal{D}(\lambda)=\mathcal{D}_{0}(I+\mathcal{D}_{0}(\lambda-\lambda_{1})+\dots+\mathcal{D}_{\kappa}(\lambda-\lambda_{1})^{\kappa})$,
where $\hbox{det}(\mathcal{D}_{0})\neq 0$ and
$[\mathcal{D}_{0},T^{(\boldsymbol{p}_{1})}]=0$. Then,
$\mathcal{D}:=\mathcal{D}_{0}(I+\mathcal{D}_{0}+\dots+\mathcal{D}_{\kappa})$.

######  Remark 11.1.

In our case, the equations $R_{l}=0$, $l=1,2,...,\kappa$ are conditions on the
entries of $A(u^{c})$. The above discussion shows that, in the isomonodromic
case, such conditions turn out to be automatically satisfied with the only
vanishing assumption $(A(u^{c}))_{ab}=0$ for $u_{a}^{c}=u_{b}^{c}$. These
conditions are equivalent to the conditions (4.24)-(4.25) of Proposition 4.2
in [13], and probably more conveninent. We will not enter into the tedious
verification of the equivalence.

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# Logarithmic Sobolev Inequalities for Bounded Domains and Applications to
Drift-Diffusion Equations

Elie Abdo  Department of Mathematics
University of California
Santa Barbara, CA 93106-3080, USA<EMAIL_ADDRESS>and  Fizay-Noah Lee
Department of Mathematics
Vanderbilt University
Nashville, TN 37235, USA<EMAIL_ADDRESS>
###### Abstract.

We prove logarithmic Sobolev inequalities on higher-dimensional bounded smooth
domains based on novel Gagliardo-Nirenberg type interpolation inequalities.
Moreover, we use them to address the long-time dynamics of some nonlinear
nonlocal drift-diffusion models and prove the exponential decay of their
solutions to constant steady states.

###### Key words and phrases:

logarithmic Sobolev inequalities, interpolation, inequalities, Nernst-Planck
equations, long-time dynamics

###### Contents

  1. 1 Introduction
  2. 2 Interpolation Inequalities
  3. 3 Logarithmic Sobolev Inequalities
  4. 4 Application: Long-Time Behavior of Solutions to Electrodiffusion Models

##  1\. Introduction

Logarithmic Sobolev inequalities, initiated by Stam in [12], have been
extensively studied in the literature over the last five decades due to their
relevance in quantum field theory, probability theory, and geometry. Such
inequalities have been initially studied for Gaussian measures [1, 8, 12] and
then generalized to Euclidean measures (see for instance [9, Chapter 8], [11],
and references therein).

In this paper, we derive Logarithmic Sobolev inequalities for bounded domains,
and we show how they can serve as a tool to study the long-time dynamics of
some nonlinear nonlocal drift-diffusion equations on $d$-dimensional bounded
connected domains with smooth boundaries. More precisely, given such a domain
$\Omega\subset\mathbb{R}^{d}$ for $2\leq d\leq 6$, and an integer $p\geq 3$
obeying $(d-2)p\leq 2d$, we prove that

$\int_{\Omega}g\ln\frac{g}{\bar{g}}dx\leq
A_{p}{\bar{g}}^{\frac{p-4}{p-2}}\left\|\nabla\left(g^{\frac{1}{p-2}}\right)\right\|_{L^{2}}^{2},\hskip
28.45274pt\bar{g}=\frac{1}{|\Omega|}\int_{\Omega}g(x)dx$ (1.1)

holds for nonnegative scalar functions $g\in H^{1}(\Omega)$, where $A_{p}$ is
a positive constant depending only on $p$, $d$, and the diameter of $\Omega$.
Our proof is based on the following three main ingredients:

  1. (1)

Novel Gagliardo-Nirenberg type interpolation inequalities of the form

$\int_{\Omega}q(x)^{p}dx\leq\bar{q}^{2}\|q\|_{L^{p-2}}^{p-2}+\sum\limits_{i=1}^{p}C_{i}\bar{q}^{\alpha_{i}}\|q-\bar{q}\|_{L^{2}}^{\beta_{i}}\|\nabla
q\|_{L^{2}}^{\gamma_{i}},\hskip
28.45274pt\bar{q}=\frac{1}{|\Omega|}\int_{\Omega}q(x)dx,$ (1.2)

that hold for nonnegative functions $q$ and integers $p\geq 2$ obeying
$(d-2)p\leq 2d$. The nonnegative constants $C_{i}$ depend only on $p$ and the
dimension $d$, and the nonnegative numbers $\alpha_{i},\beta_{i},\gamma_{i}$
satisfy $\alpha_{i}+\beta_{i}+\gamma_{i}=p$ and $\beta_{i}+\gamma_{i}\geq 2$
for any $i\in\left\\{1,\dots,p\right\\}$. The term
$\bar{q}^{2}\|q\|_{L^{p-2}}^{p-2}$ on the right-hand side of (1.2) is sharp
and is crucial for the derivation of the desired estimates (1.1).

