Construction of Complete Embedded Self-Similar Surfaces under Mean Curvature Flow. Part II

TL;DR

This paper proves the existence of certain self-similar surfaces under mean curvature flow, advancing the construction of new complete embedded solutions with specific boundary conditions and symmetries.

Contribution

It establishes the existence of solutions to the Dirichlet problem for self-similar surfaces over punctured planes, contributing to the construction of novel embedded self-similar surfaces.

Findings

Existence of solutions under small boundary conditions

Solutions satisfy prescribed symmetries

Progress towards new embedded self-similar surfaces

Abstract

We study the Dirichlet problem associated to the equation for self-similar surfaces for graphs over the Euclidean plane with a disk removed. We show the existence of a solution provided the boundary conditions on the boundary circle are small enough and satisfy some symmetries. This is the second step towards the construction of new examples of complete embedded self similar surfaces under mean curvature flow.

Equations

$$H+X\cdot \nu =0,$$

$$f(\theta )=-f(-\theta )=f(\pi /N-\theta ).$$

$$u$$

$$u=f\text{ on }\partial B_R,$$

$$u(r,\theta )=-u(r,-\theta )=u(r,\pi /N-\theta )\text{ for }r>R,\theta \in [0,2\pi )$$

$$\mathcal{E}(u)=g^{ij}(Du(\xi ))D_{ij}u(\xi )-\xi \cdot Du(\xi )+u(\xi )=0,\xi \in \Omega ,$$

$$\frac{d}{dt}X(p,t)=\mathbf{H}(p,t),p\in M,t>0$$

$$X(p,0)=X_0(p)$$

$$\frac{d}{dt}v=g^{ij}(Dv)D_{ij}v,g^{ij}(Dv)={\delta }_{ij}-\frac{D_{i}v{D}_{j}v}{1+|Dv{|}^2}$$

$$H=L^2(\Omega ,dm(\xi )),$$

$$V=\{f\in H:{\partial }_{i}f\in H\text{ for }i=1,\dots ,n\}.$$

$${\Omega }_{R,N}=\{(r,\theta ):R<r,-\pi /N<\theta <\pi /N\}$$

$$V_{0;Sym}({\Omega }_{R,N})$$

$$H_{Sym}({\Omega }_{R,N})$$

$$\mathcal{L}(u)=\Delta u-\xi \cdot Du+u.$$

$${\mathcal{L}}_{-}(u)=\Delta u-\xi \cdot Du-u$$

$$g(\theta )=-g(-\theta )=g(\pi /N-\theta ).$$

$$\mathcal{L}(f(r)g(\theta ))=f_{rr}g+\frac{1}{r}{f}_{r}g+\frac{1}{r^2}f{g}_{\theta \theta }-r{f}_{r}g+fg=\lambda fg.$$

$$g_{\theta \theta }=-m^{2}g$$

$${\mathcal{L}}_0(f)=f_{rr}+\frac{1}{r}{f}_r-r{f}_r+(1-\lambda )f-\frac{1}{r^2}{m}^{2}f=0.$$

$${\mathcal{L}}_0(r^n)=(n^2-m^2)r^{n-2}+(1-\lambda -n)r^n.$$

$$\lambda =1-(kN+2l),k\in \mathbf{N},l\in \{0\}\cup \mathbf{N},$$

$$r^{kN}{P}_{k,l}(r^2)\sin (kN\theta )$$

$$(r^{kN}{P}_{k,l}\sin (kN\theta ),r^{kN}{P}_{k,s}\sin (kN\theta ){)}_{H({\Omega }_{0,N})}=0$$

$${\int }_{\rho =0}^{\infty }{P}_{k,l}(\rho )P_{k,s}(\rho ){\rho }^{kN}{e}^{-\rho /2}d\rho =0.$$

$${\int }_a^b{e}^{r|x|}d\mu (x)$$

$$\tilde{f}(\rho ,\theta )=f(\sqrt{\rho },\theta )=\sum _{j\in \mathbf{N}}{b}_j(\rho )\sin (jN\theta )\text{ for a.e ρ∈(0,∞)}.$$

