is smooth on $\Omega$, as from (3.4) this quickly implies that $G^{\star}$ is
smooth on $\Omega^{\operatorname{inv}}$.

To that end, fix some point $(t_{0},x_{0})\in\Omega$; we must show that
$H^{\star}$ is smooth in a neighborhood of $(t_{0},x_{0})$. Let
$a^{\prime},b^{\prime}\in(a,b)$ be real numbers such that
$a^{\prime}<t_{0}<b^{\prime}$. Then, 3.3 implies that
$(\mu_{t})_{t\in[a^{\prime},b^{\prime}]}$ is the bridge-limiting measure
process on $[a^{\prime},b^{\prime}]$ with boundary data
$(\mu_{a^{\prime}};\mu_{b^{\prime}})$. Furthermore, 3.7 implies that the
densities $\varrho_{a^{\prime}}$ of $\mu_{a^{\prime}}$ and
$\varrho_{b^{\prime}}$ of $\mu_{b^{\prime}}$ are compactly supported and
uniformly bounded. Hence, 3.3, the second part of 3.11, and 3.14 (with the
$(a,b;\mu_{a},\mu_{b})$ there equal to
$(a^{\prime},b^{\prime};\mu_{a},\mu_{b})$ here) imply that
$\varrho_{t}^{\star}(x)$ and $u_{t}^{\star}(x)$ are smooth for $(t,x)$ in a
neighborhood of $(t_{0},x_{0})$. Together with (3.20) (again with 3.14), this
implies that $H^{\star}$ is smooth in a neighborhood of $(t_{0},x_{0})$,
verifying the lemma. ∎

##  Appendix D Proofs of Results From Section 8

###  D.1. Proof of 8.2

In this section we establish 8.2. We first require the following definition
from [50], which defines a class of initial data for which it is possible to
show quick convergence of Dyson Brownian motion. In what follows, we recall
the Stieltjes transform $m_{\nu}$ associated with a measure $\nu$ from (2.5),
and the empirical measure $\operatorname{emp}(\bm{a})$ associated with a
sequence $\bm{a}$ from (3.1).

######  Definition D.1 ([50, Definition 2.1]).

Fix real numbers $E_{0}\in\mathbb{R}$ and $\zeta>0$. For each positive real
number $n$, let $r=r_{n}$ and $R=R_{n}$ be two parameters satisfying
$n^{\zeta-1}\leq r\leq n^{-\zeta}$ and $n^{\zeta}r\leq R\leq n^{-\zeta}$. We
call an $n$-tuple $\bm{d}=(d_{1},d_{2},\ldots,d_{n})\in\mathbb{R}^{n}$
_$(r,R;\zeta)$ -regular with respect to $E_{0}$_ if the following holds.
Denoting $\nu=\nu^{n}=\operatorname{emp}(\bm{d})$, there exist constants
$c\in(0,1)$ and $C>1$ (both independent of $n$) such that
$\max_{j\in\llbracket 1,n\rrbracket}|d_{j}|\leq n^{C}$ and

$\displaystyle c\leq\operatorname{Im}m_{\nu^{n}}(E+\mathrm{i}\eta)\leq
C,\qquad\text{for all $n\geq C$ and}\qquad(E,\eta)\in(E-R,E+R)\times(r,10).$

######  Lemma D.2.

On the event $\mathscr{A}_{0}$, the $n^{\prime}$-tuples
$\widetilde{\bm{w}}^{-}(0)$, $\widetilde{\bm{w}}(0)$, and
$\widetilde{\bm{w}}^{+}(0)$ are
$\big{(}(n^{\prime})^{\omega/2-1};(n^{\prime})^{-\omega/2};\frac{\omega}{2}\big{)}$-regular
with respect to $0$.

###### Proof.

We only show the lemma for $\widetilde{\bm{w}}(0)$, as the proof is entirely
analogous for $\widetilde{\bm{w}}^{-}(0)$ and $\widetilde{\bm{w}}^{+}(0)$;
throughout, we restrict to the event $\mathscr{A}_{0}$ and denote
$\widetilde{\nu}=\widetilde{\nu}^{n}=\operatorname{emp}\big{(}\widetilde{\bm{w}}(0)\big{)}$.
Then, for any $z\in\mathbb{H}$,

$\displaystyle
m_{\nu}(z)=\displaystyle\frac{1}{n^{\prime}}\displaystyle\sum_{j=1}^{n^{\prime}}\displaystyle\frac{1}{\widetilde{w}_{j}(0)-z}.$

Let $z=E+\mathrm{i}\eta$ for some real numbers $E$ and $\eta$ with
$|E|\leq(n^{\prime})^{-\omega/2}$ and $(n^{\prime})^{\omega/2-1}\leq\eta\leq
10$. Then,

$\displaystyle\big{|}m_{\nu}(z)\big{|}$
$\displaystyle\leq\Bigg{|}\displaystyle\frac{1}{n^{\prime}}\displaystyle\sum_{j=1}^{n^{\prime}}\displaystyle\frac{1}{\widetilde{G}(0,jn^{\prime-1})-z}\Bigg{|}+\displaystyle\frac{1}{n^{\prime}}\displaystyle\sum_{j=1}^{n^{\prime}}\displaystyle\frac{\big{|}\widetilde{G}(0,jn^{\prime-1})-\widetilde{w}_{j}(0)\big{|}}{\big{|}\widetilde{w}_{j}(0)-z\big{|}\cdot\big{|}\widetilde{G}(0,jn^{\prime-1})-z\big{|}}$
$\displaystyle\leq\Bigg{|}\displaystyle\frac{1}{n^{\prime}}\displaystyle\sum_{j=1}^{n^{\prime}}\displaystyle\frac{1}{\widetilde{G}(0,jn^{\prime-1})-z}\Bigg{|}+\displaystyle\frac{1}{n^{\prime}\eta^{2}}\displaystyle\max_{j\in\llbracket
1,n^{\prime}\rrbracket}\bigg{|}\widetilde{G}\Big{(}0,\frac{j}{n^{\prime}}\Big{)}-\widetilde{w}_{j}(0)\bigg{|}$
$\displaystyle\leq\Bigg{|}\displaystyle\frac{1}{n^{\prime}}\displaystyle\sum_{j=1}^{n^{\prime}}\displaystyle\frac{1}{\widetilde{G}(0,jn^{\prime-1})-z}\Bigg{|}+\eta^{-2}(n^{\prime})^{\omega/2-2}\leq\Bigg{|}\displaystyle\frac{1}{n^{\prime}}\displaystyle\sum_{j=1}^{n^{\prime}}\displaystyle\frac{1}{\widetilde{G}(0,jn^{\prime-1})-z}\Bigg{|}+(n^{\prime})^{-\omega/2},$

where in the second bound we used the fact that
$\big{|}\widetilde{w}_{j}(0)-z\big{|}\cdot\big{|}\widetilde{G}(0,\frac{j}{n^{\prime}})-z\big{|}\geq(\operatorname{Im}z)^{2}=\eta^{2}$;
for the third we used the fact that we are restricting to $\mathscr{A}_{0}$,
together with (8.12); and in the fourth we used the fact that
$\eta\geq(n^{\prime})^{\omega/2-2}$.

Now let $j_{0}\in\llbracket 1,n^{\prime}\rrbracket$ be the integer (if it
exists) such that $\widetilde{G}\big{(}0,\frac{j_{0}}{n^{\prime}}\big{)}\leq
E<\widetilde{G}\big{(}0,\frac{j_{0}-1}{n^{\prime}}\big{)}$. Observe from
(8.13) that $\partial_{y}\widetilde{G}\leq-\frac{\varepsilon}{2}$ and
$\widetilde{G}(0,\frac{1}{2})=0$. Therefore, since
$|E|\leq(n^{\prime})^{-\omega/2}$, it follows for sufficiently large $n$ that
$j_{0}$ exists and

$\displaystyle\Big{|}j_{0}-\frac{n^{\prime}}{2}\Big{|}\leq
2\varepsilon^{-1}(n^{\prime})^{-\omega/2}.$

We will assume in what follows that $j_{0}\geq\frac{n^{\prime}}{2}$, as the
proof in the alternative case when $j_{0}<\frac{n^{\prime}}{2}$ is entirely
analogous. Then,

$\displaystyle\Bigg{|}$
$\displaystyle\displaystyle\frac{1}{n^{\prime}}\displaystyle\sum_{j=1}^{n^{\prime}}\displaystyle\frac{1}{\widetilde{G}(0,jn^{\prime-1})-z}\Bigg{|}$
$\displaystyle\leq\Bigg{|}\displaystyle\frac{1}{2n^{\prime}}\displaystyle\sum_{j=2j_{0}-n^{\prime}}^{n^{\prime}}\bigg{(}\displaystyle\frac{1}{\widetilde{G}(0,jn^{\prime-1})-z}+\displaystyle\frac{1}{\widetilde{G}(0,2j_{0}n^{\prime-1}-jn^{\prime-1})-z}\bigg{)}\Bigg{|}+\Bigg{|}\displaystyle\frac{1}{n^{\prime}}\displaystyle\sum_{j=1}^{2j_{0}-n^{\prime}-1}\displaystyle\frac{1}{\widetilde{G}(0,jn^{\prime-1})-z}\Bigg{|}$
$\displaystyle\leq\Bigg{|}\displaystyle\frac{1}{n^{\prime}}\displaystyle\sum_{j=2j_{0}-n^{\prime}}^{n^{\prime}}\displaystyle\frac{\widetilde{G}(0,jn^{\prime-1})+\widetilde{G}(0,2j_{0}n^{\prime-1}-jn^{\prime-1})-2z}{\big{(}\widetilde{G}(0,jn^{\prime-1})-z\big{)}\big{(}\widetilde{G}(0,2j_{0}n^{\prime-1}-jn^{\prime-1})-z\big{)}}\Bigg{|}+(n^{\prime})^{-\omega/8},$

where in the second bound we used the fact that for $j<2j_{0}-n^{\prime}\leq
4\varepsilon^{-1}(n^{\prime})^{1-\omega/2}$ we have

$\displaystyle\bigg{|}\widetilde{G}\Big{(}0,\frac{j}{n^{\prime}}\Big{)}-z\bigg{|}$
$\displaystyle\geq\widetilde{G}\Big{(}0,\displaystyle\frac{j}{n^{\prime}}\Big{)}-E$
$\displaystyle\geq\widetilde{G}\Big{(}0,\frac{j}{n^{\prime}}\Big{)}-\widetilde{G}\Big{(}0,\frac{j_{0}-1}{n^{\prime}}\Big{)}\geq\frac{\varepsilon(j_{0}-j-1)}{2n^{\prime}}\geq\frac{\varepsilon}{4}-\displaystyle\frac{10(n^{\prime})^{-\omega/2}}{\varepsilon}\geq\frac{\varepsilon}{8}\geq(n^{\prime})^{-\omega/8},$

by (8.13). We additionally have for sufficiently large $n$ that

(D.1) $\displaystyle\begin{aligned}
\bigg{|}\widetilde{G}\Big{(}0,\frac{j}{n^{\prime}}\Big{)}+\widetilde{G}\Big{(}0,\displaystyle\frac{2j_{0}-j}{n^{\prime}}\Big{)}-2z\bigg{|}&\leq\bigg{|}\widetilde{G}\Big{(}0,\displaystyle\frac{j}{n^{\prime}}\Big{)}+\widetilde{G}\Big{(}0,\displaystyle\frac{2j_{0}-j}{n^{\prime}}\Big{)}-2\widetilde{G}\Big{(}0,\displaystyle\frac{j_{0}}{n^{\prime}}\Big{)}\bigg{|}+2\eta+\displaystyle\frac{4}{\varepsilon
n^{\prime}}\\\
&\leq\displaystyle\frac{2B_{0}n^{\prime}}{n}\cdot\Big{(}\displaystyle\frac{j_{0}-j}{n^{\prime}}\Big{)}^{2}+3\eta=\displaystyle\frac{2B_{0}(j-j_{0})^{2}}{nn^{\prime}}+3\eta,\end{aligned}$

where in the first inequality we used the facts that $z=E+\mathrm{i}\eta$ with
$\widetilde{G}\big{(}0,\frac{j}{n^{\prime}}\big{)}\leq
E<\widetilde{G}\big{(}0,\frac{j-1}{n^{\prime}}\big{)}$ and
$-2\varepsilon^{-1}\leq\partial_{y}\widetilde{G}\leq-\frac{\varepsilon}{2}$
(by (8.13)), and in the second we used the facts that
$[\widetilde{G}]_{2}\leq\frac{2B_{0}n^{\prime}}{n}$ (again by (8.13)) and that
$\eta\geq(n^{\prime})^{\omega-1}$. We also have that

(D.2)
$\displaystyle\bigg{|}\widetilde{G}\Big{(}0,\displaystyle\frac{j}{n^{\prime}}\Big{)}-z\bigg{|}\geq\displaystyle\frac{\varepsilon|j-j_{0}|}{4n^{\prime}}+\displaystyle\frac{\eta}{4};\qquad\bigg{|}G\Big{(}0,\displaystyle\frac{2j_{0}-j}{n^{\prime}}\Big{)}-z\bigg{|}\geq\displaystyle\frac{\varepsilon|j-j_{0}|}{4n^{\prime}}+\displaystyle\frac{\eta}{4},$

where in both inequalities we again used the facts that
$\partial_{y}\widetilde{G}\leq-\frac{\varepsilon}{2}$. Together, (D.1) and
(D.2) yield

