# Interior gradient estimate of mean curvature equation

I'm studying by myself Mean Curvature Flow and I'm reading the paper "Interior estimates for hypersurfaces moving by mean curvature" by Klaus Ecker and Gerhard Huisken. I'm stuck in the theorem $$2.3$$.

Some definitions that are importants for the theorem are the height of $$M_t$$ with respect to the hyperplane orthogonal to $$\omega$$ by $$u = \langle x, \omega \rangle$$ and the height function $$w(y, t)$$ which is defined to be the height at time $$t$$ over a fixed point $$y$$ in this hyperplane.

$$\textbf{Theorem 2.3}$$ The gradient of the height function $$w$$ satisfies the estimate

$$\sqrt{1 + |Dw(y_0,t)|^2} \leq C_1(n) \sup_{B_R(y_0)} \sqrt{1 + |Dw_0|^2} \cdot \exp \left[ C_2(n) \ R^{-2} \ \sup_{[0,T]} \left( \sup_{B_R(y_0) \times [0,T]} w - w(y_0,t) \right)^2 \right],$$

where $$0 \leq t \leq T$$, $$B_R(y_0)$$ is a ball in the hyperplane orthogonal to $$\omega$$ and $$w_0$$ denotes the initial height function over this hyperplane.

I'm stuck on the final of the proof in two points:

$$\textbf{Point 1.}$$ The authors found

$$(15) \ v \eta \leq \sup_{\overline{M}_0} v \eta + (4 + 16 \beta) e^{64n\beta^2}.$$

The definitions and conditions necessary to understand the inequality above are $$v := \left( \langle \nu, \omega \rangle \right)^{-1}$$, $$\nu$$ is the unit normal vector of the hypersurface, $$\omega$$ is some fixed unit vector such that $$\langle \nu, \omega \rangle > 0$$, $$\eta:= -1 + \exp (\lambda \varphi)$$ (it's assumed that $$\eta \geq 0$$ and $$\eta(x,t)$$ is a smooth map which vanishes outside some compact subset of $$\mathbb{R}^{n+1}$$ in the proof), $$\lambda := 64n\beta^2$$,

$$\varphi := \left( \frac{1}{2 \beta} u + 1 - (|x|^2 - u^2) \right)_+$$, $$x \in \overline{M}_t, \beta > 0$$ to be chosen and $$\varphi := 0$$ otherwise

and $$\overline{M}_t := \{ x \in M_t \ ; \ |x|^2 - u^2 \leq 1 \}$$ for $$t \in [0,T]$$ and it's assumed $$u < 0$$ in this set, where $$u := \langle x,\omega \rangle$$.

Keeping these definitions and considerations in mind, the authors stated

As described in Sect. $$1$$ this estimate implies an equivalent bound for the height function $$w$$. At the point $$y=0$$ in the hyperplane orthogonal to $$\omega$$ we obtain in equivalence to $$(15)$$ for $$t \in [0,T]$$ and arbitrary $$\beta > 0$$

$$(16) \ [e^{64n\beta^2((2 \beta)^{-1}w(0,t) + 1)_+} - 1] \sqrt{1 + |Dw(0,t)|^2} \leq e^{64n\beta^2} \left( \sup_{B_1(0)} \sqrt{1 + |Dw_0|^2} + 4 + 16 \beta \right).$$

I didn't understand how the inequality $$(15)$$ imply the inequality $$(16)$$ since the section $$1$$ states that $$\left( \frac{d}{dt} - \triangle \right) |x|^2 = -2n$$ and $$\left( \frac{d}{dt} - \triangle \right) u = 0.$$

Furthermore, it is proved in this section that, when $$M_t = \text{graph} \ w_t$$, the evolution equation of the MCF is equivalent to the following evolution equation:

$$\frac{d}{dt} w(y,t) = \sqrt{1 + |Dw(y,t)|^2} \text{div}_y \left( \frac{Dw}{\sqrt{1 + |Dw|^2}} \right) (y,t)$$

$$\textbf{Point 2.}$$ The authors stated

Now choose $$\beta = \sup_{t \in [0,T]} -w(0,t)$$. We then infer from $$(16)$$

$$\sqrt{1 + |Dw(0,t)|^2} \leq C_1(n) \sup_{B_1(0)} \sqrt{1 + |Dw_0|^2} \exp [C_2(n) \sup_{[0,T]} (-w(0,t))^2].$$

I didn't understand how this inequality was obtained from the choose of $$\beta$$.