$${\int }_{{\Omega }_{0,N}}|f(r,\theta ){|}^2{e}^{-|\xi {|}^2/2}d\xi ={\int }_{r=0}^{\infty }{\int }_{\theta =-\pi /N}^{\pi /N}|f(r,\theta ){|}^{2}d\theta e^{-r^2/2}rdr$$

$$=\frac{\pi }{2N}{\int }_{\rho =0}^{\infty }\sum _{j\in \mathbf{N}}{b}_j^2(\rho )e^{-\rho /2}d\rho =\frac{\pi }{2N}\sum _{j\in \mathbf{N}}{\int }_{\rho =0}^{\infty }{b}_j^2(\rho )e^{-\rho /2}d\rho <\infty .$$

$$\frac{\pi }{2N}\sum _{j>J(\epsilon )}{\int }_{\rho =0}^{\infty }{b}_j^2(\rho )e^{-\rho /2}d\rho \le \epsilon /2.$$

$$\frac{\pi }{2N}{\int }_0^{\infty }|b_j{\rho }^{-jN/2}-Q_{j,\epsilon }{|}^2{\rho }^{jN}{e}^{-\rho /2}d\rho <2^{-j}\epsilon .$$

$$\Vert f-{\mathcal{Q}}_{\epsilon }{\Vert }_{H({\Omega }_{0,N})}^2$$

$$=\frac{\pi }{2N}{\int }_{\rho =0}^{\infty }\left(\sum _{j>J(\epsilon )}{b}_j^2(\rho )e^{-\rho /2}+\sum _{0<j\le J(\epsilon )}|b_j-Q_{j,\epsilon }{\rho }^{jN/2}{|}^2{e}^{-\rho /2}\right)d\rho$$

$$\le \epsilon /2+\sum _{0<j\le J(\epsilon )}{2}^{-j-1}\epsilon \le \epsilon .$$

Full text

1. Introduction

This paper is the second one of a series of three articles describing the construction of new examples of complete embedded self-similar surfaces under mean curvature flow [8][7]. Our general strategy is inspired by Kapouleas’ article [4] and is described in detail in the previous installment. Let us recall it briefly: to construct a new self-similar surface, we take two known examples suitably positioned and replace a neighborhood of their intersection with an appropriately bent scaled Scherk’s singly periodic surface. The procedure is called desingularization.

The resulting surface is not smooth; however it is a good approximate solution. The next task is to find a small function whose graph over it satisfies the self-similar surface equation

[TABLE]

where $H$ is the mean curvature and $\nu$ is the normal vector so that the mean curvature vector is $\mathbf{H}=H\nu$. The sign of $H$ is chosen so that the mean curvature of a convex surface is positive. Before considering graphs of functions on the entire surface, we have to work locally and study the Dirichlet problems with small boundary conditions corresponding to (1) for graphs of functions on the different pieces.

In the first part [8], we found that a perturbation of an appropriately scaled and bent Scherk surface satisfies (1) and can be used to desingularize the intersection of a cylinder and a plane or a sphere and a plane. In this article, we study the Dirichlet problem corresponding to (1) for functions on the outer plane, which is the Euclidean plane with a disk removed. We show that the Dirichlet problem possesses a solution if the boundary conditions on the circle are small enough. Our methods here are different from the one used in Part I. Since the equation (1) can be written with a single set of coordinates on the outer plane, our tools come mainly from partial differential equation theory, with some interior estimates established by Ecker and Huisken [1] for the Mean Curvature Flow. The third article is to discuss the gluing of the solutions on the different pieces in a manner to obtain a smooth complete embedded self-similar surface.

1.1. Main result

Let $2>R>\sqrt{3}/2$ and $N\geq 5$, consider a function $f\in C^{4}([0,2\pi])$ satisfying the symmetries

[TABLE]

Theorem 1**.**

There is an $\varepsilon_{0}>0$ depending on $R$ and $N$ such that for any $f:[0,2\pi]\to\mathbf{R}$ with $\|f\|_{C^{4}[0,2\pi]}=\varepsilon\leq\varepsilon_{0}$ and $f$ satisfying the symmetries above, there is a function $u$ on $\Omega=\mathbf{R}^{2}\setminus\bar{B}_{R}$ such that

[TABLE]

In addition, We can choose the constant $\varepsilon_{0}$ uniformly for all $R$ such that $2>R>\sqrt{3}/2$.