$\displaystyle\displaystyle\frac{1}{n^{\prime}}$
$\displaystyle\Bigg{|}\displaystyle\sum_{j=2j_{0}-n^{\prime}}^{n^{\prime}}\displaystyle\frac{\widetilde{G}(0,jn^{\prime-1})+\widetilde{G}(0,2j_{0}n^{\prime-1}-jn^{\prime-1})-2z}{\big{(}\widetilde{G}(0,jn^{\prime-1})-z\big{)}\big{(}\widetilde{G}(0,2j_{0}n^{\prime-1}-jn^{\prime-1})-z\big{)}}\Bigg{|}$
$\displaystyle\leq\displaystyle\frac{1}{n^{\prime}}\displaystyle\sum_{j=2j_{0}-n^{\prime}}^{n^{\prime}}\displaystyle\frac{\frac{2B_{0}|j-j_{0}|^{2}}{nn^{\prime}}+3\eta}{\frac{\varepsilon^{2}|j-j_{0}|^{2}}{16(n^{\prime})^{2}}+\frac{\eta^{2}}{16}}$
$\displaystyle\leq
48B_{0}\varepsilon^{-2}\displaystyle\sum_{j=0}^{n^{\prime}}\displaystyle\frac{n^{-1}j^{2}+\eta
n^{\prime}}{j^{2}+(\eta n^{\prime})^{2}}\leq
48B_{0}\varepsilon^{-2}\Bigg{(}\displaystyle\sum_{j=0}^{\eta
n^{\prime}}\Big{(}\displaystyle\frac{j^{2}}{n(\eta
n^{\prime})^{2}}+\displaystyle\frac{1}{\eta
n^{\prime}}\Big{)}+\displaystyle\sum_{j=\eta
n^{\prime}}^{n^{\prime}}\Big{(}\displaystyle\frac{1}{n}+\displaystyle\frac{\eta
n^{\prime}}{j^{2}}\Big{)}\Bigg{)}\leq C,$

for some constant $C=C(\varepsilon,\delta,B_{0})>1$; this yields the lemma. ∎

The following result of [49]121212It was stated there for Dyson Brownian
motion associated with the Gaussian Orthogonal Ensemble (GOE), but it applies
equally well to that associated with the GUE, as stated here. (see also
earlier results from [50]) states that, if one runs Dyson Brownian motion on
initial data that is $(r;R;\zeta)$-regular with respect to a point
$E_{0}\in\mathbb{R}$, then its local statistics after run for some time $t$
between $r$ and $R^{2}$ are given by the sine kernel. In what follows, we
recall notation on free convolutions from Section 2.2.

######  Lemma D.3 ([49, Theorem 2.2]).

For any integer $k\geq 1$, real numbers $\zeta,\varsigma>0$, and compactly
supported, smooth function $F\in\mathcal{C}^{\infty}(\mathbb{R}^{k})$, there
exist constants $c=c(k,\zeta,\varsigma)>0$ and $C=C(k,\zeta,\varsigma,F)>1$
such that the following holds. Adopt the notation and assumptions of
Definition D.1; fix some real number
$t\in[n^{\varsigma}r,n^{-\varsigma}R^{2}]$; define the measure
$\nu_{t}=\nu\boxplus\varrho_{\operatorname{semci}}^{(t)}$; and denote the the
density of $\nu_{t}$ by $\varrho_{t}\in L^{1}(\mathbb{R})$. Fix some real
number $E\in\big{[}E_{0}-\frac{R}{2},E_{0}+\frac{R}{2}\big{]}$; let
$\bm{d}(t)=\big{(}d_{1}(t),d_{2}(t),\ldots,d_{n}(t)\big{)}$ denote Dyson
Brownian motion, run for time $t$, with initial data $\bm{d}$; and denote its
$k$-point correlation function by $p_{\bm{d}(t)}^{(k)}$ (recall (1.5)). Then,
we have

(D.3) $\displaystyle\begin{aligned}
&\Bigg{|}\displaystyle\int_{\mathbb{R}^{k}}F(\bm{a})p_{\bm{d}(t)}^{(k)}\left(E+\displaystyle\frac{\bm{a}}{n\varrho_{t}(E)}\right)d\bm{a}-\displaystyle\int_{\mathbb{R}^{k}}F(\bm{a})p_{\sin}^{(k)}(\bm{a})d\bm{a}\Bigg{|}<Cn^{-c}.\end{aligned}$

Now we can outline the proof of 8.2.

###### Proof of 8.2 (Outline).

We only address the case when $\mathfrak{a}=0$, as the proof is entirely
analogous if $\mathfrak{a}\neq 0$. Recalling the notation from Section 8.2
(and, in particular, (8.11)), denote the measure
$\nu_{0}=(n^{\prime})^{-1}\sum_{j=1}^{n^{\prime}}\delta_{\tilde{w}_{j}(0)}\in\mathscr{P}_{0}$.
For any real number $s\geq 0$, denote its free convolution with the semicircle
law by $\nu_{s}=\nu_{0}\boxplus\mu_{\operatorname{semci}}^{(s)}$, and denote
the associated density by $\varrho_{s}\in L^{1}(\mathbb{R})$, which satisfies
$\nu_{s}(dx)=\varrho_{s}(x)dx$.

Observe that $\widetilde{s}_{0}=(n^{\prime})^{\omega-1}$ (by (8.1) and (8.11))
and that $\widetilde{G}\big{(}\widetilde{s}_{0},\frac{1}{2}\big{)}=0$ (by
(8.13)). This, together with D.2 and D.3 (with the parameters
$(n;E_{0},E;t;r,R;\zeta,\varsigma)$ there equal to
$\big{(}n^{\prime};0,0,\widetilde{s}_{0};(n^{\prime})^{\omega/2-1},(n^{\prime})^{-\omega/2};\frac{\omega}{2},\frac{\omega}{4}\big{)}$
here) then imply that

$\displaystyle\displaystyle\lim_{n\rightarrow\infty}\displaystyle\int_{\mathbb{R}^{k}}F(\bm{a})p_{\tilde{\bm{w}}(\tilde{s}_{0})}^{(k)}\bigg{(}\displaystyle\frac{\bm{a}}{n^{\prime}\varrho_{\tilde{s}_{0}}(0)}\bigg{)}d\bm{a}=\displaystyle\int_{\mathbb{R}^{k}}F(\bm{a})p_{\sin}^{(k)}(\bm{a})d\bm{a}.$

It follows from applying the change of variables replacing $\bm{a}$ with
$\frac{n^{\prime}}{n}\cdot\bm{a}$, and using (8.11), and that

$\displaystyle\displaystyle\lim_{n\rightarrow\infty}\displaystyle\int_{\mathbb{R}^{k}}F(\bm{a})p_{\breve{\bm{w}}(s_{0})}^{(k)}\bigg{(}\displaystyle\frac{\bm{a}}{n\varrho_{\tilde{s}_{0}}(0)}\bigg{)}d\bm{a}=\displaystyle\int_{\mathbb{R}^{k}}F(\bm{a})p_{\sin}^{(k)}(\bm{a})d\bm{a}.$

Since
$\theta=\partial_{y}G(t_{0},y_{0})=\partial_{y}\widetilde{G}\big{(}\widetilde{s}_{0},\frac{1}{2}\big{)}$
and since
$\lim_{n\rightarrow\infty}\partial_{x}\widetilde{G}\big{(}\widetilde{s}_{0},\frac{1}{2}\big{)}=\partial_{x}\widetilde{G}\big{(}0,\frac{1}{2}\big{)}$
(as $\lim_{n\rightarrow\infty}\widetilde{s}_{0}=0$ and
$[\widetilde{G}]_{1}=[G]_{1}\leq B_{0}$), it therefore suffices to show that
$\lim_{n\rightarrow\infty}\varrho_{\tilde{s}_{0}}(0)=\partial_{x}\widetilde{G}\big{(}0,\frac{1}{2}\big{)}$.

Stated alternatively, since the approximate density of $\widetilde{w}_{j}(0)$
around $0$ is $\partial_{x}\widetilde{G}\big{(}0,\frac{1}{2}\big{)}$, this
amounts to showing that the effect of applying the free convolution with
$\mu_{\operatorname{semci}}^{\tilde{s}_{0}}$ for $\widetilde{s}_{0}$ small has
a negligible effect on this density. This is a standard statement about free
convolutions that has been proven and applied numerous times in the random
matrix literature; we therefore do not repeat its proof here and refer, for
example, to [42, Lemmas 3.3 and 3.4] for very similar arguments. ∎

###  D.2. Proof Outline of 8.3

In this section we provide a brief outline for the proof of 8.3, using results
of [50].

###### Proof of 8.3 (Outline).

We only outline the proof of the first statement of (8.5), as that of the
second is entirely analogous. Following [50, Equation (4.10)], we define the
processes
$\bm{\xi}(s)=\big{(}\xi_{1}(s),\xi_{2}(s),\ldots,\xi_{n^{\prime}}(s)\big{)}$
and
$\bm{\xi}^{-}(s)=\big{(}\xi_{1}^{-}(s),\xi_{2}^{-}(s),\ldots,\xi_{n^{\prime}}^{-}(s)\big{)}$
by for each pair $(j,s)\in\llbracket 1,n^{\prime}\rrbracket\times[0,ns_{0}]$
setting

(D.4)
$\displaystyle\xi_{j}(s)=n\cdot\breve{w}_{j}\Big{(}\frac{s}{n}\Big{)};\qquad\xi_{j}^{-}=n\cdot\breve{w}_{j}^{-}\Big{(}\frac{s}{n}\Big{)},$

which satisfy the stochastic differential equations

(D.5)
$\displaystyle\partial_{t}\xi_{i}(t)=\displaystyle\sum_{\begin{subarray}{c}1\leq
j\leq n^{\prime}\\\ j\neq
i\end{subarray}}\displaystyle\frac{dt}{\xi_{i}(t)-\xi_{j}(t)}+dB_{i}(t);\qquad\partial_{t}\xi_{i}^{-}(t)=\displaystyle\sum_{\begin{subarray}{c}1\leq
j\leq n^{\prime}\\\ j\neq
i\end{subarray}}\displaystyle\frac{dt}{\xi_{i}^{-}(t)-\xi_{j}^{-}(t)}+dB_{i}(t).$

We further define for any distinct integers $j,k\in\llbracket
1,n^{\prime}\rrbracket$ the quantity

$\displaystyle\Delta_{jk}=(2\cdot\textbf{1}_{j<k}-1)\cdot(n^{\prime})^{-500}.$

Then following [50, Equation (4.16)] we define the processes
$\widehat{\bm{\xi}}=\big{(}\widehat{\xi}_{1}(s),\widehat{\xi}_{2}(s),\ldots,\widehat{\xi}_{n^{\prime}}(s)\big{)}$
and
$\widehat{\bm{\xi}}^{-}(s)=\big{(}\widehat{\xi}_{1}^{-}(s),\widehat{\xi}_{2}^{-}(s),\ldots,\widehat{\xi}_{n^{\prime}}^{-}(s)\big{)}$
obtained by “regularizing” the denominators on the right sides of (D.5),
namely, as the (strong) solutions to the stochastic differential equations

(D.6)
$\displaystyle\partial_{t}\widehat{\xi}_{i}(t)=\displaystyle\sum_{\begin{subarray}{c}1\leq
j\leq n^{\prime}\\\ j\neq
i\end{subarray}}\displaystyle\frac{dt}{\xi_{i}(t)-\xi_{j}(t)+\Delta_{ij}}+dB_{i}(t);\qquad\partial_{t}\widehat{\xi}_{i}^{-}(t)=\displaystyle\sum_{\begin{subarray}{c}1\leq
j\leq n^{\prime}\\\ j\neq
i\end{subarray}}\displaystyle\frac{dt}{\xi_{i}^{-}(t)-\xi_{j}^{-}(t)+\Delta_{ij}}+dB_{i}(t),$

with initial data $\widehat{\bm{\xi}}(0)=\bm{\xi}(0)$ and
$\widehat{\bm{\xi}}^{-}(0)=\bm{\xi}^{-}(0)$. It then follows from131313Let us
mention that there are two differences between the setup of [50, Lemma 4.4]
and that here. First, [50] considers the $\beta=1$ (GOE) case of Dyson
Brownian motion, while here we consider the $\beta=2$ (GUE) case. Second, the
differential equations (D.5) and (D.6) in [50] have an additional
$-\frac{1}{2n^{\prime}}\cdot\xi_{j}(t)$ term on their right sides. However, it
is quickly verified that neither point affects the proofs or statements of the
results from [50]. [50, Lemma 4.4] (together with D.2) that
$\widehat{\bm{\xi}}$ and $\widehat{\bm{\xi}}^{-}$ likely are close to
$\bm{\xi}$ and $\bm{\xi}^{-}$, respectively, namely, with probability at least
$1-(n^{\prime})^{90}$ we have

