The extension problem of the mean curvature flow (I)

Abstract

We show that the mean curvature blows up at the first finite singular time for a closed smooth embedded mean curvature flow in ${{\mathbb {R}}}^3$.

Appendices

Appendix A The parabolic Harnack inequality

In this “Appendix”, we include the parabolic Harnack inequality from Krylov-Safonov [31, 32]. First, we introduce some notations. Let $x=(x^1, x^2, \ldots , x^n)\in {{\mathbb {R}}}^n$. Denote

$$\begin{aligned} |x|= & {} \Big (\sum _{i=1}^n(x^i)^2\Big )^{\frac{1}{2}},\quad B_R(x)=\{y\in {{\mathbb {R}}}^n\;|\; |x-y|<R\},\\ Q(\theta , R)= & {} B_R(0)\times (0, \theta R^2). \end{aligned}$$

Consider the parabolic operator

$$\begin{aligned} L u=-\frac{\partial u}{\partial t}+a^{ij}(x, t)u_{ij}+b^i(x, t)u_i-c(x, t)u, \end{aligned}$$

(A.1)

where the coefficients are measurable and satisfy the conditions

$$\begin{aligned} \mu |\xi |^2\le & {} a^{ij}(x, t)\xi _i\xi _j\le \frac{1}{\mu }|\xi |^2, \end{aligned}$$

(A.2)

$$\begin{aligned} |b(x, t)|\le & {} \frac{1}{\mu },\end{aligned}$$

(A.3)

$$\begin{aligned} 0\le & {} c(x, t)\le \frac{1}{\mu }. \end{aligned}$$

(A.4)

Here $b(x, t)=(b^1(x, t), \ldots , b^n(x, t))$. Then we have

Theorem A.1

(Theorem 1.1 of [31, 32]) Suppose the operator L in (A.1) satisfies the conditions (A.2)–(A.4). Let $\theta >1, R\le 2, u\in W^{1, 2}_{n+1}(Q(\theta , R)), u\ge 0$ in $Q(\theta , R)$, and $Lu=0$ on $Q(\theta , R)$. Then there exists a constant C, depending only on $\theta , \mu$ and n, such that

$$\begin{aligned} u(0, R^2)\le C\, u(x, \theta R^2),\quad \forall \; x\in B_{\frac{R}{2}}(0). \end{aligned}$$

(A.5)

Moreover, when $\frac{1}{\theta -1}$ and $\frac{1}{\mu }$ vary within finite bounds, C also varies within finite bounds.

Note that in our case the Eq. (4.48) doesn’t satisfy the assumption that $c(x, t)\ge 0$ in (A.4). Therefore, we cannot use Theorem A.1 directly. However, the Harnack inequality still works when c(x, t) is a constant. Namely, we have

Theorem A.2

Let $\theta >1, R\le 2$. Suppose that $u(x, t)\in W^{1, 2}_{n+1}(Q(\theta , R))$ is a nonnegative solution to the equation

$$\begin{aligned} L u=-\frac{\partial u}{\partial t}+a^{ij}(x, t)u_{ij}+b^i(x, t)u_i+c u=0, \end{aligned}$$

(A.6)

where c is a constant and the coefficients satisfy (A.2)–(A.3). Then there exists a constant C, depending only on $\theta , \mu , c$ and n, such that

$$\begin{aligned} u(0, R^2)\le C\, u(x, \theta R^2),\quad \forall \; |x|<\frac{1}{2} R. \end{aligned}$$

(A.7)

Proof

Since u(x, t) is a solution of (A.6) and c is a constant, the function $v(x, t)=e^{-ct}u$ satisfies

$$\begin{aligned} -\frac{\partial v}{\partial t}+a^{ij}(x, t)v_{ij}+b^i(x, t)v_i=0. \end{aligned}$$

(A.8)

Applying Theorem A.1 to the Eq. (A.8), we have

$$\begin{aligned} v(0, R^2)\le C\, v(x, \theta R^2),\quad \forall \; |x|<\frac{1}{2} R, \end{aligned}$$

where C depends only on $\theta , \mu$ and n. Thus, for any $x\in B_{\frac{R}{2}}(0)$ we have

$$\begin{aligned} u(0, R^2)\le Ce^{-c(\theta -1)R^2}u(x, \theta R^2)\le C'u(x, \theta R^2), \end{aligned}$$

where $C'$ depends only on $\theta , \mu , c$ and n. Here we used $R\le 2$ by the assumption. The theorem is proved. $\square$

We generalize Theorem A.2 to a general bounded domain in ${{\mathbb {R}}}^n$.