Here, $u=f$ on ${\partial}B_{R}$ means that $u(R,\theta)=f(\theta)$ in polar coordinates. This is a slight abuse of notation which does not induce confusion. Note that the conditions (5) imply that $u$ is odd with respect to rotations of $180$ degrees with respect to the $x$-axis and even with respect to reflections across the planes $\theta=\pi/2N+k\pi/N$, which are exactly the symmetries imposed in Part I [8].

Computing the mean curvature and the normal vector $\nu$ of the graph of $u$ using coordinates in $\mathbf{R}^{2}$, we find that (3) is equivalent to

[TABLE]

where $g^{ij}(Du)=\delta^{ij}-\frac{D_{i}uD_{j}u}{1+|Du|^{2}}$. This is our main equation. Note that our domain is unbounded and that the coefficient in front of $u$ is positive, so the standard maximum principle is not applicable directly.

We first study the properties of the linear operator $\mathcal{L}$ associated to $\mathcal{E}$. From solutions to the linear problem, we then construct a sub and supersolution to the quasilinear equation $\mathcal{E}u=0$.

The solution to the elliptic equation (6) is found as a limit for time $\tau$ going to infinity of a solution $u$ to the parabolic equation ${\partial}_{\tau}u=\mathcal{E}u$. For initial conditions with bounded gradient, standard theory on parabolic equations assures the existence of a solution to the initial value problem on a short time interval $[0,\tau_{0})$. The constructed sub and supersolution serve as barriers for solutions of the parabolic equation ${\partial}_{\tau}u=\mathcal{E}u$ and allow us to control the growth of $u$ at infinity in the space variable for all time. Exploiting the close relation between the parabolic equation ${\partial}_{\tau}u=\mathcal{E}u$ and the mean curvature flow, we use interior estimates from the latter to bound derivatives of our solution to the former away from the boundary. Loosely speaking, this gives us control of the derivatives outside of a bounded annulus around ${\partial}\Omega$. To bridge the gap, we invoke standard parabolic theory on bounded sets and obtain uniform estimates for derivatives of $u$ for all $\tau\in[0,\tau_{0})$. This implies that the solution has to exists for all time $\tau\in[0,\infty)$. We conclude that a subsequence of $u(\cdot,\tau)$ tends to a solution of $\mathcal{E}u=0$ as $\tau$ goes to infinity by proving a monotonicity formula.

Some sections of this paper appeared in the author’s thesis, written under the direction of Sigurd Angenent at the University of Wisconsin-Madison. The author is indebted to him for invaluable discussion.

2. Definitions

2.1. Mean curvature flow

The properties of solutions to (3) are closely related to properties of solutions to the mean curvature flow (MCF), which will be used intensively. Let us therefore define it.

Let $X(\cdot,t):M^{2}\to\mathbf{R}^{3}$ be a one parameter family of immersions of $2$-dimensional smooth hypersurfaces in $\mathbf{R}^{3}$. We say that $M_{t}=X(M^{2},t)$ is a solution to the mean curvature flow if

[TABLE]

is satisfied for some initial data $X_{0}$. Here $\mathbf{H}(p,t)$ is the mean curvature vector of the hypersurface $M_{t}$ at $X(p,t)$.

If the hypersurfaces $M_{t}$ can be written locally as graphs of a function $v(\cdot,t)$ over a domain in the $xy$- plane, the quasilinear equation

[TABLE]

is equivalent to (7) up to tangential diffeomorphisms.

2.2. Weighted Hilbert Sobolev spaces $H$, $V$ and $V_{0}$

We consider the Hilbert space

[TABLE]

where $\Omega$ is an open subset of $\mathbf{R}^{n}$ (not necessarily bounded) and $m$ is the Gaussian measure $dm(\xi)=e^{-\frac{1}{2}|\xi|^{2}}d\xi.$ We write $\|f\|_{H}$ and $(f,g)_{H}$ for the norm and inner product of $H$. We also define the Hilbert space

[TABLE]

The inner product of $V$ is $(f,g)_{V}=\int_{\Omega}\{fg+Df\cdot Dg\}dm(\xi).$ Let $V_{0}$ be the closure of $C^{\infty}_{c}(\Omega)$ in $V$. Even when $\Omega$ is unbounded the inclusion $V_{0}\subset H$ is compact.