(D.7)
$\displaystyle\displaystyle\sup_{\begin{subarray}{c}t\in[0,(n^{\prime})^{1-3\omega}]\\\
j\in\llbracket
n^{\prime}/2-(n^{\prime})^{1-\omega},n^{\prime}/2+(n^{\prime})^{1-\omega}\rrbracket\end{subarray}}\Big{(}\big{|}\xi_{j}(t)-\widehat{\xi}_{j}(t)\big{|}+\big{|}\xi_{j}^{-}(t)-\widehat{\xi}_{j}^{-}(t)\big{|}\Big{)}\leq(n^{\prime})^{-15}.$

Now set141414The choice of $K$ in [50] was slightly different, but it can
quickly be verified that the below statements will valid for
$K=(n^{\prime})^{1/6}$ as well. Indeed, fixing $K=(n^{\prime})^{1/6}$
correponds in [50] to imposing that $\widetilde{\bm{w}}$ be
$\big{(}n^{-2/3-o(1)},n^{-1/3-o(1)};\frac{1}{3}+o(1)\big{)}$-regular around
$0$; this is guaranteed D.2 (which implies that $\widetilde{\bm{w}}$ is even
more regular than this). $K=(n^{\prime})^{1/6}$. Following [50, Equation
(4.24)], we define the processes of $(2K+1)$-tuples
$\widetilde{\bm{\xi}}(s)=\big{(}\widetilde{\xi}_{n^{\prime}/2-K}(s),\ldots,\widetilde{\xi}_{n^{\prime}/2+K}(s)\big{)}$
and
$\widetilde{\bm{\xi}}^{-}(s)=\big{(}\widetilde{\xi}_{n^{\prime}/2-K}^{-}(s),\ldots,\widetilde{\xi}_{n^{\prime}/2+K}^{-}(s)\big{)}$
by only retaining the middle $2K+1$ of the paths in (D.6), namely, as the
(strong) solutions to the stochastic differential equations

(D.8) $\displaystyle\begin{aligned}
&\partial_{t}\widetilde{\xi}_{i}(t)=\displaystyle\sum_{\begin{subarray}{c}n^{\prime}/2-K\leq
j\leq n^{\prime}/2+K\\\ j\neq
i\end{subarray}}\displaystyle\frac{dt}{\xi_{i}(t)-\xi_{j}(t)+\Delta_{ij}}+dB_{i}(t);\\\
&\partial_{t}\widetilde{\xi}_{i}^{-}(t)=\displaystyle\sum_{\begin{subarray}{c}n^{\prime}/2-K\leq
j\leq n^{\prime}/2+K\\\ j\neq
i\end{subarray}}\displaystyle\frac{dt}{\xi_{i}^{-}(t)-\xi_{j}^{-}(t)+\Delta_{ij}}+dB_{i}(t),\end{aligned}$

with initial data $\widetilde{\xi}_{i}(0)=\widehat{\xi}_{i}(0)$ and
$\widetilde{\xi}_{i}^{-}(0)=\widehat{\xi}_{i}^{-}(0)$, for each
$i\in\big{\llbracket}\frac{n^{\prime}}{2}-K,\frac{n^{\prime}}{2}+K\big{\rrbracket}$.
Observe from (8.4) and the fact that
$G\big{(}t_{0}-s_{0},y_{0}-\frac{n^{\prime}}{2n}+\frac{j}{n}\big{)}=G^{-}\big{(}y_{0}-\frac{n^{\prime}}{2n}+\frac{j}{n}\big{)}$
for
$j\in\big{\llbracket}\frac{n^{\prime}}{2}-K,\frac{n^{\prime}}{2}+K\big{\rrbracket}$
that

(D.9) $\displaystyle\widetilde{\bm{\xi}}(0)=\widetilde{\bm{\xi}}^{-}(0).$

Now define the processes of $(2K+1)$-tuples
$\widehat{\bm{\vartheta}}(s)=\big{(}\widehat{\vartheta}_{n^{\prime}/2-K}(s),\ldots,\widehat{\vartheta}_{n^{\prime}/2+K}(s)\big{)}$
and
$\widetilde{\bm{\vartheta}}(s)=\big{(}\widetilde{\vartheta}_{n^{\prime}/2-K}(s),\ldots,\widetilde{\vartheta}_{n^{\prime}/2+K}(s)\big{)}$
by setting

$\displaystyle\widehat{\vartheta}_{i}(s)=\widehat{\xi}_{i}(s)-\widehat{\xi}_{i}^{-}(s),\quad\text{and}\quad\widetilde{\vartheta}_{i}(s)=\widetilde{\xi}_{i}(s)-\widetilde{\xi}_{i}^{-}(s);\qquad\text{for
each
$j\in\Big{\llbracket}\displaystyle\frac{n^{\prime}}{2}-K,\displaystyle\frac{n^{\prime}}{2}+K\Big{\rrbracket}$}.$

To compare $\widehat{\bm{\vartheta}}(s)$ and $\widetilde{\bm{\vartheta}}(s)$
first observe, by (8.4) and the fact that
$\big{|}G(t_{0}-s_{0},y)-G^{-}(y)\big{|}\leq n^{\omega/2-1}$ (and that
$n^{\prime}=n^{1/2}$), that we have
$\big{|}\xi_{j}(0)-\xi_{j}^{-}(0)\big{|}\leq(n^{\prime})^{\omega}$ for each
$j\in\llbracket 1,n^{\prime}\rrbracket$. Together with 8.7, it follows that
$\big{|}\xi_{j}(s)-\xi_{j}^{-}(s)\big{|}\leq(n^{\prime})^{\omega}$ for each
$s>0$. Hence, subtracting (D.8) from (D.6) and integrating over
$t\in[0,ns_{0}]$, we obtain

(D.10) $\displaystyle\begin{aligned}
\big{|}\widetilde{\vartheta}_{i}(ns_{0})-\widehat{\vartheta}_{i}(ns_{0})\big{|}&=\Big{|}\big{(}\widetilde{\xi}_{i}(ns_{0})-\widetilde{\xi}_{i}^{-}(ns_{0})\big{)}-\big{(}\widehat{\xi}_{i}(ns_{0})-\widehat{\xi}_{i}^{-}(ns_{0})\big{)}\Big{|}\\\
&\leq
ns_{0}\cdot\displaystyle\sup_{t\in[0,ns_{0}]}\Bigg{|}\displaystyle\sum_{j:|j-n^{\prime}/2|>K}\bigg{(}\displaystyle\frac{1}{\xi_{i}(t)-\xi_{j}(t)+\Delta_{ij}}-\displaystyle\frac{1}{\xi_{i}^{-}(t)-\xi_{j}^{-}(t)+\Delta_{ij}}\bigg{)}\Bigg{|}\\\
&\leq
ns_{0}\cdot\displaystyle\sup_{t\in[0,ns_{0}]}\displaystyle\sum_{j:|j-n^{\prime}/2|>K}\displaystyle\frac{\big{|}\xi_{i}(t)-\xi_{i}^{-}(t)\big{|}+\big{|}\xi_{j}(t)-\xi_{j}^{-}(t)\big{|}}{\big{|}\xi_{i}(t)-\xi_{j}(t)+\Delta_{ij}\big{|}\big{|}\xi_{i}^{-}(t)-\xi_{j}^{-}(t)+\Delta_{ij}\big{|}},\\\
&\leq
2(n^{\prime})^{2\omega}\displaystyle\sup_{t\in[0,ns_{0}]}\sum_{j:|j-n^{\prime}/2|>K}\displaystyle\frac{1}{\big{|}\xi_{i}(t)-\xi_{j}(t)\big{|}\big{|}\xi_{i}^{-}(t)-\xi_{j}^{-}(t)\big{|}},\end{aligned}$

for any
$i\in\big{\llbracket}\frac{n^{\prime}-K}{2},\frac{n^{\prime}+K}{2}\big{\rrbracket}$,
where in the second bound we used the fact that
$ns_{0}=(n^{\prime})^{\omega}$. By 2.17, and the first statement of 8.6, there
exists a constant $c>0$ such that with probability $1-c^{-1}e^{-c(\log
n)^{2}}$ we have
$\big{|}\xi_{i}(t)-\xi_{j}(t)\big{|}\geq\frac{\varepsilon}{8}|i-j|\geq(n^{\prime})^{-\omega}K$
and
$\big{|}\xi_{i}(t)-\xi_{j}(t)\big{|}\geq\frac{\varepsilon}{8}|i-j|\geq(n^{\prime})^{-\omega}K$
whenever $\big{|}j-\frac{n^{\prime}}{2}\big{|}>K$ and
$i\in\big{\llbracket}\frac{n^{\prime}-K}{2},\frac{n^{\prime}+K}{2}\big{\rrbracket}$.
Inserting this into (D.10) yields

(D.11)
$\displaystyle\big{|}\widetilde{\vartheta}_{i}(ns_{0})-\widehat{\vartheta}_{i}(ns_{0})\big{|}\leq
2(n^{\prime})^{4\omega}K^{-2}=2(n^{\prime})^{4\omega-1/3},\qquad\text{for each
$i\in\Big{\llbracket}\displaystyle\frac{n^{\prime}-K}{2},\displaystyle\frac{n^{\prime}+K}{2}\Big{\rrbracket}$},$

where we have recalled that $K=(n^{\prime})^{1/6}$.

Following [50, Equation (4.38)], we next define the process
$\mathring{\bm{\vartheta}}(s)=\big{(}\mathring{\vartheta}_{n^{\prime}/2-K}(s),\ldots,\mathring{\vartheta}_{n^{\prime}/2+K}(s)\big{)}$
as the solution to the differential equation

(D.12)
$\displaystyle\partial_{t}\mathring{\vartheta}_{i}(t)=\displaystyle\sum_{\begin{subarray}{c}n^{\prime}/2-K\leq
j\leq n^{\prime}/2+K\\\ j\neq
i\end{subarray}}\displaystyle\frac{\big{(}\mathring{\vartheta}_{i}(s)-\mathring{\vartheta}_{j}(s)\big{)}dt}{\big{(}\xi_{i}(t)-\xi_{j}(t)+\Delta_{ij}\big{)}\big{(}\xi_{i}^{-}(t)-\xi_{j}^{-}(t)+\Delta_{ij}\big{)}},$

with initial data
$\mathring{\bm{\vartheta}}(0)=\widetilde{\bm{\vartheta}}(0)=(0,0,\ldots,0)$,
where the last equality follows from (D.9). Combining this initial condition
with the explicit form of (D.12), it follows

(D.13) $\displaystyle\mathring{\vartheta}_{i}(s)=0,\qquad\text{for each
$i\in\Big{\llbracket}\displaystyle\frac{n^{\prime}}{2}-K,\displaystyle\frac{n^{\prime}}{2}+K\Big{\rrbracket}$}.$

Next, [50, Lemma 4.8]151515It was stated there when the $\bm{\xi}^{-}$ paths
come from applying Dyson Brownian motion to the eigenvalues of a GOE random
matrix, but it is quickly verified that the proof and statement of that result
applies equally well for the $\bm{\xi}^{-}$ here (satisfying the regularity
hypothesis shown in D.2). implies (with a union bound over
$2n^{\omega/15000}+1=2(n^{\prime})^{\omega/7500}+1\leq K^{\omega/1000}$
indices) that, with probability at least $1-K^{-\omega/1000}$ we have

(D.14) $\displaystyle\displaystyle\max_{j\in\llbracket
n^{\prime}/2-n^{\omega/15000},n^{\prime}/2+n^{\omega/15000}\rrbracket}\big{|}\mathring{\vartheta}_{j}(ns_{0})-\widetilde{\vartheta}_{j}(ns_{0})\big{|}\leq
K^{4\omega-1/2}.$

Combining (D.14), (D.13), (D.11), and (D.7) yields with probability at least
$1-n^{-\omega/15000}$ that $\xi_{j}(ns_{0})-\xi_{j}^{-}(ns_{0})\leq
K^{4\omega-1/2}+2(n^{\prime})^{4\omega-1/3}+(n^{\prime})^{-15}\leq(n^{\prime})^{\omega-1/12}$.
Together with (D.4) (and the fact that $\omega\leq\frac{1}{20}$), this yields
the lemma. ∎

###  D.3. Proof Outline of 8.4

In this section we provide a brief outline for the proof of 8.4.

