$\langle\nu_{F_{i}}(-e_{n}),e_{n}\rangle\leqslant\cos\theta_{0}$. We
investigate the minimizing properties of $F_{i}$: by Theorem 3.11, each
$F_{i}$ is locally outward minimizing in $\overline{\Omega}$, hence locally
outward minimizing in $\mathbb{R}^{n}$, by direct verification. On the other
hand, by Lemma 4.2 and the scale invariance of (4.2), we obtain the inward-
minimizing property

${P(F_{i};K)}\leqslant{P(F_{i}\setminus
G;K)}+(e^{t_{i}}-1){P(G)}\qquad\forall\,G\subset\subset
K\subset\subset\mathbb{R}^{n}.$

Taking $i\to\infty$, a subsequence of $F_{i}$ converges locally to some
$F_{\infty}\subset\Omega$. Taking limit of the minimizing properties stated
above, and by a standard set replacing argument, it follows that $F_{\infty}$
is locally perimeter-minimizing in $\mathbb{R}^{n}$.

The gradient estimate in (4.1) and energy comparison implies the following:
for all $x\in\Omega$ and any competitor set $E$ with $E\Delta
E_{t}\subset\subset B(x,|x_{n}|/2)$, we have

${P\big{(}E_{t};B(x,\frac{|x_{n}|}{2})\big{)}}\geqslant{P\big{(}E;B(x,\frac{|x_{n}|}{2})\big{)}}+\frac{C}{|x_{n}|}\big{|}E\Delta
E_{t}\big{|}.$

Rescaling this, we see that $F_{i}$ satisfies the condition

${P\big{(}F_{i};B(-e_{n},\frac{1}{2})\big{)}}\geqslant{P\big{(}F;B(-e_{n},\frac{1}{2})\big{)}}+C\big{|}F\Delta
F_{i}\big{|},$

for all $F$ with $F\Delta F_{i}\subset\subset B(-e_{n},1/2)$. Thus $F_{i}$ are
uniform $(\Lambda,r_{0})$-perimeter minimizers in $B(-e_{n},1/2)$, in the
sense stated in Appendix B. Then we apply Lemma B.1(i) to obtain
$-e_{n}\in\operatorname{spt}(|D\chi_{F_{\infty}}|)$, and subsequently apply
Lemma B.3 to obtain $F_{\infty}=\big{\\{}x_{n}<-1\big{\\}}$. Finally, by Lemma
B.1(ii) we have the convergence of normal vectors
$\nu_{F_{i}}(-e_{n})\to\nu_{F_{\infty}}(-e_{n})=e_{n}$, contradicting our
hypothesis. This proves the theorem. ∎

We also establish the following approximate Liouville theorem, where the
obstacle $\\{x_{n}=0\\}$ is replaced by a “soft obstacle”, represented by the
weight function $\psi(x_{n})$ unbounded near $x_{n}=1$. See Subsection 2.3 for
the definition of weighted weak solutions.

######  Theorem 4.3.

Fix a smooth function $\psi:(-\infty,1)\to[0,\infty)$ that satisfies
$\psi|_{(-\infty,0]}\equiv 0$, and $\psi>0$, $\psi^{\prime}>0$,
$\psi^{\prime\prime}>0$, $\psi^{\prime\prime\prime}>0$ on $(0,1)$, and
$\lim_{x\to 1}\psi(x)=+\infty$. Let
$\Omega=\\{x_{n}<1\\}\subset\mathbb{R}^{n}$, and
$u\in\operatorname{Lip}_{\operatorname{loc}}(\Omega)$ be a weak solution of
the weighted IMCF

$\operatorname{div}\Big{(}e^{\psi(x_{n})}\frac{\nabla u}{|\nabla
u|}\Big{)}=e^{\psi(x_{n})}|\nabla u|$ (4.5)

in $\Omega$. Moreover, assume there is a constant $C$ so that

(1) $u\geqslant\psi(x_{n})-C$ in $\Omega$ (in particular, it holds
$\inf_{\Omega}(u)>-\infty$),

(2) $|\nabla u(x)|\leqslant\frac{C}{|x_{n}|}$ for a.e. $x\in\\{x_{n}<0\\}$.

Then $u(x^{\prime},x_{n})=\psi(x_{n})-C^{\prime}$ for some other constant
$C^{\prime}$.

Let us remark that condition (1) is not removable. Consider the (non-smooth)
weight $\psi(x)=(n-1)\log\frac{1}{1-x}$ for $x>0$. By Lemma 2.10, we may
equivalently consider the usual weak IMCF in the metric
$g^{\prime}=e^{2\psi(x_{n})/(n-1)}g$ (which is hyperbolic in
$\\{0<x_{n}<1\\}$), that initiates with the horoball $\\{x_{n}<0\\}$. Without
condition (1), the weak solution may choose to first “jump” to the totally
geodesic hypersurface
$\Sigma_{0}^{+}=\\{x_{n}=1-\sqrt{1-|x^{\prime}|^{2}}\\}$, and then continue to
evolve by
$\Sigma_{t}=\big{\\{}x:d(x,\Sigma_{0}^{+})=\operatorname{arccosh}e^{t/(n-1)}\big{\\}}$.

###### Proof.

Shifting $u$ by a constant, we may assume $\inf_{\Omega}(u)=0$. We are aimed
at proving $u(x^{\prime},x_{n})=\psi(x_{n})$. The proof consists of three
steps: we first show $u\geqslant\psi(x_{n})$ using outer barriers, and next
show that $u=0$ on $\\{x_{n}\leqslant 0\\}$ by arguing similarly as in Theorem
4.1, and finally prove $u\leqslant\psi(x_{n})$ using inner barriers.

Step 1. We show that $u\geqslant\psi(x_{n})$. The following fact can be
directly verified: if $\Sigma_{t}$ are graphs of functions $f=f(\cdot,t)$ over
$\mathbb{R}^{n-1}$, then $\Sigma_{t}$ is a solution (resp. subsolution,
supersolution) of the smooth weighted IMCF $\frac{\partial\Sigma_{t}}{\partial
t}=\frac{\nu}{H+\partial\psi/\partial\nu}$ if and only if

$\frac{\partial f}{\partial t}\Big{(}-\operatorname{div}\frac{\nabla
f}{\sqrt{1+|\nabla f|^{2}}}+\frac{\psi^{\prime}(f)}{\sqrt{1+|\nabla
f|^{2}}}\Big{)}=\ \text{(resp. $\geqslant\,,\leqslant$\,)}\ \sqrt{1+|\nabla
f|^{2}}.$ (4.6)

For $\mu>0$ and $R\geqslant 4$, consider the function

$\underline{f}(x^{\prime},t)=(1+\mu)\psi^{-1}(t)+\mu+R-\sqrt{R^{2}-|x^{\prime}|^{2}},$
(4.7)

where $\psi^{-1}$ is the inverse of $\psi$. Note that $\underline{f}$ is
smooth in the region
$\\{(x^{\prime},t)\in\mathbb{R}^{n-1}\times(0,\infty):\underline{f}(x^{\prime},t)<1\\}$.
We claim that there exists $R_{0}$ depending on $\psi$ and $\mu$, such that
$\underline{f}$ is a subsolution of (4.6) whenever $R\geqslant R_{0}$, $t>0$
and $\underline{f}<1$. To verify this, we compute

$\frac{\partial\underline{f}}{\partial
t}=\frac{1+\mu}{\psi^{\prime}(\psi^{-1}(t))}\qquad\text{and}\qquad\operatorname{div}\frac{\nabla\underline{f}}{\sqrt{1+|\nabla\underline{f}|^{2}}}=\frac{n-1}{R}.$
(4.8)

To evaluate the second term in (4.6), we note that $\underline{f}<1$ implies
$R-\sqrt{R^{2}-|x^{\prime}|^{2}}<1$, which further implies
$|x^{\prime}|^{2}\leqslant 2R-1$. Therefore, by direct estimation

$1+|\nabla\underline{f}|^{2}=\frac{R^{2}}{R^{2}-|x^{\prime}|^{2}}\leqslant\frac{R^{2}}{(R-1)^{2}}\leqslant
1+\frac{4}{R}\qquad\text{whenever $\underline{f}<1$},$ (4.9)

where we used $R\geqslant 4$. Moreover, using the convexity of $\psi$ and
$\psi^{\prime}$ we have

$\displaystyle\psi^{\prime}(\underline{f})$
$\displaystyle\geqslant\psi^{\prime}\big{(}\psi^{-1}(t)+\frac{\mu}{2}\big{)}+\psi^{\prime\prime}\big{(}\psi^{-1}(t)+\frac{\mu}{2}\big{)}\cdot\big{(}\underline{f}-\psi^{-1}(t)-\frac{\mu}{2}\big{)}$
(4.10)
$\displaystyle\geqslant\psi^{\prime}(\psi^{-1}(t))+\psi^{\prime\prime}(\frac{\mu}{2})\cdot\frac{\mu}{2}.$

Combining (4.8) (4.9) (4.10), we obtain

$\displaystyle\frac{\partial\underline{f}}{\partial
t}\Big{(}-\operatorname{div}\frac{\nabla\underline{f}}{\sqrt{1+|\nabla\underline{f}|^{2}}}+\frac{\psi^{\prime}(\underline{f})}{\sqrt{1+|\nabla\underline{f}|^{2}}}\Big{)}$
$\displaystyle\geqslant$
$\displaystyle\,\frac{1+\mu}{\psi^{\prime}(\psi^{-1}(t))}\cdot\Big{(}-\frac{n-1}{R}+\frac{\psi^{\prime}(\psi^{-1}(t))+(\mu/2)\psi^{\prime\prime}(\mu/2)}{\sqrt{1+4/R}}\Big{)}.$

As a result, if we choose $R_{0}$ sufficiently large so that

$\frac{\mu}{2}\psi^{\prime\prime}(\frac{\mu}{2})>\sqrt{1+\frac{4}{R}}\cdot\frac{n-1}{R}\quad\text{and}\quad\frac{1+\mu}{\sqrt{1+4/R}}>\sqrt{1+\frac{4}{R}}\quad\
\,\text{for all $R\geqslant R_{0}$},$

then combined with (4.9), it follows that $\underline{f}$ is a strict
subsolution of (4.6) whenever $t>0$ and $\underline{f}<1$. Switching to the
level set description, we find that the function

$\underline{u}(x^{\prime},x_{n})=\psi\Big{(}\frac{1}{1+\mu}\big{(}x_{n}-\mu-R+\sqrt{R-|x^{\prime}|^{2}}\big{)}\Big{)},$

defined such that
$\big{\\{}\underline{u}=t\big{\\}}=\operatorname{graph}\big{(}f(\cdot,t)\big{)}$,
satisfies the subsolution condition

$\operatorname{div}\Big{(}e^{\psi}\frac{\nabla\underline{u}}{|\nabla\underline{u}|}\Big{)}>e^{\psi}|\nabla\underline{u}|$

in the region
$\Omega^{\prime}=\big{\\{}\mu+R-\sqrt{R^{2}-|x^{\prime}|^{2}}<x_{n}<1\big{\\}}\subset\Omega$.

For $\varepsilon>0$, we apply the interior maximum principle (Theorem 2.8) to
the functions $u^{\prime}=u+\varepsilon$ and $\underline{u}$. By our
hypothesis we have $u\geqslant\max\\{0,\psi(x_{n})-C\\}$. Hence

$u^{\prime}(x)<\underline{u}(x)\ \ \Rightarrow\ \
\psi(x_{n})-C<\psi\big{(}\frac{x_{n}}{1+\mu}\big{)}\ \ \Rightarrow\ \
x_{n}<C^{\prime}\qquad\text{for all $x\in\Omega^{\prime}$},$

where $C^{\prime}<1$ is a constant depending only on $C,\mu,\psi$. On the
other hand, $u^{\prime}(x)<\underline{u}(x)$ implies
$\underline{u}(x)>\varepsilon$, therefore the closure of
$\\{u^{\prime}<\underline{u}\\}$ does not intersect
$\\{x_{n}=\mu+R-\sqrt{R^{2}-|x^{\prime}|^{2}}\\}$. As a result, we have
$\\{u^{\prime}<\underline{u}\\}\subset\subset\Omega^{\prime}$. By Lemma
2.10(3) and Theorem 2.8, we obtain $u^{\prime}\geqslant\underline{u}$. Taking
$\varepsilon\to 0$, $R\to\infty$ and then $\mu\to 0$, we obtain
$u\geqslant\psi(x_{n})$.

Step 2. we show that $u=0$ on $\\{x_{n}\leqslant 0\\}$, using similar methods
as in Theorem 4.1. Due to the presence of weight function, the discussion here
is more complicated. Recall that we have assumed $\inf(u)=0$ and showed that
$E_{t}\subset\\{x_{n}<\psi^{-1}(t)\\}$ for all $t>0$.

Case 2(i): suppose for any $\varepsilon>0$, there exists $\delta>0$ such that
$\partial^{*}E_{t}\subset\\{x\geqslant-\varepsilon\\}$ for all
$0<t\leqslant\delta$. Thus for each $t\in(0,\delta]$ there are only two
possibilities: either $E_{t}\supset\\{x_{n}<-\varepsilon\\}$, or
$E_{t}\subset\\{-\varepsilon<x_{n}<\psi^{-1}(t)\\}$. We claim that latter case
never occurs. Once this is proved, we have $\\{u\leqslant
0\\}\supset\\{x_{n}<-\varepsilon\\}$, which by taking $\varepsilon\to 0$
proves the goal of Step 2. It remains to prove our claim. Suppose we have
$E_{t}\subset\\{-\varepsilon<x<\psi^{-1}(t)\\}$ for some $\varepsilon,\delta$
and $t\in(0,\delta]$. Let $\pi:(x^{\prime},x_{n})\mapsto x^{\prime}$ be the
projection map. Note that
${P\big{(}E_{t};A\times\mathbb{R}\big{)}}\geqslant\mathcal{L}^{n-1}(\pi(E_{t})\cap
A)$ for any open set $A\subset\mathbb{R}^{n-1}$. Lemma 2.21 is easily
generalized to the weighted case, which implies that

$\int_{\partial^{*}E_{t}\cap
K}e^{\psi}\,d\mathcal{H}^{n-1}\leqslant\int_{\partial^{*}(E_{t}\setminus
F)\cap K}e^{\psi}\,d\mathcal{H}^{n-1}+(e^{t}-1)\int_{\partial^{*}F\cap
E_{t}}e^{\psi}\,d\mathcal{H}^{n-1},$ (4.11)

whenever $F\subset\subset K\subset\subset\Omega$. If
$\mathcal{H}^{n-1}(\partial^{*}E_{t}\cap\partial^{*}F)=0$ and $F=F^{(1)}$,
then (4.11) is reduced to

$\int_{\partial^{*}E_{t}\cap F}e^{\psi}\,d\mathcal{H}^{n-1}\leqslant
e^{t}\int_{\partial^{*}F\cap E_{t}}e^{\psi}\,d\mathcal{H}^{n-1}.$ (4.12)

Since $e^{\psi}\geqslant 1$ everywhere and $e^{\psi}\leqslant e^{t}$ in
$\overline{E_{t}}$, we have

${P(E_{t};F)}\leqslant e^{2t}{P(F;E_{t})}.$ (4.13)

Now for a radius $R>0$ we choose $F=B^{\prime}(0,R)\times(-1/2,1/2)$, where
$B^{\prime}$ denotes open balls in $\mathbb{R}^{n-1}$. For almost every $R$,
(4.13) implies

$\displaystyle\mathcal{L}^{n-1}(\pi(E_{t})\cap B^{\prime}(0,R))$
$\displaystyle\leqslant{P\big{(}E_{t};B^{\prime}(0,R)\times\mathbb{R}\big{)}}={P(E_{t};F)}\leqslant
e^{2t}{P(F;E_{t})}$ $\displaystyle\leqslant
e^{2t}\big{(}\varepsilon+\psi^{-1}(t)\big{)}\mathcal{H}^{n-2}(\partial
B^{\prime}(0,R)\cap\pi(E_{t})).$

Denoting $f(R)=\mathcal{L}^{n-1}(\pi(E_{t})\cap B^{\prime}(0,R))$, thus
$f(R)\leqslant e^{2t}\big{(}\varepsilon+\psi^{-1}(t)\big{)}f^{\prime}(R)$.
Since $\inf_{\Omega}(u)=0$, we have $f(R)>0$ for some $R$, therefore $f(R)$ is
exponentially growing as $R\to\infty$. This is impossible. From the setups
made above, this proves the desired result for Case 2(i).

For the rest of Step 2, we may assume the contrary of Case 2(i): that there is
a fixed $\varepsilon>0$ and a sequence of times $t_{i}\to 0$, such that
$\partial^{*}E_{t_{i}}\cap\\{x_{n}<-\varepsilon\\}\neq\emptyset$ for all $i$.

Case 2(ii): assume that there are angles $\theta_{i}\to 0$, such that for each
$i$ it holds

$\operatorname*{ess\,inf}_{\partial^{*}E_{t_{i}}\cap\\{x_{n}<-\varepsilon\\}}\langle\nu_{E_{t_{i}}},e_{n}\rangle\geqslant\cos\theta_{i}.$

Then arguing similarly as in Case (i) of Theorem 4.1, we have
$E_{t}\supset\\{x_{n}<0\\}$ for all $t>0$. This proves the desired result for
Step 2.

Case 2(iii): suppose the hypothesis of Case 2(ii) does not hold. Then passing
to a subsequence, there is a constant $\theta_{0}>0$ and points
$y_{i}=(y^{\prime}_{i},y_{i,n})\in\partial^{*}E_{t_{i}}\cap\\{x_{n}<-\varepsilon\\}$,
such that $\langle\nu_{E_{t_{i}}}(y_{i}),e_{n}\rangle\leqslant\cos\theta_{0}$.
We argue that this is impossible. For convenience, we denote
$E_{i}=E_{t_{i}}$. Recall from step 1 that
$E_{i}\subset\\{x_{n}<\psi^{-1}(t_{i})\\}$. We start with investigating the
minimizing properties of $E_{i}$.

1\. $E_{i}$ are local outward minimizers of the weighted perimeter
$E\mapsto\int_{\partial^{*}E}e^{\psi}\,d\mathcal{H}^{n-1}$ in $\\{x_{n}<1\\}$,
by Lemma 2.10(iii). This immediately implies that $E_{i}$ are local outward
minimizers of
$E\mapsto\int_{\partial^{*}E}e^{\min\\{\psi,t_{i}\\}}\,d\mathcal{H}^{n-1}$ in
$\big{\\{}x_{n}<\psi^{-1}(t_{i})\big{\\}}$. Then by direct verification,
$E_{i}$ locally outward minimizes the same functional
$E\mapsto\int_{\partial^{*}E}e^{\min\\{\psi,t_{i}\\}}\,d\mathcal{H}^{n-1}$ in
$\mathbb{R}^{n}$.

2\. The inward minimizing effect of $E_{i}$ is given by (4.11). Since all the
integrals in (4.11) occur inside the set $\\{\psi\leqslant t_{i}\\}$, it makes
no difference to replace each $\psi$ by $\min\\{\psi,t_{i}\\}$.