Theorem A.3

Let $\Omega$ be a bounded domain in ${{\mathbb {R}}}^n$. Suppose that $u(x, t)\in W^{1, 2}_{n+1}(\Omega \times (0, T))$ is a nonnegative solution to the equation

$$\begin{aligned} L u=-\frac{\partial u}{\partial t}+a^{ij}(x, t)u_{ij}+b^i(x, t)u_i+c u=0, \end{aligned}$$

(A.9)

where c is a constant and the coefficients satisfy (A.2)–(A.3) for a constant $\mu >0$. For any s, t satisfying $0<s<t<T$ and any $x, y\in \Omega$ with the following properties

• (1). x and y can be connected by a line segment $\gamma$ with the length $|x-y|\le l;$

• (2). Each point in $\gamma$ has a positive distance at least $\delta >0$ from the boundary of $\Omega ;$

• (3) s and t satisfy $T_1\le t-s\le T_2$ for some $T_1, T_2>0;$

we have

$$\begin{aligned} u(y, s)\le C \,u(x, t) , \end{aligned}$$

(A.10)

where C depends only on $c, n, \mu , \min \{s, \delta ^2\}, l, T_1$ and $T_2$.

Proof

Let $\gamma$ be the line segment with the property (1) and (2) connecting x and y. We set

$$\begin{aligned} p_0=y,\quad p_N=x, \quad p_i=p_0+\frac{x-y}{N}i\in \gamma \end{aligned}$$

for any $0\le i\le N$. Here we choose N to be the smallest integer satisfying

$$\begin{aligned} N>\max \Big \{\frac{2(t-s)}{s}, \frac{l}{\min \{\frac{\sqrt{s}}{4}, \frac{\delta }{4} \}}\Big \}. \end{aligned}$$

(A.11)

We define

$$\begin{aligned} R=\frac{2l}{N}, \quad \theta =1+\frac{t-s}{R^2N}. \end{aligned}$$

(A.12)

We can check that $R\le \frac{\delta }{2}$. For any $s, t\in (0, T)$, we choose $\{t_i\}_{i=0}^N$ such that $t_0=s, t_N=t$ and

$$\begin{aligned} t_{i}-t_{i-1}=\frac{t-s}{N} \end{aligned}$$

(A.13)

for all integers $1\le i\le N$. Note that (A.11)–(A.13) imply that for any $0\le i\le N-1$,

$$\begin{aligned} t_{i+1}-\theta R^2\ge s-\theta R^2=s-R^2-\frac{t-s}{N}\ge \frac{s}{4}>0 \end{aligned}$$

and

$$\begin{aligned} |p_{i+1}-p_i|=\frac{|x-y|}{N}\le \frac{l}{N}=\frac{R}{2}. \end{aligned}$$

Therefore, for any $0\le i\le N-1$ we have $(t_{i+1}-\theta R^2, t_{i+1})\subset (0, T)$ and $p_{i+1}\in B_{\frac{R}{2}}(p_i)$. Applying Theorem A.2 on $B_R(p_i)\times (t_{i+1}-\theta R^2, t_{i+1})\subset \Omega \times (0, T)$, we have

$$\begin{aligned} u(p_i, t_i)\le C\,u(p_{i+1}, t_{i+1}), \end{aligned}$$

(A.14)

where C depends only on $c, n, \mu$ and $\frac{1}{\theta -1}=\frac{R^2N}{t-s}.$ Here we used the fact that $t_i=(t_{i+1}-\theta R^2)+R^2$. Therefore,

$$\begin{aligned} u(y, s)=u(p_0, t_0)\le C^Nu(p_N, t_N)=C' u(x, t) \end{aligned}$$

(A.15)

where the constant $C'$ in (A.15) depends only on $c, n, \mu , \min \{s, \delta ^2\}, l, T_1$ and $T_2$. The theorem is proved. $\square$

A direct corollary of Theorem A.3 is the following result.