###### Proof of 8.4 (Outline).

Throughout this proof, we fix an integer
$j\in\big{\llbracket}\frac{n^{\prime}}{2}-n^{\omega/15000},\frac{n^{\prime}}{2}+n^{\omega/15000}\big{\rrbracket}$;
we may also assume that $\chi<\omega$. It suffices to show that, for any real
number $A\geq 1$,

(D.15)
$\displaystyle\mathbb{P}\Big{[}\big{|}\breve{w}_{j}^{-}(s_{0})\big{|}\geq
An^{5}\Big{]}<Ce^{-cAn};\qquad\mathbb{P}\Big{[}\big{|}\breve{w}_{j}^{+}(s_{0})\big{|}\geq
An^{5}\Big{]}<Ce^{-cAn},$

and that, for any coupling between $\breve{\bm{w}}^{-}$ and
$\breve{\bm{w}}^{+}$,

(D.16)
$\displaystyle\mathbb{P}\Big{[}\big{|}\breve{w}_{j}^{+}(s_{0})-\breve{w}_{j}^{-}(s_{0})\big{|}>n^{\chi-1}\Big{]}\leq
Ce^{-c(\log n)^{2}}.$

Indeed, given these two bounds, it would then follow that

$\displaystyle\mathbb{E}\Big{[}\big{|}\breve{w}_{j}^{+}(s_{0})-\breve{w}_{j}^{+}($
$\displaystyle
s_{0})\big{|}\cdot\textbf{1}_{|\breve{w}_{j}^{+}(s_{0})-\breve{w}_{j}^{-}(s_{0})\big{|}>n^{\chi-1}}\Big{]}$
$\displaystyle\leq
2n^{10}\cdot\mathbb{P}\Big{[}\big{|}\breve{w}_{j}^{+}(s_{0})-\breve{w}_{j}^{-}(s_{0})\big{|}>n^{\chi-1}\Big{]}+\mathbb{E}\Big{[}\big{|}\breve{w}_{j}^{+}(s_{0})\big{|}\cdot\textbf{1}_{|\breve{w}_{j}^{+}(s_{0})|>n^{10}}\Big{]}$
$\displaystyle\qquad+\mathbb{E}\Big{[}\big{|}\breve{w}_{j}^{-}(s_{0})\big{|}\cdot\textbf{1}_{|\breve{w}_{j}^{-}(s_{0})|>n^{10}}\Big{]}$
$\displaystyle\leq 2Cn^{10}e^{-c(\log
n)^{2}}+n^{5}\displaystyle\int_{n^{5}}^{\infty}\bigg{(}\mathbb{P}\Big{[}\big{|}\breve{w}_{j}^{+}(s_{0})\big{|}\geq
n^{5}A\Big{]}+\mathbb{P}\Big{[}\big{|}\breve{w}_{j}^{-}(s_{0})\big{|}\geq
n^{5}A\Big{]}\bigg{)}dA$ $\displaystyle\leq 2Cn^{10}e^{-c(\log
n)^{2}}+2c^{-1}Cn^{5}e^{-cn},$

verifying the lemma. The bound (D.15) is a quick consequence of 2.6 with the
fact that Dyson Brownian motion $\bm{\lambda}(s_{0})$, run for time $s_{0}\geq
0$ with initial data given by some $n$-tuple $\bm{a}$, has the same law as
$(1+s_{0})\cdot\widetilde{\bm{\lambda}}\big{(}\frac{s_{0}}{1+s_{0}}\big{)}$,
where $\widetilde{\bm{\lambda}}$ is a family of $n$ non-intersecting Brownian
bridges on $[0,1]$ sampled from the measure $\mathfrak{Q}^{\bm{a};\bm{0}_{n}}$
(where $\bm{0}_{n}=(0,0,\ldots,0)$ consists of $n$ zeroes).

To establish (D.16), it suffices to exhibit (deterministic) real numbers
$\theta_{j}^{-},\theta_{j}^{+}\in\mathbb{R}$ such that

(D.17)
$\displaystyle\mathbb{P}\Big{[}\big{|}\breve{w}_{j}^{+}(s_{0})-\theta_{j}^{+}\big{|}>n^{\chi/2-1}\Big{]}\leq
Ce^{-c(\log
n)^{2}};\qquad\mathbb{P}\Big{[}\big{|}\breve{w}_{j}^{-}(s_{0})-\theta_{j}^{-}\big{|}>n^{\chi/2-1}\Big{]}\leq
Ce^{-c(\log n)^{2}}.$

Indeed this, 8.3, a union bound, would yield a constant
$c_{1}=c_{1}(\chi,\varepsilon,\delta,B_{0},m)>1$ such that

$\displaystyle\mathbb{P}\big{[}|\theta_{j}^{+}-\theta_{j}^{-}|>n^{3\chi/4-1}\big{]}\geq\mathbb{P}\big{[}|\theta_{j}^{+}-\theta_{j}^{-}|>2n^{\chi/2-1}+n^{-c_{1}-1}\big{]}\geq
1-c_{1}^{-1}n^{-c_{1}}>0.$

Since $\theta_{j}^{+}$ and $\theta_{j}^{-}$ are deterministic, it follows that
$|\theta_{j}^{+}-\theta_{j}^{-}|\leq n^{3\chi/4-1}$. Together with (D.17), a
union bound, and the fact that $n^{\chi-1}>2n^{\chi/2-1}+n^{3\chi/4-1}$ for
sufficiently large $n$, this implies

$\displaystyle\mathbb{P}\Big{[}\big{|}\breve{w}_{j}^{+}(s_{0})-\breve{w}_{j}^{+}(s_{0})\big{|}>n^{\chi-1}\Big{]}$
$\displaystyle\leq\mathbb{P}\Big{[}\big{|}\breve{w}_{j}^{+}(s_{0})-\theta_{j}^{+}\big{|}>n^{\chi/2-1}\Big{]}+\mathbb{P}\Big{[}\big{|}\breve{w}_{j}^{-}(s_{0})-\theta_{j}^{-}\big{|}>n^{\chi/2-1}\Big{]}$
$\displaystyle\leq 2Ce^{-c(\log n)^{2}},$

thereby confirming (D.16).

It remains to establish (D.17); let us only explain the first bound there,
since the proof of the second is entirely analogous. This will follow from the
concentration bound 2.17. In particular, denote the measure
$\nu_{0}^{+}=(n^{\prime})^{-1}\sum_{j=1}^{n^{\prime}}\textbf{1}_{n\breve{u}_{j}^{+}/n^{\prime}}$.
For any real number $s\geq 0$, also define its free convolution
$\nu_{s}^{+}=\nu_{0}^{+}\boxplus\mu_{\operatorname{semci}}^{(s)}$ with the
semicircle distribution; let $\varrho_{s}^{+}$ denote its density with respect
to Lebesgue measure, and for each $k\in\llbracket 1,n^{\prime}\rrbracket$ let
$\gamma_{k}^{+}(s)=\gamma_{k;n^{\prime}}^{\nu_{s}^{+}}$ denote its classical
location (recall Definition 2.16). By 2.17 (and 2.13), denoting
$\widetilde{s}_{0}=\frac{ns_{0}}{n^{\prime}}=n^{\omega-1}$ (by (8.1)), there
exist constants $c_{2}>0$ and $C_{1}>1$ such that

$\displaystyle\mathbb{P}\bigg{[}\displaystyle\frac{n^{\prime}}{n}\cdot\gamma_{j+\lfloor(\log
n)^{5}\rfloor}^{+}(\widetilde{s}_{0})-n^{-2}\leq\breve{w}_{j}^{+}(s_{0})\leq\displaystyle\frac{n^{\prime}}{n}\cdot\gamma_{j-\lfloor(\log
n)^{5}\rfloor}^{+}(\widetilde{s}_{0})+n^{-2}\Big{]}\geq 1-C_{1}e^{-(\log
n)^{2}}.$

Taking
$\theta_{j}^{+}=\frac{n^{\prime}}{n}\cdot\gamma_{j}(\widetilde{s}_{0})$, to
prove (D.17) it therefore suffices to show for sufficiently large $n$ that

$\displaystyle\gamma_{j-\lfloor(\log
n)^{5}\rfloor}^{+}(\widetilde{s}_{0})-\gamma_{j+\lfloor(\log
n)^{5}\rfloor}^{+}(\widetilde{s}_{0})\leq\displaystyle\frac{(\log
n)^{10}}{n^{\prime}}.$

This follows from the fact that
$\varrho_{s_{0}}^{+}(x)\geq\frac{\varepsilon}{2}$ for
$x\in\big{[}\gamma_{j}^{+}(\widetilde{s}_{0})-(n^{\prime})^{2\omega},\gamma_{j}^{+}(\widetilde{s}_{0})+(n^{\prime})^{2\omega}\big{]}$,
which can be deduced from the fact that $\widetilde{s}_{0}=n^{1-\omega}$ is
small (by (8.1)); that
$\frac{n}{n^{\prime}}\cdot\bm{\breve{w}}^{+}(0)=\frac{n}{n^{\prime}}\cdot\bm{\breve{u}}^{+}$
approximates $G$ to within $(n^{\prime})^{\omega/2-1}\ll\widetilde{s}_{0}$,
for indices
$k\in\big{\llbracket}\frac{n^{\prime}-n^{\prime\prime}}{2},\frac{n^{\prime}+n^{\prime\prime}}{2}\big{\rrbracket}$
(by (8.4) and (8.2)); and that
$G\in\operatorname{Adm}_{\varepsilon}(\mathfrak{R})$. This last statement is a
standard one about free convolutions that has been proven and applied numerous
times in the random matrix literature; we therefore do not repeat its proof
here and refer, for example, to [42, Lemmas 3.3 and 3.4] for very similar
arguments. ∎

##  Appendix E Proofs of Results From Section 9

###  E.1. Proofs of 9.1, 9.2, and 9.3

In this section we first recall several results from [33] about regularity
properties of uniformly elliptic partial differential equations and then use
them to establish 9.1, 9.2, and 9.3. To that end, we fix an integer $d\geq 1$
and an open subset $\mathfrak{R}\subseteq\mathbb{R}^{d}$, and we recall the
notation from Section 1.3. For any function $f\in\mathcal{C}(\mathfrak{R})$;
real number $\alpha\in(0,1)$; and integer $k\in\mathbb{Z}_{\geq 0}$, define
$[f]_{\alpha}=[f]_{\alpha;\mathfrak{R}}$,
$[f]_{k,\alpha}=[f]_{k,\alpha;\mathfrak{R}}$, and
$\|f\|_{\mathcal{C}^{k,\alpha}(\mathfrak{R})}=\|f\|_{\mathcal{C}^{k,\alpha}(\overline{\mathfrak{R}})}=\|f\|_{k,\alpha}=\|f\|_{k,\alpha;\mathfrak{R}}$
by

$\displaystyle[f]_{\alpha}=\displaystyle\sup_{\begin{subarray}{c}y,z\in\mathfrak{R}\\\
y\neq
z\end{subarray}}\displaystyle\frac{\big{|}f(y)-f(z)\big{|}}{|y-z|^{\alpha}};\qquad[f]_{k,\alpha}=\displaystyle\max_{\begin{subarray}{c}\gamma\in\mathbb{Z}_{\geq
0}^{d}\\\
|\gamma|=k\end{subarray}}[\partial_{\gamma}f]_{\alpha};\qquad\|f\|_{\mathcal{C}^{k,\alpha}(\mathfrak{R})}=\|f\|_{k}+[f]_{k,\alpha}.$

Define the Hölder space $\mathcal{C}^{k,\alpha}(\mathfrak{R})$ to be the set
of $f\in\mathcal{C}^{k}(\mathfrak{R})$ such that
$[f]_{k,\alpha;\mathfrak{R}_{0}}<\infty$, for any compact subset
$\mathfrak{R}_{0}\subset\mathfrak{R}$. Also define
$\mathcal{C}^{k,\alpha}(\overline{\mathfrak{R}})=\big{\\{}f\in\mathcal{C}^{k}(\mathfrak{R}):[f]_{k,\alpha;\mathfrak{R}}<\infty\big{\\}}$.
Then $\|f\|_{k,\alpha}$ is a norm on
$\mathcal{C}^{k,\alpha}(\overline{\mathfrak{R}})$. By restriction, we have
that
$\mathcal{C}^{k,\alpha}(\mathfrak{R})\subseteq\mathcal{C}^{k,\alpha}(\mathfrak{U})$
for any subset $\mathfrak{U}\subseteq\mathfrak{R}$. Observe that, for any
$f,g\in\mathcal{C}^{0,\alpha}(\mathfrak{R})$, these norms satisfy

(E.1) $\displaystyle\begin{aligned}
\|fg\|_{\mathcal{C}^{0,\alpha}(\mathfrak{R})}&\leq\|f\|_{\mathcal{C}^{0,\alpha}(\mathfrak{R})}\|g\|_{0}.\end{aligned}$

We further say that the boundary $\partial\mathfrak{R}$ of some open set
$\mathfrak{R}\subset\mathbb{R}^{d}$ is $\mathcal{C}^{k,\alpha}$, if for any
point $z_{0}\in\partial\mathfrak{R}$, there exists a real number $r>0$ and an
injective function
$\psi=\psi_{z_{0};r}:\mathcal{B}_{r}(z_{0})\rightarrow\mathbb{R}^{d}$ such
that

$\displaystyle\mathfrak{D}_{z_{0};r;\psi;\mathfrak{R}}=\psi\big{(}\mathcal{B}_{r}(z_{0})\cap\mathfrak{R}\big{)}\subset\mathbb{R}_{>0}\times\mathbb{R}^{d-1};\qquad\psi\big{(}\mathcal{B}_{r}(z_{0})\cap\partial\mathfrak{R}\big{)}\subset\\{0\\}\times\mathbb{R}^{d-1};$

where $\psi\in\mathcal{C}^{k,\alpha}\big{(}\mathcal{B}_{r}(z_{0})\big{)}$ and
$\psi^{-1}\in\mathcal{C}^{k,\alpha}(\mathfrak{D}_{z_{0};r;\psi;\mathfrak{R}}).$
We call this boundary $\mathcal{C}^{k}$ if it is $\mathcal{C}^{k,0}$, and we
call it smooth if it is $\mathcal{C}^{k}$ for each integer $k\geq 1$.

