Volume preserving mean curvature flow for star-shaped sets

Abstract

We study the evolution of star-shaped sets in volume preserving mean curvature flow. Constructed by approximate minimizing movements, our solution preserves a strong version of star-shapedness. We also show that the solution converges to a ball as time goes to infinity. For asymptotic behavior of the solution we use the gradient flow structure of the problem, whereas a modified notion of viscosity solutions is introduced to study the geometric properties of the flow by moving planes method.

Appendices

Appendix A: Examples

1.1 A.1: Examples of unbounded total mean curvature

In this section, we present examples of unbounded total mean curvature for ${\mathbb R}^2$ in Example A.1 and ${\mathbb R}^n, n \ge 3$ in Example A.2. Here is an example of a non-convex domain in ${\mathbb R}^2$ such that the total mean curvature is unbounded although perimeter and volume are bounded.

Example A.1

Let $\{O_{i}\}_{i \in {\mathbb N}}$ be a sequence of mutually disjoint balls in ${\mathbb R}^2$ with radius $r_{i}:=\frac{1}{i^2}$ such that $O_{i} \subset \subset B_{10}$ for all $i \in {\mathbb N}$. Define a domain $\Omega$ by

$$\begin{aligned} \Omega := B_{10} - \bigcup _{i=1}^{\infty } O_i. \end{aligned}$$

(A.1)

Gauss–Bonnet theorem implies $\int _{\partial O_i} H = -2\pi$ for all $i \in {\mathbb N}$. Thus, total mean curvature is unbounded although perimeter and volume are positive and bounded as follows,

$$\begin{aligned} |\Omega | = w_n \left( 10^2 - \sum _{i=1}^{\infty } \frac{1}{i^4} \right) , \quad \text {Per}(\Omega ) = n w_n \left( 10- \sum _{i=1}^{\infty } \frac{1}{i^2} \right) . \end{aligned}$$

(A.2)

Based on Example A.1, we can construct a simply connected and star-shaped set, whose total mean curvature is unbounded for $n\ge 3$ (see Example A.2 and Lemma A.3).

Example A.2

Let $\phi :{\mathbb R}^n \rightarrow {\mathbb R}^+$ be a smooth function such that $\phi$ is concave,

$$\begin{aligned} 0< \phi <1 \hbox { in }B_1^{n-1}, \quad \phi = 0 \hbox { in }(B_1^{n-1})^c \hbox { and } |D\phi | \le 1 \hbox { in } {\mathbb R}^n. \end{aligned}$$

(A.3)

Denote

$$\begin{aligned} {\mathcal D}&:= \{ [0,\phi (x')] \times x' : x' \in B_1^{n-1} \} \subset {\mathbb R}^n, \quad \nonumber \\ \partial ^- D&:= B_1^{n-1} \text { and } \partial ^+ D:= \{ (\phi (x'),x') : x' \in B_1^{n-1} \}. \end{aligned}$$

(A.4)

Here, $B_1^{n-1}$ is a ball of radius 1 and center 0 in ${\mathbb R}^{n-1}$.

As Example A.1, choose $\{ x_i \}_{ i \in {\mathbb N}} \subset {\mathbb R}^{n-1}$ such that $B_{r_i}^{n-1}(x_i)$ are a sequence of disjoint balls such that

$$\begin{aligned} r_{i}:=\frac{1}{i^{\frac{1}{n-2}}} \hbox { and } x_i \in B_{10}^{n-1}. \end{aligned}$$

(A.5)

Define a domain $\Omega \subset {\mathbb R}^n$ by

$$\begin{aligned} \Omega := \Omega _1 \cup \Omega _2 \hbox { where } \Omega _1 := [-40,40]^n \quad \hbox { and } \quad \Omega _2 := \bigcup _{i=1}^{\infty } (r_i {\mathcal D}+ (40, x_i)) . \end{aligned}$$

(A.6)

where $e_1$ is a unit vector of the first axis. From Lemma A.3, $\Omega$ is star-shaped with respect to some ball.