Combining the inward and outward minimizing properties, we conclude that

$\displaystyle\int_{\partial^{*}E_{i}\cap
K}e^{\min\\{\psi,t_{i}\\}}\,d\mathcal{H}^{n-1}$
$\displaystyle\leqslant\int_{\partial^{*}F\cap
K}e^{\min\\{\psi,t_{i}\\}}\,d\mathcal{H}^{n-1}$
$\displaystyle\qquad\qquad+(e^{t_{i}}-1)\int_{\partial^{*}(E_{i}\setminus
F)}e^{\min\\{\psi,t_{i}\\}}\,d\mathcal{H}^{n-1}$

whenever $F\Delta E_{i}\subset\subset K\subset\subset\mathbb{R}^{n}$. This
directly implies

${P(E_{i};K)}\leqslant
e^{t_{i}}{P(F;K)}+e^{t_{i}}(e^{t_{i}}-1){P(E_{i}\setminus F)}.$

Next, consider the rescaled sets $F_{i}=(E_{i}-y^{\prime}_{i})/|y_{i,n}|$. It
follows that $F_{i}\subset\\{x_{n}<1/\varepsilon\\}$,
$-e_{n}\in\partial^{*}F_{i}$, and
$\langle\nu_{F_{i}}(-e_{n}),e_{n}\rangle\leqslant\cos\theta_{0}$, and $F_{i}$
satisfy the minimizing property

${P(F_{i};K)}\leqslant
e^{t_{i}}{P(F;K)}+e^{t_{i}}(e^{t_{i}}-1){P(F_{i}\setminus F)}$ (4.14)

whenever $F\Delta F_{i}\subset\subset K\subset\subset\mathbb{R}^{n}$.
Moreover, since $\psi_{i}=0$ in $\\{x_{n}<0\\}$, and by the gradient estimate
$|\nabla u|\leqslant C/|x_{n}|$, we have the uniform
$(\Lambda,r_{0})$-minimizing property

${P\big{(}F_{i};B(-e_{n},\frac{1}{2})\big{)}}\leqslant{P\big{(}F;B(-e_{n},\frac{1}{2})\big{)}}+C|F\Delta
F_{i}|$ (4.15)

whenever $F\Delta F_{i}\subset\subset B(-e_{n},\frac{1}{2})$.

We are in a position to pass to the limit: there is a subsequence of $F_{i}$
that converge to a set $F_{\infty}$ in $L^{1}_{\operatorname{loc}}$. By (4.15)
and Lemma B.1(i) we have $-e_{n}\in\partial^{*}F_{\infty}$. By (4.14) and the
fact $t_{i}\to 0$ and the standard set replacing argument, the limit
$F_{\infty}$ locally minimizes the perimeter in $\mathbb{R}^{n}$. By Lemma B.3
and the fact $F_{\infty}\subset\\{x_{n}<1/\varepsilon\\}$, we have
$F_{\infty}=\\{x_{n}<-1\\}$. Finally, by Lemma B.1(ii) we have convergence of
normal vectors $\nu_{F_{i}}(-e_{n})\to\nu_{F_{\infty}}(-e_{n})=e_{n}$,
contradicting our hypothesis. This completes the proof of step 2.

Step 3. We have shown that $u\geqslant\psi(x_{n})$ and $u=0$ on
$\\{x_{n}<0\\}$. In this step we use inner barriers to show that
$u\leqslant\psi(x_{n})$, which completes the proof of the theorem. For
constants $\mu\ll 1$, $R\gg 1$ and for $t>0$, we consider the function

$\overline{f}(x^{\prime},t)=(1-\mu)\int_{0}^{t}\frac{ds}{\psi^{\prime}(\psi^{-1}(s))+(n-1)/R}-R+\sqrt{R^{2}-|x^{\prime}|^{2}}.$

We claim that $\overline{f}$ is a supersolution of (4.6) whenever $t>0$ and
$\overline{f}>0$. Note that
$\overline{f}\leqslant(1-\mu)\int_{0}^{t}\frac{ds}{\psi^{\prime}(\psi^{-1}(s))}<(1-\mu)\psi^{-1}(t)$.
In particular, $\overline{f}<1$ always holds. To verify the supersolution
property, according to (4.6) we compute

$\frac{\partial\overline{f}}{\partial
t}=\frac{1-\mu}{\psi^{\prime}(\psi^{-1}(t))+(n-1)/R}\qquad\text{and}\qquad\operatorname{div}\Big{(}\frac{\nabla
f}{\sqrt{1+|\nabla f|^{2}}}\Big{)}=-\frac{n-1}{R}.$

By the convexity of $\psi$, we have
$\psi^{\prime}(\overline{f})\leqslant\psi^{\prime}(\psi^{-1}(t))$. Therefore,

$\displaystyle\,\frac{\partial\overline{f}}{\partial
t}\Big{(}-\operatorname{div}\frac{\nabla\overline{f}}{\sqrt{1+|\nabla\overline{f}|^{2}}}+\frac{\psi^{\prime}(\overline{f})}{\sqrt{1+|\nabla\overline{f}|^{2}}}\Big{)}$
$\displaystyle\leqslant$
$\displaystyle\,\frac{1-\mu}{\psi^{\prime}(\psi^{-1}(t))+(n-1)/R}\cdot\Big{(}\frac{n-1}{R}+\psi^{\prime}(\psi^{-1}(t))\Big{)}$
$\displaystyle=$ $\displaystyle\,1-\mu<\,\sqrt{1+|\nabla\overline{f}|^{2}}.$

Thus $\overline{f}$ is a supersolution of (4.6).

Consider the unique positive function $\overline{u}$ on $\\{0<x_{n}<1\\}$,
such that
$E_{t}(\overline{u})=\big{\\{}(x^{\prime},x_{n}):0<x_{n}<\overline{f}(x^{\prime},t)\big{\\}}$
for all $t>0$. Then $\overline{u}$ is a smooth supersolution of IMCF in the
region $\Omega^{\prime}:=\\{0<x_{n}<1,\overline{u}<\infty\\}$. For
$\varepsilon>0$ we wish to compare $u$ with the function
$\overline{u}+\varepsilon$. Assembling the facts: $u=0$ on $\\{x_{n}=0\\}$,
and
$\\{\overline{u}<\infty\\}\subset\big{\\{}x_{n}<1-\mu-R+\sqrt{R^{2}-|x^{\prime}|^{2}}\big{\\}}$,
and $u\in\operatorname{Lip}_{\operatorname{loc}}(\\{x_{n}<1\\})$, we conclude
that $\\{u>\overline{u}+\varepsilon\\}\subset\subset\Omega^{\prime}$. By Lemma
2.10 (3) and Theorem 2.8, we obtain $u\leqslant\overline{u}+\varepsilon$.
Taking $\varepsilon\to 0$ and then $R\to\infty$ and then $\mu\to 0$, we
eventually obtain $u\leqslant\psi(x_{n})$. ∎

##  5 Parabolic estimates near smooth obstacles

In this section, we compute several parabolic evolution equations for the
smooth IMCF near a smooth obstacle. The aim is to show the following: if
$\Sigma_{t}$ evolves under the IMCF in a smooth domain $\Omega$, such that (1)
$\nu_{\Sigma_{t}}$ is approximately equal to $\nu_{\Omega}$ on
$\partial\Omega$ (where $\nu$ denote the outer unit normal of the
corresponding objects), and (2) in some neighborhood of $\partial\Omega$ we
have $\langle\nu_{\Sigma_{t}},\partial_{r}\rangle\geqslant\frac{1}{2}$ (where
$\partial_{r}$ is an extention of $\nu_{\Omega}$, see below), then it holds
$\langle\nu_{\Sigma_{t}},\partial_{r}\rangle\geqslant 1-Cr^{\gamma}-o(1)$ in
some smaller neighborhood of $\partial\Omega$. This estimate enters the proofs
in Section 6 in showing the $C^{1,\alpha}$ regularity of level sets and
boundary regularity of blow-up limits.

The following notations are used consistently in the present and the next
sections. Suppose $\Omega$ is a smooth domain in a Riemannian manifold $M$.
Define the signed distance function

$r(x):=\left\\{\begin{aligned} &-d(x,\partial\Omega)\qquad x\in\Omega,\\\
&d(x,\partial\Omega)\qquad x\notin\Omega,\end{aligned}\right.$

For $\delta\in\mathbb{R}$, we denote $\Omega_{\delta}:=\big{\\{}x\in
M:r(x)<\delta\big{\\}}$. We use $r_{0}$ to denote a radius such that $r(x)$ is
smooth in $\Omega\setminus\overline{\Omega_{-r_{0}}}$. In Lemma
5.1$\,\sim\,$5.3, the existence of such a radius is implicitly assumed. In
$\Omega\setminus\overline{\Omega_{-r_{0}}}$ we define the outer radial vector
field $\partial_{r}:=\nabla r$.

For a hypersurface $\Sigma\subset\Omega\setminus\overline{\Omega_{-r_{0}}}$,
we use $\nu,A,H$ to denote the unit normal vector, second fundamental form and
mean curvature. We use $|\cdot|_{\Sigma},\nabla_{\Sigma},\Delta_{\Sigma}$ to
denote the (Hilbert-Schmidt) norm, gradient and Laplacian on $\Sigma_{t}$, and
use $|\cdot|,\nabla,\Delta$ for objects with respect to the ambient manifold
$M$. Also, denote

$p:=\langle\nu,\partial_{r}\rangle.$

The following algebraic fact is useful:
$|\nu-p\partial_{r}|=|\partial_{r}-p\nu|=|\nabla_{\Sigma}r|=\sqrt{1-p^{2}}$.

When there is a family of hypersurfaces $\\{\Sigma_{t}\\}$, we often omit the
dependence on $t$ in the above notations. When $\Sigma_{t}$ evolves under the
IMCF, we denote $\Box=\partial_{t}-\frac{1}{H^{2}}\Delta_{\Sigma}$ the
associated heat operator.

We start with evaluating $\Delta_{\Sigma}p$. The main formula (5.1) is
arranged such that each term vanishes when $\Sigma=\partial\Omega_{-r}$ for
some $r$, and the terms with unfavorable sign become small when $M$ and
$\partial\Omega$ are close to being flat.

######  Lemma 5.1.

Let $\Sigma$ be a smooth hypersurface in
$\Omega\setminus\overline{\Omega_{-r_{0}}}$, such that $p>0$ on $\Sigma$. Then
we have

$\displaystyle\Delta_{\Sigma}p$
$\displaystyle\leqslant\langle\nabla_{\Sigma}H,\partial_{r}\rangle-p\big{|}A-p^{-1}\nabla^{2}r\big{|}^{2}_{\Sigma}$
(5.1)
$\displaystyle\qquad+|\nabla^{2}r|^{2}p^{-1}(1-p^{2})+\big{(}|\operatorname{Ric}|+n|\nabla^{2}\partial_{r}|\big{)}\sqrt{1-p^{2}}.$

(In the second term, the expression $\nabla^{2}r$ means the restriction of
$\nabla^{2}_{M}r$ to $\Sigma$.)

###### Proof.

Near a given point $x\in\Sigma$, we let $\\{e_{i}\\}_{1\leqslant i\leqslant
n-1}$ be a local orthonormal frame on $\Sigma$, such that
$\nabla_{e_{i}}^{\Sigma}e_{j}=0$ at $x$. We compute

$\displaystyle\Delta_{\Sigma}p$
$\displaystyle=\sum_{i}e_{i}e_{i}\langle\nu,\partial_{r}\rangle=\sum_{i}e_{i}\Big{[}\langle
A(e_{i}),\partial_{r}\rangle+\langle\nu,\nabla_{e_{i}}\partial_{r}\rangle\Big{]}$
$\displaystyle=\sum_{i}\Big{[}\langle\nabla_{e_{i}}(A(e_{i})),\partial_{r}\rangle+2\langle
A(e_{i}),\nabla_{e_{i}}\partial_{r}\rangle+\langle\nu,\nabla_{e_{i}}\nabla_{e_{i}}\partial_{r}\rangle\Big{]}.$
(5.2)

Here $A(e_{i})$ is understood as a tangent vector field of $\Sigma$. First
observe that

$2\sum_{i}\langle
A(e_{i}),\nabla_{e_{i}}\partial_{r}\rangle=2\sum_{i}\nabla^{2}r(A(e_{i}),e_{i})=2\langle
A,\nabla^{2}r\rangle_{\Sigma}.$ (5.3)

Next, the first term in (5.2) is calculated as

$\displaystyle\sum_{i}\langle\nabla_{e_{i}}A(e_{i}),\partial_{r}\rangle$
$\displaystyle=\sum_{i}\Big{\langle}(\nabla_{e_{i}}^{\Sigma}A)(e_{i})+A(\nabla_{e_{i}}^{\Sigma}e_{i})-A(e_{i},A(e_{i}))\nu,\,\partial_{r}\Big{\rangle}$
$\displaystyle=\langle\operatorname{div}_{\Sigma}A-|A|^{2}_{\Sigma}\,\nu,\partial_{r}\rangle$
$\displaystyle=\langle\operatorname{div}_{\Sigma}A,\partial_{r}-p\nu\rangle-p|A|^{2}$
$\displaystyle\leqslant\langle\nabla_{\Sigma}H,\partial_{r}\rangle-p|A|^{2}_{\Sigma}+|\operatorname{Ric}|\sqrt{1-p^{2}},$
(5.4)

where we use the traced Codazzi equation and note that
$\operatorname{div}_{\Sigma}A$, $\nabla_{\Sigma}H$ are tangent to $\Sigma$. It
remains to simplify the last term in (5.2). We calculate

$\displaystyle\sum_{i}\langle\nu,\nabla_{e_{i}}\nabla_{e_{i}}\partial_{r}\rangle$
$\displaystyle=\sum_{i}\Big{[}\langle\nu-p\partial_{r},\nabla_{e_{i}}\nabla_{e_{i}}\partial_{r}\rangle+p\langle\partial_{r},\nabla_{e_{i}}\nabla_{e_{i}}\partial_{r}\rangle\Big{]}$
$\displaystyle\leqslant
n|\nabla^{2}\partial_{r}|\sqrt{1-p^{2}}-p\sum_{i}\big{|}\nabla_{e_{i}}\partial_{r}\big{|}^{2},$
(5.5)

Continuing to evaluate the second term in (5.5):

$\sum_{i}\big{|}\nabla_{e_{i}}\partial_{r}\big{|}^{2}=\sum_{ij}\langle\nabla_{e_{i}}\partial_{r},e_{j}\rangle^{2}+\sum_{i}\langle\nabla_{e_{i}}\partial_{r},\nu\rangle^{2}\geqslant\sum_{ij}\langle\nabla_{e_{i}}\partial_{r},e_{j}\rangle^{2}=|\nabla^{2}r|^{2}_{\Sigma}.$
(5.6)

Inserting (5.3)$\,\sim\,$(5.6) into (5.2), we obtain

$\displaystyle\Delta_{\Sigma}p$
$\displaystyle\leqslant\langle\nabla_{\Sigma}H,\partial_{r}\rangle-p|A|^{2}_{\Sigma}+2\langle
A,\nabla^{2}r\rangle_{\Sigma}-p|\nabla^{2}r|^{2}_{\Sigma}$
$\displaystyle\hskip
144.0pt+|\operatorname{Ric}|\sqrt{1-p^{2}}+n|\nabla^{2}\partial_{r}|\sqrt{1-p^{2}}.$

This implies (5.1) by completing the square and noting that
$|\nabla^{2}r|_{\Sigma}\leqslant|\nabla^{2}r|$. ∎

######  Lemma 5.2.

Suppose $\\{\Sigma_{t}\\}$ evolves under the IMCF in
$\Omega\setminus\overline{\Omega_{-r_{0}}}$, such that $p>0$ holds everywhere.
Then we have

$\displaystyle\Box p$
$\displaystyle\geqslant\frac{|A-p^{-1}\nabla^{2}r|^{2}_{\Sigma}}{2H^{2}}p-2n\frac{|\nabla^{2}r|^{2}}{H^{2}p}(1-p^{2})-\frac{|\operatorname{Ric}|+n|\nabla^{2}\partial_{r}|}{H^{2}}\sqrt{1-p^{2}}.$
(5.7)

###### Proof.

We compute

$\displaystyle\frac{\partial p}{\partial t}$
$\displaystyle=\partial_{t}\langle\nu,\partial_{r}\rangle=\langle
H^{-2}\nabla_{\Sigma}H,\partial_{r}\rangle+\langle\nu,\nabla_{H^{-1}\nu}\partial_{r}\rangle.$

For the second term, we use the fact $\nabla^{2}r(\partial_{r},X)=0$ for all
$X$ to evaluate

$\displaystyle\langle\nu,\nabla_{H^{-1}\nu}\partial_{r}\rangle=\frac{1}{H}\nabla^{2}r(\nu,\nu)=\frac{1}{H}\nabla^{2}r(\nu-p\partial_{r},\nu-p\partial_{r})\geqslant-\frac{1}{H}|\nabla^{2}r|(1-p^{2}).$

Combined with (5.1), we obtain

$\displaystyle\Box p$
$\displaystyle\geqslant\frac{|A-p^{-1}\nabla^{2}r|_{\Sigma}^{2}}{H^{2}}p-\frac{1}{H}|\nabla^{2}r|(1-p^{2})$
(5.8)
$\displaystyle\qquad\qquad\qquad-\frac{|\nabla^{2}r|^{2}}{H^{2}p}(1-p^{2})-\frac{|\operatorname{Ric}|+n|\nabla^{2}\partial_{r}|}{H^{2}}\sqrt{1-p^{2}}.$

We bound the second term in this expression as follows:

$\displaystyle\frac{1}{H}|\nabla^{2}r|(1-p^{2})$
$\displaystyle=\frac{H-p^{-1}\operatorname{tr}_{\Sigma}\nabla^{2}r+p^{-1}\operatorname{tr}_{\Sigma}\nabla^{2}r}{H^{2}}|\nabla^{2}r|(1-p^{2})$
$\displaystyle\leqslant\sqrt{n-1}\frac{|A-p^{-1}\nabla^{2}r|_{\Sigma}+p^{-1}|\nabla^{2}r|_{\Sigma}}{H^{2}}|\nabla^{2}r|(1-p^{2}).$

Using Young’s inequality, we continue the estimate:

$\displaystyle\frac{1}{H}|\nabla^{2}r|(1-p^{2})$
$\displaystyle\leqslant\Big{[}\frac{|A-p^{-1}\nabla^{2}r|_{\Sigma}^{2}}{2H^{2}}p+\frac{n}{2H^{2}p}|\nabla^{2}r|^{2}(1-p^{2})^{2}\Big{]}+n\frac{|\nabla^{2}r|^{2}}{H^{2}p}(1-p^{2}).$

The result follows by combining this into (5.8) and noting that $0<p\leqslant
1$. ∎

######  Lemma 5.3.

Assume the same conditions as in Lemma 5.2. Let $\eta=\eta(r)$ be a smooth
radial function, and denote $\eta^{\prime}=d\eta/dr$,
$\eta^{\prime\prime}=d^{2}\eta/dr^{2}$. Then we have

$\displaystyle\Box\eta$
$\displaystyle\leqslant\frac{2p\eta^{\prime}}{H^{2}}\big{(}H-p^{-1}\operatorname{tr}_{\Sigma}\nabla^{2}r\big{)}-\frac{\eta^{\prime\prime}}{H^{2}}(1-p^{2})+\frac{n|\eta^{\prime}|}{H^{2}}|\nabla^{2}r|.$
(5.9)

and

$\Box\eta\leqslant\frac{2p\eta^{\prime}}{H}-\frac{\eta^{\prime\prime}}{H^{2}}(1-p^{2})+\frac{n|\eta^{\prime}|}{H^{2}}|\nabla^{2}r|.$
(5.10)

###### Proof.

This is obtained by combining

$\frac{\partial\eta}{\partial
t}=\frac{\langle\nabla\eta,\nu\rangle}{H}=\frac{p\eta^{\prime}}{H}=\frac{p\eta^{\prime}}{H^{2}}\big{(}H-p^{-1}\operatorname{tr}_{\Sigma}\nabla^{2}r+p^{-1}\operatorname{tr}_{\Sigma}\nabla^{2}r\big{)}$

with

$\displaystyle\Delta_{\Sigma}\eta$
$\displaystyle=\eta^{\prime}\Delta_{\Sigma}r+\eta^{\prime\prime}|\nabla_{\Sigma}r|^{2}=\eta^{\prime}(\operatorname{tr}_{\Sigma}\nabla^{2}r-pH)+\eta^{\prime\prime}(1-p^{2}),$

and noting that
$\operatorname{tr}_{\Sigma}\nabla^{2}r\leqslant\sqrt{n-1}|\nabla^{2}r|_{\Sigma}\leqslant
n|\nabla^{2}r|$. ∎

######  Lemma 5.4.

Suppose $\Omega$ is a smooth domain, and $r_{0}>0$ is a radius such that
$r(x)$ is smooth in $\Omega\setminus\overline{\Omega_{-r_{0}}}$. Assume that

$|\operatorname{Ric}|\leqslant r_{0}^{-2},\qquad|\nabla^{2}r|\leqslant
r_{0}^{-1},\qquad|\nabla^{2}\partial_{r}|\leqslant r_{0}^{-2}$ (5.11)

hold inside $\Omega\setminus\overline{\Omega_{-r_{0}}}$. Let
$\big{\\{}\Sigma_{t}\subset\Omega\setminus\overline{\Omega_{-r_{0}}}\big{\\}}$
be a smooth family of hypersurfaces evolving under the IMCF. Assume
additionally that $p\geqslant\frac{1}{2}$ on all $\Sigma_{t}$. Then there
exist constants $\gamma\in(0,1/2)$ and $C_{1},C_{2}>0$ depending only on $n$,
such that the following holds: if we set

$\eta(r)=\big{(}b-r/r_{0}\big{)}^{-\gamma},\qquad F=(1-p)\eta,$ (5.12)

with any choice $b\in(0,1]$, then we have the evolution inequality

$\displaystyle(1-p)^{\frac{2}{\gamma}-1}H^{2}\Box
F+\langle\nabla_{\Sigma}F,X\rangle\leqslant\frac{C_{1}}{r_{0}^{2}}\big{(}-F^{\frac{\gamma+2}{\gamma}}+C_{2}\,\big{)}$
(5.13)

on $\Sigma_{t}$, where $X$ is a vertain smooth vector field.