  2. (2)

The concavity of the logarithm function, which allows for the application of
Jensen’s inequality and reduces the task of establishing (1.1) to a problem of
seeking good control of some $L^{p}$ norm.

  3. (3)

Basic logarithmic estimates

$\ln\left(1+\sum\limits_{i=1}^{n}C_{i}x^{i}\right)\leq Cx$ (1.3)

that hold for any $x\geq 0$ and some positive universal constant
$C=C(C_{1},\dots,C_{n})$.

One of the main challenges is the derivation of (1.2), which implies (from
applications of the classical $L^{2}$ Poincaré inequality) that the $p$-th
power of the $L^{p}$ norm of $q$ is controlled by a linear combination of
powers of $\|\nabla q\|_{L^{2}}$ plus the sharp term
$\bar{q}^{2}\|q\|_{L^{p-2}}^{p-2}$ (which is simply bounded by $1$ for
specific choices of $q$). Additionally, we must ensure that the linear
combination of powers of $\|\nabla q\|_{L^{2}}$ does not include low powers;
in particular, the exponents must be at least $2$. To this end, we seek a
special decomposition of the $L^{p}$ norm of $q$, which allows us to make use
of standard interpolation inequalities while preserving the aforementioned
desired properties.

This paper is organized as follows. In Section 2, we derive sharp Gagliardo-
Nirenberg type interpolation inequalities (1.2), which apply to nonnegative
functions in $H^{1}(\Omega)$. Then, in Section 3, we use the interpolation
inequalities from Section 2 to derive the logarithmic Sobolev inequalities for
bounded domains (1.1). Lastly, in Section 4, we show how the logarithmic
Sobolev inequalities from Section 3 can be used to prove exponential
convergence to equilibrium for the Nernst-Planck system, which is a nonlinear,
nonlocal drift-diffusion system. We show that the rate of convergence depends
naturally only on the domain and the diffusion coefficients.

##  2\. Interpolation Inequalities

In this section, we prove a class of sharp Gagliardo-Nirenberg type
inequalities. The inequality (2.2) is sharp in the sense that if $q$ is taken
to be a positive constant $c$, then we have

$\int_{\Omega}q(x)^{p}dx=c^{p}|\Omega|=c^{2}c^{p-2}|\Omega|=\bar{q}^{2}\|q\|_{L^{p-2}}^{p-2}.$
(2.1)

Indeed, (2.2) gives the exact corrector arising from interpolating a scalar
function with a non-vanishing average. This will be used to establish a
logarithmic Sobolev inequality that could serve as a main tool to study the
long-time behavior of solutions to a large class of nonlinear nonlocal drift-
diffusion equations.

######  Proposition 2.1.

Let $d\geq 2$. Let $\Omega\subset\mathbb{R}^{d}$ be a bounded, connected
domain with smooth boundary. If $d=2$, then let $p\geq 2$ be an integer. If
$d\geq 3$, then let $p$ be an integer such that $2\leq p\leq\frac{2d}{d-2}$.
Then the following interpolation inequality

$\displaystyle\int_{\Omega}q(x)^{p}dx$
$\displaystyle\leq\bar{q}^{2}\|q\|_{L^{p-2}}^{p-2}+C\|q-\bar{q}\|_{L^{2}}^{\frac{2d+2p-pd}{2}}\|\nabla
q\|_{L^{2}}^{\frac{d(p-2)}{2}}+C\bar{q}^{p-2}\|q-\bar{q}\|_{L^{2}}^{\frac{2d+2p-dp}{p}}\|\nabla
q\|_{L^{2}}^{\frac{d(p-2)}{p}}$ (2.2)
$\displaystyle+C\sum\limits_{k=3}^{p-1}\bar{q}^{k-2}\|q-\bar{q}\|_{L^{2}}^{(p-k+2)-\frac{d(p-k)}{2}}\|\nabla
q\|_{L^{2}}^{\frac{d(p-k)}{2}}+C\bar{q}^{p-2}\|\nabla q\|_{L^{2}}^{2}$

holds for any nonnegative scalar function $q\in H^{1}(\Omega)$. Here $\bar{q}$
denotes the average of $q$ over $\Omega$, and $C$ is a positive constant that
depends only on $d$, $p$ and the diameter of $\Omega$. (We define
$\|q\|_{L^{0}}^{0}:=|\Omega|$, and we assume the summation is empty if $p\leq
3$.)