The boundary is divided by two parts,

$$\begin{aligned} {\partial \Omega _t}&= \Gamma _1 \cup \Gamma _2 \hbox { where } \Gamma _1:= \left( \partial \Omega _1 {\setminus } \left( \bigcup _{i=1}^{\infty } r_i \partial {\mathcal D}^-+ (40,x_i) \right) \right) \hbox { and }\nonumber \\ \Gamma _2&:= \bigcup _{i=1}^{\infty } (r_i\partial {\mathcal D}^+ + (40,x_i)). \end{aligned}$$

(A.7)

Note that $\Gamma _1$ has bounded total mean curvature, but total mean curvature of $\Gamma _2$ is unbounded as follows. From (A.5) and the change of variables, it holds that

$$\begin{aligned} \int _{\Gamma _2} H d\sigma = \sum _{i=1}^{\infty } \int _{r_i\partial {\mathcal D}^+ + x_i} H d\sigma = \left( \int _{\partial {\mathcal D}^+} H d\sigma \right) \sum _{i=1}^{\infty } r_{i}^{n-2}. \end{aligned}$$

(A.8)

Then, (A.7) implies that

$$\begin{aligned} \int _{\Gamma _2} H d\sigma = \left( \int _{\partial {\mathcal D}^+} H d\sigma \right) \sum _{i=1}^{\infty } \frac{1}{i}, \end{aligned}$$

(A.9)

while volume and perimeter are bounded as follows.

$$\begin{aligned} |\Omega | = \left( 80^n + |{\mathcal D}| \sum _{i=1}^{\infty } r_i^{n} \right) , \quad \text {Per}(\Omega ) = \left( 80^{n-1} n - \sum _{i=1}^{\infty } w_{n-1} r_i^{n-1} + |\partial ^+ {\mathcal D}| \sum _{i=1}^{\infty } r_i^{n-1} \right) . \end{aligned}$$

(A.10)

Lemma A.3

There exists $r>0$ such that $\Omega$ given in Example A.2 is star-shaped with respect to $B_r$.

Proof

As $\Omega _1$ given in (A.6) is star-shaped, it is enough to consider a point in $\Omega _2$. Note that $\Gamma _2$ given in (A.7) is smooth. From Lemma C.1, it is enough to show that

$$\begin{aligned} x \cdot \mathbf {n}_x \ge r \end{aligned}$$

(A.11)

for some $r>0$ and all $x \in \Gamma _2$. Denote $x = (x^1,x') \in {\mathbb R}^n$. For $x \in r_i \partial {\mathcal D}^+ + (40,x_i)$, there exists $y_i \in B_1^{n-1}$ such that

$$\begin{aligned} x = \left( r_i \phi (y_i), r_i y_i\right) + (40,x_i) \end{aligned}$$

(A.12)

and thus

$$\begin{aligned} \mathbf {n}_x = \frac{ (1, - D\phi (y_i))}{\sqrt{ 1 + |D\phi (y_i)|^2}}. \end{aligned}$$

(A.13)

From (A.3) and (A.5), it holds that

$$\begin{aligned} x \cdot \mathbf {n}_x \ge \frac{40}{\sqrt{ 1 + |D\phi (y_i)|^2}} - \frac{|D\phi (y_i) | (| r_i y_i| + |x_i|) }{\sqrt{ 1 + |D\phi (y_i)|^2}} \ge 10. \end{aligned}$$

(A.14)

$\square$

1.2 A.2: A non-convex set satisfying the $\rho \textit{-reflection}$ property

Example A.4

Consider a non-convex set $O_r$ for $r> 2\rho$ defined by

$$\begin{aligned} O_r := (B_r(0) + \rho e_1) \cup (B_r(0) - \rho e_1) \end{aligned}$$

(A.15)

where $e_1$ denotes the first unit vector. Then, $O_r$ has $\rho \textit{-reflection}$ for all $r>2\rho$ (see Lemma A.5). In particular, we can find a non-convex set with unit volume satisfying the assumption (1.5) in Theorem 1 (see Lemma A.6).

Lemma A.5

For $r> 2\rho$, $O_r$ given in (A.15) has $\rho \textit{-reflection}$.

Proof

First of all, as $r> 2\rho$, we have $\overline{B_\rho (0)} \subset B_r(0) \pm \rho e_1$ and thus $O_r$ satisfies the first condition of $\rho \textit{-reflection}$ in Definition 2.1.