###### Proof.

Note that $r<0$ inside $\Omega$, hence $\eta>0$ and increasing when
approaching $\partial\Omega$. Combining Lemma 5.2 and equation (5.9) in Lemma
5.3, then using (5.11) with $\frac{1}{2}\leqslant p\leqslant 1$ to simplify
the resulting expressions, we have

$\displaystyle\Box F$ $\displaystyle=-\eta\Box
p+(1-p)\Box\eta-\frac{2}{H^{2}}\langle\nabla_{\Sigma}(1-p),\nabla_{\Sigma}\eta\rangle$
$\displaystyle\leqslant-\frac{|A-p^{-1}\nabla^{2}r|^{2}_{\Sigma}}{2H^{2}}\eta
p+2n\frac{|\nabla^{2}r|^{2}}{H^{2}p}\eta(1-p^{2})+\frac{|\operatorname{Ric}|+n|\nabla^{2}\partial_{r}|}{H^{2}}\eta\sqrt{1-p^{2}}$
$\displaystyle\qquad+\frac{2p\eta^{\prime}}{H^{2}}(1-p)\big{(}H-p^{-1}\operatorname{tr}_{\Sigma}\nabla^{2}r\big{)}-\frac{\eta^{\prime\prime}}{H^{2}}(1-p)(1-p^{2})+n\frac{|\eta^{\prime}|}{H^{2}}|\nabla^{2}r|(1-p)$
$\displaystyle\qquad-\frac{2}{H^{2}\eta}\langle\nabla_{\Sigma}F,\nabla_{\Sigma}\eta\rangle+\frac{2}{H^{2}\eta}(1-p)|\nabla_{\Sigma}\eta|^{2}$
$\displaystyle\leqslant-\frac{|A-p^{-1}\nabla^{2}r|^{2}_{\Sigma}}{2H^{2}}\eta
p+\frac{8n}{H^{2}r_{0}^{2}}\eta(1-p)+\frac{4n\eta}{H^{2}r_{0}^{2}}\sqrt{1-p}$
$\displaystyle\qquad+\frac{2p\eta^{\prime}}{H^{2}}(1-p)\big{(}H-p^{-1}\operatorname{tr}_{\Sigma}\nabla^{2}r\big{)}-\frac{\eta^{\prime\prime}}{H^{2}}(1-p)^{2}(1+p)+\frac{n|\eta^{\prime}|}{H^{2}r_{0}}(1-p)$
$\displaystyle\qquad-\frac{2}{H^{2}\eta}\langle\nabla_{\Sigma}F,\nabla_{\Sigma}\eta\rangle+\frac{2}{H^{2}}\frac{(\eta^{\prime})^{2}}{\eta}(1-p)^{2}(1+p).$

We use Young’s inequality to estimate the fourth term:

$\displaystyle\frac{2p\eta^{\prime}}{H^{2}}(1-p)\big{(}H-p^{-1}\operatorname{tr}_{\Sigma}\nabla^{2}r\big{)}$
$\displaystyle\leqslant\frac{(H-p^{-1}\operatorname{tr}_{\Sigma}\nabla^{2}r)^{2}}{2(n-1)H^{2}}\eta
p+8(n-1)\frac{(\eta^{\prime})^{2}}{H^{2}\eta}p(1-p)^{2}$
$\displaystyle\leqslant\frac{|A-p^{-1}\nabla^{2}r|^{2}_{\Sigma}}{2H^{2}}\eta
p+8n\frac{(\eta^{\prime})^{2}}{H^{2}\eta}p(1-p)^{2}.$

Further, we calculate

$\eta^{\prime}=\frac{\gamma}{r_{0}}\eta^{\frac{\gamma+1}{\gamma}},\qquad\eta^{\prime\prime}=\frac{\gamma(\gamma+1)}{r_{0}^{2}}\eta^{\frac{\gamma+2}{\gamma}}.$

Inserting these to the main estimate, we obtain

$\displaystyle H^{2}\Box F$
$\displaystyle\leqslant\frac{8n}{r_{0}^{2}}\eta(1-p)+\frac{4n\eta}{r_{0}^{2}}\sqrt{1-p}+8n\frac{\gamma^{2}}{r_{0}^{2}}\eta^{\frac{\gamma+2}{\gamma}}p(1-p)^{2}$
$\displaystyle\qquad-\frac{\gamma(\gamma+1)}{r_{0}^{2}}\eta^{\frac{\gamma+2}{\gamma}}(1-p)^{2}(1+p)+\frac{n\gamma}{r_{0}^{2}}\eta^{\frac{\gamma+1}{\gamma}}(1-p)$
$\displaystyle\qquad-\frac{2}{\eta}\langle\nabla_{\Sigma}F,\nabla_{\Sigma}\eta\rangle+\frac{2\gamma^{2}}{r_{0}^{2}}\eta^{\frac{\gamma+2}{\gamma}}(1-p)^{2}(1+p).$

Multiplying both sides by $(1-p)^{\frac{2}{\gamma}-1}$, and using the facts
$0<p\leqslant 1$, $1-p\leqslant\sqrt{1-p}$, we obtain

$\displaystyle(1-p)^{\frac{2}{\gamma}-1}H^{2}\Box F$
$\displaystyle\leqslant\frac{12n}{r_{0}^{2}}\eta(1-p)^{\frac{2}{\gamma}-\frac{1}{2}}+\frac{n\gamma}{r_{0}^{2}}\eta^{\frac{\gamma+1}{\gamma}}(1-p)^{\frac{2}{\gamma}}-\langle\nabla_{\Sigma}F,X\rangle$
(5.14)
$\displaystyle\qquad+\frac{1}{r_{0}^{2}}\Big{[}8n\gamma^{2}+4\gamma^{2}-\gamma(\gamma+1)\Big{]}\eta^{\frac{\gamma+2}{\gamma}}(1-p)^{\frac{\gamma+2}{\gamma}},$

where $X$ is a certain smooth vector field. Choosing $\gamma$ sufficiently
small (depending only on $n$), we can realize

$8n\gamma^{2}+4\gamma^{2}-\gamma(\gamma+1)<-3\gamma^{2}.$ (5.15)

Using Hölder’s inequality, we have

$12n\eta(1-p)^{\frac{2}{\gamma}-\frac{1}{2}}\leqslant\gamma^{2}\eta^{\frac{\gamma+2}{\gamma}}(1-p)^{\frac{\gamma+2}{\gamma}}+C(n)\eta^{-\frac{(4-3\gamma)(2+\gamma)}{3\gamma^{2}}}$
(5.16)

and

$n\gamma\eta^{\frac{\gamma+1}{\gamma}}(1-p)^{\frac{2}{\gamma}}\leqslant\gamma^{2}\eta^{\frac{\gamma+2}{\gamma}}(1-p)^{\frac{\gamma+2}{\gamma}}+C(n)\eta^{-\frac{(1-\gamma)(2+\gamma)}{\gamma^{2}}}$
(5.17)

Notice that $\eta\geqslant 2^{-\gamma}$ in
$\Omega\setminus\overline{\Omega_{-r_{0}}}$, which bounds the last terms
uniformly by constants. Inserting (5.15)$\,\sim\,$(5.17) into (5.14), we
finally obtain

$(1-p)^{\frac{2}{\gamma}-1}H^{2}\Box
F+\langle\nabla_{\Sigma}F,X\rangle\leqslant\frac{1}{r_{0}^{2}}\Big{(}-\gamma^{2}F^{\frac{\gamma+2}{\gamma}}+C(n)\Big{)}.\qed$

######  Lemma 5.5.

Suppose $b,\gamma\in(0,1/2)$ are constants, $\Omega$ is a smooth domain, and
$r_{0}>0$ is a radius such that $r(x)$ is smooth in
$\Omega\setminus\overline{\Omega_{-r_{0}}}$. Assume that

$|\operatorname{Ric}|\leqslant r_{0}^{-2},\qquad|\nabla^{2}r|\leqslant
r_{0}^{-1},\qquad|\nabla^{2}\partial_{r}|\leqslant r_{0}^{-2}$ (5.18)

holds inside $\Omega\setminus\Omega_{-r_{0}}$. Suppose
$\big{\\{}\Sigma_{t}\subset\Omega\setminus\overline{\Omega_{-r_{0}}}\big{\\}}$
evolves by the IMCF. Additionally, assume $p\geqslant\frac{1}{2}$ and
$1-p\leqslant C_{3}(b-r/r_{0})^{\gamma}$ on each $\Sigma_{t}$. Set

$\eta(r)=\big{(}b-r/r_{0}\big{)}^{1-\gamma}-(2b)^{1-\gamma},\qquad
G=r_{0}H\eta(r).$ (5.19)

Then in $\Omega_{-br_{0}}\setminus\overline{\Omega_{-r_{0}}}$ we have the
evolution inequality

$\displaystyle\Box G+\langle\nabla_{\Sigma}G,X\rangle$
$\displaystyle\leqslant-\frac{G}{n}+\frac{4n}{G}+C_{4}\big{(}b-r/r_{0}\big{)}^{-\gamma}\Big{(}\frac{C_{5}}{G}-1\Big{)},$
(5.20)

where $X$ is a certain smooth vector field, and the constants $C_{4},C_{5}$
depend on $n,C_{3}$.

###### Proof.

Recall the evolution equation of the mean curvature:

$\Box
H=-\frac{2}{H^{3}}|\nabla_{\Sigma}H|^{2}-\Big{(}|A|^{2}+\operatorname{Ric}(\nu,\nu)\Big{)}\frac{1}{H}.$

Combining (5.10) and (5.18) we have

$\displaystyle r_{0}^{-1}\Box G$ $\displaystyle=\eta\Box
H+H\Box\eta-\frac{2}{H^{2}}\langle\nabla_{\Sigma}H,\nabla_{\Sigma}\eta\rangle$
$\displaystyle\leqslant-\frac{2\eta}{H^{3}}|\nabla_{\Sigma}H|^{2}-\frac{\eta
H}{n-1}+\frac{\eta}{r_{0}^{2}H}+2p\eta^{\prime}-\frac{\eta^{\prime\prime}}{H}(1-p^{2})+\frac{n|\eta^{\prime}|}{r_{0}H}$
$\displaystyle\qquad-\frac{2}{H^{3}}\langle\nabla_{\Sigma}H,\nabla_{\Sigma}(H\eta)-\eta\nabla_{\Sigma}H\rangle$
$\displaystyle=-\frac{G}{(n-1)r_{0}}+\frac{\eta^{2}}{r_{0}G}+2p\eta^{\prime}-r_{0}\frac{\eta\eta^{\prime\prime}}{G}(1-p^{2})+\frac{n\eta|\eta^{\prime}|}{G}-\langle\nabla_{\Sigma}G,X\rangle.$

Note that

$\eta^{\prime}=-(1-\gamma)r_{0}^{-1}(b-r/r_{0})^{-\gamma},\qquad\eta^{\prime\prime}=-\gamma(1-\gamma)r_{0}^{-2}(b-r/r_{0})^{-1-\gamma}.$

These imply that $|\eta\eta^{\prime}|\leqslant 2r_{0}^{-1}$ in
$\Omega_{-br_{0}}\setminus\overline{\Omega_{-r_{0}}}$, since $\gamma<1/2$. In
addition, we have $\eta\leqslant 2$ and $1-p^{2}\leqslant 2(1-p)$. Hence

$\Box
G\leqslant-\frac{G}{n}+\frac{4+2n}{G}-2p(1-\gamma)(b-r/r_{0})^{-\gamma}+2\gamma(1-\gamma)\frac{1-p}{G}(b-r/r_{0})^{-2\gamma}.$

Inserting our assumption $1-p\leqslant C_{3}(b-r/r_{0})^{\gamma}$, we obtain

$\Box
G\leqslant-\frac{G}{n}+\frac{4n}{G}+(b-r/r_{0})^{-\gamma}\Big{[}-2p(1-\gamma)+\frac{C}{G}\Big{]}-\langle\nabla_{\Sigma}G,X\rangle.$

Since
$p\geqslant\frac{1}{2},\gamma\leqslant\frac{1}{2}\,\Rightarrow-2p(1-\gamma)\leqslant-\frac{1}{2}$,
the result follows. ∎

##  6 Initial value problems in smooth domains

In this section we prove the main existence and regularity theorem, which is
stated below. In Subsection 6.1, we set up some definitions and preliminary
estimates, then we define the approximation scheme mentioned in the
introduction, which gives a candidate solution $u$ (it is temporarily only an
interior solution). In the end of Subsection 6.1 we state the remining tasks
needed to conclude that $u$ satisfies Theorem 6.1. Then in Subsection 6.2 we
make an outline of the blow up strategy, and finally prove these results in
Subsection 6.3.

######  Theorem 6.1.

Let $\Omega\subset M$ be a precompact domain with smooth boundary, and
$E_{0}\subset\subset\Omega$ be a $C^{1,1}$ domain. Then there exists a
solution $u$ of
$\mathsf{IVP}\boldsymbol{(}{\Omega;E_{0}}\boldsymbol{)}\texttt{\emph{+}}\mathsf{OBS}\boldsymbol{(}{\partial\Omega}\boldsymbol{)}$,
unique up to equivalence. There exists $\gamma\in(0,1)$ depending on $n$, such
that the following holds.

(i) We have
$u\in\operatorname{Lip}_{\operatorname{loc}}(\Omega)\cap\operatorname{BV}(\Omega)\cap
C^{0,\gamma}(\overline{\Omega})$. More precisely, it holds

$|\nabla u(x)|\leqslant\sup_{\partial E_{0}\cap
B(x,r)}H_{+}+\frac{C(n)}{r},\qquad x\in\Omega\setminus E_{0},\ \
r\leqslant\sigma(x;\Omega,g),$ (6.1)

where $\sigma(x;\Omega,g)$ is as in Definition 2.14, and

$|\nabla u(x)|\leqslant Cd(x,\partial\Omega)^{\gamma-1},\qquad
x\in\Omega\setminus\overline{E_{0}},$ (6.2)

for some constant $C>0$.

(ii) The solution $u$ is calibrated in $\Omega\setminus\overline{E_{0}}$ by a
vector field $\nu$, which satisfies

$\langle\nu,\partial_{r}\rangle(x)\geqslant 1-Cd(x,\partial\Omega)^{\gamma}$
(6.3)

in some small neighborhood of $\partial\Omega$. Here $C>0$ is a constant, and
$\partial_{r}:=-\nabla d(\cdot,\partial\Omega)$ is the outpointing unit vector
perpendicular to $\partial\Omega$.

(iii) Each level set $\partial E_{t}$ is a $C^{1,\gamma/2}$ hypersurface in
some small neighborhood of $\partial\Omega$.

(iv) If $v\in\operatorname{Lip}_{\operatorname{loc}}(\Omega)$ is another
solution of $\mathsf{IVP}\boldsymbol{(}{\Omega;E_{0}}\boldsymbol{)}$, then
$u\geqslant v$ in $\Omega\setminus E_{0}$.

###  6.1 Setups, notations, and the approximating scheme

We make the following setups and constructions.

The signed distance $r(x)$ and regular radius $r_{I}$.

Fix $\Omega$, $E_{0}$ as in Theorem 6.1. Let $r(x)$ be the signed distance
function to $\partial\Omega$, taking negative values in $\Omega$ and positive
values in $M\setminus\Omega$. For $\delta\in\mathbb{R}$, we set
$\Omega_{\delta}=\big{\\{}x\in M:r(x)<\delta\big{\\}}$. Let $r_{I}$ be a
sufficiently small radius, such that $d(\partial
E_{0},\partial\Omega)>3r_{I}$, and $r(x)$ is smooth in
$\Omega_{3r_{I}}\setminus\overline{\Omega_{-3r_{I}}}$. Define the radial
vector field $\partial_{r}:=\nabla r$, which is smooth in the same region. We
further decrease $r_{I}$ to achieve the following: in
$\Omega_{3r_{I}}\setminus\overline{\Omega_{-3r_{I}}}$ we have

$|\operatorname{Ric}|\leqslant\frac{1}{100n^{2}r_{I}^{2}},\qquad|\nabla\partial_{r}|\leqslant\frac{1}{100n^{2}r_{I}},\qquad|\nabla^{2}\partial_{r}|\leqslant\frac{1}{100n^{2}r_{I}^{2}},$
(6.4)

and also for all $x\in\Omega\setminus\Omega_{-r_{I}}$, $r,s\leqslant r_{I}$ we
have $\mathcal{H}^{n-1}\big{(}B(x,r)\cap\partial\Omega_{-s}\big{)}\leqslant
2|B^{n-1}|r^{n-1}$, and finally for all $x\in\Omega\setminus\Omega_{-r_{I}}$
we have $\sigma(x;\Omega,g)\geqslant\frac{1}{2}|r(x)|$ (see Definition 2.14).

The weight functions $\psi_{\delta}$.

We fix a function $\psi_{0}:(-\infty,1)\to\mathbb{R}_{+}$ such that:

(1) the conditions of Theorem 4.3 are satisfied (in particular, $\psi_{0}$ is
strictly increasing and convex in $(0,1)$, and $\psi_{0}|_{[-\infty,0]}\equiv
0$, $\lim_{x\to 1}\psi_{0}(x)=+\infty$),

(2) $\psi_{0}(\frac{1}{2})>\frac{1}{2}$, and $\psi^{\prime}_{0}(x)>1$ for all
$x\in[\frac{1}{2},1)$, and $\psi_{0}(x)>\psi_{0}(\frac{1}{2})+3$ for all
$x\in[\frac{3}{4},1)$.

(3) $\psi_{0}(x)\geqslant(n-1)\log\frac{1}{1-x}$ for all
$x\in[\frac{1}{2},1)$.

For each $\delta\in(0,r_{I})$, we define the (smooth) weight function

$\psi_{\delta}(x)=\left\\{\begin{aligned} &0,\qquad x\in\Omega,\\\
&\psi_{0}\big{(}\delta^{-1}r(x)\big{)},\qquad
x\in\Omega_{\delta}\setminus\Omega.\end{aligned}\right.$

Note that $\lim_{x\to\partial\Omega_{\delta}}\psi_{\delta}(x)=+\infty$.

The approximating equations.

