$\displaystyle\big{\|}F_{1}^{\prime}\big{(}\varphi_{k}(t)\big{)}\big{\|}_{L^{q}(\Omega)}\leq
c(t)<\infty$ (6.27)

for all $q\geq 4$ and all $t\in[0,T]\setminus\mathcal{N}$, after possibly
replacing $\mathcal{N}$ by a larger null set. As $c(t)$ is independent of $q$,
this directly entails

$\displaystyle\big{\|}F_{1}^{\prime}\big{(}\varphi_{k}(t)\big{)}\big{\|}_{L^{\infty}(\Omega)}\leq
c(t)<\infty$ (6.28)

for all $t\in[0,T]\setminus\mathcal{N}$.

As for all $t\in[0,T]\setminus\mathcal{N}$ and all $k\in\mathbb{N}$, the
function $F_{1}^{\prime}(\varphi_{k}(t))$ is measurable and non-negative, we
apply Fatou’s lemma to derive the estimate

$\displaystyle\int_{\Omega}\underset{k\to\infty}{\lim\inf}\;\big{|}F_{1}^{\prime}\big{(}\varphi_{k}(t)\big{)}\big{|}^{2}\;\mathrm{d}\mathbf{x}\leq\underset{n\to\infty}{\lim\inf}\;\int_{\Omega}\big{|}F_{1}^{\prime}\big{(}\varphi_{k}(t)\big{)}\big{|}^{2}\;\mathrm{d}\mathbf{x}\leq\left|\Omega\right|c(t).$
(6.29)

Moreover, it is straightforward to check that $\varphi_{k}(t)\to\varphi(t)$
a.e. in $\Omega$ for all $t\in[0,T]\setminus\mathcal{N}$, after possibly
replacing $\mathcal{N}$ by a larger null set.

Let now $t\in[0,T]\setminus\mathcal{N}$ be arbitrary. Since
$\left|\varphi_{k}(t)\right|<1$ a.e. in $\Omega$ for all $k\in\mathbb{N}$,
this already implies that $\varphi(t)\in[-1,1]$ a.e. in $\Omega$. However, if
$\left|\varphi(t,\mathbf{x})\right|=1$ for some $x\in\Omega$, we have
$\left|\varphi_{k}(t,\mathbf{x})\right|\to 1$ and thus
$\left|F_{1}^{\prime}(\varphi_{k}(t,\mathbf{x}))\right|\to+\infty$ as
$k\to\infty$. We thus conclude that the set

$\displaystyle\mathcal{M}_{t}:=\big{\\{}x\in\Omega\,\big{|}\,\left|\varphi(t,\mathbf{x})\right|=1\big{\\}}$

has Lebesgue measure zero as otherwise, this would contradict (6.29).
Consequently, we have $\varphi(t)\in(-1,1)$ a.e. in $\Omega$ and

$\displaystyle F_{1}^{\prime}\big{(}\varphi_{k}(t)\big{)}\to
F_{1}^{\prime}\big{(}\varphi(t)\big{)}\quad\text{a.e.\leavevmode\nobreak\ in
$\Omega$ as $k\to\infty$}.$ (6.30)

In combination with (6.28), this proves

$\displaystyle\big{\|}F_{1}^{\prime}\big{(}\varphi(t)\big{)}\big{\|}_{L^{\infty}(\Omega)}\leq
c(t)<\infty$ (6.31)

for all $t\in[0,T]\setminus\mathcal{N}$. We now recall (Proof of Theorem 2.9.)
as well as the regularities $\varphi,\mu\in L^{2}(0,T;H^{2}(\Omega))$. Taking
the square on both sides of (6.31) and integrating with respect to time from
$0$ to $T$, we eventually conclude $F_{1}^{\prime}(\varphi)\in
L^{2}(0,T;L^{\infty}(\Omega))$. As a direct consequence, we have
$F^{\prime}(\varphi)\in L^{2}(0,T;L^{\infty}(\Omega))$.