I will be grateful if someone can explain me how obtain these two inequalities.

$$\textbf{EDIT:}$$

After do some research on the internet, I found some clue about my doubt in point $$1$$ in this paper by Colding and Minicozzi where explain that the theorem $$2.3$$ is an application of Koorevar's argument to mean curvature flow. Keeping the work of Koorevar in mind, I wouldn't be able to find the work of Koorevar, but I find this other paper which states that the porpuse is generalize Koorevar's argument for Killing graphs and I figure out that the Koorevar's argument is known as $$\textbf{interior gradient estimate for mean curvature equation}$$ on the parabolic PDE's literature by this last paper too. Knowing this, I found this presentation which tells about the history of the study of the interior gradient estimate for mean curvature equation.

I will read the proof given on the last paper that I cited and I will try do the argument for the $$\mathbb{R}^{n+1}$$, where the argument is applied on Ecker and Huisken's paper, but I don't know if I'm able to understand the argument since I don't have familiarity with Killing graphs, so I will appreciate if anyone who knows about the argument of the interior gradient estimate for mean curvature equation can give me any hints.

I managed to develop something for the point $$2$$, but it is not the estimate pointed out on paper yet.

My attempt:

Firstly, I did some observations:

$$\textbf{(a)}$$ $$1 \leq \sup_{B_1(0)} \sqrt{1 + |Dw_0|^2}$$,

$$\textbf{(b)}$$ $$e^x \geq x + 1$$ for every $$x \geq 0$$,

$$\textbf{(c)}$$ $$x < x^2 + 1 \Longrightarrow e^x < e^{x^2 + 1}$$ for every $$x \geq 0$$,

By $$b$$ and $$c$$, follows that

$$\textbf{(d)}$$ $$x + 1 < e^{x^2 + 1}$$ for every $$x \geq 0$$,

Using $$d$$, $$(16)$$ and $$w(0,t) \geq -\beta$$ by definition of $$\beta$$, we observe that

\begin{align} [e^{32n\beta^2} - 1] \sqrt{1 + |Dw(0,t)|^2} & = [e^{64n\beta^2((2 \beta)^{-1}(-\beta) + 1)_+} - 1] \sqrt{1 + |Dw(0,t)|^2} \\ & = [e^{64n\beta^2((2 \beta)^{-1}(-\beta) + 1)_+} - 1] \sqrt{1 + |Dw(0,t)|^2} \\ & \leq [e^{64n\beta^2((2 \beta)^{-1}w(0,t) + 1)_+} - 1] \sqrt{1 + |Dw(0,t)|^2} \\ & \leq e^{64n\beta^2} \left( \sup_{B_1(0)} \sqrt{1 + |Dw_0|^2} + 4 + 16 \beta \right) \\ & < e^{64n\beta^2} \left( \sup_{B_1(0)} \sqrt{1 + |Dw_0|^2} + 15 + 16 \beta \right) \\ & \overset{\textbf{(a)}}{<} e^{64n\beta^2} \left( \sup_{B_1(0)} \sqrt{1 + |Dw_0|^2} + (15 + 16 \beta) \sup_{B_1(0)} \sqrt{1 + |Dw_0|^2} \right) \\ & = e^{64n\beta^2} 16 \sup_{B_1(0)} \sqrt{1 + |Dw_0|^2} \left( 1 + \beta \right) \\ & \overset{\textbf{(d)}}{<} e^{64n\beta^2} 16 \sup_{B_1(0)} \sqrt{1 + |Dw_0|^2} \exp(\beta^2 + 1) \\ & = e^{64n\beta^2 + 1} 16 \sup_{B_1(0)} \sqrt{1 + |Dw_0|^2} \exp(\beta^2) \\ & = e^{64n\beta^2 + 1} 16 \sup_{B_1(0)} \sqrt{1 + |Dw_0|^2} \exp \left( \left( \sup_{t \in [0,T]} -w(0,t) \right)^2 \right), \end{align}

which imply

$$\sqrt{1 + |Dw(0,t)|^2} < \left( 16 \frac{e^{64n\beta^2 + 1}}{[e^{32n\beta^2} - 1]} \right) \sup_{B_1(0)} \sqrt{1 + |Dw_0|^2} \exp \left( \left( \sup_{t \in [0,T]} -w(0,t) \right)^2 \right)$$