For $\varepsilon>0$, we consider the weighted IMCF

$\operatorname{div}\Big{(}e^{\psi_{\delta}}\frac{\nabla u}{|\nabla
u|}\Big{)}=e^{\psi_{\delta}}|\nabla u|,$ (6.5)

and its elliptic regularization (see Remark 2.20):

$\operatorname{div}\Big{(}e^{\psi_{\delta}}\frac{\nabla
u}{\sqrt{\varepsilon^{2}e^{2\psi_{\delta}/(n-1)}+|\nabla
u|^{2}}}\Big{)}=e^{\psi_{\delta}}\sqrt{\varepsilon^{2}e^{2\psi_{\delta}/(n-1)}+|\nabla
u|^{2}}.$ (6.6)

######  Lemma 6.2 (barrier functions).

Suppose $\delta\in(0,r_{I})$. Then the function

${\underline{u}_{\delta}}(x)=\psi_{\delta}(x)-\psi_{0}\big{(}\frac{1}{2}\big{)}-\Big{(}\frac{r(x)}{\delta}-\frac{1}{2}\Big{)}$
(6.7)

is negative in $\Omega_{\delta/2}\setminus\Omega$ and positive in
$\Omega\setminus\overline{\Omega_{\delta/2}}$. Moreover,
${\underline{u}_{\delta}}$ is a strict subsolution of (6.5) with nonvanishing
gradient in $\Omega_{\delta}\setminus\Omega_{\delta/2}$. The function

${\overline{u}_{\delta}}(x)=\psi_{\delta}(x)+\frac{r(x)}{r_{I}}$ (6.8)

is a strict supersolution of (6.5) in the region
$\Omega_{\delta}\setminus\Omega_{-r_{I}}$.

###### Proof.

The claim on the sign of ${\underline{u}_{\delta}}$ follows from
$\psi_{0}(1/2)>1/2$ and the strict convexity of $\psi_{0}$. In
$\Omega_{\delta}\setminus\Omega_{\delta/2}$ we have

$|\nabla{\underline{u}_{\delta}}|=\Big{|}\frac{\partial\psi_{\delta}}{\partial
r}-\frac{1}{\delta}\Big{|}=\frac{1}{\delta}\big{|}\psi^{\prime}_{0}(r(x))-1\big{|}=\frac{1}{\delta}\Big{(}\psi^{\prime}_{0}(\delta^{-1}r(x))-1\Big{)}=\frac{\partial\psi_{\delta}}{\partial
r}-\frac{1}{\delta}.$

The third equality is because $\psi^{\prime}_{0}>1$ on $[1/2,1)$. The same
facts imply that $|\nabla{\underline{u}_{\delta}}|\neq 0$ everywhere in
$\Omega_{\delta}\setminus\Omega_{\delta/2}$. On the other hand, we calculate

$\displaystyle\operatorname{div}\Big{(}e^{\psi_{\delta}}\frac{\nabla{\underline{u}_{\delta}}}{|\nabla{\underline{u}_{\delta}}|}\Big{)}$
$\displaystyle=\operatorname{div}\big{(}e^{\psi_{\delta}}\partial_{r}\big{)}=e^{\psi_{\delta}}\Big{(}\frac{\partial\psi_{\delta}}{\partial
r}+\operatorname{div}\partial_{r}\Big{)}>e^{\psi_{\delta}}\Big{(}\frac{\partial\psi_{\delta}}{\partial
r}-\frac{1}{100r_{I}}\Big{)},$

where the last inequality comes from (6.4). Hence ${\underline{u}_{\delta}}$
is a strict subsolution on $\Omega_{\delta}\setminus\Omega_{\delta/2}$.

Next, inside $\Omega_{\delta}\setminus\Omega_{-r_{I}}$ we calculate

$\operatorname{div}\Big{(}e^{\psi_{\delta}}\frac{\nabla{\overline{u}_{\delta}}}{|\nabla{\overline{u}_{\delta}}|}\Big{)}\leqslant
e^{\psi_{\delta}}\Big{(}\frac{\partial\psi_{\delta}}{\partial
r}+\frac{1}{100r_{I}}\Big{)}<e^{\psi_{\delta}}\Big{(}\frac{\partial\psi_{\delta}}{\partial
r}+\frac{1}{r_{I}}\Big{)}=|\nabla{\overline{u}_{\delta}}|,$

confirming the supersolution property. ∎

######  Lemma 6.3 (approximating solutions).

For each $\delta<r_{I}$ and $\lambda\in(3/4,1)$, there exists
$\varepsilon(\delta,\lambda)>0$ such that the boundary value problem

$\displaystyle[left=\empheqlbrace]$
$\displaystyle\operatorname{div}\Big{(}e^{\psi_{\delta}}\frac{\nabla
u_{\varepsilon,\delta,\lambda}}{\sqrt{\varepsilon^{2}e^{2\psi_{\delta}/(n-1)}+|\nabla
u_{\varepsilon,\delta,\lambda}|^{2}}}\Big{)}=e^{\psi_{\delta}}\sqrt{\varepsilon^{2}e^{2\psi_{\delta}/(n-1)}+|\nabla
u_{\varepsilon,\delta,\lambda}|^{2}}$ (6.9) $\displaystyle\hskip
320.0pt\text{in $\Omega_{\lambda\delta}\setminus\overline{E_{0}}$},$
$\displaystyle u_{\varepsilon,\delta,\lambda}=0\qquad\text{on $\partial
E_{0}$},$ (6.10) $\displaystyle
u_{\varepsilon,\delta,\lambda}={\underline{u}_{\delta}}-2\qquad\text{on
$\partial\Omega_{\lambda\delta}$}$ (6.11)

admits a solution $u_{\varepsilon,\delta,\lambda}\in
C^{\infty}\big{(}\overline{\Omega_{\lambda\delta}}\setminus E_{0}\big{)}$ for
all $0<\varepsilon\leqslant\varepsilon(\delta,\lambda)$. We have the $C^{0}$
bounds

$\max\big{\\{}\\!-\varepsilon,\psi_{\delta}(x)-C\big{\\}}\leqslant
u_{\varepsilon,\delta,\lambda}(x)\leqslant\psi_{\delta}(x)+C\qquad\forall
x\in\Omega_{\lambda\delta}\setminus E_{0},$ (6.12)

for some constant $C>0$ independent of $\varepsilon,\delta,\lambda$. In
particular, $-\varepsilon\leqslant u_{\varepsilon,\delta,\lambda}\leqslant C$
in $\overline{\Omega}\setminus E_{0}$.

We also have the gradient estimate

$|\nabla u_{\varepsilon,\delta,\lambda}(x)|\leqslant\sup_{\partial E_{0}\cap
B(x,r)}H_{+}+2\varepsilon+\frac{C(n)}{r}$ (6.13)

for all $x\in\Omega\setminus E_{0}$ and $0<r\leqslant\sigma(x;\Omega,g)$,
where $H_{+}=\max\big{\\{}H_{\partial E_{0}},0\big{\\}}$.

###### Proof.

In $\Omega_{\delta}$ we consider the conformally transformed metric
$g^{\prime}=e^{2\psi_{\delta}/(n-1)}g$. The fact
$\psi_{0}\geqslant(n-1)\log\frac{1}{1-x}$ in $[1/2,1)$ ensures that
$g^{\prime}$ is a complete metric in $\Omega_{\delta}$. Let
${\underline{u}_{\delta}}$ be as in (6.7). For convenience, we modify
${\underline{u}_{\delta}}$ inside $\Omega$, so that it is smooth with negative
value there (this does not affect any argument below). Then
${\underline{u}_{\delta}}$ is smooth and proper in $\Omega_{\delta}$, with
$\\{{\underline{u}_{\delta}}<0\\}=\Omega_{\delta/2}$. By Lemma 6.2 and Lemma
2.10(3), ${\underline{u}_{\delta}}$ is a smooth subsolution of IMCF in the
region
$\big{(}\Omega_{\delta}\setminus\overline{\Omega_{\delta/2}},g^{\prime}\big{)}$,
with nonvanishing gradient there. Finally, note that $\Omega_{\lambda\delta}$
is a sub-level set of ${\underline{u}_{\delta}}$ (namely,
$\Omega_{\lambda\delta}=\\{{\underline{u}_{\delta}}<L\\}$ for some $L>2$).
Thus we may invoke Theorem 2.17 on $(\Omega_{\delta},g^{\prime})$ to obtain
that: there is $\varepsilon(\delta,\lambda)>0$ such that the regularized
equation (2.18) $\sim$ (2.20) admits a solution for all
$\varepsilon\leqslant\varepsilon(\delta,\lambda)$. But under the present
settings, the regularized equation is exactly (6.9) $\sim$ (6.11), by Remark
2.20. This shows the existence of the solution
$u_{\varepsilon,\delta,\lambda}$.

To obtain (6.13), we note that the gradient estimate in Theorem 2.17 states
that

$\big{|}\nabla_{g^{\prime}}u_{\varepsilon,\delta,\lambda}(x)\big{|}\leqslant\max\Big{\\{}\sup_{B_{g^{\prime}}(x,r)\cap\partial
E_{0}}H_{+},\
\sup_{B_{g^{\prime}}(x,r)\cap\partial\Omega_{\lambda\delta}}|\nabla_{g^{\prime}}u_{\varepsilon,\delta,\lambda}|_{g^{\prime}}\Big{\\}}+2\varepsilon+\frac{C(n)}{r}$

for all $x\in\overline{\Omega_{\lambda\delta}}\setminus E_{0}$ and
$r\leqslant\sigma(x;\Omega_{\delta},g^{\prime})$. Since $g^{\prime}=g$ inside
$\Omega$, and
$\sigma(x;\Omega_{\delta},g^{\prime})\geqslant\sigma(x;\Omega,g)$ for all
$x\in\Omega$, and
$B_{g^{\prime}}(x,r)\cap\partial\Omega_{\lambda\delta}=\emptyset$ when
$r\leqslant\sigma(x;\Omega,g)$, this gradient estimate directly implies
(6.13). Next, the lower bound in (6.12) follows from (2.23) and (6.7). The
upper bound is derived as follows: by (6.13) we have

$\sup\big{\\{}u_{\varepsilon,\delta,\lambda}(x):x\in\overline{\Omega_{-r_{I}}}\,\big{\\}}\leqslant
C,$ (6.14)

where $C>0$ is independent of $\varepsilon,\delta,\lambda$. We compare
$u_{\varepsilon,\delta,\lambda}$ with ${\overline{u}_{\delta}}+C+1$ inside
$\overline{\Omega_{\lambda\delta}}\setminus\Omega_{-r_{I}}$, where
${\overline{u}_{\delta}}$ is as in (6.8). On $\partial\Omega_{-r_{I}}$ we have

$u_{\varepsilon,\delta,\lambda}\leqslant C={\overline{u}_{\delta}}+C+1.$

On $\partial\Omega_{\lambda\delta}$ we have

$u_{\varepsilon,\delta,\lambda}={\underline{u}_{\delta}}-2<{\overline{u}_{\delta}}<{\overline{u}_{\delta}}+C+1.$

Since ${\overline{u}_{\delta}}$ is a strict supersolution of (6.5) in the
compact set $\overline{\Omega_{\lambda\delta}}\setminus\Omega_{-r_{I}}$, by
continuity, we may further decrease $\varepsilon(\delta,\lambda)$ so that
${\overline{u}_{\delta}}$ is a strict supersolution of (6.9) in the same
region. By the maximum principle, we obtain
$u_{\varepsilon,\delta,\lambda}\leqslant{\overline{u}_{\delta}}+C+1$ in
$\overline{\Omega_{\lambda\delta}}\setminus\Omega_{-r_{I}}$. This implies the
desired upper bound along with (6.14). ∎

Convergence to an interior solution.

######  Lemma 6.4.

There is a sequence $\delta_{i}\to 0$, $\lambda_{i}\to 1$, $\varepsilon_{i}\to
0$, such that there are solutions
$u_{i}:=u_{\varepsilon_{i},\delta_{i},\lambda_{i}}$ of (6.9) $\sim$ (6.11)
that converge in $C^{0}_{\operatorname{loc}}(\Omega\\!\setminus\\!E_{0})$ to a
function
$u\in\operatorname{Lip}_{\operatorname{loc}}(\Omega\\!\setminus\\!E_{0})$. The
function $u$ solves $\mathsf{IVP}\boldsymbol{(}{\Omega;E_{0}}\boldsymbol{)}$
and is calibrated by some vector field $\nu$ in
$\Omega\setminus\overline{E_{0}}$.

###### Proof.

Set $\delta_{i}=1/i$, $\lambda_{i}=1-1/4i$, and select $\varepsilon_{i}\to 0$
with $\varepsilon_{i}<\varepsilon(\delta_{i},\lambda_{i})$. The approximate
solutions are thus given by Theorem 6.3. With an application of Theorem 2.18
to the data $\Omega_{i}=\Omega\setminus\overline{E_{0}}$, $g_{i}=g$ (note that
$\psi_{\delta_{i}}\equiv 0$ in $\Omega$, so the weighted equation is reduced
to the ordinary IMCF), we find a subsequence of $u_{i}$ that converges in
$C^{0}_{\operatorname{loc}}(\Omega\setminus\overline{E_{0}})$ to a calibrated
solution
$u\in\operatorname{Lip}_{\operatorname{loc}}(\Omega\setminus\overline{E_{0}})$.
Additionally, due to (6.13), the functions $u_{i}$ are uniformly Lipschitz up
to $\partial E_{0}$. Therefore, the uniform convergence and Lipschitz
regularity of $u$ holds up to $\partial E_{0}$. Then by (6.12) (6.10), we have
$u\geqslant 0$ in $\Omega\setminus E_{0}$ and $u|_{\partial E_{0}}=0$.
Extending $u$ with negative values in $E_{0}$, we obtain a solution of
$\mathsf{IVP}\boldsymbol{(}{\Omega;E_{0}}\boldsymbol{)}$. ∎

The remaining tasks.

For the clarity of proofs, we split the remaining part of Theorem 6.1 as
follows:

######  Proposition 6.5.

Let $u$, $\nu$ be the solution and calibration obtained in Lemma 6.4. Then
there is a radius $r_{0}\in(0,r_{I})$ and dimensional constants
$\gamma=\gamma(n)\in(0,1)$, $C=C(n)>0$, such that

$1-\langle\nu,\partial_{r}\rangle\leqslant Cr_{0}^{-\gamma}|r|^{\gamma}$
(6.15)

and

$|\nabla u|\leqslant Cr_{0}^{-\gamma}|r|^{\gamma-1}$ (6.16)

hold inside $\Omega\setminus\overline{\Omega_{-r_{0}}}$. In particular, $u$
solves
$\mathsf{IMCF}\boldsymbol{(}{\Omega\setminus\overline{E_{0}}}\boldsymbol{)}\texttt{\emph{+}}\mathsf{OBS}\boldsymbol{(}{\partial\Omega}\boldsymbol{)}$.

######  Proposition 6.6.

$u$ can be extend continuously to $\partial\Omega$, and we have $u\in
C^{0,\gamma}(\overline{\Omega}\setminus\Omega_{-r_{0}/2})$.

######  Proposition 6.7.

There exists a sufficiently small radius $r_{1}\in(0,r_{0})$ such that: for
all $t>0$, the set $\partial E_{t}\setminus\overline{\Omega_{-r_{1}}}$ is a
$C^{1,\gamma/2}$ hypersurface.

###### Proof of Theorem 6.1.

It is stated in Proposition 6.5 that $u$ respects the obstacle
$\partial\Omega$. The condition
$u\in\operatorname{Lip}_{\operatorname{loc}}(\Omega)\cap BV(\Omega)\cap
C^{0,\gamma}(\overline{\Omega})$ follows by interior regularity and Lemma 2.24
and Poposition 6.6. The gradient bound (6.1) follows by taking limit of
(6.13). The boundary regularity (6.2) follows from (6.16) inside
$\Omega\setminus\Omega_{-r_{0}}$, and follows from the interior regularity
inside $\Omega_{-r_{0}}$. The bound on calibration follows from (6.15).
Finally, the regularity of level sets follows from Proposition 6.7, and the
maximality follows from Corollary 3.22. ∎

###  6.2 The spacetime foliation; strategies of the proof

The approximation process described above, which essentially invokes the
elliptic regularization in Theorem 2.18, provides an additional set of data.
For each $i$, the family of hypersurfaces
$\Sigma_{t}^{i}:=\operatorname{graph}\big{(}\varepsilon_{i}^{-1}(u_{i}-t)\big{)}$
forms a downward translating soliton of the IMCF in the product domain
$\big{(}\Omega_{\lambda_{i}\delta_{i}}\times\mathbb{R},\
\exp(\frac{2\psi_{\delta_{i}}}{n-1})g+dz^{2}\big{)}$, due to Remark 2.20 and
the geometric meaning of elliptic regularization. Equivalently, the function
$U_{i}(x,z)=u_{i}(x)-\varepsilon_{i}z$ solves the smooth IMCF (2.3) in the
same region. The proof of Proposition 6.5 $\sim$ 6.7 is done by obtaining the
corresponding estimates on $\Sigma_{t}^{i}$, and then passing to the limit.

To distinguish from objects on $M$, we will use bold symbols to denote objects
on $M\times\mathbb{R}$. We denote by $\boldsymbol{r}(x,z)=r(x)$ the signed
distance to $\partial\Omega\times\mathbb{R}$. Note that $\boldsymbol{r}$ is
smooth in
$(\Omega_{3r_{I}}\setminus\overline{\Omega_{-3r_{I}}})\times\mathbb{R}$. Then
denote the product metric $\boldsymbol{g}=g+dz^{2}$ and radial vector field
$\boldsymbol{\partial_{r}}=\nabla_{\boldsymbol{g}}\boldsymbol{r}$. In the
region $(\overline{\Omega}\setminus\overline{E_{0}})\times\mathbb{R}$, let

$\boldsymbol{\nu}_{i}:=\frac{\nabla_{\boldsymbol{g}}U_{i}}{|\nabla_{\boldsymbol{g}}U_{i}|_{\boldsymbol{g}}}=\frac{\nabla
u_{i}-\varepsilon_{i}\partial_{z}}{\sqrt{\varepsilon_{i}+|\nabla u_{i}|^{2}}}$
(6.17)

be the downward unit normal vector field of the foliation $\Sigma_{t}^{i}$ (we
remind that $\psi_{\delta_{i}}=0$ in $\overline{\Omega}$, so the weighted IMCF
is reduced to the usual one). According to Theorem 2.18, the vector fields
$\boldsymbol{\nu}_{i}$ converge to some $\boldsymbol{\nu}$ in the weak sense
in $L^{1}_{\operatorname{loc}}(\Omega\setminus\overline{E_{0}})$, and the
projection of $\boldsymbol{\nu}$ onto the $\Omega$ factor is the calibration
$\nu$ given in Lemma 6.4.

The above propositions are proved via the following steps:

1\. We show that when $i\to\infty$, we have
$\inf_{\partial\Omega\times\mathbb{R}}\langle\boldsymbol{\nu}_{i},\boldsymbol{\partial_{r}}\rangle_{\boldsymbol{g}}\to
1$ (Lemma 6.8). The proof uses a blow-up argument: if there is a sequence of
exceptional points $x_{i}\in\partial\Omega\times\mathbb{R}$, then we may
rescale the solutions $u_{i}$ near $x_{i}$ and obtain an exceptional blow-up
limit. The limit function will be a weak solution of

$\operatorname{div}\Big{(}e^{\psi_{0}(x_{n})}\frac{\nabla u}{|\nabla
u|}\Big{)}=e^{\psi_{0}(x_{n})}|\nabla u|.$

However, Theorem 4.3 implies that this limit must have the form
$u=\psi(x_{n})-C$. Then, standard geometric measure theory is used to show
that the space-time graphs $\Sigma_{t}^{i}$ converges to a hyperplane at
$x_{i}$ in $C^{1}$ sense, contradicting our hypotheses for bad points.