Let us now consider the case where $F$ is a regular potential. After fixing
representatives $\varphi$ and $\mu$ of their equivalence class, we introduce
the truncation

$\displaystyle\sigma_{k}:\mathbb{R}\to\mathbb{R},\quad\sigma_{k}(s):=\begin{cases}k&\text{if
$s\geq k$},\\\ s&\text{if $\left|s\right|<k$},\\\ -k&\text{if
$s\leq-k$},\end{cases}$

and we set $\varphi_{k}:=\sigma_{k}\circ\varphi$ for all $k\in\mathbb{N}$.
Proceeding exactly as for the logarithmic potential, we show that
$F_{1}^{\prime}\big{(}\varphi_{k}(t)\big{)}$ satisfies (6.28) for all
$t\in[0,T]\setminus\mathcal{N}$, where $\mathcal{N}\subset[0,T]$ is some null
set and $c(t)$ is the constant defined in (Proof of Theorem 2.9.). As the
regular potential is assumed to fulfill ($\mathbf{F3}$), we know that
$F_{1}^{\prime}:\mathbb{R}\to\mathbb{R}$ is continuous. Hence, since
$\varphi_{k}(t)\to\varphi(t)$ a.e. in $\Omega$ for all
$t\in[0,T]\setminus\mathcal{N}$, we deduce the convergence (6.30). This
entails (6.31) and by proceeding as above, we conclude the claim.

Step 5: From now on, let $F$ be the logarithmic potential. We now show that
the strict separation property (2.15) holds for almost all $t\in[0,T]$.

As a consequence of (6.31), there exist null sets $\mathcal{N}\subset[0,T]$
and $\mathcal{O}\subset\Omega$ such that for all
$(t,\mathbf{x})\in\Omega_{T}\setminus(\mathcal{N}\times\mathcal{O})$, it holds
$\varphi(t,\mathbf{x})\in(-1,1)$ and

$\displaystyle\big{|}\ln\big{(}1+\varphi(t,\mathbf{x})\big{)}-\ln\big{(}1-\varphi(t,\mathbf{x})\big{)}\big{|}=\Big{|}\frac{2}{\theta}\,F_{1}^{\prime}\big{(}\varphi(t,\mathbf{x})\big{)}\Big{|}\leq\frac{2c(t)}{\theta}.$
(6.32)

Let now $(t,\mathbf{x})\in\Omega_{T}\setminus(\mathcal{N}\times\mathcal{O})$
be arbitrary.

If $\varphi(t,\mathbf{x})\in[0,1)$, we have
$\ln\big{(}1+\varphi(t,\mathbf{x})\big{)}\geq 0$ and
$-\ln\big{(}1-\varphi(t,\mathbf{x})\big{)}\geq 0$. Hence,

$\displaystyle-\ln\big{(}1-\varphi(t,\mathbf{x})\big{)}\leq\frac{2c(t)}{\theta}.$

By a straightforward computation, we thus infer

$\displaystyle 0\leq\varphi(t,\mathbf{x})\leq
1-\delta(t),\quad\text{with}\quad\delta(t):=\exp\left(-\frac{2c(t)}{\theta}\right)\in(0,1].$
(6.33)

If $\varphi(t,\mathbf{x})\in(-1,0)$, a similar computation yields
$-\big{(}1-\delta(t)\big{)}\leq\varphi(t,\mathbf{x})<0$.

In summary, we have thus shown that

$\displaystyle\|\varphi(t)\|_{L^{\infty}(\Omega)}\leq 1-\delta(t)$

for all $t\in[0,T]\setminus\mathcal{N}$. This verifies the strict separation
property (2.15) for almost all $t\in[0,T]$.

Step 6: As the final step, we verify the uniform separation property (2.16) in
the case $d=2$.

Recalling the definition of $F_{1}$ in the context of the logarithmic
potential (see (2.8)), we observe

$\displaystyle\left|F_{1}^{\prime\prime}(s)\right|\leq\theta\mathrm{e}^{\frac{2}{\theta}\left|F^{\prime}(s)\right|}\quad\text{for
all $s\in\mathbb{R}$}.$ (6.34)

Since $d=2$, we know that for all $u\in H^{1}(\Omega)$ and $r\in[2,\infty)$,
it holds

$\displaystyle\|u\|_{L^{r}(\Omega)}$ $\displaystyle\leq
C_{\Omega}\sqrt{r}\,\|u\|_{H^{1}(\Omega)}$

for some constant $C_{\Omega}>0$ depending only on $\Omega$. This inequality
can, for instance, be found in [29, p. 479]. Consequently, we infer from
(6.24) that