Note that a ball $B_r(x_0)$ satisfies (2.1) if the half space $\Pi ^-_{\nu }(s)$ contains $x_0$. For $\nu \in S^{n-1}$ and all $s>\rho$, $\Pi ^-_{\nu }(s)$ contains $\pm \rho e_1$ and $\pm \rho e_1$ are the centers of balls $B_r(0) \pm \rho e_1$. Using this fact and the aforementioned reflection property of the balls, we have

$$\begin{aligned} \Psi _{\Pi _\nu (s)} ((B_r(0) + \rho e_1) \cap \Pi ^+_{\nu }(s)) \subset (B_r(0) + \rho e_1) \cap \Pi ^-_{\nu }(s) \end{aligned}$$

and

$$\begin{aligned} \Psi _{\Pi _\nu (s)} ((B_r(0) - \rho e_1) \cap \Pi ^+_{\nu }(s)) \subset (B_r(0) - \rho e_1) \cap \Pi ^-_{\nu }(s). \end{aligned}$$

Therefore, $O_r$ satisfies the second condition of $\rho \textit{-reflection}$ in Definition 2.1. $\square$

Lemma A.6

For all $\rho \in [0, (5 c_n)^{-1})$, there exists $r > 2\rho$ such that $|O_r| = 1$. Here, $c_n = |B_1(0)|^{1/n}$.

Proof

Note that $r \mapsto |O_r|$ is a increasing function in $r>2\rho$. Also, as $|O_r| > |B_r(0)|$, $|O_r|$ goes to infinity as $r \rightarrow \infty$. Furthermore, it holds that

$$\begin{aligned} |O_{2\rho } | \le 2 |B_{2\rho }(0)| = 2 (2\rho c_n)^n \le \frac{2^{n+1}}{5^n} < 1 \end{aligned}$$

(A.16)

and thus we conclude. $\square$

Appendix B: Proof of Proposition 3.5

In this section, we prove Proposition 3.5. First, in Lemma B.3, we show short-time star-shapedness based on the Hölder continuity in Lemma B.1. The remaining arguments are parallel to Theorem 6.5 in [20]. For simplicity, we fix $r_0$ and $R_0$ given in (3.11) and $\delta \in (0,\delta _0)$ for $\delta _0$ given in (3.4). Also, let ${\Omega _t}$ be a viscosity solution of $V = -H + \gamma _{\delta }(|\Xi _t|)$ where $\Xi _t$ is a flat flow given in Definition B.4.

First, let us recall several properties of solutions from [20].

Lemma B.1

[20, Corollary 3.7] Assume that $\Omega _{0} \in S_{r,R}$. Then, there exists

$$\begin{aligned} {\mathcal M}_1={\mathcal M}_1(r,R,\Vert \gamma _{\delta }(|\Xi _t|)\Vert _{L^\infty ([0,T])}) \end{aligned}$$

such that we have

$$\begin{aligned} \sup _{x \in {\partial \Omega _t}} d(x, {\partial \Omega _0}) \le {\mathcal M}_1|t|^{\frac{1}{2}} \hbox { for } t \in [0,T]. \end{aligned}$$

(B.1)

Theorem B.2

[20, Theorem 3.6] Let $I = [0,t_0)$ be the maximal interval satisfying $\overline{B_\rho } \subset \Omega _t$. Then, $\Omega _t$ has $\rho \textit{-reflection}$ in I.

Lemma B.1 and Theorem B.2 imply the following lemma.

Lemma B.3

(Short-time star-shapedness) For $r>r_0>0$ and $0<R<R_0$, suppose that ${\overline{B}}_{(1+\beta )\rho } \subset \Omega _0$ and ${\Omega _0}\in S_{r,R}$ for $r = \rho (\beta ^2 + 2\beta )$. Then, for all $t \in [0,t_1]$, it holds that for some ${\hat{r}} > r_0$ and ${\hat{R}} < R_0$

$$\begin{aligned} {\Omega _t}\in S_{{\hat{r}},{\hat{R}}}. \end{aligned}$$

(B.2)

where

$$\begin{aligned} t_1=t_1(r,R,\Vert \gamma _{\delta }(|\Xi _t|)\Vert _{L^\infty ([0,T])}) := \frac{1}{2} \left( \min \left\{ \frac{ \sqrt{r^2 + \rho ^2} - \sqrt{r_0^2 + \rho ^2}}{{\mathcal M}_1}, \frac{R_0 - R}{{\mathcal M}_1} \right\} \right) ^2. \end{aligned}$$

(B.3)

Here, ${\mathcal M}_1$ is given in Lemma B.1.