2\. Let $r_{i}$ be the largest radius so that
$\langle\boldsymbol{\nu}_{i},\boldsymbol{\partial_{r}}\rangle_{\boldsymbol{g}}\geqslant
1/2$ holds inside $(\Omega\setminus\Omega_{-r_{i}})\times\mathbb{R}$. We show
that $r_{i}$ is uniformly bounded below (Lemma 6.9). If this is false, then we
find a subsequence of radii $r_{i}\to 0$ and exceptional points
$x_{i}\in\partial\Omega_{-r_{i}}\times\mathbb{R}$, with
$\langle\boldsymbol{\nu}_{i},\boldsymbol{\partial_{r}}\rangle_{\boldsymbol{g}}(x_{i},0)=1/2$.
Using the result of step 1 and the parabolic estimate in Lemma 5.4, we will
obtain the bounds

$1-\langle\boldsymbol{\nu}_{i},\boldsymbol{\partial_{r}}\rangle_{\boldsymbol{g}}\leqslant
C\big{(}o(1)+|r|/r_{i}\big{)}^{\gamma}$ (6.18)

for some uniform constants $C,\gamma$. Then consider a blow-up sequence
centered at $x_{i}$. The limit is an exceptional weak solution $u^{\prime}$ on
$\\{x_{n}<0\\}\subset\mathbb{R}^{n+1}$. The bound (6.18) passes to the limit
and implies that $u^{\prime}$ respects the obstacle $\\{x_{n}=0\\}$, due to
Corollary 3.19. This eventually contradicts Theorem 4.1, by looking at the
convergence of the space-time graphs $\Sigma_{t}^{i}$ (this is more involved
compared to the previous step).

3\. Combining the above two steps, we eventually obtain (6.15) through another
application of parabolic estimate (Lemma 5.4). The boundary gradient estimate
(6.16) and Hölder regularity follows by applying Lemma 5.5. Finally,
Proposition 6.7 follows by an $\varepsilon$-regularity theorem for almost
perimeter-minimizers (in the sense of (6.41) below).

###  6.3 Proof of Propositions 6.5 $\sim$ 6.7

We assume all the setups and notations made in subsection 6.1 and 6.2.

######  Lemma 6.8.

For any $\theta<1$, there exists $i_{0}\in\mathbb{N}$ such that for all
$i\geqslant i_{0}$ we have

$\inf_{\partial\Omega\times\mathbb{R}}\langle\boldsymbol{\nu}_{i},\boldsymbol{\partial_{r}}\rangle_{\boldsymbol{g}}\geqslant\theta.$

###### Proof.

Suppose that the conclusion does not hold. Then passing to a subsequence
(which we do not relabel), we can find a constant $\theta_{0}<1$ and a
sequence of points $q_{i}=(x_{i},z_{i})\in\partial\Omega\times\mathbb{R}$,
such that
$\langle\boldsymbol{\nu}_{i},\boldsymbol{\partial_{r}}\rangle_{\boldsymbol{g}}(x_{i},z_{i})\leqslant\theta_{0}$.
Since $\boldsymbol{\nu}_{i}$ is invariant under vertical translation, we may
assume $z_{i}=0$. Passing to a further subsequence, we may assume $x_{i}\to
x_{0}\in\partial\Omega$.

Consider the domains

$\Omega_{i}:=B_{g}(x_{i},\sqrt{\delta_{i}})\cap\Omega_{\lambda_{i}\delta_{i}},\qquad\boldsymbol{\Omega}_{i}:=\Omega_{i}\times(-1,1)$

with the rescaled and conformally transformed metrics

$h_{i}:=\delta_{i}^{-2}e^{2\psi_{\delta_{i}}/(n-1)}g,\qquad\boldsymbol{h}_{i}:=h_{i}+\delta_{i}^{-2}dz^{2}.$

Set the normalized function $u^{\prime}_{i}(x):=u_{i}(x)-u_{i}(x_{i})$. By the
equation (6.9), Remark 2.20 and the scaling invariance of IMCF, the functions
$U^{\prime}_{i}(x,z):=u^{\prime}_{i}(x)-\varepsilon_{i}z$ are smooth solutions
of the IMCF in $(\boldsymbol{\Omega}_{i},\boldsymbol{h}_{i})$.

To properly state the blow-up process, we define some suitable coordinate
maps. In each tangent space $T_{x_{i}}M$, fix an orthonormal frame
$\\{e_{i}\\}$ such that $e_{1},\cdots,e_{n-1}$ are tangential to
$\partial\Omega$, and $e_{n}=\partial_{r}$. With respect to this frame, the
$g$-exponential map at $x_{i}$ (denoted by $\exp_{x_{i}}$) is a diffeomorphism
from a small ball in $\mathbb{R}^{n}$ to its image. Define the maps

$\Phi_{i}:x\mapsto\exp_{x_{i}}(\delta_{i}x),\qquad\boldsymbol{\Phi}_{i}:(x,z)\mapsto\big{(}\\!\exp_{x_{i}}(\delta_{i}x),\delta_{i}z\big{)},$

and set the preimages111The rule for setting up the notations: boldface $\to$
objects in $M\times\mathbb{R}$; tilde $\to$ objects in the pulled-back spaces;
prime $\to$ normalized functions (so that the value at $x_{i}$ is zero).
$\widetilde{\Omega}_{i}:=\Phi_{i}^{-1}(\Omega_{i})\subset\mathbb{R}^{n}$,
$\widetilde{\boldsymbol{\Omega}}_{i}:=\boldsymbol{\Phi}_{i}^{-1}(\boldsymbol{\Omega}_{i})=\widetilde{\Omega}_{i}\times(-\delta_{i}^{-1},\delta_{i}^{-1})$.
Also, set the pullbacks $\widetilde{h}_{i}:=\Phi_{i}^{*}h_{i}$,
$\widetilde{\boldsymbol{h}}_{i}:=\boldsymbol{\Phi}_{i}^{*}\boldsymbol{h}_{i}=\widetilde{h}_{i}+dz^{2}$,
and $\widetilde{u}_{i}:=u^{\prime}_{i}\circ\Phi_{i}$,
$\widetilde{U}^{\prime}_{i}:=U^{\prime}_{i}\circ\boldsymbol{\Phi}_{i}$. Note
that $\widetilde{U}^{\prime}_{i}(0)=U^{\prime}_{i}(x_{i},0)=0$ and
$\widetilde{U}_{i}(x,z)=\widetilde{u}^{\prime}_{i}(x)-\varepsilon_{i}\delta_{i}z$.
By diffeomorphism invariance, $\widetilde{U}^{\prime}_{i}$ are smooth
solutions of IMCF in
$(\widetilde{\boldsymbol{\Omega}}_{i},\widetilde{\boldsymbol{h}}_{i})$.
Therefore, $\widetilde{u}_{i}$ solves the
$(\varepsilon_{i}\delta_{i})$-regularized equation

$\operatorname{div}\left(\frac{\nabla_{\widetilde{h}_{i}}\widetilde{u}_{i}}{\big{(}\varepsilon_{i}^{2}\delta_{i}^{2}+|\nabla_{\widetilde{h}_{i}}\widetilde{u}_{i}|^{2}\big{)}^{1/2}}\right)=\sqrt{\varepsilon_{i}^{2}\delta_{i}^{2}+|\nabla_{\widetilde{h}_{i}}\widetilde{u}_{i}|^{2}}\qquad\text{in
}(\widetilde{\Omega}_{i},\widetilde{h}_{i}).$

Next, we take limit of the rescaled objects. Note that
$\widetilde{\Omega}_{i}\to\\{x_{n}<1\\}\subset\mathbb{R}^{n}$ locally, since
$\delta_{i}\to 0$ and $\lambda_{i}\to 1$. Also,
$\widetilde{h}_{i}\to\widetilde{h}:=e^{2\psi_{0}(x_{n})/(n-1)}(dx_{1}^{2}+\cdots+dx_{n}^{2})$
locally smoothly. Apply Theorem 2.18 with the data $\widetilde{\Omega}_{i}$,
$\widetilde{h}_{i}$ and $\widetilde{u}_{i}$. As a result, there is a
subsequence such that $\widetilde{u}^{\prime}_{i}$ converges in
$C^{0}_{\operatorname{loc}}$ to a function
$\widetilde{u}^{\prime}\in\operatorname{Lip}_{\operatorname{loc}}(\\{x_{n}<1\\})$,
and $\widetilde{u}^{\prime}$ solves
$\mathsf{IMCF}\boldsymbol{(}{\\{x_{n}<1\\},\widetilde{h}}\boldsymbol{)}$. By
Lemma 2.10(3), $\widetilde{u}^{\prime}$ is a weak solution of the weighted
IMCF

$\operatorname{div}\Big{(}e^{\psi_{0}(x_{n})}\frac{\nabla_{0}\widetilde{u}^{\prime}}{|\nabla_{0}\widetilde{u}^{\prime}|}\Big{)}=e^{\psi_{0}(x_{n})}|\nabla_{0}\widetilde{u}^{\prime}|,$

where $\nabla_{0}$ denotes the Euclidean gradient. The gradient estimate
(2.25) implies

$|\nabla_{0}\widetilde{u}^{\prime}(x)|=|\nabla_{\widetilde{h}}\widetilde{u}^{\prime}(x)|\leqslant\frac{C(n)}{\sigma\big{(}x;\\{x_{n}<1\\},\widetilde{h}\big{)}}\leqslant\frac{C(n)}{|x_{n}|},\qquad\forall\,x\in\\{x_{n}<0\\}.$

Furthermore, the $C^{0}$ bounds in Theorem 6.3 gives (recall
$u^{\prime}_{i}=u_{i}-u_{i}(x_{i})$)

$u^{\prime}_{i}\geqslant\psi_{\delta_{i}}-2C\ \ \Rightarrow\ \
\widetilde{u}^{\prime}_{i}(x)\geqslant\psi_{\delta_{i}}\big{(}\\!\exp_{x_{i}}(\delta_{i}x)\big{)}-2C=\psi_{0}(x_{n})+o(1)-2C.$

This passes to the limit and gives the bound
$\widetilde{u}^{\prime}(x)\geqslant\psi_{0}(x_{n})-2C$. Now all the
assumptions of Theorem 4.3 are met, and we obtain
$\widetilde{u}^{\prime}(x)=\psi_{0}(x_{n})-C^{\prime}$ for some other
$C^{\prime}$.

Set
$\widetilde{U}^{\prime}(x,z):=\widetilde{u}^{\prime}(x)=\psi_{0}(x_{n})-C^{\prime}$,
which is the $C^{0}_{\operatorname{loc}}$ limit of
$\widetilde{U}^{\prime}_{i}$. The sub-level sets
$E_{0}(\widetilde{U}^{\prime}_{i})$ have locally uniformly bounded perimeter,
by Lemma 2.24(i), Thus passing to a further sequence, we may assume
$E_{0}(\widetilde{U}^{\prime}_{i})\to E$ in $L^{1}_{\operatorname{loc}}$, for
some set $E\subset\big{\\{}(x,z):x_{n}<1\big{\\}}$. Since
$\widetilde{U}^{\prime}_{i}\to\widetilde{U}^{\prime}$ in
$C^{0}_{\operatorname{loc}}$, we have (see Remark 2.2)

$E_{0}(\widetilde{U}^{\prime})\subset E\subset
E_{0}^{+}(\widetilde{U}^{\prime})$ (6.19)

up to zero measure. Next, by Corollary 2.16 and the nice convergence of
$\widetilde{\boldsymbol{h}}_{i}$, the gradients
$|\nabla_{\widetilde{\boldsymbol{h}}_{i}}\widetilde{U}^{\prime}_{i}|$ are
uniformly bounded in $B_{\mathbb{R}^{n+1}}(0,1/2)$, for all large $i$. Then by
(2.6), $E_{0}(\widetilde{U}^{\prime}_{i})$ are uniform
$(\Lambda,r_{0})$-perimeter minimizers in
$\big{(}B_{0}(0,1/2),\widetilde{\boldsymbol{h}}_{i}\big{)}$. By Lemma B.1(i)
and the fact $0\in\partial E_{0}(\widetilde{U}^{\prime}_{i})$, we have
$0\in\operatorname{spt}(|D\chi_{E}|)$. Combined with (6.19) and the expression
$\widetilde{U}^{\prime}=\psi(x_{n})-C$, we see that there are only two
possibilities:

(1) $C^{\prime}>0$ and $E=\\{x_{n}<0\\}$, or

(2) $C^{\prime}=0$, $\widetilde{U}^{\prime}(x,z)=\psi_{0}(x_{n})$, and
$E\subset\\{x_{n}<0\\}$.

Suppose the second case holds. By Theorem 2.9, the limit set $E$ locally
minimizes the energy $J_{\widetilde{U}^{\prime}}$ in
$\big{\\{}(x,z):x_{n}<1\big{\\}}$. As $\widetilde{U}^{\prime}\equiv 0$ in
$\big{\\{}(x,z):x_{n}<0\big{\\}}$, this implies that $E$ is locally inward-
minimizing. On the other hand, $E$ is an outward minimizer of the weighted
perimeter $\int_{\partial^{*}E}e^{\psi_{0}}$. By direct verification, this
implies that $E$ is locally outward perimeter-minimizing in
$\mathbb{R}^{n+1}$. As a result, $E$ is locally perimeter-minimizing in
$\mathbb{R}^{n+1}$. By Lemma B.3 and the facts $E\subset\\{x_{n}<0\\}$,
$0\in\operatorname{spt}(|D\chi_{E}|)$, we have $E=\\{x_{n}<0\\}$.

So in either case we conclude that $E=\\{x_{n}<0\\}$. Then by Lemma B.1(ii),
we have

$\nu_{E_{0}(\widetilde{U}^{\prime}_{i})}(0)\to\nu_{E}(0)=\partial_{x_{n}},$

where $\nu_{E_{0}(\widetilde{U}^{\prime}_{i})}$ is the outer unit normal of
$E_{0}(\widetilde{U}^{\prime}_{i})$ with respect to
$\widetilde{\boldsymbol{h}}_{i}$. On the other hand, we may evaluate by
unraveling the pullbacks

$\big{\langle}\nu_{E_{0}(\widetilde{U}^{\prime}_{i})},\partial_{x_{n}}\big{\rangle}_{\widetilde{\boldsymbol{h}}_{i}}(0)=\langle\boldsymbol{\nu}_{i},\boldsymbol{\partial_{r}}\rangle_{\boldsymbol{g}}(x_{i},0).$

This contradicts our initial assumption
$\langle\boldsymbol{\nu}_{i},\boldsymbol{\partial_{r}}\rangle_{\boldsymbol{g}}(x_{i},0)\leqslant\theta_{0}$,
thus proves the lemma. ∎

######  Lemma 6.9.

There exists $r_{0}<r_{I}$ and $i_{0}\in\mathbb{N}$, such that for all
$i\geqslant i_{0}$ we have

$\inf_{(\Omega\setminus\Omega_{-r_{0}})\times\mathbb{R}}\langle\boldsymbol{\nu}_{i},\boldsymbol{\partial_{r}}\rangle_{\boldsymbol{g}}\geqslant\frac{1}{2}.$

###### Proof.

By Lemma 6.8, we have
$\inf_{\partial\Omega\times\mathbb{R}}\langle\boldsymbol{\nu}_{i},\boldsymbol{\partial_{r}}\rangle_{\boldsymbol{g}}\geqslant
3/4$ for all sufficiently large $i$. Suppose that the lemma is false. Then
passing to a subsequence, we may find radii $r_{i}\in(0,r_{I})$, $r_{i}\to 0$,
and a sequence of points
$q_{i}=(x_{i},z_{i})\in\partial\Omega_{-r_{i}}\times\mathbb{R}$, such that

$\langle\boldsymbol{\nu}_{i},\boldsymbol{\partial_{r}}\rangle_{\boldsymbol{g}}(x_{i},z_{i})\leqslant\frac{1}{2}.$
(6.20)

By translation invariance, we may assume $z_{i}=0$. By re-selecting each
$x_{i}$ to have the smallest distance to $\partial\Omega$ (and decreasing
$r_{i}$ correspondingly), we can further assume that

$\langle\boldsymbol{\nu}_{i},\boldsymbol{\partial_{r}}\rangle_{\boldsymbol{g}}\geqslant\frac{1}{2}\quad\text{inside}\quad(\Omega\setminus\Omega_{-r_{i}})\times\mathbb{R}.$
(6.21)

Finally, passing to a further subsequence, we may assume $x_{i}\to
x_{0}\in\partial\Omega$. By Lemma 6.8, there exist numbers $\theta_{i}\to 1$
such that

$\inf_{\partial\Omega\times\mathbb{R}}\langle\boldsymbol{\nu}_{i},\boldsymbol{\partial_{r}}\rangle_{\boldsymbol{g}}\geqslant\theta_{i}.$
(6.22)

Step 1. We establish a uniform bound on $\boldsymbol{\nu}_{i}$ by parabolic
estimates.

Recall that
$\Sigma_{t}^{i}=\operatorname{graph}\big{(}\varepsilon_{i}^{-1}(u_{i}-t)\big{)}$
forms a downward translating soliton of the usual IMCF in
$(\overline{\Omega}\setminus\overline{E_{0}})\times\mathbb{R}$. We apply Lemma
5.4 to the flow $\Sigma_{t}^{i}$, with $\Omega$ replaced by
$\Omega\times\mathbb{R}$ and with the choice $r_{0}=r_{i}$. The small
curvature condition (5.11) is implied by (6.4), and the condition
$p\geqslant\frac{1}{2}$ there comes from (6.21). Choosing the parameter
$b_{i}=(1-\theta_{i})^{1/\gamma}$ there, it follows that the quantity
$F=\big{(}1-\langle\boldsymbol{\nu}_{i},\boldsymbol{\partial_{r}}\rangle_{\boldsymbol{g}}\big{)}(b_{i}-r/r_{i})^{-\gamma}$
satisfies

$\big{(}1-\langle\boldsymbol{\nu}_{i},\boldsymbol{\partial_{r}}\rangle_{\boldsymbol{g}}\big{)}^{\frac{2}{\gamma}-1}H^{2}\Box
F+\langle\nabla_{\Sigma}F,X\rangle\leqslant\frac{C_{1}}{r_{i}^{2}}\big{(}\\!-F^{\frac{\gamma+2}{\gamma}}+C_{2}\big{)}$

inside $(\Omega\setminus\overline{\Omega_{-r_{i}}})\times\mathbb{R}$, where
$\Box=\partial_{t}-H^{-2}\Delta_{\Sigma}$ is the heat operator associated to
the IMCF, and the subscripts $\Sigma$ are shorthands for $\Sigma_{t}^{i}$.
Since $\Sigma_{t}^{i}$ evolves as translating soliton in the $z$-direction, we
have $\partial_{t}F=\langle\nabla_{\Sigma}F,Y\rangle$ for some smooth vector
field $Y$. Hence

$-\big{(}1-\langle\boldsymbol{\nu}_{i},\boldsymbol{\partial_{r}}\rangle_{\boldsymbol{g}}\big{)}^{\frac{2}{\gamma}-1}\Delta_{\Sigma}F+\langle\nabla_{\Sigma}F,Z\rangle\leqslant\frac{C_{1}}{r_{i}^{2}}\big{(}\\!-F^{\frac{\gamma+2}{\gamma}}+C_{2}\big{)},$

for some $\gamma\in(0,1/2)$ and $C_{1},C_{2}>0$ depending only on $n$. Since
$\Sigma_{t}^{i}$ is graphical, the intersection
$\Sigma_{t}^{i}\cap\big{(}(\overline{\Omega}\setminus\Omega_{-r_{I}})\times\mathbb{R}\big{)}$
is compact. By the maximum principle and invariance in the $z$-direction, we
obtain

$\displaystyle\max_{(\overline{\Omega}\setminus\Omega_{-r_{i}})\times\mathbb{R}}(F)$
$\displaystyle\leqslant\max\Big{\\{}\max_{\partial\Omega\times\mathbb{R}}(F),\
\max_{\partial\Omega_{-r_{i}}\times\mathbb{R}}(F),\
C_{2}^{\frac{\gamma}{\gamma+2}}\Big{\\}}$
$\displaystyle\leqslant\max\Big{\\{}1,\ \frac{1}{2},\
C_{2}^{\frac{\gamma}{\gamma+2}}\Big{\\}}.$

From this we have

$1-\langle\boldsymbol{\nu}_{i},\boldsymbol{\partial_{r}}\rangle_{\boldsymbol{g}}\leqslant
C_{3}(b_{i}-\boldsymbol{r}/r_{i})^{\gamma}\qquad\text{in}\ \
(\overline{\Omega}\setminus\Omega_{-r_{i}})\times\mathbb{R}.$ (6.23)

Step 2. We establish an inward minimizing property of the level sets of
$U_{i}$, where recall $U_{i}(x,z)=u_{i}(x)-\varepsilon_{i}z$. Denote
$\boldsymbol{\Omega}=\Omega\times\mathbb{R}$. Fix $t\in\mathbb{R}$, and
consider $E=E_{t}(U_{i})\cap\boldsymbol{\Omega}$. Suppose $F\subset\subset
K\subset\subset(M\setminus\overline{E_{0}})\times\mathbb{R}$. Using the
boundary condition (6.22), the fact $\nu_{E}=\boldsymbol{\nu}_{i}$ in
$\boldsymbol{\Omega}$, and the smooth IMCF equation
$\operatorname{div}(\boldsymbol{\nu}_{i})=|\nabla_{\boldsymbol{g}}U_{i}|$, we
evaluate by the divergence theorem as follows. In the expressions, we write
$|\cdot|=\mathcal{H}^{n-1}(\cdot)$.