###### Proof.

We decompose the integral $\int_{\Omega}q^{p}$ into the sum of three terms,
$I_{1},I_{2}$ and $I_{3}$, where

$\displaystyle I_{1}$ $\displaystyle=\int_{\Omega}(q-\bar{q})^{2}q^{p-2}dx,$
(2.3) $\displaystyle I_{2}$
$\displaystyle=-\bar{q}^{2}\int_{\Omega}q^{p-2}dx,$ (2.4) $\displaystyle
I_{3}$ $\displaystyle=2\bar{q}\int_{\Omega}q^{p-1}dx.$ (2.5)

We apply Hölder’s inequality and make use of Sobolev embeddings and Gagliardo-
Nirenberg interpolation inequalities to estimate $I_{1}$ as follows,

$\displaystyle I_{1}$
$\displaystyle\leq\|q-\bar{q}\|_{L^{p}}^{2}\|q^{p-2}\|_{L^{\frac{p}{p-2}}}=\|q-\bar{q}\|_{L^{p}}^{2}\|q\|_{L^{p}}^{p-2}$
(2.6) $\displaystyle\leq
C\|q-\bar{q}\|_{L^{p}}^{2}\left(\|q-\bar{q}\|_{L^{p}}^{p-2}+|\bar{q}|^{p-2}\right)$
$\displaystyle\leq C\|q-\bar{q}\|_{L^{2}}^{\frac{2p-dp+2d}{p}}\|\nabla
q\|_{L^{2}}^{\frac{d(p-2)}{p}}\left(\|q-\bar{q}\|_{L^{2}}^{\frac{(p-2)(2p-dp+2d)}{2p}}\|\nabla
q\|_{L^{2}}^{\frac{d(p-2)^{2}}{2p}}+|\bar{q}|^{p-2}\right)$ $\displaystyle\leq
C\|q-\bar{q}\|_{L^{2}}^{\frac{2d+2p-pd}{2}}\|\nabla
q\|_{L^{2}}^{\frac{(p-2)d}{2}}+C|\bar{q}|^{p-2}\|q-\bar{q}\|_{L^{2}}^{\frac{2p-dp+2d}{p}}\|\nabla
q\|_{L^{2}}^{\frac{d(p-2)}{p}}.$

Now we decompose $I_{3}$ into the sum of four terms, $I_{3,1},I_{3,2}$,
$I_{3,3}$, and $I_{3,4}$, where

$\displaystyle I_{3,1}$
$\displaystyle=2\sum\limits_{k=3}^{p-1}\bar{q}^{k-2}\int_{\Omega}(q-\bar{q})^{2}q^{p-k}dx\left(=0\text{
if }p\leq 3\right),$ (2.7) $\displaystyle I_{3,2}$
$\displaystyle=2\bar{q}^{p-2}\int_{\Omega}(q-\bar{q})^{2}dx,$ (2.8)
$\displaystyle I_{3,3}$
$\displaystyle=2\bar{q}^{p-1}\int_{\Omega}(q-\bar{q})dx,$ (2.9) $\displaystyle
I_{3,4}$ $\displaystyle=2\bar{q}^{2}\int_{\Omega}q^{p-2}dx.$ (2.10)

It is evident that the quantity $I_{3,3}$ vanishes. An application of the
Poincaré inequality to the mean-free function $q-\bar{q}$ yields the bound

$I_{3,2}\leq C\bar{q}^{p-2}\|\nabla q\|_{L^{2}}^{2}.$ (2.11)

As for the term $I_{3,1}$, (assuming $p>3$) we apply Hölder’s inequality,
interpolate, and obtain

$\displaystyle I_{3,1}$ $\displaystyle\leq
C\sum\limits_{k=3}^{p-1}\bar{q}^{k-2}\int_{\Omega}(q-\bar{q})^{2}\left((q-\bar{q})^{p-k}+\bar{q}^{p-k}\right)$
(2.12) $\displaystyle\leq
C\sum\limits_{k=3}^{p-1}\bar{q}^{k-2}\|q-\bar{q}\|_{L^{p-k+2}}^{p-k+2}+C\bar{q}^{p-2}\|q-\bar{q}\|_{L^{2}}^{2}$
$\displaystyle\leq
C\sum\limits_{k=3}^{p-1}\bar{q}^{k-2}\|q-\bar{q}\|_{L^{2}}^{(p-k+2)-\frac{d(p-k)}{2}}\|\nabla
q\|_{L^{2}}^{\frac{d(p-k)}{2}}+C\bar{q}^{p-2}\|\nabla q\|_{L^{2}}^{2}$

Putting all these estimates together, we infer that (2.2) holds, ending the
proof of Proposition 2.1. ∎

##  3\. Logarithmic Sobolev Inequalities

Prior to proving the logarithmic Sobolev inequalities, we establish the
following lemma, which, roughly speaking, gives us linear control of
logarithmic terms.