Proof

From Lemma B.1, it holds that in $[0,t_1]$

$$\begin{aligned} {\overline{B}}_{(1+\beta )\rho - {\mathcal M}_1 t_1^{\frac{1}{2}}} \subset \Omega _t \subset B_{R + {\mathcal M}_1 t_1^{\frac{1}{2}}}. \end{aligned}$$

(B.4)

Theorem B.2 implies that $\Omega _t$ has $\rho \textit{-reflection}$ for all $t \in [0,t_1]$

By (B.3), it holds that

$$\begin{aligned} R + {\mathcal M}_1 t_1^{\frac{1}{2}}< R_0 \text{ and } r_0 < \left( ((1+\beta )\rho - {\mathcal M}_1 t_1^{\frac{1}{2}})^2 - \rho ^2 \right) ^{\frac{1}{2}}. \end{aligned}$$

(B.5)

From (C.9), we conclude. $\square$

Recall definition of flat flows and comparison principle from [20].

Definition B.4

[20, Definition 5.2] Let $(\Xi _t)_{t \ge 0}$ be a flat flow of (1.3) if there exists a sequence $h_k \rightarrow 0$ such that

$$\begin{aligned} d_H(\Xi _t, E_{t}(h_k)) \rightarrow 0. \end{aligned}$$

Here, $E_t=E_t(h)$ is given in (3.12).

Lemma B.5

[20, Proposition 6.1] Suppose that ${\Omega _t}\in S_{r,R}$ in [0, T] for some $r>r_0$ and $R<R_0$.

$$\begin{aligned} \hbox { If } \Omega _0 \subset \subset \Xi _0, \hbox { then } \Omega _t \subset \subset \Xi _t \hbox { in } [0,T]. \hbox { Also, if } \Xi _0 \subset \subset \Omega _0, \hbox { then }\Xi _t \subset \subset \Omega _t \hbox { in } [0,T]. \end{aligned}$$

(B.6)

Proof of Proposition 3.5

Let ${\Omega _t}^{\varepsilon ,+}$ and ${\Omega _t}^{\varepsilon ,-}$ be viscosity solutions of $V = -H + \gamma _{\delta }(|\Xi _t|)$ starting from ${\Omega _0}^{\varepsilon ,-} := (1-\varepsilon ){\Omega _0}$ and ${\Omega _0}^{\varepsilon ,+}:= (1+\varepsilon ){\Omega _0}$, respectively. Note that

$$\begin{aligned} {\overline{B}}_{(1+\beta _1)\rho } \subset {\Omega _0}\in S_{r_1,R_1} \end{aligned}$$

(B.7)

where $r_1 (= \rho (\beta _1^2 + 2\beta _1)^{\frac{1}{2}})$ and $\beta _1$ are given in (3.7) and $R_1$ is given in (3.8). From (B.7), there exist $\varepsilon _0>0$, $\beta \in (0, \beta _1)$ and $R \in (R_1,R_0)$ such that

$$\begin{aligned} {\overline{B}}_{(1+\beta )\rho } \subset {\Omega _0}^{\varepsilon ,\pm } \in S_{r,R} \hbox { for all } \varepsilon \in [0,\varepsilon _0], \hbox { and } r := \rho (\beta ^2 + 2\beta )^{\frac{1}{2}}>r_0. \end{aligned}$$

(B.8)

Here, $r_0$ and $R_0$ is given in (3.11). $\square$

Let us show that $\Xi _t = {\Omega _t}$ in $[0,t_1]$ for $t_1=t_1(r,R,K_1)$ given in (B.3) and $K_1 := \frac{1}{\delta }\max \{ 1, |B_{R_0}|\}$. As $|\gamma _{\delta }(|\Xi _t|) | \le K_1$, we can apply Lemma B.3 combining with (B.8) and conclude that for some ${\hat{r}} > r_0$ and ${\hat{R}} < R_0$

$$\begin{aligned} {\Omega _t}^{\varepsilon ,\pm } \in S_{{\hat{r}},{\hat{R}}} \hbox { in } [0,t_1]. \end{aligned}$$

(B.9)

By Lemma B.5, we conclude that

$$\begin{aligned} {\Omega _t}^{\varepsilon ,-} \subset \Xi _t \subset {\Omega _t}^{\varepsilon ,+}. \end{aligned}$$