$\displaystyle\quad\ \ {P\big{(}E;\boldsymbol{\Omega}\cap
K\big{)}}-{P\big{(}E\\!\setminus\\!F;\boldsymbol{\Omega}\cap
K\big{)}}+\theta_{i}\Big{(}\big{|}\partial^{*}E\cap\partial\boldsymbol{\Omega}\cap
K\big{|}-\big{|}\partial^{*}(E\\!\setminus\\!F)\cap\partial\boldsymbol{\Omega}\cap
K\big{|}\Big{)}$
$\displaystyle\leqslant\int_{\partial^{*}E\cap\boldsymbol{\Omega}\cap
K}\langle\nu_{E},\boldsymbol{\nu}_{i}\rangle_{\boldsymbol{g}}-\int_{\partial^{*}(E\setminus
F)\cap\boldsymbol{\Omega}\cap K}\langle\nu_{E\setminus
F},\boldsymbol{\nu}_{i}\rangle_{\boldsymbol{g}}+\int_{\partial\boldsymbol{\Omega}\cap
K}\big{(}\chi_{\partial^{*}E}-\chi_{\partial^{*}(E\setminus
F)}\big{)}\langle\nu_{\boldsymbol{\Omega}},\boldsymbol{\nu}_{i}\rangle_{\boldsymbol{g}}$
$\displaystyle=\int_{E\cap F}|\nabla_{\boldsymbol{g}}U_{i}|_{\boldsymbol{g}}.$

Next, by the coarea formula we have

$\int_{E\cap
F}|\nabla_{\boldsymbol{g}}U_{i}|_{\boldsymbol{g}}=\int_{\inf_{F\cap\boldsymbol{\Omega}}(U_{i})}^{t}\big{|}\partial^{*}E_{s}(U_{i})\cap\boldsymbol{\Omega}\cap
F\big{|}\,ds.$

By a perimeter decomposition and Gronwall argument, as in Lemma 2.21, these
imply

$\displaystyle{P\big{(}E;\boldsymbol{\Omega}\cap
K\big{)}}+\theta_{i}\big{|}\partial^{*}E\cap\partial\boldsymbol{\Omega}\cap
K\big{|}$
$\displaystyle\leqslant{P\big{(}E\\!\setminus\\!F;\boldsymbol{\Omega}\cap
K\big{)}}+\theta_{i}\big{|}\partial^{*}(E\\!\setminus\\!F)\cap\partial\boldsymbol{\Omega}\cap
K\big{|}$
$\displaystyle\qquad\quad+\int_{\inf_{F\cap\boldsymbol{\Omega}}(U_{i})}^{t}e^{t-s}{P\big{(}F;E_{s}(U_{i})\cap\boldsymbol{\Omega}\big{)}}\,ds.$

Notice that ${P(E;K)}={P(E;\boldsymbol{\Omega}\cap
K)}+|\partial^{*}E\cap\partial\boldsymbol{\Omega}\cap K|$, since
$E\subset\boldsymbol{\Omega}$. So the above inequality clearly implies (where
we have plugged in the definition of $E$):

$\theta_{i}{P\big{(}E_{t}(U_{i})\cap\boldsymbol{\Omega};K\big{)}}\leqslant{P\big{(}(E_{t}(U_{i})\cap\boldsymbol{\Omega})\setminus
F;K\big{)}}+\big{(}e^{t-\inf_{F\cap\boldsymbol{\Omega}}(U_{i})}-1\big{)}{P(F)}.$
(6.24)

We remark that, by the spirit of Theorem 3.8, outward-minimizing properties
are sort of automatic. On the other hand, the inward-minimizing property is
closely related to the boundary condition. This explains why (6.22) is invoked
in this step.

Step 3. We are ready for the blow-up argument. Let $y_{i}\in\partial\Omega$ be
the unique point with the smallest distance from $x_{i}$, thus
$d(x_{i},y_{i})=r_{i}$. Consider the domains

$\Omega_{i}:=B_{g}(y_{i},\sqrt{r_{i}})\cap\Omega,\qquad\boldsymbol{\Omega}_{i}:=\Omega_{i}\times(-1,1),$

with the metrics

$h_{i}:=r_{i}^{-2}g,\qquad\boldsymbol{h}_{i}:=r_{i}^{-2}\big{(}g+dz^{2}\big{)}.$

Define $u^{\prime}_{i}(x):=u(x)-u(x_{i})$,
$U^{\prime}_{i}(x,z):=u^{\prime}_{i}(x)-\varepsilon z$. Near $y_{i}$ we set up
a geodesic normal coordinate in the same way as in Lemma 6.8. Let
$\exp_{y_{i}}$ be the induced exponential map; thus we have
$\exp_{y_{i}}(-r_{i}e_{n})=x_{i}$. Consider the scaled coordinate maps

$\Phi_{i}:x\mapsto\exp_{y_{i}}(r_{i}x),\qquad\boldsymbol{\Phi}_{i}:(x,z)\mapsto(\exp_{y_{i}}(r_{i}x),r_{i}z).$

Denote
$\widetilde{\Omega}_{i}:=\Phi_{i}^{-1}(\Omega_{i})\subset\mathbb{R}^{n}$ and
$\widetilde{\boldsymbol{\Omega}}_{i}:=\boldsymbol{\Phi}_{i}^{-1}(\boldsymbol{\Omega}_{i})=\Omega_{i}\times(-r_{i}^{-1},r_{i}^{-1})\subset\mathbb{R}^{n+1}$,
equipped with the metrics and functions

$\displaystyle\widetilde{h}_{i}=\Phi_{i}^{*}h_{i},\qquad\ \ \
\widetilde{\boldsymbol{h}}_{i}:=\boldsymbol{\Phi}_{i}^{*}\boldsymbol{h}_{i}=\widetilde{h}_{i}+dz^{2},$
$\displaystyle\widetilde{u}^{\prime}_{i}:=u^{\prime}_{i}\circ\Phi_{i},\qquad\widetilde{U}^{\prime}_{i}:=U^{\prime}_{i}\circ\boldsymbol{\Phi}_{i}.$

So $\widetilde{\Omega}_{i}$ is the intersection of
$B_{\mathbb{R}^{n}}(0,1/\sqrt{r_{i}})$ with a set that locally approximates
$\\{x_{n}<0\\}$ as $i\to\infty$. Also, notice that
$\Phi_{i}^{-1}(\partial\Omega_{-r_{i}})$ locally approaches $\\{x_{n}=-1\\}$
when $i\to\infty$. Let us view $\widetilde{u}^{\prime}_{i}$,
$\widetilde{U}^{\prime}_{i}$ as functions defined only on
$\widetilde{\Omega}_{i}$, $\widetilde{\boldsymbol{\Omega}}_{i}$ (thus we are
discarding the values outside these domains).

As usual, denote $e_{n}=(0,\cdots,0,1)\in\mathbb{R}^{n}$. Further denote
$\boldsymbol{e}_{n}=(e_{n},0)\in\mathbb{R}^{n+1}$. Note the following facts:
$x_{i}=\Phi_{i}(-e_{n})$, and
$\widetilde{U}^{\prime}_{i}(x,z)=\widetilde{u}^{\prime}_{i}(x)-\varepsilon_{i}r_{i}z$,
so $\widetilde{U}^{\prime}_{i}(-\boldsymbol{e}_{n})=0$. Also,
$\widetilde{U}^{\prime}_{i}$ solves the smooth IMCF in
$(\widetilde{\boldsymbol{\Omega}}_{i},\widetilde{\boldsymbol{h}}_{i})$, by
scaling and diffeomorphism invariance.

We have
$\widetilde{\boldsymbol{\Omega}}_{i}\to\\{(x,z):x_{n}<0\\}\subset\mathbb{R}^{n+1}$
and
$\widetilde{\boldsymbol{h}}_{i}\to\textbf{euc}:=dx_{1}^{2}+\cdots+dx_{n+1}^{2}$
locally smoothly. By Corollary 2.16, we have the gradient estimate

$\big{|}\nabla_{\widetilde{\boldsymbol{h}}_{i}}\widetilde{U}^{\prime}_{i}\big{|}_{\widetilde{\boldsymbol{h}}_{i}}(x,z)\leqslant\frac{C(n)}{|x_{n}|}+o(1),$
(6.25)

where the $o(1)$ term is uniform in all compact sets and goes to zero as
$i\to\infty$.

By Theorem 2.18 (applied with the data $\widetilde{\boldsymbol{\Omega}}_{i}$,
$\widetilde{\boldsymbol{h}}_{i}$, $\widetilde{U}^{\prime}_{i}$), up to a
subsequence, we have $\widetilde{U}^{\prime}_{i}\to\widetilde{U}^{\prime}$ in
$C^{0}_{loc}$ for some
$\widetilde{U}^{\prime}\in\operatorname{Lip}_{\operatorname{loc}}(\\{(x,z):x_{n}<0\\})$,
and the calibrations
$\widetilde{\boldsymbol{\nu}}_{i}:=\nabla_{\widetilde{\boldsymbol{h}}_{i}}\widetilde{U}^{\prime}_{i}/|\nabla_{\widetilde{\boldsymbol{h}}_{i}}\widetilde{U}^{\prime}_{i}|$
converges to some $\widetilde{\boldsymbol{\nu}}$ in the weak
$L^{1}_{\operatorname{loc}}$ topology. Finally, $\widetilde{\boldsymbol{\nu}}$
calibrates $\widetilde{U}^{\prime}$ as a solution of
$\mathsf{IMCF}\boldsymbol{(}{\\{x_{n}<0\\};\textbf{euc}}\boldsymbol{)}$.

Taking limit of (6.25) and the $C^{0}$ bounds in Theorem 6.3, we have

$|\nabla_{\widetilde{\boldsymbol{h}}_{0}}\widetilde{U}^{\prime}|\leqslant\frac{C(n)}{|x_{n}|},\qquad\widetilde{U}(x,z)\geqslant-C.$
(6.26)

For $(x,z)\in\widetilde{\boldsymbol{\Omega}}_{i}$ with $-1\leqslant x_{n}<0$,
we may calculate by unraveling the pullbacks and using the asymptotics of the
exponential map:

$\big{\langle}\widetilde{\boldsymbol{\nu}}_{i},\,\partial_{x_{n}}\big{\rangle}_{\widetilde{\boldsymbol{h}}_{i}}(x,z)=\langle\boldsymbol{\nu}_{i},\boldsymbol{\partial_{r}}\rangle_{\boldsymbol{g}}(\Phi_{i}(x),r_{i}z)+o(1),$
(6.27)

where the $o(1)$ term locally uniformly converges to zero as $\delta\to 0$. By
(6.23) this implies

$\big{\langle}\widetilde{\boldsymbol{\nu}}_{i},\,\partial_{x_{n}}\big{\rangle}_{\widetilde{\boldsymbol{h}}_{i}}(x,z)\geqslant
1-C_{3}\big{(}b_{i}-r_{i}^{-1}r(\Phi_{i}(x))\big{)}^{\gamma}-o(1).$ (6.28)

Note that $\lim_{i\to\infty}r_{i}^{-1}r(\Phi_{i}(x))=x_{n}$ and
$\lim_{i\to\infty}b_{i}=0$. Thus taking the limit $i\to\infty$, we obtain

$\big{\langle}\widetilde{\boldsymbol{\nu}},\,\partial_{x_{n}}\big{\rangle}_{\textbf{euc}}(x,z)\geqslant
1-C_{3}|x_{n}|^{\gamma}\qquad\text{a.e. when}\ \ -1<x_{n}<0.$ (6.29)

Then by Corollary 3.19, $\widetilde{U}^{\prime}$ actually solves of
$\mathsf{IMCF}\boldsymbol{(}{\\{x_{n}<0\\},\textbf{euc}}\boldsymbol{)}\texttt{+}\mathsf{OBS}\boldsymbol{(}{\\{x_{n}=0\\}}\boldsymbol{)}$.
Then applying Theorem 4.1 with (6.26) with
$\widetilde{U}^{\prime}(-\boldsymbol{e}_{n})=0$, it follows that
$\widetilde{U}^{\prime}\equiv 0$.

Step 4. We argue similarly as in Lemma 6.8, to obtain a contradiction from the
level sets. Recall $-\boldsymbol{e}_{n}\in\partial
E_{0}(\widetilde{U}^{\prime}_{i})$. Applying Lemma 2.24(i) to
$\widetilde{\boldsymbol{\Omega}}_{i}\subset\mathbb{R}^{n+1}$, we have the
uniform bound (recall that we are treating $E_{0}(\widetilde{U}^{\prime}_{i})$
as subsets of $\widetilde{\boldsymbol{\Omega}}_{i}$)

${P\big{(}E_{0}(\widetilde{U}^{\prime}_{i});K\big{)}}\leqslant{P\big{(}\widetilde{\boldsymbol{\Omega}}_{i}\cap
K\big{)}},\qquad\forall K\subset\subset\mathbb{R}^{n+1}.$

Thus, a subsequence of $E_{0}(\widetilde{U}^{\prime}_{i})$ converges to some
$E\subset\big{\\{}(x,z):x_{n}<0\big{\\}}$ in
$L^{1}_{\operatorname{loc}}(\mathbb{R}^{n+1})$.

The gradient bound (6.25) and (2.6) implies the uniform
$(\Lambda,r_{0})$-minimizing effect

${P\big{(}E_{0}(\widetilde{U}^{\prime}_{i});B(-\boldsymbol{e}_{n},\frac{1}{2})\big{)}}\leqslant{P(F;B(-\boldsymbol{e}_{n},\frac{1}{2}))}+C\big{|}E_{0}(\widetilde{U}^{\prime}_{i})\Delta
F\big{|},$ (6.30)

whenever $E_{0}(\widetilde{U}^{\prime}_{i})\Delta F\subset\subset
B(-\boldsymbol{e}_{n},1/2)$. Then by Lemma B.1(i), we obtain
$(-e_{n},0)\in\operatorname{spt}(|D\chi_{E}|)$. By Theorem 2.9, $E$ is a local
minimizer of $J_{\widetilde{U}^{\prime}}$ in
$\big{\\{}(x,z):x_{n}<0\big{\\}}$, and since $\widetilde{U}^{\prime}$ is shown
to be constant, $E$ is actually a local perimeter minimizer in
$\big{\\{}(x,z):x_{n}<0\big{\\}}$ (with respect to the Euclidean metric). At
this point, the minimizing is unknown to hold up to the boundary. However, one
may show that $E$ is locally outward-minimizing in $\mathbb{R}^{n+1}$, by an
approximation argument as in Theorem 3.8.

If we can show that $E$ is locally inward perimeter-minimizing in
$\mathbb{R}^{n+1}$, then Lemma B.3 implies $E=\\{(x,z):x_{n}<-1\\}$, and then
Lemma B.1(ii) implies the convergence of normal vectors

$\widetilde{\boldsymbol{\nu}}_{i}(-\boldsymbol{e}_{n})=\nu_{E_{0}(\widetilde{U}^{\prime}_{i})}(-\boldsymbol{e}_{n})\to\partial_{x_{n}}.$

This would contradict (6.20) due to (6.27), thus proving the lemma. It remains
to show the (non-trivial) inward-minimizing of $E$.

For each fixed $R$, we denote $Q_{R}=\\{|x^{\prime}|\leqslant
R,|x_{n}|\leqslant R,|z|\leqslant R\\}\subset\mathbb{R}^{n+1}$. Let us prove
the following almost constancy conclusion: for each $R\geqslant 2$ we have

$\lim_{i\to\infty}\min\Big{\\{}\widetilde{U}^{\prime}_{i}(x,z):(x,z)\in
Q_{R}\cap\widetilde{\boldsymbol{\Omega}}_{i}\Big{\\}}=0.$ (6.31)

Recall the direction bound (6.21) and the definition
$\boldsymbol{\nu}_{i}=\nabla_{\boldsymbol{g}}U^{\prime}_{i}/|\nabla_{\boldsymbol{g}}U^{\prime}_{i}|_{\boldsymbol{g}}$.
Pulling back to $\widetilde{\boldsymbol{\Omega}}_{i}$, these imply that

$\frac{\partial\widetilde{U}^{\prime}_{i}}{\partial x_{n}}\geqslant
0\quad\text{in}\quad\Big{\\{}\\!-1/2\leqslant x_{n}<0,|x^{\prime}|\leqslant
R,|z|\leqslant R\Big{\\}}\cap\widetilde{\boldsymbol{\Omega}}_{i}.$

for sufficiently large $i$. Therefore, the minimum in (6.31) is attained at a
point with $-R\leqslant x_{n}\leqslant-1/2$, for each large $i$. Since
$\widetilde{U}^{\prime}_{i}\to\widetilde{U}^{\prime}\equiv 0$ uniformly on the
compact set $\big{\\{}|x^{\prime}|\leqslant R,-R\leqslant
x_{n}\leqslant-1/2,|z|\leqslant R\big{\\}}$, our claim (6.31) immediately
follows.