######  Lemma 3.1.

Let $f:[0,\infty)\to[0,\infty)$ be a $C^{1}$ function satisfying

$\displaystyle f(0)$ $\displaystyle=0$ (3.1)
$\displaystyle\frac{f^{\prime}(x)}{1+f(x)}$ $\displaystyle\leq A\quad\text{for
some constant $A$, for all }x\geq 0.$ (3.2)

Then $\ln(1+f(x))\leq Ax$ for any $x\geq 0.$ In particular, for any
$\alpha_{1},...,\alpha_{n}\geq 1$, it holds that

$\displaystyle\ln\left(1+\sum_{i=1}^{n}C_{i}x^{\alpha_{i}}\right)\leq Ax$
(3.3)

for any $x\geq 0$, where $A$ is a positive constant depending on the
nonnegative constants $C_{1},\dots,C_{n}$ and $\alpha_{1},\dots,\alpha_{n}$.

###### Proof.

Define $G(x)=\ln(1+f(x))-Ax$. We have $G(0)=0$. And,

$\displaystyle G^{\prime}(x)=\frac{f^{\prime}(x)}{1+f(x)}-A\leq 0,$ (3.4)

so it follows that $G(x)\leq 0$ for all $x\geq 0$, and the lemma follows.

∎

######  Theorem 1.

Let $2\leq d\leq 6$ be an integer. Let $p$ be an integer such that
$p\in[3,\frac{2d}{d-2}]$ if $d>2$ and $p\in[3,\infty)$ if $d=2$. Let
$\Omega\subset\mathbb{R}^{d}$ be a bounded, connected domain with smooth
boundary. Let $g\in H^{1}(\Omega)$ be a nonnegative scalar function. Then
there exists a positive constant $A_{p}$ depending only on $p$, $d$, and the
diameter of $\Omega$ such that

$\int_{\Omega}g\ln\frac{g}{\bar{g}}dx\leq
A_{p}{\bar{g}}^{\frac{p-4}{p-2}}\left\|\nabla\left(g^{\frac{1}{p-2}}\right)\right\|_{L^{2}}^{2}.$
(3.5)

holds.

###### Proof.

As $\frac{g}{|\Omega|\bar{g}}dx$ is a probability measure and $\ln x$ is a
concave function, it follows from Jensen’s inequality that

$\displaystyle\int_{\Omega}g\ln\frac{g}{\bar{g}}dx=\frac{(p-2)|\Omega|\bar{g}}{2}\int_{\Omega}\frac{g}{|\Omega|\bar{g}}\ln\left(\frac{g}{\bar{g}}\right)^{\frac{2}{p-2}}dx$
(3.6)
$\displaystyle\leq\frac{(p-2)|\Omega|\bar{g}}{2}\ln\left(\int_{\Omega}\frac{g^{\frac{p}{p-2}}}{|\Omega|\bar{g}^{\frac{p}{p-2}}}dx\right)=\frac{p-2}{2}\|g\|_{L^{1}}\ln\left(\left\|\frac{1}{|\Omega|^{\frac{1}{p}}}\left(\frac{g}{\bar{g}}\right)^{\frac{1}{p-2}}\right\|_{L^{p}}^{p}\right).$

We let
$q=\frac{1}{|\Omega|^{\frac{1}{p}}}\left(\frac{g}{\bar{g}}\right)^{\frac{1}{p-2}}$.
A straightforward application of Hölder’s inequality with exponents $p-2$ and
$\frac{p-2}{p-3}$ ($=\infty$ if $p=3$) yields the estimate

$\bar{q}^{2}=\left(\frac{1}{|\Omega|}\int_{\Omega}\frac{1}{|\Omega|^{\frac{1}{p}}}\left(\frac{g}{\bar{g}}\right)^{\frac{1}{p-2}}dx\right)^{2}\leq\frac{1}{|\Omega|^{2+\frac{2}{p}}}\left(\int_{\Omega}\frac{g}{\bar{g}}dx\right)^{\frac{2}{p-2}}|\Omega|^{\frac{2p-6}{p-2}}=|\Omega|^{-2-\frac{2}{p}}|\Omega|^{\frac{2}{p-2}}|\Omega|^{\frac{2p-6}{p-2}}=|\Omega|^{-\frac{2}{p}},$
(3.7)

and a direct computation gives the equality

$\|q\|_{L^{p-2}}^{p-2}=\int_{\Omega}\frac{g}{\bar{g}|\Omega|^{\frac{p-2}{p}}}dx=|\Omega|^{\frac{2}{p}}.$
(3.8)

Applying Proposition 2.1, the Poincaré inequality $\|q-\bar{q}\|_{L^{2}}\leq
C\|\nabla q\|_{L^{2}}$, and estimate (3.7), we deduce that

$\|q\|_{L^{p}}^{p}\leq 1+C_{1}\|\nabla q\|_{L^{2}}^{2}+C_{2}\|\nabla
q\|_{L^{2}}^{p}+\sum\limits_{k=3}^{p-1}C^{\prime}_{k}\|\nabla
q\|_{L^{2}}^{p-k+2}$ (3.9)

holds, where $C_{1},C_{2}$ and the $C^{\prime}_{k}$’s depend only on $\Omega$
and $p$. As before, the summation is considered empty if $p\leq 3$.