The following lemma provides the _Schauder estimates_ for solutions to
uniformly elliptic partial differential equations with Hölder continuous
coefficients; as in 9.2 and 9.4, we state a form of it that addresses interior
($\mathfrak{R}^{\prime}$ empty) and boundary regularity
($\mathfrak{R}^{\prime}=\mathfrak{R}$) simultaneously. Although these bounds
hold in all dimensions $d\geq 2$, we only state them for $d=2$.

######  Lemma E.1 ([33, Theorem 6.6]).

Fix real numbers $r>0$, $B>1$, and $\alpha\in(0,1)$, and bounded open sets
$\mathfrak{R}^{\prime}\subseteq\mathfrak{R}\subset\mathbb{R}^{2}$ such that
$\mathfrak{R}^{\prime}$ has $\mathcal{C}^{2,\alpha}$ boundary. There exists a
constant $C=C(r,B,\alpha,\mathfrak{R}^{\prime})>1$ such that the following
holds. For each $j,k\in\\{1,2\\}$, let
$a_{jk},b_{j}\in\mathcal{C}^{0,\alpha}(\mathfrak{R})$ denote functions such
that $a_{xy}=a_{yx}$ and

$\displaystyle\displaystyle\max_{j,k\in\\{1,2\\}}\|a_{jk}\|_{0,\alpha}<B;\qquad\displaystyle\max_{j\in\\{1,2\\}}\|b_{j}\|_{0,\alpha}<B;\qquad\displaystyle\inf_{z\in\mathfrak{R}}\displaystyle\sum_{j,k\in\\{1,2\\}}a_{jk}(z)\xi_{i}\xi_{j}\geq
B^{-1}|\xi|^{2},$

all hold for any $\xi=(\xi_{1},\xi_{2})\in\mathbb{R}^{2}$. Let
$g\in\mathcal{C}^{0,\alpha}(\mathfrak{R})$ denote a function, and suppose that
$F\in\mathcal{C}^{2,\alpha}(\mathfrak{R})$ satisfies the elliptic partial
differential equation

(E.2)
$\displaystyle\displaystyle\sum_{j,k\in\\{1,2\\}}a_{jk}(z)\partial_{j}\partial_{k}F(z)+\displaystyle\sum_{j\in\\{1,2\\}}b_{j}(z)\partial_{j}F(z)=g,\quad\text{for
each $z\in\mathfrak{R}$.}$

Assume that there exists some function
$\varphi\in\mathcal{C}^{2,\alpha}(\overline{\mathfrak{R}})$ such that
$F(z)=\varphi(z)$, for each
$z\in\partial\mathfrak{R}^{\prime}\cap\partial\mathfrak{R}$. Then, recalling
the set $\mathfrak{D}_{r}\subseteq\mathfrak{R}$ from (9.1), we have

$\displaystyle\|F\|_{\mathcal{C}^{2,\alpha}(\mathfrak{D}_{r})}\leq
C\big{(}\|F\|_{\mathcal{C}^{0}(\mathfrak{R})}+|\varphi\|_{\mathcal{C}^{2,\alpha}(\mathfrak{R})}+\|g\|_{\mathcal{C}^{0,\alpha}(\mathfrak{R})}\big{)}.$

We will be interested in properties of solutions to certain families of non-
linear, uniformly elliptic partial differential equations. The following
result provides a comparison principle for such solutions.

######  Lemma E.2 ([33, Theorem 10.1]).

Fix a constant $B>0$ and a bounded open subset
$\mathfrak{R}\subset\mathbb{R}^{2}$. For each $j,k\in\\{1,2\\}$, let
$a_{jk}\in\mathcal{C}^{1}(\mathbb{R}^{2})$ denote functions such that
$a_{xy}=a_{yx}$ and

(E.3) $\displaystyle
B^{-1}|\xi|^{2}\leq\displaystyle\sum_{j,k\in\\{1,2\\}}a_{jk}(w)\xi_{j}\xi_{k}\leq
B|\xi|^{2},\qquad\text{for any $\xi=(\xi_{1},\xi_{2})\in\mathbb{R}^{2}$ and
$w\in\mathbb{R}^{2}$.}$

Moreover, for each $i\in\\{1,2\\}$, let
$f_{i}:\partial\mathfrak{R}\rightarrow\mathbb{R}$ denote a continuous
function, and suppose that
$F_{i}:\overline{\mathfrak{R}}\rightarrow\mathbb{R}$ with
$F_{i}\in\mathcal{C}^{2}(\mathfrak{R})\cap\mathcal{C}(\overline{\mathfrak{R}})$
is a solution to the partial differential equation

(E.4) $\displaystyle\displaystyle\sum_{j,k\in\\{1,2\\}}a_{jk}\big{(}\nabla
F(z)\big{)}\partial_{j}\partial_{k}F(z)=0,\quad\text{for each
$z\in\mathfrak{R}$},$

with boundary data $F_{i}|_{\partial\mathfrak{R}}=f_{i}$. If $f_{1}(z)\geq
f_{2}(z)$ for each $z\in\partial\mathfrak{R}$, then $F_{1}(z)\geq F_{2}(z)$
for each $z\in\mathfrak{R}$.

The following lemma provides interior and boundary regularity estimates (which
are stated simultaneously, as in 9.2) for solutions $F$ to equations of the
form (E.4); it is known from [33, Corollary 6.7], [33, Theorem 8.29], [33,
Equation (13.41)], together with [5, Chapter 3.6].

######  Lemma E.3 ([5, 33]).

For a fixed integer $m\in\mathbb{Z}_{>1}$; real numbers
$r,B\in\mathbb{R}_{>0}$; and bounded open subsets
$\mathfrak{R}^{\prime}\subseteq\mathfrak{R}\subset\mathbb{R}^{2}$ with
$\mathcal{C}^{m+1}$ boundaries, there exists a constant
$C=C(r,B,m,\mathfrak{R}^{\prime})>1$ such that the following holds. For each
$j,k\in\\{1,2\\}$, let $a_{jk}\in\mathcal{C}^{m-1}(\mathbb{R}^{2})$ denote
functions with $a_{xy}=a_{yx}$ and

(E.5)
$\displaystyle\displaystyle\max_{j,k\in\\{1,2\\}}\|a_{jk}\|_{\mathcal{C}^{m-1}(\mathbb{R}^{2})}\leq
B;\qquad\displaystyle\inf_{z\in\mathbb{R}^{2}}\displaystyle\sum_{j,k\in\\{1,2\\}}a_{jk}(z)\xi_{j}\xi_{k}\geq
B^{-1}|\xi|^{2},$

for any $\xi=(\xi_{1},\xi_{2})\in\mathbb{R}^{2}$. Then for any continuous
function $f\in\mathcal{C}(\partial\mathfrak{R})$ such that $\|f\|_{0}\leq B$,
the following two statements hold.

  1. (1)

There exists a unique
$F\in\mathcal{C}^{2}(\mathfrak{R})\cap\mathcal{C}(\overline{\mathfrak{R}})$
satisfying the partial differential equation (E.4) with boundary data
$F|_{\partial\mathfrak{R}}=f$.

  2. (2)

Suppose there exists a function
$\varphi\in\mathcal{C}^{m+1}(\overline{\mathfrak{R}}^{\prime})$ such that
$\|\varphi|\|_{\mathcal{C}^{m+1}(\mathfrak{R}^{\prime})}\leq B$ and
$f(z)=\varphi(z)$ for each
$z\in\partial\mathfrak{R}^{\prime}\cap\partial\mathfrak{R}$. Letting
$\mathfrak{D}_{r}=\big{\\{}z\in\mathfrak{R}:d(z,\partial\mathfrak{R}\setminus\partial\mathfrak{R}^{\prime})>r\big{\\}}$,
we have $\|F\|_{\mathcal{C}^{m}(\mathfrak{D}_{r})}\leq C$.

From the above statements, we can quickly establish 9.1, 9.2, and 9.3.

###### Proof of 9.1.

Observe that the second statement of the lemma follows from the first. Indeed,
set
$G^{-}(z)=F_{2}(z)-\sup_{w\in\partial\mathfrak{R}}\big{|}F_{1}(z)-F_{2}(z)\big{|}$
and
$G^{+}(z)=F_{2}(z)+\sup_{z\in\partial\mathfrak{R}}\big{|}F_{1}(z)-F_{2}(z)\big{|}$,
for each $z\in\overline{\mathfrak{R}}$. Then, $G^{-}$ and $G^{+}$ solve (1.2)
on $\mathfrak{R}$ (since $F_{2}$ does), and $G^{-}\leq F_{1}\leq G^{+}$ on
$\partial\mathfrak{R}$; hence, $G^{-}\leq F_{1}\leq G^{+}$ on $\mathfrak{R}$,
implying
$\sup_{z\in\mathfrak{R}}\big{|}F_{1}(z)-F_{2}(z)\big{|}\leq\sup_{z\in\partial\mathfrak{R}}\big{|}F_{1}(z)-F_{2}(z)\big{|}$.
Taking $(F_{1},F_{2})=(F,0)$ yields
$\sup_{z\in\mathfrak{R}}\big{|}F(z)\big{|}\leq\sup_{z\in\partial\mathfrak{R}}\big{|}F(z)\big{|}$.

Now we establish the first statement of the lemma, which would follow from
E.2, except that (E.3) might not be valid for a uniform constant $B>1$. To
circumvent this, we restrict to a subset of $\mathfrak{R}$. More precisely,
assume to the contrary that $F_{1}(z_{0})>F_{2}(z_{0})$ for some
$z_{0}\in\mathfrak{R}$. Then, since
$F_{1},F_{2}\in\mathcal{C}(\overline{\mathfrak{R}})$ and
$F_{1}|_{\partial\mathfrak{R}}\leq F_{2}|_{\partial\mathfrak{R}}$, there
exists some real number $\varepsilon>0$ and open subset
$\mathfrak{R}^{\prime}$ with
$\overline{\mathfrak{R}^{\prime}}\subset\mathfrak{R}$, such that
$F_{1}(z)-\varepsilon\leq F_{2}(z)$ for each
$z\in\partial\mathfrak{R}^{\prime}$, and $F_{1}(z)-\varepsilon>F_{2}(z)$ for
each $z\in\mathfrak{R}^{\prime}$.

Observe that $F_{1}-\varepsilon$ solves (1.2). Furthermore, since
$F_{1},F_{2}\in\mathcal{C}^{2}(\mathfrak{R})\cap\operatorname{Adm}(\mathfrak{R})$,
we have by compactness that
$\big{|}\partial_{y}F_{j}(z)\big{|}+\big{|}\partial_{y}F_{j}(z)\big{|}^{-1}\leq
B$ is uniformly bounded by some constant $B>1$, over
$z\in\overline{\mathfrak{R}^{\prime}}$. It follows from the explicit form
(1.3) of the coefficients $\mathfrak{d}_{ij}$ that (E.3) holds (for a larger
value of $B$, given by $\pi^{2}B^{4}$). Applying E.2 then indicates that
$F_{1}(z)-\varepsilon\leq F_{2}(z)$ for all $z\in\mathfrak{R}^{\prime}$. This
is a contradiction for $z=z_{0}$, confirming the lemma. ∎

###### Proof of 9.2.

For each $i,j\in\\{t,y\\}$, let
$\widetilde{\mathfrak{d}}_{ij}:\mathbb{R}^{2}\rightarrow\mathbb{R}$ be smooth
functions such that
$\widetilde{\mathfrak{d}}_{ij}(u,v)=\mathfrak{d}_{ij}(u,v)$ whenever
$v\in[-\varepsilon^{-1},-\varepsilon]$, and such that (E.5) holds for some
constant $\widetilde{B}=\widetilde{B}(\varepsilon)>1$. Then, E.3 yields a
constant $C=C(\varepsilon,r,B,m,\mathfrak{R}^{\prime})>1$ such that there is a
unique solution $\widetilde{F}$ to the equation

(E.6)
$\displaystyle\displaystyle\sum_{i,j\in\\{t,y\\}}\widetilde{\mathfrak{d}}_{ij}\big{(}\nabla\widetilde{F}(z)\big{)}\partial_{i}\partial_{j}\widetilde{F}(z)=0,\qquad\text{for
$z\in\mathfrak{R}$, with $\widetilde{F}|_{\partial\mathfrak{R}}=f$},$

and that this $\widetilde{F}$ satisfies
$\|\widetilde{F}\|_{\mathcal{C}^{m}(\mathfrak{D}_{r})}\leq C$. Observe since
$\partial_{x}F(z)\in[-\varepsilon^{-1},\varepsilon]$ for each
$z\in\mathfrak{R}$ (as $F\in\operatorname{Adm}_{\varepsilon}(\mathfrak{R})$);
since $F$ satisfies (1.2); and since
$\mathfrak{d}_{ij}(u,v)=\widetilde{\mathfrak{d}}_{ij}(u,v)$ for
$v\in[-\varepsilon^{-1},\varepsilon]$, that $F$ also satisfies (E.6). Hence,
$F=\widetilde{F}$, and so
$\|F\|_{\mathcal{C}^{m}(\mathfrak{D}_{r})}=\|\widetilde{F}\|_{\mathcal{C}^{m}(\mathfrak{D}_{r})}\leq
C$. ∎

We can also establish 9.3 as a quick consequence of results from [30].