Now we pull back (6.24) via $\boldsymbol{\Phi}_{i}$. In this process we notice
that: $U_{i}$ and $U^{\prime}_{i}$ differs only by additive constants, so
(6.24) holds for $\widetilde{U}^{\prime}_{i}$ as well. Also, the inequality
(6.24) is invariant under scaling. Moreover, notice a subtlety in notations:
in our setting, $E_{0}(\widetilde{U}^{\prime}_{i})$ is the pull back of
$E_{0}(U^{\prime}_{i})\cap\boldsymbol{\Omega}$ (i.e. the intersection with
$\boldsymbol{\Omega}$ is already done via $\boldsymbol{\Phi}_{i}$). So for any
$F\subset\subset Q_{R}\subset\subset B_{\mathbb{R}^{n+1}}(0,1/\sqrt{r_{i}})$,
we have

$\theta_{i}{P\big{(}E_{0}(\widetilde{U}^{\prime}_{i});Q_{R}\big{)}}\leqslant{P\big{(}E_{0}(\widetilde{U}^{\prime}_{i})\setminus
F;Q_{R}\big{)}}+\big{(}e^{-\inf_{F\cap\widetilde{\boldsymbol{\Omega}}_{i}}(\widetilde{U}^{\prime}_{i})}-1\big{)}{P(F)},$

where the perimeters are with respsect to the rescaled and pulled-back metric
$\widetilde{\boldsymbol{h}}_{i}$. Taking $i\to\infty$, using (6.31),
$\theta_{i}\to 1$, and the convergence
$\widetilde{\boldsymbol{h}}_{i}\to\textbf{euc}$ in
$C^{0}_{\operatorname{loc}}(\mathbb{R}^{n+1})$, we obtain by the standard set
replacing argument

${P(E;Q_{R})}\leqslant{P(E\setminus F;Q_{R})},$

where the perimeters are with respect to the Euclidean metric. This proves the
inward minimizing of $E$ in $\mathbb{R}^{n+1}$, thus completes the proof. ∎

###### Proof of Proposition 6.5.

Combining the results of Lemma 6.8 and 6.9, by taking a subsequence we can
assume the following: there is a radius $r_{0}<r_{I}$ such that

$\inf_{\partial\Omega\times\mathbb{R}}\langle\boldsymbol{\nu}_{i},\boldsymbol{\partial_{r}}\rangle_{\boldsymbol{g}}\geqslant
1-\frac{1}{i},\qquad\inf_{(\Omega\setminus\Omega_{-r_{0}})\times\mathbb{R}}\langle\boldsymbol{\nu}_{i},\boldsymbol{\partial_{r}}\rangle_{\boldsymbol{g}}\geqslant\frac{1}{2},\qquad\text{for
all }i.$ (6.32)

Apply Lemma 5.4 with the domain $\Omega\times\mathbb{R}$ in place of $\Omega$,
and with the smooth solutions
$\Sigma_{t}^{i}=\operatorname{graph}\big{(}\varepsilon_{i}^{-1}(u_{i}-t)\big{)}$
and the parameter $b=i^{-1/\gamma}$. The small curvature and
$p\geqslant\frac{1}{2}$ condition there are satisfied by (6.4) (6.32). Arguing
verbatim as in Step 1 of Lemma 6.9, we obtain

$1-\langle\boldsymbol{\nu}_{i},\boldsymbol{\partial_{r}}\rangle_{\boldsymbol{g}}\leqslant
C_{3}\big{(}i^{-1/\gamma}-r/r_{0}\big{)}^{\gamma}\qquad\text{in
}(\Omega\setminus\overline{\Omega_{-r_{0}}})\times\mathbb{R}.$ (6.33)

We next apply Lemma 5.5 in the region
$(\Omega\setminus\overline{\Omega_{-r_{0}}})\times\mathbb{R}$. Consider the
quantity

$G=r_{0}\Big{(}(i^{-1/\gamma}-r/r_{0})^{1-\gamma}-(2i^{-1/\gamma})^{1-\gamma}\Big{)}H,$

where $H$ is the mean curvature of $\Sigma_{t}^{i}$. The conditions in Lemma
5.5 are satisfied by (6.4) (6.32) (6.33) respectively. By the result (5.20)
and the fact that $\Sigma_{t}^{i}$ forms a translating soliton, we obtain the
inequality

$-\frac{1}{H^{2}}\Delta_{\Sigma}G+\langle\nabla_{\Sigma}G,Y\rangle\leqslant-\frac{G}{n}+\frac{4n}{G}+C_{4}\big{(}i^{-1/\gamma}-r/r_{0}\big{)}^{-\gamma}\big{(}\frac{C_{5}}{G}-1\big{)}$

in $(\Omega\setminus\overline{\Omega_{-r_{0}}})\times\mathbb{R}$, for some
$C_{4},C_{5}>0$ independent of $i$. By the maximum principle, this implies

$\displaystyle\max_{(\overline{\Omega_{-br_{0}}}\setminus\Omega_{-r_{0}})\times\mathbb{R}}(G)$
$\displaystyle\leqslant\max\Big{\\{}\max_{\partial\Omega_{-r_{0}}\times\mathbb{R}}(G),\
\max_{\partial\Omega_{-br_{0}}\times\mathbb{R}}(G),2n+C_{5}\Big{\\}}$
$\displaystyle\leqslant\max\Big{\\{}C(n),0,2n+C_{5}\Big{\\}}.$ (6.34)

Here, the control of $G$ on $\partial\Omega_{-r_{0}}\times\mathbb{R}$ follows
by Corollary 2.16 and the assumptions for $r_{I}$:

$G(x)\leqslant r_{0}\cdot 2\cdot
H(x)\leqslant\frac{2C(n+1)r_{0}}{\sigma(x;\Omega\times\mathbb{R},\boldsymbol{g})}\leqslant\frac{2C(n+1)r_{0}}{\sigma(x;\Omega,g)}\leqslant
C,\quad x\in\partial\Omega_{-r_{0}}\times\mathbb{R}.$

So (6.34) implies

$\sqrt{\varepsilon_{i}^{2}+|\nabla
u_{i}^{2}|}=H\leqslant\frac{C(n)r_{0}^{-1}}{(i^{-1/\gamma}-r/r_{0})^{1-\gamma}-(2i^{-1/\gamma})^{1-\gamma}}.$
(6.35)

As $i\to\infty$, we pass (6.33) to the limit and project onto the $\Omega$
factor, and obtain

$1-\langle\nu,\partial_{r}\rangle\leqslant
C_{3}\big{(}|r|/r_{0}\big{)}^{\gamma}\qquad\text{a.e. in}\ \
\Omega\setminus\Omega_{-r_{0}},$ (6.36)

see below (6.17) for the convergence to $\nu$. Passing (6.35) to the limit, we
obtain

$|\nabla u|\leqslant C(n)r_{0}^{-\gamma}|r|^{\gamma-1}\qquad\text{in
}(\Omega\setminus\overline{\Omega_{-r_{0}}})\times\mathbb{R}.$

Finally, by Corollary 3.19 and (6.36), the solution $u$ respects the obstacle
$\partial\Omega$. ∎

###### Proof of Proposition 6.6.

Let $g_{0}$ be the product metric on $\partial\Omega\times(0,r_{0})$, and
$\Phi:\partial\Omega\times(0,r_{0})\to\Omega\setminus\overline{\Omega_{-r_{0}}}$
be the normal exponential map. By (6.4), we have
$\frac{1}{2}g_{0}\leqslant\Phi^{*}g\leqslant 2g_{0}$. By (6.16), (pulling back
via $\Phi$,) we may view $u=u(y,r)$ as a function on
$\partial\Omega\times(0,r_{0})$ with $|\nabla_{g_{0}}u|\leqslant
Cr_{0}^{-\gamma}r^{\gamma-1}$.

For $0<r_{1},r_{2}\leqslant r_{0}/2$, from the gradient bound we have

$\big{|}u(y,r_{1})-u(y,r_{2})\big{|}\leqslant
C\gamma^{-1}r_{0}^{-\gamma}|r_{1}^{\gamma}-r_{2}^{\gamma}|\leqslant
C\gamma^{-1}r_{0}^{-\gamma}|r_{1}-r_{2}|^{\gamma}.$ (6.37)

Note that this implies $u\in
L^{\infty}\big{(}\partial\Omega\times(0,r_{0}/2)\big{)}$. Next, for
$r\leqslant r_{0}/2$ and $d_{\partial\Omega}(y_{1},y_{2})\leqslant r$, we have

$\big{|}u(y_{1},r)-u(y_{2},r)\big{|}\leqslant
Cr_{0}^{-\gamma}r^{\gamma-1}d_{\partial\Omega}(y_{1},y_{2})\leqslant
Cr_{0}^{-\gamma}d_{\partial\Omega}(y_{1},y_{2})^{\gamma}.$ (6.38)

Finally, suppose $r\leqslant r_{0}/2$ and
$d_{\partial\Omega}(y_{1},y_{2})\geqslant r$. If
$d_{\partial\Omega}(y_{1},y_{2})\geqslant r_{0}/2$, then we have

$\big{|}u(y_{1},r)-u(y_{2},r)\big{|}\leqslant
2||u||_{L^{\infty}(\partial\Omega\times(0,r_{0}/2))}\cdot(r_{0}/2)^{-\gamma}\cdot
d_{\partial\Omega}(y_{1},y_{2})^{\gamma}.$ (6.39)

If $r\leqslant d_{\partial\Omega}(y_{1},y_{2})\leqslant r_{0}/2$, then we set
$s=d_{\partial\Omega}(y_{1},y_{2})$ and estimate

$\displaystyle\big{|}u(y_{1},r)-u(y_{2},r)\big{|}$
$\displaystyle\leqslant\big{|}u(y_{1},r)-u(y_{1},s)\big{|}+\big{|}u(y_{2},r)-u(y_{2},s)\big{|}+\big{|}u(y_{1},s)-u(y_{2},s)\big{|}$
$\displaystyle\leqslant
2C\gamma^{-1}r_{0}^{-\gamma}|s-r|^{\gamma}+Cr_{0}^{-\gamma}s^{\gamma-1}d_{\partial\Omega}(y_{1},y_{2})$
$\displaystyle\leqslant(2C\gamma^{-1}+C)r_{0}^{-\gamma}d_{\partial\Omega}(y_{1},y_{2})^{\gamma}.$

Combined with (6.37) $\sim$ (6.39), it follows that $u$ can be extend to a
$C^{0,\gamma}$ function on $\partial\Omega\times[0,r_{0}/2]$. Finally, by the
smallness of $r_{0}$, there is a constant $C$ such that
$d_{\partial\Omega\times[0,r_{0}/2]}(x,y)\leqslant Cd_{g}(\Phi(x),\Phi(y))$.
This shows that $u$ extends to a $C^{0,\gamma}$ function on
$\overline{\Omega}\setminus\Omega_{-r_{0}/2}$. ∎

###### Proof of Proposition 6.7.

For any $x\in\Omega\setminus\Omega_{-r_{0}/3}$ and $r<r_{0}/3$, by (6.16) and
the choice of $r_{I}$ we have

$\displaystyle\int_{B(x,r)\cap\Omega}|\nabla u|$
$\displaystyle\leqslant\int_{\max\\{0,|r(x)|-r\\}}^{|r(x)|+r}\mathcal{H}^{n-1}\big{(}B(x,r)\cap\Omega_{-s}\big{)}Cr_{0}^{-\gamma}s^{\gamma-1}\,ds$
(6.40) $\displaystyle\leqslant C^{\prime}r_{0}^{-\gamma}r^{n-1+\gamma}.$

It is already known that $u$ respects the obstacle $\partial\Omega$. For all
$t>0$ and each competitor set $E\subset M$ satisfying $E\Delta
E_{t}\subset\subset B(x,r)$, we compare the energy
$\widetilde{J}_{u}^{B(x,r)}(E_{t})\leqslant\widetilde{J}_{u}^{B(x,r)}(E\cap\Omega)$
and obtain (note that $E_{t}\Delta(E\cap\Omega)\subset\subset B(x,r)$ since
$E_{t}\subset\Omega$)

$\displaystyle{P\big{(}E_{t};B(x,r)\big{)}}$
$\displaystyle\leqslant{P\big{(}E\cap\Omega;B(x,r)\big{)}}+\int_{E_{t}\Delta(E\cap\Omega)}|\nabla
u|$
$\displaystyle\leqslant{P\big{(}E;B(x,r)\big{)}}+{P\big{(}\Omega;B(x,r)\big{)}}-{P\big{(}E\cup\Omega;B(x,r)\big{)}}+Cr_{0}^{-\gamma}r^{n-1+\gamma}.$

It is directly verifiable, using the $C^{2}$ smoothness of $\partial\Omega$
(see [56, Section 1.6]), that

${P\big{(}\Omega;B(x,r)\big{)}}\leqslant{P\big{(}E\cup\Omega;B(x,r)\big{)}}+Cr_{0}^{-2}r^{n+1}.$

Thus we obtain the almost-minimizing condition

${P\big{(}E_{t};B(x,r)\big{)}}\leqslant{P\big{(}E;B(x,r)\big{)}}+Cr_{0}^{-\gamma}r^{n-1+\gamma}.$
(6.41)

Next, combining (6.15) and Remark 2.12, we obtain that almost every $E_{t}$
satisfies

$1-\langle\nu_{E_{t}},\partial_{r}\rangle\leqslant
C_{3}r_{0}^{-\gamma}|r|^{\gamma}.$ (6.42)

In addition, on $\partial^{*}E_{t}\cap\partial^{*}\Omega$ it is clear that
$\nu_{E_{t}}=\partial_{r}$. Since $\partial_{r}$ is smooth, this has the
following consequence: for any $l>0$ there exists a sufficiently small radius
$r(l)$, such that for all $x\in\overline{\Omega}\setminus\Omega_{-r(l)}$,
$s\leqslant r(l)$, the set $E_{t}\cap B(x,s)$ is representable as the sub-
graph of a $l$-Lipschitz function in some geodesic normal coordinate near $x$.
Conbining (6.41) and this small slope condition, the classical small excess
regularity theorem (see [55] and [4, 56]) implies the following: there is a
sufficiently small $r_{1}<r_{0}$, such that almost every
$E_{t}\setminus\overline{\Omega_{-r_{1}}}$ is a $C^{1,\gamma/2}$ hypersurface.
By the Arzela-Ascoli theorem, the same conclusion holds for all $t>0$. This
proves the desired result. ∎

##  7 On maximal solutions

The following is a restatement of Theorem 1.7.

######  Theorem 7.1.

Let $(M,g)$ be a (possibly incomplete) smooth Riemannian manifold, and
$E_{0}\subset M$ be a (possibly unbounded) $C^{1,1}$ domain. Then there
exists, up to equivalence, a unique maximal solution of
$\mathsf{IVP}\boldsymbol{(}{M;E_{0}}\boldsymbol{)}$, in the sense of
Definition 2.5.

###### Proof.

Find a sequence of precompact $C^{1,1}$ domains $E_{0}^{i}$, such that
$E_{0}^{1}\subset E_{0}^{2}\subset\cdots\subset E_{0}$, and
$\bigcup_{i\geqslant 1}E_{0}^{i}=E_{0}$, and that for all $K\subset\subset M$,
the sets $E_{0}^{i}\cap K$ eventually stablize (i.e. $E_{0}^{i}\cap
K=E_{0}^{i+1}\cap K=E_{0}^{i+2}\cap K=\cdots$ for large enough $i$). Next,
find a sequence of smooth precompact domains $\Omega_{i}\supset\supset
E_{0}^{i+1}$ such that
$\Omega_{1}\subset\subset\Omega_{2}\subset\subset\cdots\subset M$ and
$\bigcup_{i\geqslant 1}\Omega_{i}=M$. Let $u_{i}$ be the (unique) solution of
$\mathsf{IVP}\boldsymbol{(}{\Omega_{i};E_{0}^{i}}\boldsymbol{)}\texttt{+}\mathsf{OBS}\boldsymbol{(}{\partial\Omega_{i}}\boldsymbol{)}$,
given by Theorem 6.1.

Applying Corollary 3.22 to each $\Omega_{i}$, we have $u_{1}\geqslant
u_{2}\geqslant u_{3}\geqslant\cdots$ outside $E_{0}$. By the interior gradient
estimates (6.1) and Arzela-Ascoli theorem, some subsequence of $u_{i}$
converge in $C^{0}_{\operatorname{loc}}$ to a function
$u\in\operatorname{Lip}_{\operatorname{loc}}(M\setminus\overline{E_{0}})$. By
Theorem 2.13, $u$ is a calibrated weak solution in
$M\setminus\overline{E_{0}}$. Since (6.1) provides a uniform gradient estimate
up to $\partial E_{0}$, the resulting function $u$ is Lipschitz up to
$\partial E_{0}$, and satisfies $u\geqslant 0$, $u|_{\partial E_{0}}=0$.
Extending $u$ by negative values inside $E_{0}$, we obtain a solution of
$\mathsf{IVP}\boldsymbol{(}{M;E_{0}}\boldsymbol{)}$.

It remains to show that $u$ is maximal. Suppose
$v\in\operatorname{Lip}_{\operatorname{loc}}(M)$ is another solution of
$\mathsf{IVP}\boldsymbol{(}{M;E_{0}}\boldsymbol{)}$. For each $i$ we find
functions $v_{i}\in\operatorname{Lip}_{\operatorname{loc}}(\Omega_{i})$, such
that $v_{i}=v$ on $\Omega_{i}\setminus E_{0}$, and $v_{i}=0$ on
$E_{0}\setminus E_{0}^{i}$, and $v_{i}<0$ on $E_{0}^{i}$. By Definition
2.4(ii) it can be verified that $v_{i}$ is a subsolution of
$\mathsf{IVP}\boldsymbol{(}{\Omega_{i};E_{0}^{i}}\boldsymbol{)}$. Then by
Corollary 3.22, we obtain $v_{i}\leqslant u_{i}$ on $\Omega_{i}\setminus
E_{0}^{i}$, hence $v\leqslant u_{i}$ on $\Omega_{i}\setminus E_{0}$. As $u$ is
the descending limit of $u_{i}$, it follows that $v\leqslant u$ on $M\setminus
E_{0}$. ∎

When $\Omega$ is a non-compact domain, the fact that $u$ solves
$\mathsf{IVP}\boldsymbol{(}{\Omega;E_{0}}\boldsymbol{)}\texttt{+}\mathsf{OBS}\boldsymbol{(}{\partial\Omega}\boldsymbol{)}$
does not imply that it is the maximal solution of
$\mathsf{IVP}\boldsymbol{(}{\Omega;E_{0}}\boldsymbol{)}$. Intuitively, the
maximal solution respects not only the obstacle $\partial\Omega$, but also the
“invisible obstacle” at infinity. Applying Corollary 3.13 to each $u_{i}$ in
the proof above, and taking $i\to\infty$, we find by lower semi-continuity

${P(E_{t})}\leqslant e^{t}{P(E_{0})}$

for the maximal solution. In comparison, if $u$ is a proper solution then we
have equality. Thus, a portion of the perimeter may be lost at infinity.

Finally, we include the following examples of maximal solutions.

1\. In Choi-Daskalopoulos [13], one considers the (smooth) IMCF with with a
convex, non-compact, $C^{1,1}$ initial domain $E_{0}\subset\mathbb{R}^{n}$. A
compact approximation argument is employed to obtain a solution: one finds
bounded convex domains $E_{0}^{1}\subset E_{0}^{2}\subset\cdots$ with
$\bigcup_{i\geqslant 1}E_{0}^{i}=E_{0}$, and then take $u_{i}$ to be the
proper (hence maximal) solution of
$\mathsf{IVP}\boldsymbol{(}{\mathbb{R}^{n};E_{0}^{i}}\boldsymbol{)}$, and
finally take the descending limit $u=\lim_{i\to\infty}u_{i}$. Arguing
similarly as above, it follows that $u$ is the maximal solution of
$\mathsf{IVP}\boldsymbol{(}{\mathbb{R}^{n};E_{0}}\boldsymbol{)}$.

2\. In the setting of Lemma 2.7, the function

$u_{0}(r)=(n-1)\log\big{[}f(r)/f(r_{0})\big{]}$

is the maximal solution of
$\mathsf{IVP}\boldsymbol{(}{\Omega;E_{0}}\boldsymbol{)}$.

3\. Consider $E_{0}$ a half-space in the hyperbolic space, and let $u$ be the
maximal solution of
$\mathsf{IVP}\boldsymbol{(}{\mathbb{H}^{n};E_{0}}\boldsymbol{)}$. By
uniqueness, the function $u|_{\mathbb{H}^{n}\setminus E_{0}}$ must be
invariant under any isometry that preserves $E_{0}$. Therefore, the level sets
of $u$ are equi-distance sets from $E_{0}$. This reduces the flow to a one-
dimension problem: the maximal solution must be

$u=(n-1)\log\cosh d(x,E_{0}).$

##  Appendix A Soliton examples in $\mathbb{R}^{2}$

In this section, we collect several smooth solitons of IMCF in the plane. We
also refer to the previous works [9, 10, 16, 30, 31]. In these works, it was
already noticed that planar solitons of IMCF are usually incomplete. However,
due to the nature of singularities at their endpoints, the soliton curves
contact tangentially with the boundary of the region that they sweep out.
Therefore, the solutions that they generate would respect the boundary
obstacle. This observation provides us the first class of nontrivial explicit
examples.

We expect that the solitons presented here, as well as their analogies in more
general circumstances, appear as the blow-down limits at infinity or the blow-
up limits at non-smooth boundary points. See Question A.7, A.8 for more
elaborations. Another role of the examples is to provide insights in forming
the main theory in Section 3 and 4.