Since $\nabla q=\frac{\nabla
g^{\frac{1}{p-2}}}{|\Omega|^{\frac{1}{p}}\bar{g}^{\frac{1}{p-2}}}$, it follows
from Lemma 3.1 that

$\ln(\|q\|_{L^{p}}^{p})\leq\frac{A^{\prime}_{p}}{\bar{g}^{\frac{2}{p-2}}}\left\|\nabla\left(g^{\frac{1}{p-2}}\right)\right\|_{L^{2}}^{2},$
(3.10)

where $A^{\prime}_{p}$ is a positive constant depending only on $p$, $d$, and
the diameter of $\Omega$. Returning to (3.6), we obtain the desired estimate
(3.5). ∎

######  Remark 1.

We note that the inequality (3.5) becomes

$\int_{\Omega}g\ln\frac{g}{\bar{g}}\,dx\leq
A_{4}\|\nabla\sqrt{g}\|_{L^{2}}^{2}$ (3.11)

when $d=2,3,4$ and $p=4$. We can compare this inequality to Gross’s
logarithmic Sobolev inequality [8], which in particular for $0\leq f\in
H^{1}(\mathbb{R}^{d})$ says

$\int_{\mathbb{R}^{d}}f\ln\frac{f}{\bar{f}}\,d\nu\leq
C\int_{\mathbb{R}^{d}}|\nabla\sqrt{f}|^{2}\,d\nu$ (3.12)

where $\nu$ is the standard Gaussian measure on $\mathbb{R}^{d}$ and

$\bar{f}=\int_{\mathbb{R}^{d}}f\,d\nu.$

The Bakry-Emery criteria together with the perturbation lemma of Holley and
Strook give conditions on probability measures $\mu$ on $\mathbb{R}^{d}$ for
which (3.12) holds with $\nu$ replaced by $\mu$ (see [2] and references
therein). In addition, by making using of a convex cutoff function, one can
deduce from (3.12) a Poincaré-type inequality

$\int_{D}f\ln\frac{f}{\bar{f}}\,d\mu\leq C\int_{D}|\nabla\sqrt{f}|^{2}\,d\mu$
(3.13)

for uniformly convex domains $D$ and admissible probability measures $\mu$ on
$D$. Essentially, the inequality (3.11) treats the special case of the measure
$|\Omega|^{-1}dx$ for $d=2,3,4$ and at least in this context, generalizes
(3.13) to domains that need not be convex. Let us note, however, that the
original proofs of (3.12) and (3.13) follow very different approaches from
ours.

##  4\. Application: Long-Time Behavior of Solutions to Electrodiffusion
Models

The Nernst-Planck (NP) system is a mathematical model for the electrodiffusion
of charged particles and is given by

$\displaystyle\partial_{t}c_{i}$ $\displaystyle=D_{i}{\mbox{div}\,}(\nabla
c_{i}+z_{i}c_{i}\nabla\Phi),\quad i=1,\dots,N$ (4.1)
$\displaystyle-\epsilon\Delta\Phi$
$\displaystyle=\rho=\sum_{i=1}^{N}z_{i}c_{i}$ (4.2)

The variable $c_{i}$ represents the local ionic concentration of the $i$-th
ionic species and its time evolution is modeled by the Nernst-Planck equations
(4.1). The dynamics of $c_{i}$ are dictated by diffusion due to its own
concentration gradient and by the presence of an electrical field,
$\nabla\Phi$. The electrical potential $\Phi$, in turn, is generated by the
charge density $\rho$ via a Poisson equation.

The $D_{i}>0$ are the constant diffusion coefficients, $\epsilon>0$ represents
the dielectric permittivity of the medium, and the $z_{i}$ are the ionic
valences, which may take on positive or negative values. We note that
$\epsilon$ is proportional to the square of the Debye length and, in most
physical regimes, is a very small constant. This fact motivates the study of
the NP system in the context of singular perturbation theory where $\epsilon$
is considered in the limit of $\epsilon\to 0$ [7, 14].