###### Proof of 9.3.

This is a special case of [30, Theorem 1]; let us explain how to match our
setup to the one there. First some point $z_{0}\in\mathfrak{R}$; it suffices
to show that $F$ is real analytic in some neighborhood of $z_{0}$. We may
assume by translation in what follows that $z_{0}=0$; let $r>0$ be a real
number such that $\mathcal{B}_{4r}\subseteq\mathfrak{R}$. By 9.2, there exists
a constant $C>1$ such that $\|F\|_{\mathcal{C}^{2}(\mathcal{B}_{2r})}\leq C$
and $F\in\mathcal{C}^{3}(\mathcal{B}_{2r})$.

Now, define the function $\Phi:\mathbb{R}^{3}\rightarrow\mathbb{R}$ by setting
$\Phi(v_{1},v_{2},v_{3})=v_{2}+\pi^{2}v_{1}^{-4}v_{3}$; also define the set
$\mathfrak{E}=\big{\\{}(v_{1},v_{2},v_{3})\in\mathbb{R}^{3}:\varepsilon\leq-
v_{1}\leq\varepsilon^{-1},|v_{2}|\leq C,|v_{3}|\leq C\big{\\}}$. Then we can
reformulate the equation (1.2) as
$\Phi(\partial_{y}F,\partial_{tt}F,\partial_{yy}F)=0$; since
$F\in\operatorname{Adm}_{\varepsilon}(\mathfrak{R})$ and
$\|F\|_{\mathcal{C}^{2}(\mathcal{B}_{2r})}\leq C$, we also have
$\big{(}\partial_{y}F(z),\partial_{tt}F(z),\partial_{yy}F(z)\big{)}\in\mathfrak{E}$
for any $z\in\mathcal{B}_{2r}$. On $\mathfrak{E}$, $\Phi$ is real analytic; in
particular, for any point $(v_{1},v_{2},v_{3})\in\mathfrak{E}$ and integers
$k_{1},k_{2},k_{3}\geq 0$, we have

$\displaystyle\left|\partial_{v_{1}}^{k_{1}}\partial_{v_{2}}^{k_{2}}\partial_{v_{3}}^{k_{3}}\Phi(v_{1},v_{2},v_{3})\right|\leq\pi^{2}C\varepsilon^{-k-4}(k+3)!,$

where we have denoted $k=k_{1}+k_{2}+k_{3}\geq 0$. Together with the fact that
$F\in\mathcal{C}^{3}(\mathcal{B}_{2r})$, this verifies the assumptions in [30,
Theorem 1] (in particular, see [30, Equation (5)]), with the $M_{k}$ there
equal to $k!$ here. It follows that there exists some constant $A>1$ such
that, for any point $(t,y)\in\mathcal{B}_{r}$ and integers $k_{1},k_{2}\geq
0$, we have

(E.7)
$\displaystyle\big{|}\partial_{t}^{k_{1}}\partial_{y}^{k_{2}}F(t,y)\big{|}\leq
A^{k_{1}+k+2}(k_{1}+k_{2})!\leq A^{2}\cdot(2A)^{k_{1}+k_{2}}k_{1}!k_{2}!,$

and hence $F$ is real analytic on $\mathcal{B}_{r}$. ∎

###  E.2. Proof of 9.4

In this section we establish 9.4. To that end, we show the following
proposition (which quickly implies 9.4) bounding the derivatives of
$F_{1}-F_{2}$ assuming $F_{1}$ and $F_{2}$ satisfy the same nonlinear,
elliptic partial differential equation with close boundary data. As in 9.4, we
state its interior and boundary forms simultaneously; its proof will be
similar to that of [2, Proposition A.5], which provides its interior variant
(at $m=1$).

######  Lemma E.4.

For any integer $m\geq 1$; real numbers $r>0$ and $B>1$; and bounded open
subsets $\mathfrak{R}^{\prime}\subseteq\mathfrak{R}\subset\mathbb{R}^{2}$ such
that $\mathfrak{R}^{\prime}$ has $\mathcal{C}^{m+5}$ boundary, there exists a
constants $C=(B,m,r,\mathfrak{R}^{\prime})>1$ such that the following holds.
For each $j,k\in\\{1,2\\}$, let $a_{jk}\in\mathcal{C}^{m+4}(\mathbb{R}^{2})$
denote functions satisfying $a_{xy}=a_{yx}$ and

(E.8)
$\displaystyle\displaystyle\max_{j,k\in\\{1,2\\}}\|a_{jk}\|_{\mathcal{C}^{m+4}(\mathbb{R}^{2})}\leq
B;\qquad\displaystyle\inf_{z\in\mathbb{R}^{2}}\displaystyle\sum_{j,k\in\\{1,2\\}}a_{jk}(z)\xi_{j}\xi_{k}\geq
B^{-1}|\xi|^{2},$

for any $\xi=(\xi_{x},\xi_{y})\in\mathbb{R}^{2}$. For each $i\in\\{1,2\\}$,
let $f_{i}\in\mathcal{C}(\partial\mathfrak{R})$ denote a continuous function
such that $\sup_{z\in\partial\mathfrak{R}}\big{|}f_{i}(z)\big{|}\leq B$, and
let
$F_{i}\in\mathcal{C}^{m+4}(\mathfrak{R})\cap\mathcal{C}(\overline{\mathfrak{R}})$
denote the solution to

(E.9) $\displaystyle\displaystyle\sum_{j,k\in\\{1,2\\}}a_{jk}\big{(}\nabla
F_{i}(x)\big{)}\partial_{j}\partial_{k}F_{i}(z)=0,\quad\text{for each
$z\in\mathfrak{R}$},$

with boundary data $F_{i}|_{\partial\mathfrak{R}}=f_{i}$. Assume that there
exist functions $\varphi_{i}\in\mathcal{C}^{m+5}(\overline{\mathfrak{R}})$
such that $\|\varphi_{i}\|_{\mathcal{C}^{m+5}(\mathfrak{R})}<B$ and
$F_{i}(z)=\varphi_{i}(z)$ for each
$z\in\partial\mathfrak{R}^{\prime}\cap\partial\mathfrak{R}$. Then, letting
$\mathfrak{D}_{r}\subseteq\mathfrak{R}$ be as in (9.1), we have

(E.10) $\displaystyle\|F_{1}-F_{2}\|_{\mathcal{C}^{m}(\mathfrak{D}_{r})}\leq
C\big{(}\|F_{1}-F_{2}\|_{\mathcal{C}^{0}(\mathfrak{R})}+\|\varphi_{1}-\varphi_{2}\|_{\mathcal{C}^{m+2}(\mathfrak{R})}\big{)}.$

###### Proof of 9.4 (Outline).

This follows from E.4, very similarly to how 9.2 followed from E.3; we omit
further details. ∎

To establish E.4, we will use the following lemma, which is an interpolation
estimate that bounds derivatives of a function in terms of its higher (and
lower) derivatives; its proof is similar to that of [33, Lemma 6.35].

######  Lemma E.5.

For any integers $k>j>0$ and bounded open subset
$\mathfrak{R}\subset\mathbb{R}^{2}$ with $\mathcal{C}^{k+1}$ boundary, there
exists a constant $C=C(k,\mathfrak{R})>1$ such that the following holds. Let
$F\in\mathcal{C}^{k}(\mathfrak{R})$ denote a function, and suppose that
$0<A\leq B$ are real numbers such that $A\geq\|F\|_{0}$ and
$B\geq[F]_{k;0;\mathfrak{R}}$. Then,

(E.11) $\displaystyle[F]_{j,0;\mathfrak{R}}\leq CA^{1-j/k}B^{j/k}.$

###### Proof.

Since the proof of this lemma is similar to that of [33, Lemma 6.35], we only
establish it in the case $(j,k)=(1,2)$; the proof for general $k>j\geq 1$ is
entirely analogous.

Fix some constant $C_{0}=C_{0}(\mathfrak{R})>1$ to be determined later, and
assume to the contrary that there exists some $i\in\\{1,2\\}$ and
$z_{0}\in\mathfrak{R}$ such that
$\big{|}\partial_{i}F(z_{0})\big{|}>C_{0}(AB)^{1/2}$. Assume that $i=x$ and
$\partial_{i}F(z_{0})>C_{0}(AB)^{1/2}$, as the proofs in the alternative cases
are entirely analogous. Further set $w=(1,0)\in\mathbb{R}^{2}$ and
$w^{\prime}=(2^{-1/2},2^{-1/2})\in\mathbb{R}^{2}$. Since the boundary of
$\mathfrak{R}$ is $\mathcal{C}^{k+1}$, there exists a constant
$t_{0}=t_{0}(\mathfrak{R})>0$ such that either $z_{0}+tw\in\mathfrak{R}$ for
each $t\in[0,t_{0}]$; $z_{0}-tw\in\mathfrak{R}$ for each $t\in[0,t_{0}]$;
$z_{0}+tw^{\prime}\in\mathfrak{R}$ for each $t\in[0,t_{0}]$; or
$z_{0}-tw^{\prime}\in\mathfrak{R}$ for each $t\in[0,t_{0}]$. Let us assume
that the third case holds, as the proofs in the remaining ones are very
similar. Then, since $[F]_{2,0;\mathfrak{R}}\leq B$, we have
$\big{|}\partial_{x}F(z_{0}+tw^{\prime})-\partial_{x}F(z_{0})\big{|}\leq Bt$
for each $t\in[0,t_{0}]$.

Thus, $\partial_{x}F(z_{0}+tw^{\prime})>2^{-1/2}C_{0}(AB)^{1/2}-Bt$ whenever
$t\in[0,t_{0}]$, and so integration over $t$ yields
$\big{|}F(z_{0})-F(z_{0}+tw^{\prime})\big{|}>2^{-1/2}C_{0}(AB)^{1/2}t-\frac{Bt^{2}}{2}$,
for any $t\in[0,t_{0}]$. Selecting $t=\big{(}\frac{A}{B}\big{)}^{1/2}t_{0}\leq
t_{0}$ yields
$\|F(z_{0})-F(z_{0}+t_{0}w^{\prime})\big{|}>\big{(}2^{-1/2}C_{0}t_{0}-\frac{t_{0}^{2}}{2}\big{)}A$.
In particular, selecting $C_{0}$ sufficiently large so that
$2^{-1/2}C_{0}t_{0}-t_{0}^{2}>3$, this implies that
$\|F\|_{0;\mathfrak{R}}>A$, which is a contradiction, verifying (E.11). ∎

Now we can establish E.4.

###### Proof of E.4.

The proof of this proposition is similar to that of [2, Proposition A.5],
obtained by replacing the interior regularity, Schauder, and interpolation
estimates (given by [2, Lemma A.3], [2, Lemma A.1], and [2, Lemma A.6],
respectively), by the boundary ones (given by E.3, E.1, and E.5,
respectively). We only address the case $m=1$ here (so that it suffices to
bound $|\nabla F_{1}-\nabla F_{2}|$), as bounding the difference of the higher
derivatives follows similarly by differentiating the equations below.
Throughout, we assume that $\mathfrak{R}^{\prime}=\mathfrak{R}$ for notational
simplicity; the proof in the case of more general $\mathfrak{R}^{\prime}$ is
entirely analogous.