(B.10)

Note that ${\Omega _t}^{\varepsilon ,+}$ and ${\Omega _t}^{\varepsilon ,-}$ converges to ${\Omega _t}$ as $\varepsilon \rightarrow 0$ from the uniqueness in [20, Theorem 4.3]. We conclude that $\Xi _t = {\Omega _t}$ in $[0,t_1]$. As the proof of Theorem 6.5 in [20], we can iterate this step to conclude that $\Xi _t = {\Omega _t}$ for all $t\ge 0$. Lastly, from Proposition 3.3, ${({\Omega ^\delta _t})_{t\ge 0}}$ is uniquely determined by its initial data. Now we conclude (3.13). $\Box$

Appendix C: Geometric properties

In this section, we consider geometric properties of $\rho \textit{-reflection}$ and $S_{r}$. First, let us recall a local property of $S_{r}$ from [20].

Lemma C.1

[20, Lemma 3.2] For a continuously differentiable and bounded function $\phi : {\mathbb {R}}^n \rightarrow {\mathbb {R}}$, let us denote the positive set of $\phi$ by $\Omega (\phi )$. Let us assume that $\Omega (\phi )$ contains $B_r(0)$ and $D\phi \ne 0$ on $\partial \Omega (\phi )$. Then the set $\Omega (\phi )$ is in $S_r$ if and only if

$$\begin{aligned} x \cdot \mathbf {n}_x = x \cdot \left( - \frac{D\phi }{|D\phi |}(x)\right) \ge r \hbox { for all } x\in \partial \Omega (\phi ), \end{aligned}$$

where $\mathbf {n}_x$ denotes the outward normal of $\partial \Omega (\phi )$ at x.

Here are several properties of $\rho \textit{-reflection}$ and $S_{r}$ from [15].

Lemma C.2

[15, Lemma 20] Suppose that $\Omega$ has $\rho \textit{-reflection}$. Then, we have

$$\begin{aligned} \sup _{x \in \partial \Omega } |x| - \inf _{x \in \partial \Omega } |x| \le 4\rho . \end{aligned}$$

Lemma 4.3 can be shown by Lemma C.3

Lemma C.3

[15, Lemma 3] For a bounded domain $\Omega$ containing $B_r(0)$, the followings are equivalent:

• (1) $\Omega \in S_r$.

• (2) For all $x\in \partial \Omega$, there is an interior cone to $\Omega$:

  $$\begin{aligned} IC(x,r):=\left( (x+C(-x,\theta _{x,r})) \cap C(x,\frac{\pi }{2} - \theta _x ) \right) \cup B_r(0) \subset \Omega \end{aligned}$$

  (C.1)

  where where $C(x,\theta )$ is a cone in the direction x with opening angle $\theta$ for $x \in {\mathbb R}^n$ and $\theta \in [0, \pi ]$,

  $$\begin{aligned} C(x,\theta ):=\{ y \mid \langle x,y \rangle > \cos \theta |x| |y| \} \hbox { and }\theta _{x,r} := \arcsin \frac{r}{|x|} \in \left[ 0,\frac{\pi }{2}\right] . \end{aligned}$$

• (3) For all $x\in \partial \Omega$, there is an exterior cone to $\Omega$:

  $$\begin{aligned} EC(x,r):=\left( x+C(x,\theta _{x,r})\right) \cap B_\epsilon (x) \subset \Omega ^c \text { where } \theta _{x,r} := \arcsin \frac{r}{|x|} \in \left[ 0,\frac{\pi }{2}\right] . \end{aligned}$$

  (C.2)

Proof of Lemma 4.3

From Lemma C.3, it holds that for all $x \in \partial E$,

$$\begin{aligned} IC(x,r) \subset E \hbox { and } EC(x,r) \subset E^c \end{aligned}$$

(C.3)

where IC is an interior cone given in (C.1), and EC is an exterior cone given in (C.2). Note that as $|x| \le R$, the angle of both the interior cone and exterior cone, $\theta _x$, is bounded from below as follows,

$$\begin{aligned} \theta _x := \arcsin \frac{r}{|x|} \ge \arcsin \frac{r}{R}. \end{aligned}$$

(C.4)