We consider the above system on a bounded domain $\Omega\subset\mathbb{R}^{d}$
for $d\geq 2$ with smooth boundary. A wide range of physically relevant
boundary conditions may be considered (insert ref). Here, we take homogeneous
Neumann boundary conditions for $\Phi$ and all $c_{i}$:

$\displaystyle{\partial_{n}c_{i}}_{|\partial\Omega}$ $\displaystyle=0,\quad
i=1,\dots,N$ (4.3) $\displaystyle{\partial_{n}\Phi}_{|\partial\Omega}$
$\displaystyle=0.$

This choice of boundary conditions necessitates a few comments. First,
strictly speaking $\Phi$ is determined uniquely only up to a constant from
(4.2) and (4.3). Thus for concreteness, we impose the additional condition
that

$\displaystyle\int_{\Omega}\Phi\,dx=0$ (4.4)

for all time.

Second, ionic concentration is conserved in the sense that

$\displaystyle\frac{d}{dt}\int_{\Omega}c_{i}\,dx=0$ (4.5)

for each $i$. We note, formally integrating (4.2), we find that the net charge
density must be zero for all time

$\displaystyle\int_{\Omega}\rho\,dx=0\quad t\geq 0.$ (4.6)

Therefore, to be self consistent, we restrict ourselves to initial conditions
such that $\int_{\Omega}\rho(0)\,dx=0$. Incidentally, this also requires us to
have both positive and negative valences $z_{i}$.

For our choice of boundary conditions, it is well known that the following
dissipative energy equality holds:

$\displaystyle\frac{d}{dt}\mathcal{E}+\mathcal{D}=0$ (4.7)

where

$\displaystyle\mathcal{E}=\sum_{i=1}^{N}\int_{\Omega}c_{i}\ln\frac{c_{i}}{\bar{c}_{i}}\,dx+\frac{\epsilon}{2}\int_{\Omega}|\nabla\Phi|^{2}\,dx$
(4.8)

where $\bar{c}_{i}=\frac{1}{|\Omega|}\int_{\Omega}c_{i}\,dx$, and

$\displaystyle\mathcal{D}=\sum_{i=1}^{N}D_{i}\int_{\Omega}c_{i}|\nabla(\ln
c_{i}+z_{i}\Phi)|^{2}\,dx.$ (4.9)

For the purpose of this paper, we do not directly address the issue of global
regularity of solutions and instead assume that we are working with globally
regular solutions. We could instead study global weak solutions, in which case
the energy equality (4.7) is instead an inequality; however we shall not
pursue this path. For the state of the art of global wellposedness and
regualrity, we refer the reader to [4, 5, 6].

For regular solutions, the derivation of the above energy equality follows
straightforwardly from multiplying (4.1) by the quantity $\ln
c_{i}+z_{i}\Phi$, summing in $i$, and integrating by parts. Indeed, we have

$\displaystyle\sum_{i=1}^{N}D_{i}\int_{\Omega}{\mbox{div}\,}(\nabla
c_{i}+z_{i}c_{i}\nabla\Phi)(\ln c_{i}+z_{i}\Phi)\,dx$
$\displaystyle=-\sum_{i=1}^{N}D_{i}\int_{\Omega}(\nabla
c_{i}+z_{i}c_{i}\nabla\Phi)\cdot\nabla(\ln c_{i}+z_{i}\Phi)\,dx$ (4.10)
$\displaystyle=-\sum_{i=1}^{N}D_{i}\int_{\Omega}c_{i}|\nabla(\ln
c_{i}+z_{i}\Phi)|^{2}\,dx$

where in the first equality, no boundary terms arise due to the boundary
conditions (4.3). On the other hand, on the left hand side of (4.1), we get

$\displaystyle\sum_{i=1}^{N}\int_{\Omega}\partial_{t}c_{i}(\ln
c_{i}+z_{i}\Phi)\,dx$
$\displaystyle=\sum_{i=1}^{N}\int_{\Omega}\partial_{t}(c_{i}\ln
c_{i}-c_{i})+\int_{\Omega}(\partial_{t}\rho)\Phi\,dx$ (4.11)
$\displaystyle=\sum_{i=1}^{N}\frac{d}{dt}\int_{\Omega}c_{i}\ln
c_{i}\,dx-\epsilon\int_{\Omega}(\partial_{t}\Delta\Phi)\Phi\,dx$
$\displaystyle=\sum_{i=1}^{N}\frac{d}{dt}\int_{\Omega}c_{i}\ln
c_{i}\,dx+\epsilon\int_{\Omega}(\partial_{t}\nabla\Phi)\cdot\nabla\Phi\,dx$
$\displaystyle=\sum_{i=1}^{N}\frac{d}{dt}\int_{\Omega}c_{i}\ln
c_{i}\,dx+\frac{\epsilon}{2}\frac{d}{dt}\int_{\Omega}|\nabla\Phi|^{2}\,dx$

where in the second line we used (4.5). Then, we observe that due to (4.5)
again, we have
$\frac{d}{dt}\int_{\Omega}c_{i}\ln\frac{c_{i}}{\bar{c}_{i}}\,dx=\frac{d}{dt}\int_{\Omega}c_{i}\ln
c_{i}\,dx$, and this completes the derivation of (4.7).