3. The Linear Operator

Let us define sections $\Omega_{R,N}$ of the outer plane by

[TABLE]

and their corresponding Hilbert spaces

[TABLE]

In this section, we study the properties of $\mathcal{L}$, the linear operator associated to $\mathcal{E}$ defined in (6),

[TABLE]

3.1. Eigenvalues and eigenfunctions

Consider the operator

[TABLE]

on $\Omega_{0,N}$. The compactness of the inclusion $V_{0}(\Omega_{0,N})\subset H(\Omega_{0,N})$ yields the following theorem:

Theorem 2**.**

The operators $\mathcal{L}_{-}$ and $\mathcal{L}$ have a countable set of eigenvalues having no limit point except possibly $\lambda=\infty$ and a corresponding $H$-orthogonal basis of eigenfunctions $\varphi_{i}\in V_{0}(\Omega_{0,N})$.

Proof.

Theory on compact bounded self-adjoint operators in Hilbert spaces gives us the result for $\mathcal{L}_{-}$. The result for $\mathcal{L}$ then follows immediately. ∎

Consider eigenfunctions of the form $f(r)g(\theta)$ satisfying the symmetries

[TABLE]

There is an eigenvalue $\lambda\in\mathbf{R}$ such that

[TABLE]

Separation of variables and the symmetry conditions (10) on $g$ imply that the only possibilities for $g$ are $g=C\sin(m\theta)$ with $m=kN$, $k\in\mathbf{N}$. Therefore

[TABLE]

We have

[TABLE]

$\mathcal{L}_{0}$ is a linear map on the vector space $V=\{r^{m}F(r^{2})\mid F\textrm{ polynomial }\}$ and it is upper diagonal in the basis $\{r^{m},r^{m+2},r^{m+4},\ldots\}$. The operator $\mathcal{L}_{0}$ has a zero eigenvalue if one of the entry on the diagonal vanishes, i.e. if $1-\lambda-(m+2l)=0$. We therefore found some of the eigenvalues of our original operator $\mathcal{L}$,

[TABLE]

where $\mathbf{N}=\{1,2,\ldots\}$ is the set of natural numbers. For each value of $k,l$ the corresponding eigenfunction is of the form

[TABLE]

where $P_{k,l}$ is a polynomial of degree $l$.

We now show that the eigenvalues values given by (11) are in fact all the possible eigenvalues of $\mathcal{L}$. This is done by proving the following theorem:

Theorem 3**.**

The set $S$ of eigenfunctions (12) corresponding to the eigenvalues (11) forms a basis for the Hilbert space $H_{Sym}(\Omega_{0,N})$.

Before we start the proof, note that for each fixed $k\in\mathbf{N}$ and $s\neq l$,

[TABLE]

since they are eigenfunctions corresponding to different eigenvalues. Hence, $\int_{r=0}^{\infty}P_{k,l}(r^{2})P_{k,s}(r^{2})r^{2kN}e^{-r^{2}/2}r\ dr=0.$ By the change of variable $\rho=r^{2}$,

[TABLE]

The polynomials $P_{k,l}$, which are closely related to the general Laguerre polynomials, are therefore orthogonal in $L^{2}(0,\infty;\rho^{kN}e^{-\rho/2}d\rho)$, the $L^{2}$-space weighted by $\rho^{kN}e^{-\rho/2}d\rho$. In addition, they span the functions $\rho^{n}$, $n\in\mathbf{N}$. The fact that $\{P_{k,l}\}_{l\in\{0\}\cup\mathbf{N}}$ is a maximal orthogonal set in $L^{2}(0,\infty;\rho^{kN}e^{-\rho/2}d\rho)$ follows from the theorem p. 333 in [9] which is recalled below.

Theorem 4** (Theorem p. 333 [9]).**

The orthonormal sequence of polynomials attached to a mass distribution function $\mu(x)$ on the interval $(a,b)$, is complete in $L^{2}(a,b;\mu)$ whenever there exists a number $r>0$ such that the integral

[TABLE]

exists.

Proof of Theorem 3.