Theorem A.4

Let $\Omega$ be a bounded domain in ${{\mathbb {R}}}^n$. Suppose that $u(x, t)\in W^{1, 2}_{n+1}(\Omega \times (0, T))$ is a nonnegative solution to the equation

$$\begin{aligned} L u=-\frac{\partial u}{\partial t}+a^{ij}(x, t)u_{ij}+b^i(x, t)u_i+c u=0, \end{aligned}$$

(A.16)

where c is a constant and the coefficients satisfy (A.2)–(A.3) for a constant $\mu >0$, and $\Omega ', \Omega ''$ are subdomains in $\Omega$ satisfying the following properties:

• (1). $\Omega '\subset \Omega ''\subset \Omega$, and $\Omega ''$ has a positive distance $\delta >0$ from the boundary of $\,\Omega$;

• (2). $\Omega '$ can be covered by k balls with radius r, and all balls are contained in $\Omega ''$.

Then for any s, t satisfying $0<s<t<T$ and any $x, y\in \Omega '$, we have

$$\begin{aligned} u(y, s)\le C \,u(x, t) , \end{aligned}$$

(A.17)

where C depends only on $c, n, \mu , \min \{s, \delta ^2\}, t-s, r$ and k.

Proof

By the assumption, we can find finite many points ${{\mathcal {A}}}=\{q_1, q_2, \ldots , q_k\}$ such that

$$\begin{aligned} \Omega '\subset \cup _{q\in {{\mathcal {A}}}}B_r(q)\subset \Omega ''. \end{aligned}$$

(A.18)

For any $x, y\in \Omega '$, there exists two points in ${{\mathcal {A}}}$, which we denote by $q_1$ and $q_2$, such that $x\in B_{r}(q_1)$ and $y\in B_{r}(q_2)$. Then x and y can be connected by a polygonal chain $\gamma$, which consists of two line segments $\overline{xq_1}, \overline{yq_2}$ and a polygonal chain with vertices in ${{\mathcal {A}}}$ connecting $q_1$ and $q_2$. Clearly, the number of the vertices of $\gamma$ is bounded by $k+2$ and the total length of $\gamma$ is bounded by $(k+2) r$. Moreover, by the assumption we have $\gamma \subset \Omega ''$ and each point in $\gamma$ has a positive distance at least $\delta >0$ from the boundary of $\Omega$.

Assume that the polygonal chain $\gamma$ has consecutive vertices $\{p_0, p_1, \ldots , p_N\}$ with $p_0=y, p_N=x$ and $1\le N\le k+2$. We apply Theorem A.3 for each line segment $\overline{p_ip_{i+1}}$ and the interval $[t_i, t_{i+1}]$, where $\{t_i\}$ is chosen as in (A.13). Note that

$$\begin{aligned} \frac{t-s}{k+2}\le t_{i+1}-t_i=\frac{t-s}{N}\le t-s. \end{aligned}$$

Thus, for any $0\le i\le N-1$ we have

$$\begin{aligned} u(p_i, t_i)\le Cu(p_{i+1}, t_{i+1}), \end{aligned}$$

(A.19)

where C depends only on $c, n, \mu , \min \{s, \delta ^2\}, r, k$ and $t-s$, and (A.19) implies (A.17). This finishes the proof of Theorem A.4. $\square$

Appendix B The interior estimates of parabolic equations

In this “Appendix”, we present the interior estimates of parabolic equations from G. Lieberman’s book [37] for the reader’s convenience.