Denoting

(E.12)
$\displaystyle\varsigma=\max\Big{\\{}\|F_{1}-F_{2}\|_{\mathcal{C}^{0}(\mathfrak{R})},\|\varphi_{1}-\varphi_{2}\|_{\mathcal{C}^{3}(\mathfrak{R})}\Big{\\}}$

we have

(E.13)
$\displaystyle\|F_{1}-F_{2}\|_{\mathcal{C}^{0}(\mathfrak{R})}\leq\varsigma;\qquad\displaystyle\max_{i\in\\{1,2\\}}\displaystyle\sup_{z\in\mathfrak{R}}\big{|}F_{i}(z)\big{|}\leq
B,$

where the second bound follows from E.2 and the fact that
$\big{|}f_{i}(z)\big{|}\leq B$, for each $i\in\\{1,2\\}$ and
$z\in\partial\mathfrak{R}$. Moreover, the $m=5$ case of E.3 yields a constant
$C_{1}=C_{1}(B,\mathfrak{R})>1$ such that

(E.14)
$\displaystyle\displaystyle\max_{i\in\\{1,2\\}}\|F_{i}\|_{\mathcal{C}^{5}(\mathfrak{R})}<C_{1}.$

Thus (E.11) (with the $F$ there equal to $F_{1}-F_{2}$ here; the $A$ there
equal to $\varsigma$ here; the $B$ there equal to $C_{1}$ here; and the
$(j,k)$ there equal to either $(1,5)$ or $(2,5)$), together with (E.13) and
(E.14), yields a constant $C_{2}=C_{2}(B,\mathfrak{R})>1$ such that

(E.15) $\displaystyle\begin{aligned}
\displaystyle\sup_{z\in\mathfrak{R}}\big{|}\nabla F_{1}(z)-\nabla
F_{2}(z)\big{|}&\leq
C_{2}\varsigma^{4/5};\qquad\|F_{1}-F_{2}\|_{\mathcal{C}^{2}(\mathfrak{R})}\leq
C_{2}\varsigma^{3/5}.\end{aligned}$

Thus, we may assume in what follows that $\varsigma<\delta$, for some
sufficiently small constant $\delta=\delta(\varepsilon,B)>0$ (to be specified
later), for otherwise the result follows from the fact that
$\sup_{z\in\mathfrak{R}}\big{|}\nabla F_{1}(z)-\nabla F_{2}(z)\big{|}\leq
C_{2}\varsigma^{4/5}$ from (E.15). To improve the first bound in (E.15) to
(E.10), define $g:\mathfrak{R}\rightarrow\mathbb{R}$ by setting

$\displaystyle
g(z)=\varsigma^{-1/2}\big{(}F_{2}(z)-F_{1}(z)\big{)},\qquad\text{for each
$z\in\overline{\mathfrak{R}}$.}$

Then the first bound in (E.13) and (E.15) together yield

(E.16)
$\displaystyle\|g\|_{\mathcal{C}^{0}(\mathfrak{R})}\leq\varsigma^{1/2};\qquad\|g\|_{\mathcal{C}^{1}(\mathfrak{R})}\leq
2C_{2}\varsigma^{3/10};\qquad\|g\|_{\mathcal{C}^{2}(\mathfrak{R})}\leq
C_{2}\varsigma^{1/10}.\qquad.$

Next, setting $i=2$ and using the fact that $F_{2}=F_{1}+\varsigma^{1/2}g$ in
(E.9), we obtain

(E.17) $\displaystyle\displaystyle\sum_{1\leq j,k\leq 2}a_{jk}\big{(}\nabla
F_{1}(z)+\varsigma^{1/2}\nabla
g(z)\big{)}\big{(}\partial_{j}\partial_{k}F_{1}(z)+\varsigma^{1/2}\partial_{j}\partial_{k}g(z)\big{)}=0.$

Then, subtracting the $i=1$ case of (E.9) from (E.17), we deduce that

(E.18) $\displaystyle\begin{aligned}
&\displaystyle\sum_{j,k\in\\{1,2\\}}\Big{(}\mu_{jk}(z)\partial_{j}\partial_{k}g(z)+\nu_{jk}(z)\kappa_{jk}(z)\cdot\nabla
g(z)\Big{)}\\\
&\qquad\quad=-\varsigma^{1/2}\displaystyle\sum_{j,k\in\\{1,2\\}}\Big{(}\big{(}\nabla
g(z)\cdot\kappa_{jk}(z)\big{)}\partial_{j}\partial_{k}g(z)+\nu_{jk}(z)h_{jk}(z)+\varsigma^{1/2}h_{jk}(z)\partial_{j}\partial_{k}g(z)\Big{)},\end{aligned}$

where for each $j,k\in\\{1,2\\}$, we have defined
$\mu_{jk},\nu_{jk},h_{jk}\in\mathcal{C}(\mathfrak{R})$ and
$\kappa_{jk}:\mathfrak{R}\rightarrow\mathbb{R}^{2}$ by setting

(E.19) $\displaystyle\begin{aligned} &\mu_{jk}(z)=a_{jk}\big{(}\nabla
F_{1}(z)\big{)};\qquad\kappa_{jk}(z)=\nabla a_{jk}\big{(}\nabla
F_{1}(z)\big{)};\qquad\nu_{jk}(z)=\partial_{j}\partial_{k}F_{1}(z);\\\
&h_{jk}(z)=\varsigma^{-1}\Big{(}a_{jk}\big{(}\nabla
F_{1}(z)+\varsigma^{1/2}\nabla g(z)\big{)}-a_{jk}\big{(}\nabla
F_{1}(z)\big{)}-\varsigma^{1/2}\nabla g(z)\cdot\nabla a_{jk}\big{(}\nabla
F_{1}(z)\big{)}\Big{)},\\\ \end{aligned}$

for each $z\in\mathfrak{R}$. Now, observe that (E.1), (E.14), and the fact
that $\|a_{jk}\|_{\mathcal{C}^{4}(\mathbb{R}^{2})}\leq B$ together imply the
existence of a constant $C_{3}=C_{3}(B,\mathfrak{R})>1$ such that

(E.20) $\displaystyle\|\mu_{jk}\|_{\mathcal{C}^{0,1/2}(\mathfrak{R})}\leq
C_{3};\qquad\|\kappa_{jk}(z)\|_{\mathcal{C}^{0,1/2}(\mathfrak{R})}\leq
C_{3};\qquad\|\nu_{jk}(z)\|_{\mathcal{C}^{0,1/2}(\mathfrak{R})}\leq C_{3}.$

Since the $a_{jk}$ satisfy (E.8), it follows from (E.18), (E.12), and the
$\alpha=\frac{1}{2}$ case of the Schauder estimate E.1 that there exists a
constant $C_{4}=C_{4}(B,\mathfrak{R})>1$ such that

$\displaystyle\|g\|_{\mathcal{C}^{2,1/2}(\mathfrak{R})}$ $\displaystyle\leq
C_{4}\|g\|_{\mathcal{C}^{0}(\mathfrak{R})}+C_{4}\varsigma^{1/2}\displaystyle\max_{j,k\in\\{1,2\\}}\Big{\|}\big{(}\nabla
g(z)\cdot\kappa_{jk}(z)\big{)}\partial_{j}\partial_{k}g(z)\Big{\|}_{\mathcal{C}^{0,1/2}(\mathfrak{R})}$
$\displaystyle\qquad+C_{4}\varsigma^{1/2}\displaystyle\max_{j,k\in\\{1,2\\}}\big{\|}\nu_{jk}(z)h_{jk}(z)\big{\|}_{\mathcal{C}^{0,1/2}(\mathfrak{R})}+C_{4}\varsigma^{1/2}$
$\displaystyle\qquad+C_{4}\varsigma\displaystyle\max_{j,k\in\\{1,2\\}}\big{\|}h_{jk}(z)\partial_{j}\partial_{k}g(z)\big{\|}_{\mathcal{C}^{0,1/2}(\mathfrak{R})}.$

Thus, by (E.1) and (E.20), there exists a constant
$C_{5}=C_{5}(B,\mathfrak{R})>1$ such that

(E.21) $\displaystyle\begin{aligned}
\|g\|_{\mathcal{C}^{2,1/2}(\mathfrak{R})}&\leq
C_{5}\varsigma^{1/2}\Big{(}\|g\|_{\mathcal{C}^{2,1/2}(\mathfrak{R})}\|g\|_{\mathcal{C}^{1}(\mathfrak{R})}+\|g\|_{1,1/2;\mathfrak{R}}\|g\|_{\mathcal{C}^{2}(\mathfrak{R})}+\|h\|_{\mathcal{C}^{0,1/2}(\mathfrak{R})}+1\Big{)}\\\
&\quad+C_{5}\varsigma\Big{(}\|h\|_{\mathcal{C}^{0}(\mathfrak{R})}\|g\|_{\mathcal{C}^{2,1/2}(\mathfrak{R})}+\|h\|_{\mathcal{C}^{0,1/2}(\mathfrak{R})}\|g\|_{\mathcal{C}^{2}(\mathfrak{R})}\Big{)}+C_{5}\|g\|_{\mathcal{C}^{0}(\mathfrak{R})}.\end{aligned}$

Now, (E.16) yields a constant $C_{6}=C_{6}(B,\mathfrak{R})>1$ such that

(E.22) $\displaystyle\begin{aligned}
\max\big{\\{}\|g\|_{\mathcal{C}^{1}(\mathfrak{R})},\|g\|_{1,1/2;\mathfrak{R}},\|g&\|_{\mathcal{C}^{2}(\mathfrak{R})},\|h\|_{\mathcal{C}^{0}(\mathfrak{R})},\|h\|_{\mathcal{C}^{0,1/2}(\mathfrak{R})}\big{\\}}\\\
&\quad\leq\max\big{\\{}\|g\|_{\mathcal{C}^{2}(\mathfrak{R})},\|h\|_{\mathcal{C}^{1}(\mathfrak{R})}\big{\\}}<C_{6},\end{aligned}$

where the bound on $\|h\|_{\mathcal{C}^{1}(\mathfrak{R})}$ follows from
differentiating the definition (E.19) of $h$; (E.14); (E.16); the fact that
$\|a_{jk}\|_{\mathcal{C}^{4}(\mathbb{R}^{2})}\leq B$ for each
$j,k\in\\{1,2\\}$; and a Taylor expansion.

Inserting the first bound of (E.16); (E.22); and the fact that
$\|g\|_{\mathcal{C}^{2,1/2}(\mathfrak{R})}\geq\|g\|_{\mathcal{C}^{2}(\mathfrak{R})}$
into (E.21) then yields a constant $C_{7}=C_{7}(B,\mathfrak{R})>1$ such that

$\displaystyle(1-C_{7}\varsigma^{1/2})\|g\|_{\mathcal{C}^{2,1/2}(\mathfrak{R})}\leq
C_{7}\varsigma^{1/2}.$

Thus, if $\delta<\min\big{\\{}\frac{1}{4C_{7}^{2}},1\big{\\}}$ and
$\varsigma<\delta$, we obtain that

(E.23)
$\displaystyle\|F_{1}-F_{2}\|_{\mathcal{C}^{1}(\mathfrak{R})}\leq\|F_{1}-F_{2}\|_{\mathcal{C}^{2,1/2}(\mathfrak{R})}=\varsigma^{1/2}\|g\|_{\mathcal{C}^{2,1/2}(\mathfrak{R})}\leq
2C_{7}\varsigma,$

from which the proposition follows. ∎

###  E.3. Proof of 9.6

In this section we establish 9.6.

###### Proof of 9.6.

We first perform a rescaling, to which end we define the open rectangles

$\displaystyle\widehat{\mathfrak{R}}=L\cdot\mathfrak{R}=(0,1)\times($
$\displaystyle-L,L);\qquad\widehat{\mathfrak{S}}_{r}=L\cdot\mathfrak{S}_{r}=(r,1-r)\times(r+1-L,L-r-1);$
$\displaystyle\quad\widehat{\mathfrak{S}}=L\cdot\mathfrak{S}=(0,1)\times(1-L,L-1),$

as well as the functions
$\widehat{F}\in\mathcal{C}^{m}(\widehat{\mathfrak{R}})$ and
$\widehat{f}_{i},\widehat{g}_{i}:[-L,L]\rightarrow\mathbb{R}$ for each
$i\in\\{0,1\\}$ by, for any $w\in[0,1]\times[-L,L]$ and $x\in[-L,L]$, setting

(E.24) $\displaystyle\widehat{F}(w)=L\cdot
F(L^{-1}w);\quad\widehat{f}_{i}(x)=L\cdot
f_{i}(L^{-1}x);\quad\widehat{g}_{i}(x)=L\cdot g_{i}(L^{-1}x),\quad\text{so
$\nabla\widehat{F}(w)=\nabla F(L^{-1}w)$},$

which in particular implies that
$\widehat{F}\in\operatorname{Adm}_{\varepsilon}(\widehat{\mathfrak{R}})$.
Further define the functions
$\widehat{H}\in\mathcal{C}^{m}(\widehat{\mathfrak{R}})$ and
$\widehat{h}_{i}:[-L,L]\rightarrow\mathbb{R}$ for each $i\in\\{0,1\\}$ by, for
any $(t,x)\in[0,1]\times[-L,L]$, setting

(E.25)
$\displaystyle\widehat{h}_{i}(x)=\widehat{g}_{i}(x)-\widehat{F}(i,x)=\widehat{g}_{i}(x)-\widehat{f}_{i}(x);\qquad\widehat{H}(t,x)=\widehat{F}(t,x)+(1-t)\widehat{h}_{0}(x)+t\widehat{h}_{1}(x),$

so that $\widehat{H}(i,x)=\widehat{g}_{i}(x)$ for each $i\in\\{0,1\\}$ and
$x\in[-L,L]$. The scalings (E.24) (together with the assumptions of the lemma)
then imply for any index $i\in\\{0,1\\}$ and integer $j\in\llbracket
0,m\rrbracket$ that

(E.26) $\displaystyle[\widehat{F}]_{j}\leq
BL^{1-j};\quad[\widehat{g}_{i}]_{j}\leq BL^{1-j};\quad[\widehat{H}]_{j}\leq
3BL^{1-j};\quad\|\widehat{F}-\widehat{H}\|_{\mathcal{C}^{0}(\widehat{\mathfrak{R}})}\leq\|\widehat{h}_{0}\|_{\mathcal{C}^{0}}+\|\widehat{h}_{1}\|_{\mathcal{C}^{0}}\leq
2\delta.$

It suffices to show the existence of a solution
$\widehat{G}\in\operatorname{Adm}_{\varepsilon/2}(\widehat{\mathfrak{S}})$ to
the equation (1.2) such that