We note that the main reason we define $\mathcal{E}$ with
$\int_{\Omega}c_{i}\ln\frac{c_{i}}{\bar{c}_{i}}\,dx$ instead of
$\int_{\Omega}c_{i}\ln c_{i}\,dx$ is that (as we will see below)
$c_{i}\to\bar{c}_{i}$ in the time asymptotic limit, and thus we view
$\int_{\Omega}c_{i}\ln\frac{c_{i}}{\bar{c}_{i}}\,dx$ as an expression of
relative entropy, which in a way measures the distance between $c_{i}$ and
$\bar{c}_{i}$.

For our choice of boundary conditions (4.3), we have the following lower bound
for $\mathcal{D}$

$\displaystyle\mathcal{D}$ $\displaystyle\geq
D\sum_{i=1}^{N}\int_{\Omega}c_{i}|\nabla(\ln c_{i}+z_{i}\Phi)|^{2}\,dx$ (4.12)
$\displaystyle=D\sum_{i=1}^{N}\int_{\Omega}c_{i}\left|\frac{\nabla
c_{i}}{c_{i}}+z_{i}\nabla\Phi\right|^{2}\,dx$
$\displaystyle=D\sum_{i=1}^{N}\int_{\Omega}\frac{|\nabla
c_{i}|^{2}}{c_{i}}+2z_{i}\nabla
c_{i}\cdot\nabla\Phi+z_{i}^{2}c_{i}|\nabla\Phi|^{2}\,dx$
$\displaystyle=D\sum_{i=1}^{N}\int_{\Omega}4|\nabla\sqrt{c_{i}}|^{2}+2z_{i}\nabla
c_{i}\cdot\nabla\Phi+z_{i}^{2}c_{i}|\nabla\Phi|^{2}\,dx$
$\displaystyle=D\sum_{i=1}^{N}\int_{\Omega}4|\nabla\sqrt{c_{i}}|^{2}+z_{i}^{2}c_{i}|\nabla\Phi|^{2}\,dx+\frac{2D}{\epsilon}\int_{\Omega}\rho^{2}\,dx$
$\displaystyle=:\tilde{\mathcal{D}}$

where $D=\min_{i}D_{i}$, and the second to last line follows from integrating
by parts using the boundary conditions (4.3) and the Poisson equation (4.2).
In summary, we have shown,

######  Proposition 4.1.

Let $c_{1},...,c_{N}$ be a classical solution of (4.1)-(4.2) with initial
concentrations satisfying boundary conditions (4.3) and net electroneutrality,

$\sum_{i=1}^{N}z_{i}\bar{c}_{i}=0$

where $\bar{c}_{i}=\frac{1}{|\Omega|}\int_{\Omega}c_{i}(0)\,dx.$ Then
$c_{1},...,c_{N}$ satisfy

$\displaystyle\frac{d}{dt}\mathcal{E}+\mathcal{D}=0$ (4.13)

where

$\displaystyle\mathcal{E}=\sum_{i=1}^{N}\int_{\Omega}c_{i}\ln\frac{c_{i}}{\bar{c}_{i}}\,dx+\frac{\epsilon}{2}\int_{\Omega}|\nabla\Phi|^{2}\,dx$
(4.14)

and

$\displaystyle\mathcal{D}=\sum_{i=1}^{N}D_{i}\int_{\Omega}c_{i}|\nabla(\ln
c_{i}+z_{i}\Phi)|^{2}\,dx.$ (4.15)

In particular, $c_{1},...,c_{N}$ satisfy

$\displaystyle\frac{d}{dt}\mathcal{E}+\tilde{\mathcal{D}}\leq 0$ (4.16)

where

$\displaystyle\tilde{\mathcal{D}}=D\left(\sum_{i=1}^{N}\int_{\Omega}4|\nabla\sqrt{c_{i}}|^{2}\,dx+\int_{\Omega}\frac{2}{\epsilon}\rho^{2}+z_{i}^{2}c_{i}|\nabla\Phi|^{2}\,dx\right)$
(4.17)

and $D=\min_{i}D_{i}$.