It suffices to show that the set of all finite linear combinations of members of $S$ is dense. Let $f(r,\theta)\in H_{Sym}(\Omega_{0,N})$. Since $f$ is periodic in $\theta$ and $f(r,\cdot)\in L^{2}(-\pi/N,\pi/N)$ for almost every $r$, we can expand $f(r,\cdot)$ in Fourier series for almost every $r\in(0,\infty)$. The symmetries of $f$ imply $f(r,\theta)=\sum_{j\in\mathbf{N}}a_{j}(r)\sin(jN\theta)$. Using the change of variable $\rho=r^{2}$, we write

[TABLE]

The $H$-norm of $f$ is finite, therefore

[TABLE]

Hence, the functions $b_{j}$’s are in $L^{2}(0,\infty;e^{-\rho/2}d\rho)$ for $j\in\mathbf{N}$. For every $\varepsilon>0$, there is a $J(\varepsilon)\in\mathbf{N}$ so that

[TABLE]

We can approximate $b_{j}\rho^{-jN/2}\in L^{2}(0,\infty;\rho^{jN}e^{-\rho/2}d\rho)$ by a linear combination of $P_{j,l}$, since $\{P_{j,l}\}_{l\in\{0\}\cup\mathbf{N}}$ is a maximal orthogonal set . More precisely, for every $\varepsilon>0$ and every $j$, there is a linear combination of $P_{j,l}$’s denoted by $Q_{j,\varepsilon}$ so that

[TABLE]

Define $\mathcal{Q}_{\varepsilon}$ to be the linear combination of elements of $S$ given by $\mathcal{Q}_{\varepsilon}=\sum_{0<j\leq J(\varepsilon)}Q_{j,\varepsilon}\rho^{jN/2}\sin(jN\theta)$. Then

[TABLE]

This shows that every function in $H_{Sym}(\Omega_{0,N})$ can be approximated arbitrarily closely by a linear combination of functions of $S$. ∎

3.2. Poincaré inequality

As an immediate consequence of Theorem 3, we have the following inequality for $u\in C_{c}^{\infty}(\Omega_{0,N})$:

[TABLE]

where $\lambda_{0}$ is the lowest eigenvalue of the operator $-\mathcal{L}_{-}$. This implies a Poincaré inequality

[TABLE]

for every domain $\Omega^{\prime}\subset\Omega_{0,N}$.

3.3. Dirichlet problem

Let $\Omega_{R,N}$ be defined as in (9).

Lemma 5**.**

For any $g\in H(\Omega_{R,N})$, the equation

[TABLE]

possesses a weak solution $u\in V_{0}(\Omega_{R,N})$.

Proof.

It is a classical variational argument for the functional J(u)=\int_{\Omega_{R,N}}\Bigl{\{}\frac{1}{2}|Du(\xi)|^{2}-\frac{1}{2}u(\xi)^{2}+g(\xi)u(\xi)\Bigr{\}}\;dm(\xi), over $u\in V_{0}(\Omega_{R,N})$ and using the Poincaré inequality (15). ∎

3.4. Maximum Principle.

Let $\Omega^{\prime}$ be any domain that is a (not necessarily proper) subset of $\Omega_{R,N}$.

Definition 6**.**

We say that $u\in V(\Omega^{\prime})$ satisfies $u\leq 0$ on $\partial\Omega^{\prime}$ if its positive part $u^{+}=\max(u,0)\in V_{0}(\Omega^{\prime})$.

Theorem 7** (Maximum Principle).**

Let $N\geq 5$. Suppose $u\in V(\Omega^{\prime})$ satisfies

[TABLE]

then $u\leq 0$ in $\Omega^{\prime}$.

Proof.