Thus, for $\eta _1(r,R):= |IC(Re_1,r) \cap B_\varepsilon (Re_1)|$, it holds that for $\varepsilon \in (0,r)$

$$\begin{aligned} \eta _1 \varepsilon ^{n} \le |IC(x,r) \cap B_\varepsilon (x)| \le |E \cap B_\varepsilon (x)|. \end{aligned}$$

(C.5)

Here, $e_1$ is a unit vector in the positive $x_1$ direction. Similarly, it holds that

$$\begin{aligned} |B_\varepsilon (x) {\setminus } E| \ge |B_\varepsilon (x) \cap EC(x,r)| \ge \eta _1 \varepsilon ^{n}. \end{aligned}$$

(C.6)

As $E \in S_{r,R}$, there exists $\varepsilon _0 = \varepsilon (r,R) < r$ such that for all $\varepsilon \in (0,\varepsilon _0)$

$$\begin{aligned} B_\varepsilon (x) \cap \partial E = (U, f(U)) \end{aligned}$$

(C.7)

up to rotation for some Lipschitz function $f = f_{x,\varepsilon } : U \subset B_\varepsilon ^{n-1}(x) \rightarrow {\mathbb R}$. Note that as $E \in S_{r,R}$, the Lipschitz constant of f is uniformly bounded by some constant ${\mathcal L}= {\mathcal L}(r,R)$.

From Theorem 9.1 in [22],

$$\begin{aligned} {\mathcal H}^{n-1}(B_\varepsilon (x) \cap \partial E) = \int _{U} \sqrt{ 1+ | \nabla f |^2 } dx \le |U| \sqrt{ 1+ {\mathcal L}^2} \le n w_n \varepsilon ^{n-1} \sqrt{ 1+ {\mathcal L}^2}. \end{aligned}$$

(C.8)

Thus, (4.3) holds with $\eta _2(r,R):=n w_n \sqrt{ 1+ {\mathcal L}^2}$. Here, $w_n$ is a volume of a unit ball in ${\mathbb R}^n$. On the other hand, from the isoperimetric inequality in [22, Proposition 12.37] and (4.2), we get the lower bound of (4.3). $\square$

Lemma C.4

[15, Lemma 21] Suppose that $\Omega$ has $\rho \textit{-reflection}$. Moreover, $\Omega \in S_r$ with

$$\begin{aligned} r=(\inf _{x\in \partial \Omega } |x|^2 - \rho ^2 )^{1/2}. \end{aligned}$$

(C.9)

For $E, F \subset {\mathbb R}^n$, define the Hausdorff distance by

$$\begin{aligned} d_H(E,F) := \max \left\{ \sup _{x \in E} d(x,F), \sup _{x \in F} d(x,E) \right\} . \end{aligned}$$

(C.10)

We say that $A \lesssim _{r,R} B$ if there exists a constant $C=C(r,R)>0$ depending on r, R such that $A \le CB$.

Lemma C.5

[15, Lemmas 23, 24] Consider sets $\Omega _1$ and $\Omega _2$ in $S_{r,R}$ for $R>r>0$. Then the following holds:

$$\begin{aligned} d_H(\partial \Omega _1, \partial \Omega _2) \lesssim _{r,R} d_H(\Omega _1, \Omega _2), \quad |\Omega _1 \Delta \Omega _2| \lesssim _{r,R} d_H(\Omega _1, \Omega _2). \end{aligned}$$

Lemma C.6

[15, Lemma 24] The metric space $(\partial S_{r,R}, d_H)$ is compact:

• (1) Suppose that $\Gamma _j \in (\partial S_{r,R}, d_H)$ for some $r,R>0$ and all $j \in {\mathbb N}$. Then $\{\Gamma _j\}_{j \in {\mathbb N}}$ has a subsequence that converges and any subsequential limit is also in $\partial S_{r,R}$.

• (2) Let I be a compact interval in ${\mathbb R}$ and $\Gamma _j : I \rightarrow \partial S_{r,R}$ for $j \in {\mathbb N}$ is an equicontinuous sequence of paths in $(\partial S_{r,R}, d_H)$. Then, there is a subsequence of the $\Gamma _j(\cdot )$ that converges uniformly on I on a path $\Gamma : I \rightarrow (\partial S_{r,R}, d_H)$.

Lastly, let us show the following property of characteristic functions.