$\displaystyle\big{\|}F_{1}^{\prime}\big{(}\varphi_{k}(t)\big{)}\big{\|}_{L^{q}(\Omega)}\leq
C\sqrt{q}\big{(}\|\mu\|_{L^{\infty}(0,T;H^{1}(\Omega))}+c_{*}\|\mu\|_{L^{\infty}(0,T;H^{1}(\Omega))}\big{)}\leq
C\sqrt{q}$

for almost all $t\in[0,T]$, all $q\geq 2$, and all $k\in\mathbb{N}$ with
$k\geq 2$. Arguing similarly as in Step 5, we conclude

$\displaystyle\big{\|}F_{1}^{\prime}\big{(}\varphi(t)\big{)}\big{\|}_{L^{q}(\Omega)}\leq
C\sqrt{q}$ (6.35)

for almost all $t\in[0,T]$ and all $q\geq 2$. Based on this inequality, we
proceed as in the proof of [14, Theorem 3.1] to obtain the estimate

$\displaystyle\int_{\Omega}\left|\mathrm{e}^{\frac{2}{\theta}\left|F^{\prime}(\varphi(t))\right|}\right|^{m}\;\mathrm{d}\mathbf{x}\leq
C(m)$ (6.36)

for all $m\in\mathbb{N}$, almost all $t\in[0,T]$ and some constant $C(m)$
depending only on $m$ and the constant $C$ from (6.35) (cf. [14, Eq. (26)]
with the constants therein being chosen as $C_{1}=\frac{2}{\theta}$, $C_{2}=0$
and $\beta=1$). In view of (6.34), inequality (6.36) written for $m=6$
directly implies

$\displaystyle\big{\|}F_{1}^{\prime\prime}(\varphi)\big{\|}_{L^{\infty}(0,T;L^{6}(\Omega))}\leq
C.$ (6.37)

Employing the chain rule and the continuous embedding
$H^{2}(\Omega)\hookrightarrow W^{1,6}(\Omega)$, we thus obtain

$\displaystyle\big{\|}\nabla
F_{1}^{\prime}(\varphi)\big{\|}_{L^{\infty}(0,T;L^{3}(\Omega))}$
$\displaystyle\leq\big{\|}F_{1}^{\prime\prime}(\varphi)\big{\|}_{L^{\infty}(0,T;L^{6}(\Omega))}\|\nabla\varphi\|_{L^{\infty}(0,T;L^{6}(\Omega))}$
$\displaystyle\leq C\|\varphi\|_{L^{\infty}(0,T;H^{2}(\Omega))}\leq C.$ (6.38)

Using (6.35) written for $q=3$ as well as the continuous embedding
$W^{1,3}(\Omega)\hookrightarrow C(\overline{\Omega})$, we infer

$\displaystyle\big{\|}F_{1}^{\prime}\big{(}\varphi(t)\big{)}\big{\|}_{L^{\infty}(\Omega)}\leq\|F_{1}^{\prime}(\varphi)\|_{L^{\infty}(0,T;W^{1,3}(\Omega))}\leq
C=:C^{*}$ (6.39)

for almost all $t\in[0,T]$. As we have already shown $\varphi\in
C(\overline{\Omega_{T}})$, we know that (6.39) actually holds true for all
$t\in[0,T]$. By arguing as in Step 5, we eventually conclude

$\displaystyle\left|\varphi(t,\mathbf{x})\right|\leq
1-\delta^{*}\quad\text{with}\quad\delta^{*}:=\exp\left(-\frac{2C^{*}}{\theta}\right)\in(0,1],$

for all $(t,\mathbf{x})\in[0,T]$, where $C^{*}$ is the constant from (6.39).
Therefore, the uniform separation property (2.16) is verified.

This means that all claims are established and thus, the proof is complete. ∎

## Acknowledgement

The authors gratefully acknowledge the support by the RTG 2339 “Interfaces,
Complex Structures, and Singular Limits” of the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation). The authors also
acknowledge continuous inspiration by the work of Pierluigi Colli who,
according to MathSciNet, published 47 papers related to the Cahn–Hilliard
equation. We also thank the anonymous referee for valuable comments which
helped to improve the paper significantly.

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