(E.27) $\displaystyle\begin{aligned}
&\widehat{G}(i,x)=\widehat{g}_{i}(x),\quad\text{for each $i\in\\{1,2\\}$ and
$x\in[1-L,L-1]$};\\\
&\|\widehat{F}-\widehat{G}\|_{\mathcal{C}^{k}(\widehat{\mathfrak{S}}_{r})}\leq
C_{1}\cdot\big{(}\|\widehat{h}_{0}\|_{\mathcal{C}^{0}}+\|\widehat{h}_{1}\|_{\mathcal{C}^{0}}\big{)};\\\
&\|\widehat{F}-\widehat{G}\|_{\mathcal{C}^{m-5}(\widehat{\mathfrak{S}})}\leq
C_{2}\cdot\big{(}\|\widehat{h}_{0}\|_{\mathcal{C}^{m-3}}+\|\widehat{h}_{1}\|_{\mathcal{C}^{m-3}}\big{)},\end{aligned}$

for some constants $C_{1}=C_{1}(\varepsilon,r,B,k)>1$ and
$C_{2}=C_{2}(\varepsilon,B,m)>1$. Then, the function
$G\in\operatorname{Adm}_{\varepsilon/2}(\mathfrak{S})$ defined by
$G(w)=L^{-1}\cdot\widehat{G}(Lw)$ for each
$w\in[0,L^{-1}]\times[L^{-1}-1,1-L^{-1}]$ would satisfy the conditions of the
lemma (where here, we again used the fact that $\nabla
G(w)=\nabla\widehat{G}(Lw)$, as in (E.24); to deduce the last bound in the
third part of the lemma, we also used
$\|\widehat{h}_{i}\|_{\mathcal{C}^{m-3}}\leq L\|h_{i}\|_{\mathcal{C}^{m-3}}$
for each $i\in\\{0,1\\}$, $\|F-G\|_{\mathcal{C}^{m-5}}\leq
L^{m-6}\|\widehat{F}-\widehat{G}\|_{\mathcal{C}^{m-5}}$, and the interpolation
estimate E.5 to deduce that $\|f_{i}-g_{i}\|_{\mathcal{C}^{m-3}}\leq
C_{2}^{\prime}\|f_{i}-g_{i}\|_{\mathcal{C}^{0}}^{3/m}$ for each
$i\in\\{0,1\\}$ and some constant
$C_{2}^{\prime}=C_{2}^{\prime}(\varepsilon,B,m)>1$, which holds since
$\|f_{i}-g_{i}\|_{\mathcal{C}^{m}}\leq\|f_{i}\|_{\mathcal{C}^{m}}+\|g_{i}\|_{\mathcal{C}^{m}}\leq
2B$).

Figure 14. Shown on the left is the open set
$(0,1)\times\big{[}-\frac{1}{2},\frac{1}{2}\big{]}\subset\mathfrak{U}\subset(0,1)\times(-1,1)$.
Shown on the right is the open set $\mathfrak{V}$ as in (E.28).

To that end, let us fix an open set $\mathfrak{U}\subset(0,1)\times(-1,1)$
with smooth boundary, such that
$(0,1)\times\big{[}-\frac{1}{2},\frac{1}{2}\big{]}\subset\mathfrak{U}$; see
the left side of Figure 14. For any real number $a\in\mathbb{R}$, we also
denote its vertical translate $\mathfrak{U}_{a}=\mathfrak{U}+(0,a)$, and we
define the domain
$[0,1]\times[1-L,L-1]\subset\mathfrak{V}\subset[0,1]\times[-L,L]$ by

(E.28)
$\displaystyle\mathfrak{V}=\big{(}[0,1]\times[1-L,L-1]\big{)}\cup\mathfrak{U}_{1-L}\cup\mathfrak{U}_{L-1}.$

See the right side of Figure 14. As in the proof of 9.2, for any indices
$i,j\in\\{t,x\\}$ we also fix fuctions
$\widetilde{\mathfrak{d}}_{ij}\in\mathcal{C}^{m+3}(\mathbb{R}^{2})$ so that
$\widetilde{\mathfrak{d}}_{ij}(u,v)=\mathfrak{d}_{ij}(u,v)$ whenever
$v\in\big{[}-\frac{2}{\varepsilon},-\frac{\varepsilon}{2}\big{]}$, and such
that (E.5) holds for some constant
$\widetilde{B}=\widetilde{B}(\varepsilon)>1$.

Then, E.3 yields a solution
$\widehat{G}\in\mathcal{C}^{m-1}(\overline{\mathfrak{V}})$ to the equation

(E.29)
$\displaystyle\displaystyle\sum_{i,j\in\\{t,x\\}}\widetilde{\mathfrak{d}}_{ij}\big{(}\nabla\widehat{G}(z)\big{)}\partial_{i}\partial_{j}\widehat{G}(z)=0,\qquad\text{for
$z\in\mathfrak{V}$, with
$\widehat{G}|_{\partial\mathfrak{V}}=\widehat{H}|_{\partial\mathfrak{V}}$}.$

We first verify that $\widehat{G}$ satisfies the three properties in (E.27),
and then we confirm that
$\widehat{G}\in\operatorname{Adm}_{\varepsilon/2}(\mathfrak{V})\subseteq\operatorname{Adm}_{\varepsilon/2}(\widehat{\mathfrak{S}})$
(where the last inclusion holds by restricting $\widehat{G}$, as
$\widehat{\mathfrak{S}}\subset\mathfrak{V}$) and that $\widehat{G}$ solves the
original equation (1.2). To confirm the first condition in (E.27), observe
from the definition (E.28) of $\mathfrak{V}$ that
$\\{0,1\\}\times[1-L,L-1]\subset\partial\mathfrak{V}$, so for any
$(i,x)\in\\{0,1\\}\times[1-L,L-1]$ we have
$\widehat{G}(i,x)=\widehat{H}(i,x)=\widehat{g}_{i}(x)$.

The second and third statements will follow from a suitable application of
E.4. To implement this, first observe that

(E.30)
$\displaystyle\displaystyle\sup_{w\in\mathfrak{V}}\big{|}\widehat{F}(w)-\widehat{G}(w)\big{|}\leq\displaystyle\sup_{w\in\partial\mathfrak{V}}\big{|}\widehat{F}(w)-\widehat{G}(w)\big{|}=\displaystyle\sup_{w\in\partial\mathfrak{V}}\big{|}\widehat{F}(w)-\widehat{H}(w)\big{|}\leq\|\widehat{h}_{0}\|_{\mathcal{C}^{0}}+\|\widehat{h}_{1}\|_{\mathcal{C}^{0}}\leq
2\delta.$

Here, the first inequality follows from the comparison principle E.2 (applied
twice, as in the proof of 9.1, first with $(F_{1},F_{2})$ there equal to
$(\widehat{F},\widehat{G}-\sup_{w\in\partial\mathfrak{V}}\big{|}\widehat{F}(w)-\widehat{G}(w)\big{|})$
and next with $(F_{1},F_{2})$ there equal to
$(\widehat{F},\widehat{G}+\sup_{w\in\partial\mathfrak{V}}\big{|}\widehat{F}(w)-\widehat{G}(w)\big{|})$
here); the second follows from the boundary data (E.29) for $\widehat{G}$; and
the third and fourth follow from the last bounds in (E.26).

Now, fix some $w_{0}=(t_{0},x_{0})\in\widehat{\mathfrak{S}}_{r}$ so that
$\mathcal{B}_{r}(t_{0},x_{0})\subseteq\mathfrak{V}$, and apply E.4 with the
$(F_{1},F_{2})$ there equal to
$\big{(}\widehat{F}-\widehat{F}(0,x_{0});\widehat{G}-\widehat{F}(0,x_{0})\big{)}$
here; $(B,m,r;\mathfrak{R})$ there equal to
$\big{(}8B+\widetilde{B},k;\frac{r}{2};\mathfrak{U}_{x_{0}}\big{)}$ here; and
$\mathfrak{R}^{\prime}$ there empty here (so we may take the
$(\varphi_{1},\varphi_{2})$ there to be $0$ here). This yields a constant
$C_{1}=C_{1}(\varepsilon,r,B,k)>1$ (recalling that $\widetilde{B}$ only
depends on $\varepsilon$) such that

$\displaystyle\big{\|}\widehat{F}-\widehat{G}\|_{\mathcal{C}^{k}(\mathcal{B}_{r/2}(w_{0}))}\leq
C_{1}\cdot\displaystyle\sup_{w\in\partial\mathfrak{U}_{x_{0}}}\big{|}\widehat{F}(w)-\widehat{G}(w)\big{|}\leq
C_{1}\Big{(}\|\widehat{h}_{0}\|_{\mathcal{C}^{0}}+\|\widehat{h}_{1}\|_{\mathcal{C}^{0}}\Big{)},$

where the last bound follows from (E.30). Ranging over
$w_{0}\in\widehat{\mathfrak{S}}_{r}$ gives the second part of (E.27).

To establish the third part of (E.27), fix some
$w_{0}\in(t_{0},x_{0})\in\mathfrak{V}$, and let $s\in[1-L,L-1]$ be such that
$\mathcal{B}_{r/4}(w_{0})\subset\mathfrak{U}_{s}$ (which is guaranteed to
exist by (E.28)). Then, apply E.4 with the $(F_{1},F_{2})$ equal to
$\big{(}\widehat{F}-\widehat{F}(0,x_{0}),\widehat{G}-\widehat{F}(0,x_{0})\big{)}$
here; $(B,m,r;\mathfrak{R},\mathfrak{R}^{\prime})$ there equal to
$\big{(}8B+\widetilde{B},m-5,\frac{r}{8};\mathfrak{U}_{s},\mathfrak{U}_{s}\big{)}$
here; and $\varphi_{1}(t,x)$ and $\varphi_{2}(t,x)$ there equal to
$\widehat{F}(t,x)-\widehat{F}(0,x_{0})$ and
$\widehat{H}(t,x)-\widehat{F}(0,x_{0})$ here, respectively. This yields a
constant $C_{2}=C_{2}(\varepsilon,r,B,m)>1$ (which also depends on the open
set $\mathfrak{U}$, but we omit this dependence since $\mathfrak{U}$ is fixed)
such that

$\displaystyle\big{\|}\widehat{F}-\widehat{G}\big{\|}_{\mathcal{C}^{m-5}(\mathfrak{U}_{s})}\leq
C_{2}\bigg{(}\displaystyle\sup_{w\in\partial\mathfrak{U}_{s}}\big{|}\widehat{F}(w)-\widehat{G}(w)\big{|}+\|\widehat{F}-\widehat{H}\|_{\mathcal{C}^{m-3}}\bigg{)}\leq
2C_{2}\Big{(}\|\widehat{h}_{0}\|_{\mathcal{C}^{m-3}}+\|\widehat{h}_{1}\|_{\mathcal{C}^{m-3}}\Big{)},$

where in the last bound we used (E.25) and (E.30). Ranging over all
$w\in\mathfrak{V}$ yields

(E.31)
$\displaystyle\big{\|}\widehat{F}-\widehat{G}\big{\|}_{\mathcal{C}^{m-5}(\mathfrak{V})}\leq
2C_{2}\Big{(}\|\widehat{h}_{0}\|_{\mathcal{C}^{m-3}}+\|\widehat{h}_{1}\|_{\mathcal{C}^{m-3}}\Big{)},$

which since $\mathfrak{S}\subset\mathfrak{V}$ implies the third part of
(E.27).

It remains to verify that
$\widehat{G}\in\operatorname{Adm}_{\varepsilon/2}(\mathfrak{V})\subseteq\operatorname{Adm}_{\varepsilon/2}(\widehat{\mathfrak{S}})$
and that $\widehat{G}$ satisfies (1.2). Observe that it suffices to prove the
former, as it implies the latter since $\widehat{G}$ satisfies (E.29), and
$\widetilde{\mathfrak{d}}_{ij}(\nabla\widehat{G})=\mathfrak{d}_{ij}(\nabla\widehat{G})$
on $\mathfrak{V}$ if
$\widehat{G}\in\operatorname{Adm}_{\varepsilon/2}(\mathfrak{V})$. To show the
former, observe from (E.31), (E.30), and the
($(F;j,k;\mathfrak{R})=(\widehat{F}-\widehat{G};1,2;\mathfrak{U}_{s})$ case of
the) interpolation estimate E.5 that there exists a constant
$C_{3}=C_{3}(\varepsilon,B,m)>1$ (also implicitly dependent on $\mathfrak{U}$)
such that the following holds. For sufficiently small
$\delta=\delta(\varepsilon,B,m)>0$ we have

$\displaystyle\displaystyle\sup_{w\in\mathfrak{V}}\big{|}\nabla\widehat{F}(w)-\nabla\widehat{G}(w)\big{|}\leq
C_{3}\delta^{1/2}.$

Together with the fact that
$\widehat{F}\in\operatorname{Adm}_{\varepsilon}(\widehat{\mathfrak{R}})$, it
follows (after decreasing $\delta$ again if necessary) that
$\widehat{G}\in\operatorname{Adm}_{\varepsilon/2}(\mathfrak{V})$, thereby
establishing the lemma. ∎

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