Let $X=(x, t)$ be a point in ${{\mathbb {R}}}^{n+1}$ and $x=(x_1, x_2, \ldots , x_n)\in {{\mathbb {R}}}^n$. The norms on ${{\mathbb {R}}}^n$ and ${{\mathbb {R}}}^{n+1}$ are given by

$$\begin{aligned} |x|=\Big (\sum _{i=1}^n(x^i)^2\Big )^{\frac{1}{2}},\quad |X|=\max \{|x|, |t|^{\frac{1}{2}}\}. \end{aligned}$$

Let $\Omega$ be a domain in ${{\mathbb {R}}}^{n+1}$. Let $d(X, Y)=\min \{d(X), d(Y)\}$ where $d(X)=\mathrm {dist}(X, {{\mathcal {P}}}\Omega \cap \{t<t_0\})$. Here ${{\mathcal {P}}}\Omega$ denotes the parabolic boundary of $\Omega$. We define

$$\begin{aligned} |f|_0=\sup _{\Omega } |f|. \end{aligned}$$

If $b\ge 0$, we define

$$\begin{aligned} |f|_0^{(b)}=\sup _{X\in \Omega }d(X)^b|f(X)|. \end{aligned}$$

If $a=k+\alpha >0$ and $a+b\ge 0$, where k is a nonnegative integer and $\alpha \in (0, 1]$, we define

$$\begin{aligned}{}[f]_a^{(b)}= & {} \sup \Big \{\sum _{|\beta |+2j=k}d(X, Y)^{a+b}\frac{|D_x^{\beta }D_t^j f(X)-D_x^{\beta }D_t^j f(Y)|}{|X-Y|^{\alpha }}: \\&\qquad X\ne Y \; \hbox {in}\;\Omega \Big \},\\ \langle f\rangle _a^{(b)}= & {} \sup \Big \{\sum _{|\beta |+2j=k-1}d(X, Y)^{a+b}\frac{|D_x^{\beta }D_t^j f(X)-D_x^{\beta }D_t^j f(Y)|}{|X-Y|^{1+\alpha }}: \\&\qquad X\ne Y \; \hbox {in}\;\Omega ,\;x=y\Big \}, \\ |f|_a^{(b)}= & {} \sum _{|\beta |+2j\le k}|D_x^{\beta }D_t^jf|_0^{(|\beta |+2j+b)}+[f]_a^{(b)}+\langle f\rangle _a^{(b)}. \end{aligned}$$

We also define $|f|_a^*=|f|_a^{(0)}$, and define the spaces

$$\begin{aligned} H_a^*=\{f:\;|f|_a^*<\infty \},\quad H_a^{(b)}=\{f:\; |f|_a^{(b)}<\infty \}. \end{aligned}$$

With these notations, we have the following result.

Theorem B.1

(Theorem 4.9 of [37]) Let $\Omega$ be a bounded domain in ${{\mathbb {R}}}^{n+1}$, and let $a^{ij}\in H_{\alpha }^{(0)}$ and $b^i\in H_{\alpha }^{(1)}$ satisfy

$$\begin{aligned} \lambda |\xi |^2\le & {} a^{ij}\xi _i\xi _j\le \Lambda |\xi |^2, \quad [a^{ij}]_{\alpha }^{(0)}\le A, \end{aligned}$$

(B.1)

$$\begin{aligned} |b^i|_{\alpha }^{(1)}\le & {} B \end{aligned}$$

(B.2)

for some constants $A, B, \lambda$ and $\Lambda$. Let $c\in H_{\alpha }^{(2)}$ satisfy

$$\begin{aligned} |c|_{\alpha }^{(2)}\le c_1 \end{aligned}$$

(B.3)

for some constant $c_1$ and let $f\in H_{\alpha }^{(2)}$. If $u\in H^*_{2+\alpha }$ is a solution of

$$\begin{aligned} -\frac{\partial u}{\partial t}+a^{ij}u_{ij}+b^i u_i+c u=f \end{aligned}$$

(B.4)

in $\Omega$, then there is a constant C determined only by $A, B, c_1, n, \lambda$ and $\Lambda$ such that

$$\begin{aligned} |u|_{2+\alpha }^*\le C(|u|_0+|f|_{\alpha }^{(2)}). \end{aligned}$$

(B.5)