For the second term in the dissipative term $\tilde{\mathcal{D}}$, we remark
that from (4.2) and the fact that $\Phi$ satisfies homogeneous Neumann
boundary conditions, we have the following elliptic estimate [10]

$\displaystyle\frac{2}{\epsilon}\int_{\Omega}\rho^{2}\,dx=2\epsilon\int_{\Omega}\left(\frac{\rho}{\epsilon}\right)^{2}\,dx\geq
C\epsilon\int_{\Omega}|\nabla\Phi|^{2}\,dx$ (4.18)

where $C>0$ depends only on $\Omega$. In addition, for the first term in
$\tilde{\mathcal{D}}$, we have the following bounds,

######  Proposition 4.2.

For dimensions $d=2,3,4$, we have

$\displaystyle\int_{\Omega}c_{i}\ln\frac{c_{i}}{\bar{c}_{i}}\,dx\leq
C\int_{\Omega}|\nabla\sqrt{c_{i}}|^{2}\,dx$ (4.19)

for each $i$, where $C$ depends only on $\Omega$.

###### Proof.

This follows from Theorem 1 by taking $g=c_{i}$ and $d=2,3,4$, $p=4$. ∎

Now we state and prove our main result of this section:

######  Theorem 2.

Under the same assumptions as Proposition 4.1, the concentrations
$c_{1},...,c_{N}$ converge to $\bar{c}_{1},...,\bar{c}_{N}$ in relative
entropy and $\nabla\Phi$ converges to $0$ in $L^{2}$, that is,

$\mathcal{E}(t)\leq\mathcal{E}(0)e^{-rt}$ (4.20)

where $r>0$ depends only on the dimension, $D_{i}$ and $\Omega$. In
particular, $c_{i}$ converges to $\bar{c}_{i}$ in $L^{1}$:

$\sum_{i=1}^{N}\frac{\|c_{i}(t)-\bar{c}_{i}\|_{L^{1}}^{2}}{\bar{c}_{i}}\leq
C_{\Omega}\mathcal{E}(0)e^{-rt}$ (4.21)

where $C_{\Omega}>0$ depends only on $\Omega$.

###### Proof.

It follows from (4.16), (4.18), and Proposition 4.2 that

$\displaystyle\frac{d}{dt}\mathcal{E}+CD\mathcal{E}\leq 0$ (4.22)

where $C$ depends only on $\Omega$. Then, (4.20) follows from the above
inequality. Next, using the Csiszar-Kullback-Pinsker inequality (see e.g.
[13]), we have for each $i$,

$\displaystyle\|c_{i}-\bar{c}_{i}\|_{L^{1}}^{2}\leq
2|\Omega|\bar{c}_{i}\int_{\Omega}c_{i}\ln\frac{c_{i}}{\bar{c}_{i}}\,dx.$
(4.23)

Then, (4.21) follows from (4.20) and the above inequality. ∎

######  Remark 2.

In light of this theorem, we briefly mention a closely related work, which
also addresses the question of exponential convergence to equilibrium in the
context of the Nernst-Planck system. In [3], the authors establish the
exponential convergence to equilibrium for the Nernst-Planck system for
nonlinear no-flux boundary conditions. For this choice of boundary conditions,
an analog of the dissipative equality from Proposition 4.1 holds, and the
authors use logarithmic Sobolev inequalities derived in [2] to control the
dissipative term corresponding to $\mathcal{D}$. One main difference with our
work is that the logarithmic Sobolev inequality from [2] is obtained from a
more general result that holds on Euclidean spaces (as opposed to bounded
domains); and, the restriction from Euclidean spaces to bounded domains
necessitates the use of a convex cut-off function, which restricts the
applicability of the inequality to functions defined on uniformly convex
domains (see Remark 1). On the other hand, with our logarithmic Sobolev
inequality, we only require that the domain is connected (in particular, it
need not even be simply connected).

Another difference has to do with the rate of convergence $r$. In [3], the
logarithmic Sobolev inequality is used where the underlying probability
measure is $e^{-z_{i}\Phi}/\left(\int_{\Omega}e^{-z_{i}\Phi}\,dx\right)dx,$ in
analogy with the classical logarithmic Sobolev inequality for Gaussian
measures. The use of this measure is what allows for the application of the
inequality from [2]. On the other hand, this means that the rate of
convergence ends up depending on pointwise lower and upper bounds on the
potential $\Phi$. While uniform in time $L^{\infty}$ bounds on $\Phi$ are
obtainable (see e.g. [4]), sharp bounds are unknown, and all known bounds
depend on negative powers of the parameter $\epsilon$. In particular, these
bounds are unbounded in the electroneutral (or quasi-neutral) limit of
$\epsilon\to 0$. In contrast, the rate $r>0$ obtained in our theorem depends
naturally on just the diffusion coefficients $D_{i}$ and the domain $\Omega$.
In particular, this rate is independent of $\epsilon$, which is significant in
the context of the study of the electroneutral limit. We point out, however,
that our ability to derive this $\epsilon$-independent rate depended crucially
on our choice of boundary conditions (4.3), and it is not obvious how to
generalize our proof to more complex, nonlinear boundary conditions, which
preserve the dissipative structure.

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