Lemma C.7

Let $\{ {({\Omega ^{k}_{t}})_{t\ge 0}}\}_{k \in {\mathbb N}}$ be a sequence of sets in $S_{r,R}$ for $0<r<R$. Suppose that ${\Omega ^{k}_{t}}$ converges locally uniformly to ${\Omega ^\infty _t}$ on ${[0, \infty )}$. For a sequence of functions $\{ u_k \}_{k \in {\mathbb N}\cup \{ + \infty \} }$ defined by

$$\begin{aligned} u_k := \chi _{{\Omega ^{k}_{t}}} - \chi _{({\Omega ^{k}_{t}})^C} \hbox { for } k \in {\mathbb N}\cup \{ + \infty \}, \end{aligned}$$

(C.11)

it holds that

$$\begin{aligned} u_\infty ^* = \mathop {\lim \,\sup {}^*}\limits _{k \rightarrow \infty }u_k \hbox { and } (u_\infty )_* = \mathop {\lim \,\inf {}_*}\limits _{k \rightarrow \infty } u_k. \end{aligned}$$

(C.12)

Here, $\mathop {\lim \,\sup {}^*}\limits$ and $\mathop {\lim \,\inf {}_*}\limits$ are given in (2.4).

Proof

Let us show the first equation in (C.12) only. The second one can be shown by the parallel arguments.

By uniform convergence of ${\Omega ^{k}_{t}}$ in a finite interval, for any $j \in {\mathbb N}$, there exists $k_1>0$ such that for all $k>k_1$

$$\begin{aligned} d_H({\Omega ^{k}_{t}}, {\Omega ^\infty _t}) < \frac{1}{j}. \end{aligned}$$

(C.13)

Thus, for any $x \in {\Omega ^\infty _t}$ and $k>k_1$, there exists $y \in {\Omega ^{k}_{t}}$ such that $|x-y| < \frac{1}{j}$. Thus, we conclude that

$$\begin{aligned} \mathop {\lim \,\sup {}^*}\limits _{k \rightarrow \infty }u_k(x,t) = \lim \limits _{j \rightarrow \infty } \sup \left\{ u_k(y,s) : k \ge j,\quad |y-x| \le \frac{1}{j}, \quad |s-t| \le \frac{1}{j} \right\} = 1 \end{aligned}$$

(C.14)

and $u_\infty ^*(x) = \mathop {\lim \,\sup {}^*}\limits _{k \rightarrow \infty }u_k(x)$ for $x \in {\Omega ^\infty _t}$.

Note that we have for any sets $\Omega _1, \Omega _2 \in S_{r,R}$

$$\begin{aligned} d_H(\Omega _1^C, \Omega _2^C) \le d_H(\partial \Omega _1, \partial \Omega _2). \end{aligned}$$

(C.15)

Combining this with Lemma C.5, we conclude that $({\Omega ^{k}_{t}})^C$ converges locally uniformly to $({\Omega ^\infty _t})^C$. By parallel arguments, for any $x \in ({\Omega ^\infty _t})^C$, we conclude that $\mathop {\lim \,\sup {}^*}\limits _{k \rightarrow \infty }u_k(x,t) = -1$. As $\mathop {\lim \,\sup {}^*}\limits _{k \rightarrow \infty }u_k$ is upper semicontinuous, we conclude (C.12). $\square$

Lemma C.8

For any function $u : Q \rightarrow {\mathbb R}$ and $\Theta \in C({[0, \infty )})$, it holds that

$$\begin{aligned} \widehat{u^*}(\cdot ;\Theta ) = {\widehat{u}}(\cdot ;\Theta )^* \end{aligned}$$

(C.16)

and

$$\begin{aligned} \widetilde{u_*}(\cdot ;\Theta ) = {\widetilde{u}}(\cdot ;\Theta )_*. \end{aligned}$$

(C.17)

Proof

Let us only show (C.16). The parallel arguments imply (C.17).