The following setups are assumed. Suppose $\gamma\subset\mathbb{R}^{2}$ is a
smooth curve with nonvanishing curvature, with unit speed parametrization $s$.
Let $\tau:=d\gamma/ds$ be its unit tangential vector, and $\theta$ be the
angle function so that $\tau=(\cos\theta,\sin\theta)$, and
$\nu:=(\sin\theta,-\cos\theta)$ be the normal vector, finally
$\kappa:=d\theta/ds$ be the curvature. Set the support function $h=x\cdot\nu$,
where $x$ is the position vector. The following formula is well-known:

$\gamma=h\nu+\frac{dh}{d\theta}\tau,$ (A.1)

where $dh/d\theta=\frac{dh/ds}{d\theta/ds}=\frac{dh/ds}{\kappa}$ is a well-
defined quantity.

######  Example A.1 (cycloids).

By [16, Theorem 8], any translating solition $\gamma$ of IMCF in
$\mathbb{R}^{2}$ is isometric to a cycloid. Indeed, after a scaling and rigid
motion, we may assume that $\gamma$ is translating in the positive
$x$-direction with speeed 1. Then the soliton equation reads
$\frac{1}{\kappa}=\langle\nu,\partial_{x}\rangle=\sin\theta$. Solving this
(see [16]), we obtain a cycloid equation

$\gamma(\theta)=\frac{1}{4}\big{(}1-\cos 2\theta,2\theta-\sin
2\theta\big{)},\qquad\theta\in(0,\pi).$ (A.2)

The curve $\gamma$ is embedded into the strip region
$\Omega=\mathbb{R}\times(0,\pi/2)$, and its endpoints $\gamma(0)$,
$\gamma(\pi)$ lie on $\partial\Omega$. By Taylor expansion, near $\gamma(0)$
the curve is the graph of

$y=\frac{2\sqrt{2}}{3}x^{3/2}+O(x^{5/2}),\qquad x>0.$ (A.3)

A similar result holds near $\gamma(\pi)$ by symmetry. By Corollary 3.19, the
translations of $\gamma$ generate a solution of
$\mathsf{IMCF}\boldsymbol{(}{\Omega}\boldsymbol{)}\texttt{+}\mathsf{OBS}\boldsymbol{(}{\partial\Omega}\boldsymbol{)}$
(which is actually smooth).

In the solution generated, each sub-level set $E_{t}$ is the unbounded region
enclosed by the translation $\gamma+(t,0)$ and the two lines
$(-\infty,t)\times\\{0\\}$, $(-\infty,t)\times\\{\pi/2\\}$. By the expansion
(A.3), each $\partial E_{t}$ is globally a $C^{1,1/2}$ curve.

Figure 3: translating cycloids in a strip region. The bold curve represents
$\partial E_{t}$.

######  Remark A.2 (homothetic solitons).

Let $c\in\mathbb{R}\setminus\\{0\\}$. If a curve $\gamma$ satisfies

$h\kappa=c^{-1},$ (A.4)

then the family of curves $\gamma_{t}=e^{ct}\gamma$ solves the IMCF. Urbas
[58] computed that (A.4) is equivalent to

$\frac{d^{2}h}{d\theta^{2}}=(c-1)h.$ (A.5)

For a full classification result based on this equation, see [9, 16, 58].
Furthermore, we refer to [10] for a classification of more complex solitons
with simultaneous scaling and rotating behaviors.

We are interested in the case $c<1$. Solving (A.5), we obtain up to a
normalization $h(\theta)=\sin\big{(}\sqrt{1-c}\,\theta\big{)}$. Since the
curve $\gamma$ becomes singular when $h(\theta)=0$, due to (A.4), we restrict
the parameter to a single period $\theta\in(0,\pi/\sqrt{1-c})$. Then we insert
the expression into (A.1) and obtain the curve $\gamma(\theta)$ in complex
form:

$\gamma(\theta)=-\sin\big{(}\sqrt{1-c}\,\theta\big{)}ie^{i\theta}+\sqrt{1-c}\cos\big{(}\sqrt{1-c}\,\theta\big{)}e^{i\theta}.$
(A.6)

Setting $\mu=1-\sqrt{1-c}$ and $k=\frac{1+\sqrt{1-c}}{1-\sqrt{1-c}}$, we can
rewrite (A.6) as

$\gamma(\theta)=\frac{1}{2}\mu\big{(}ke^{i\mu\theta}-e^{ik\mu\theta}\big{)}.$
(A.7)

Comparing with the classical equations [36, Section 6.3, 6.5], we find that
(A.7) describes a hypocycloid ($c<0$) or epicycloid ($c>0$). This was
previously observed in [9, 16].

Figure 4: shrinking hypocycloids and expanding epicycloids, expanding
nephroids. These correspond to Examples A.3, A.4, A.6 respectively.

######  Example A.3 ($c<0$, shrinking hypocycloids).

Assume $c<0$, and set $T=\pi/\sqrt{1-c}$. From (A.6) we have
$\gamma(0)=\sqrt{1-c}$ and $\gamma(T)=\sqrt{1-c}\,e^{i\pi/\sqrt{1-c}-i\pi}$.
Hence $\gamma$ is supported in a planar cone $\Omega$ with angle
$\big{(}1-\frac{1}{\sqrt{1-c}}\big{)}\pi$. Now the family of curves
$\gamma_{t}=e^{ct}\gamma$ form a smooth solution of IMCF in $\Omega$. By the
fact $\lim_{\theta\to 0}h(\theta)=\lim_{\theta\to T}h(\theta)=0$, the normal
vector of $\gamma$ converges to the outer normal vector of $\partial\Omega$
when $\theta\to 0,T$. Then by Corollary 3.19 and scaling invariance, the
solution generated by $\gamma$ respects the obstacle $\partial\Omega$. In such
a solution, the sub-level set $E_{t}$ is the unbounded region enclosed by
$\gamma_{t}$ and the two radial lines emeneting from $\gamma_{t}(0)$ and
$\gamma_{t}(T)$. The boundary of $E_{t}$ is a $C^{1,1/2}$ curve, by similar
calculations as above.

This provides examples of solutions of IMCF that approach $+\infty$ near
$\partial\Omega$.

######  Example A.4 ($c>0$, expanding epicycloids).

Assume $c>0$. Similarly as above, the family of curves
$\gamma_{t}=e^{ct}\gamma$ form a solution of IMCF in a cone domain $\Omega$
with angle $\big{(}\frac{1}{\sqrt{1-c}}-1\big{)}\pi$, which respects the
obstacle $\partial\Omega$. The sub-level set $E_{t}$ is the bounded region
enclosed by $\gamma_{t}$ and the two line segments joining 0 and
$\gamma_{t}(0)$, $\gamma_{t}\big{(}\pi/\sqrt{1-c}\big{)}$. Note that $\gamma$
is embedded in $\mathbb{R}^{2}$ only when $c\leqslant 8/9$. For the case
$c>8/9$, we may view $\gamma$ as embeddedd in the universal cover of
$\mathbb{R}^{2}\setminus\\{0\\}$.

This provides examples of solutions that approach $-\infty$ near
$\partial\Omega$.

######  Example A.5 (solution in bounded domains).

Let $u$ be a solution in Example A.4 generated by an epicycloid. The
restriction of $u$ inside $E_{t}(u)$ is another solution that respects the
obstacle $\partial E_{t}(u)$. This provides an example of nontrivial solutions
in precompact domains. See Remark 3.5(vi) and Theorem 3.20 for related
discussions.

######  Example A.6 ($c=3/4$, nephroid).

In the special case $c=\frac{3}{4}$ in Example A.4, the supporting domain
$\Omega$ becomes a half plane. The corresponding curve $\gamma$ is often
called a nephroid. This particular example shows that the condition
$\inf_{\Omega}(u)>-\infty$ in Theorem 4.1 cannot be removed.

In [24, Proposition 7.2], it is proved that for a proper weak solution $u$ in
$\mathbb{R}^{n}$, the rescaled limit at infinity is the unique proper eternal
solution in $\mathbb{R}^{n}\setminus\\{0\\}$, which is
$u_{\infty}=(n-1)\log|x|$. It is natural to ask whether similar facts hold
under the present of obstacles. Below we say that $Q\subset\mathbb{R}^{n}$ is
a conic domain with smooth link, if $\lambda Q=Q$ for all $\lambda>0$ and
$\partial Q\setminus\\{0\\}$ is a smooth hypersurface.

######  Question A.7.

Let $Q\subset\mathbb{R}^{n}$ be a lically Lipschitz conic domain with smooth
link. Then there is a unique expanding soliton that generates a solution of
$\mathsf{IMCF}\boldsymbol{(}{Q}\boldsymbol{)}\texttt{\emph{+}}\mathsf{OBS}\boldsymbol{(}{\partial
Q}\boldsymbol{)}$. Suppose $\Omega\subset\mathbb{R}^{n}$ is smooth and
asymptotic to $Q$ at infinity. Consider the maximal solution of
$\mathsf{IVP}\boldsymbol{(}{\Omega;E_{0}}\boldsymbol{)}$ for some
$E_{0}\subset\subset\Omega$. Then at spatial infinity, a suitably normalized
blow-down sequence converges to the expanding soliton supported in $Q$.

On the other hand, shrinking solitons are related to the asymptotic limit near
a rough boundary point, when the boundary converges nicely to a convex tangent
cone. In this respect, we raise the following

######  Question A.8.

Let $Q\subset\mathbb{R}^{n}$ be a convex conic domain with smooth link. Then
there is a unique shrinking soliton that generates a solution of
$\mathsf{IMCF}\boldsymbol{(}{Q}\boldsymbol{)}\texttt{\emph{+}}\mathsf{OBS}\boldsymbol{(}{\partial
Q}\boldsymbol{)}$. Let $\Omega\subset\mathbb{R}^{n}$ be a bounded domain, with
$0\in\partial\Omega$, and $\partial\Omega\setminus\\{0\\}$ is smooth, and
$\Omega$ is asymptotic to $Q$ near $0$. Consider the maximal solution of
$\mathsf{IVP}\boldsymbol{(}{\Omega;E_{0}}\boldsymbol{)}$ for some
$E_{0}\subset\subset\Omega$. Then at $x$, a normalized blow-up sequence
converges to the shrinking soliton supported in $Q$.

The solution $u$ in both questions are characterized as the maximal solution.
Since the main existence Theorem 1.6 requires smoothness of the domain, we did
not state that $u$ is a solution of
$\mathsf{IVP}\boldsymbol{(}{\Omega;E_{0}}\boldsymbol{)}\texttt{+}\mathsf{OBS}\boldsymbol{(}{\partial\Omega}\boldsymbol{)}$,
although this is expected.

##  Appendix B Backgrounds in geometric measure theory

We collect relevant results in geometric measure theory, which are used in the
main text. We assume that the reader is familiar with the notions of sets with
locally finite perimeter and reduced boundary. For a detailed introduction on
this topic, we refer to the books [5, 17, 41]. Given a set $E$ with locally
finite perimeter, we denote by $\partial^{*}E$ the reduced boundary of $E$,
and $\nu_{E}$ the measure-theoretic outer unit normal, and $|D\chi_{E}|$ the
perimeter measure. De Giorgi’s structure theorem implies that
$|D\chi_{E}|=\mathcal{H}^{n-1}\llcorner\partial^{*}E$. For an open set
$\Omega$, we denote ${P(E;\Omega)}=|D\chi_{E}|(\Omega)$ the perimeter of $E$
inside $U$.

Given a measurable set $E$, and $\alpha\in[0,1]$, we define

$\displaystyle E^{(\alpha)}=\Big{\\{}x\in M:\lim_{r\to 0}\frac{|E\cap
B(x,r)|}{|B(x,r)|}=\alpha\Big{\\}}.$

The set $E^{(1)}$ is often called the measure-theoretic interior. Federer
proved that if $E$ has locally finite perimeter, then we have a disjoint union

$M=E^{(1)}\cup E^{(0)}\cup\partial^{*}E\cup Z$ (B.1)

for some $Z$ with $\mathcal{H}^{n-1}(Z)=0$. See [41, Theorem 16.2]. For sets
$E,F$ with locally finite perimeters, such that
$\mathcal{H}^{n-1}\big{(}\partial^{*}E\cap\partial^{*}F\big{)}=0$, we have the
decomposition of perimeters

$\displaystyle{P(E\cap F;K)}$
$\displaystyle=\mathcal{H}^{n-1}\big{(}\partial^{*}E\cap F^{(1)}\cap
K\big{)}+\mathcal{H}^{n-1}\big{(}\partial^{*}F\cap E^{(1)}\cap K\big{)},$
(B.2) $\displaystyle{P(E\cup F;K)}$
$\displaystyle=\mathcal{H}^{n-1}\big{(}\partial^{*}E\cap F^{(0)}\cap
K\big{)}+\mathcal{H}^{n-1}\big{(}\partial^{*}F\cap E^{(0)}\cap K\big{)},$
$\displaystyle{P(E\setminus F;K)}$
$\displaystyle=\mathcal{H}^{n-1}\big{(}\partial^{*}E\cap F^{(0)}\cap
K\big{)}+\mathcal{H}^{n-1}\big{(}\partial^{*}F\cap E^{(1)}\cap K\big{)},$

see [41, Theorem 16.3].

Let $\Omega$ be a domain in a Riemannian manifold $(M,g)$, and
$\Lambda,r_{0}>0$ be two constants. We call $E$ a $(\Lambda,r_{0})$-perimeter
minimizer in $(\Omega,g)$, if for all balls $B(x,r)\subset\subset M$ with
$r\leqslant r_{0}$, and all comparison sets $F$ with $E\Delta F\subset\subset
B(x,r)\cap\Omega$, we have

${P\big{(}E;B(x,r)\big{)}}\leqslant{P\big{(}F;B(x,r)\big{)}}+\Lambda|E\Delta
F|.$ (B.3)

This definition follows [41, Chapter 21] except that we do not assume
$\operatorname{spt}(|D\chi_{E}|)=\partial E$. It is well-known [4, 41, 55]
that the measure-theoretic boundary of a $(\Lambda,r_{0})$-perimeter minimizer
is a $C^{1,\alpha}$ ($\alpha<1/2$) hypersurface except for a codimension 8
singular set.

The following regularity and convergence results follow by combining (the
Riemannian analogue of) Theorem 21.11, 21.14 and 26.6 in [41], see also [55,
56].

######  Lemma B.1.

Let $(M,g)$ be a Riemannian manifold. Suppose the domains $\Omega_{i}$
converges to $\Omega$ locally, and the metrics $g_{i}\to g$ locally smoothly.
Suppose $E_{i}\subset\Omega_{i}$ are $(\Lambda,r_{0})$-perimeter minimizers in
$(\Omega_{i},g_{i})$, and $x_{i}\in\operatorname{spt}(|D\chi_{E_{i}}|)$, such
that $E_{i}\to E$ in $L^{1}_{\operatorname{loc}}$ and $x_{i}\to x$. Then:

(i) $E$ is a $(\Lambda,r_{0})$-perimeter minimizer in $(\Omega,g)$, and
$x\in\operatorname{spt}(|D\chi_{E}|)$, and the upper density of $E$ at $x$ is
bounded above by a dimensional constant $c<1$.

(ii) If $x\in\partial^{*}E$, then $x_{i}\in\partial^{*}E_{i}$ for sufficiently
large $i$, and we have $\nu_{E_{i}}(x_{i})\to\nu_{E}(x)$.

Finally, we prove two technical lemmas that are used in the main text. We
denote by $\\{e_{i}\\}$ the standard basis of $\mathbb{R}^{n}$, and for a
point $x=(x_{1},\cdots,x_{n-1},x_{n})\in\mathbb{R}^{n}$, we write
$x=(x^{\prime},x_{n})$ for short.

######  Lemma B.2.

Let $\Omega\subset\mathbb{R}^{n}$ be a convex domain. Fix
$\theta\in(0,\pi/2)$. Suppose $E\subset\mathbb{R}^{n}$ has locally finite
perimeter. Moreover, assume that

$\operatorname*{ess\,inf}_{\partial^{*}E\cap\Omega}\langle\nu_{E},e_{n}\rangle\geqslant\cos\theta.$

If $E$ has nonzero lower density at a point $x=(x^{\prime},x_{n})\in\Omega$,
then $E$ has density 1 on the set

$\Omega\cap\Big{\\{}(y^{\prime},y_{n})\in\mathbb{R}^{n}:y_{n}<x_{n}-|y^{\prime}-x^{\prime}|\tan\theta\Big{\\}}.$
(B.4)

###### Proof.

Let $\rho_{\varepsilon}$ ($\varepsilon>0$) be a family of standard mollifiers,
and set $f_{\varepsilon}=\chi_{E}*\rho_{\varepsilon}$. Let
$\Omega^{\prime}\subset\subset\Omega$ be another convex domain. For any smooth
vector field $X$ with $\operatorname{supp}(X)\subset\Omega^{\prime}$ and any
$\varepsilon<d(\partial\Omega^{\prime},\partial\Omega)$, we compute

$\displaystyle-\int_{\mathbb{R}^{n}}\nabla f_{\varepsilon}\cdot X$
$\displaystyle=\int_{\mathbb{R}^{n}}f_{\varepsilon}\operatorname{div}X=\int_{E}\operatorname{div}(\rho_{\varepsilon}*X)=\int_{\partial^{*}E}(\rho_{\varepsilon}*X)\cdot\nu_{E}\,d\mathcal{H}^{n-1}.$

Let $w$ be any constant vector field with $\langle
w,e_{n}\rangle\geqslant\sin\theta$ (thus it holds
$\langle\nu_{E},w\rangle\geqslant 0$ a.e.). Setting $X=\varphi w$ for
arbitrary $\varphi\in C^{\infty}_{0}(\Omega^{\prime})$, $\varphi\geqslant 0$,
we obtain

$-\int_{\mathbb{R}^{n}}\varphi\frac{\partial f_{\varepsilon}}{\partial
w}=\int_{\partial^{*}E}(\rho_{\varepsilon}*\varphi)\langle
w,\nu_{E}\rangle\geqslant 0.$

This implies $\frac{\partial f_{\varepsilon}}{\partial w}\leqslant 0$ in
$\Omega^{\prime}$. Let $\Theta>0$ be the lower density of $E$ at the given
point $x$. For all sufficiently small $\varepsilon$ we have
$f_{\varepsilon}(x)\geqslant\frac{1}{2}\Theta$, and it follows by the
convexity of $\Omega^{\prime}$ that
$f_{\varepsilon}\geqslant\frac{1}{2}\Theta$ in the region

$Q:=\Omega^{\prime}\cap\Big{\\{}(y^{\prime},y_{n}):y_{n}<x_{n}-|y^{\prime}-x^{\prime}|\tan\theta\Big{\\}}.$

Taking $\varepsilon\to 0$, we conclude that $E$ has density at least
$\frac{1}{2}\Theta$ almost everywhere in $Q$. Therefore $E$ occupies full
measure in $Q$. The result follows by taking $\Omega^{\prime}\to\Omega$. ∎

######  Lemma B.3.

Let $E\subset\\{x_{n}<0\\}$ be nonempty and locally perimeter-minimizing in
$\mathbb{R}^{n}$. Then $E$ must be a half space, i.e. there exists $c>0$ such
that $E=\big{\\{}(x^{\prime},x_{n}):x_{n}<-c\big{\\}}$.

###### Proof.

Choose $x\in\partial^{*}E$. There exists a sequence $\lambda_{i}\to\infty$,
such that the limit $E_{\infty}=\lim_{i\to\infty}\lambda_{i}(E-x)$ exists and
is a perimeter-minimizing cone. Since $E_{\infty}\subset\\{x_{n}\leqslant
0\\}$ and $0\in\operatorname{spt}\big{(}|D\chi_{E_{\infty}}|\big{)}$, it
follows by a strong maximum principle [59, Theorem 4] that $\partial
E_{\infty}=\\{x_{n}=0\\}$. Then by the classical monotonicity formula (see
[41, Theorem 28.17] for details), $\partial E$ itself is a cone centered at
$x$. Thus $\partial E$ must be a flat hyperplane. ∎

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Department of mathematics, Duke University, Durham NC, USA.

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