Let us assume that both sides are finite at $(x_0,t_0) \in Q$. We claim that

$$\begin{aligned} \widehat{u^*}(x_0,t_0;\Theta ) \le {\widehat{u}}(x_0,t_0;\Theta )^*. \end{aligned}$$

(C.18)

By the upper semicontinuity of $({\widehat{u}})^*$, for $\varepsilon >0$ there exists $\delta \in (0, t_0)$ such that

$$\begin{aligned} {\widehat{u}}(\cdot , \cdot ;\Theta ) < {\widehat{u}}(x_0,t_0;\Theta )^* + \varepsilon \hbox { in } B_{\delta }(x_0) \times (t_0 - \delta , t_0 + \delta ). \end{aligned}$$

(C.19)

From the definition of ${\widehat{u}}$ in (2.14), it holds that

$$\begin{aligned} u(\cdot , \cdot )< {\widehat{u}}(x_0,t_0;\Theta )^* + \varepsilon \hbox { in } {\mathcal N}_1:= \{ (x,t) \in Q : |t - t_0|< \delta \hbox { and } |x - x_0 | < \delta + \Theta (t)\}. \end{aligned}$$

(C.20)

Furthermore, by the continuity of $\Theta$, for any $y \in {\overline{B}}_{\Theta (t_0)}(x_0)$ there exists a small neighborhood ${\mathcal N}_2$ of $(y,t_0)$ in Q such that ${\mathcal N}_2 \subset {\mathcal N}_1$. From (C.20), we have

$$\begin{aligned} u(y,t_0)^* \le {\widehat{u}}(x_0,t_0;\Theta )^* + \varepsilon \hbox { for all } y \in {\overline{B}}_{\Theta (t_0)}(x_0). \end{aligned}$$

(C.21)

As $\varepsilon$ is arbitrary, we conclude (C.18).

Next, let us show the opposite inequality,

$$\begin{aligned} \widehat{u^*}(x_0,t_0;\Theta ) \ge {\widehat{u}}(x_0,t_0;\Theta )^*. \end{aligned}$$

(C.22)

For any $\varepsilon >0$, let us show that there exists $\delta \in (0, t_0)$ such that

$$\begin{aligned} \widehat{u^*}(x_0,t_0;\Theta ) > u(\cdot , \cdot ) - \varepsilon \hbox { in } B_{\delta + \Theta (t_0)}(x_0) \times (t_0 - \delta , t_0 + \delta ). \end{aligned}$$

(C.23)

If not, then there exists $\{ (y_k,s_k) \}_{k \in {\mathbb N}} \subset Q$ such that

$$\begin{aligned} |s_k - t_0 | \le \frac{1}{k}, \ \ |y_k - x_0| < \frac{1}{k} + \Theta (t_0) \hbox { and } \widehat{u^*}(x_0,t_0) \le u(y_k,s_k) - \varepsilon . \end{aligned}$$

(C.24)

Then, $\{ s_k \}_{k \in {\mathbb N}}$ converges to $t_0$. Also, by compactness of ${\overline{B}}_{\Theta (t_0)+1}(x_0)$, there exists a subsequence $\{ k_i \}_{i \in {\mathbb N}}$ and $y^* \in {\overline{B}}_{\Theta (t_0)+1}(x_0)$ such that $\{ y_{k_i} \}_{ i \in {\mathbb N}}$ converges to $y^*$. Thus, it holds that

$$\begin{aligned} \widehat{u^*}(x_0,t_0) \le \mathop {\lim \,\sup }\limits _{k \rightarrow \infty } u(y_k,s_k) - \varepsilon \le u^*(y^*,t_0) - \varepsilon . \end{aligned}$$

(C.25)

On the other hand, (C.24) implies $y^* \in {\overline{B}}_{\Theta (t_0)}(x_0)$, which contradicts to (C.25). As a consequence, we get (C.23).

By (C.23) and the continuity of $\Theta$, there exists $\delta _1 \in (0, \delta )$ such that

$$\begin{aligned} \widehat{u^*}(x_0,t_0;\Theta ) > u(\cdot , \cdot ) - \varepsilon \hbox { in } \{(x,t) \in Q : |t - t_0|< \delta _1 \hbox { and } |x - x_0 | < \delta _1 + \Theta (t) \}. \end{aligned}$$

(C.26)

As $\varepsilon$ is arbitrary, we get (C.22) and conclude (C.16).

Similar arguments can be applied if either the left hand side or right hand side in (C.16) is infinity at $(x_0,t_0) \in Q$. In particular, for any $\varepsilon \in (0, t_0)$, there exists a sequence $\{ x_k, t_k \}_{ k \in {\mathbb N}}$ in $(t_0 - \varepsilon , t_0 + \varepsilon ) \times B_{\Theta (t_0) + \varepsilon }(x_0)$ such that $u(x_k, t_k)$ converging to infinity. This implies that the other side is infinity. $\square$