First note that $\psi:=r\cos\theta$ is a solution of $\mathcal{L}\psi=0$. By the hypotheses, we have

[TABLE]

Define the function $\varphi$ such that $u=\psi\varphi.$ Since $1/\psi$ and $|D\psi|$ are uniformly bounded, $\varphi\in V$. Moreover, $\varphi$ satisfies

[TABLE]

Since $\mathcal{L}\psi=0$ and $\varphi v\in V_{0}$, we have $\int_{\Omega^{\prime}}D\psi\cdot D(\varphi v)-\psi\varphi v\ dm=0.$ Hence,

[TABLE]

Take $v=\max(\varphi,0)$ in (17). Note that we can do this since $u^{+}\in V_{0}$ and $\psi\neq 0$, so $v\in V_{0}$. Therefore

[TABLE]

since $Dv=D\varphi$ on the support of $v$. Using $R\cos\frac{\pi}{2N}\leq\psi$ and $|D\psi|\leq 1$, we get

[TABLE]

and

[TABLE]

by the Cauchy Schwarz inequality. Our Poincaré inequality (15) implies that

[TABLE]

A simple computation shows that for $N\geq 5$, $R\cos\frac{\pi}{2N}-\frac{1}{\sqrt{N}}\geq\frac{1}{\sqrt{2}}\cos\frac{\pi}{2N}-\frac{1}{\sqrt{N}}>0$. Therefore

[TABLE]

But $v\in V_{0}(\Omega^{\prime})$, so $v\equiv 0$; this means that $\varphi\leq 0$ and $u\leq 0$ in $\Omega^{\prime}$. ∎

3.5. Boundary value problem

Let $F\in C^{4}([0,2\pi])$ with the symmetries (2): $F(\theta)=-F(-\theta)=F(\pi/N-\theta)$. We will use the notation $K_{0}=\|F\|_{C^{4}([0,2\pi])}$ throughout Section 3.

Definition 8**.**

We say that $u\in V(\Omega)$ $(V(\Omega_{R,N}))$ is a solution to the problem

[TABLE]

if $u$ satisfies

[TABLE]

where $\psi$ is a function in $C^{4}(\bar{\Omega})$ ($C^{4}(\bar{\Omega}_{R,N})$ resp.) with $\psi(R,\theta)=F(\theta)$ and $\psi(r,\theta)=-\psi(r,-\theta)=\psi(r,\pi/N-\theta)$ for $r\geq R,\theta\in[0,2\pi)$, ($r\geq R,\theta\in[-\pi/N,\pi/N]$ resp.).

It follows from Lemma 5 and the Maximum Principle (Theorem 7) that the solution $u$ to $\mathcal{L}u=0$ in $\Omega_{R,N}$, $u=F$ on ${\partial}\Omega_{R,N}$ exists and is well defined, in other words it is unique and independent of the choice of $\psi$.

Denote by $U_{1}$ the extension of $u$ to $\Omega$ given by

[TABLE]

The function $U_{1}$ is a solution to boundary value problem

[TABLE]

Moreover, it is smooth by standard elliptic theory.

3.6. Behavior of solution at infinity

Let $u$ be the solution to the problem $\mathcal{L}u=0$ in $\Omega_{R,N}$, $u=F$ on ${\partial}\Omega_{R,N}$. We can choose $\psi$ in Definition 8 satisfying $\|\psi\|_{\infty}\leq\|F\|_{\infty}$ with $\psi=0$ in $\Omega_{R,N}\setminus B_{R+2}.$ Let $a=3\|F\|_{\infty}$ and consider the function

[TABLE]

Since $N\geq 5$, $a=3\|F\|_{\infty}\geq\frac{2\|F\|_{\infty}}{\cos(\pi/N)}$ therefore $w_{1}\leq 0$ on $\partial\Omega_{R,N}$. It follows from the Maximum Principle that $w_{1}\leq 0$ in $\Omega_{R,N}$, hence $u(r,\theta)\leq ar\cos\theta\textrm{ in }\Omega_{R,N}.$ An similar argument with $w_{2}=-ar\cos\theta-u$ gives a lower bound on $u(r,\theta)$, so

[TABLE]

Therefore, we also have

[TABLE]

3.7. Bounds on the first and second derivatives of the solution

Let $U_{1}$ be the solution to (18) from section 3.5.

Lemma 9**.**

There is a constant $K_{1}$ independent of $U_{1}$ and $F$ so that

[TABLE]

Proof.

Step 1: Estimate away from the boundary.

The function $U_{1}$ is smooth in $\Omega$ by elliptic theory. A computation shows that the function

[TABLE]

satisfies the heat equation $v_{t}=\Delta v.$ The theory for parabolic equations gives us the following estimates on the derivatives of $v$,

[TABLE]

where $C(m)$ is a constant independent of $v$ or $s$ and as long as $v$ exists in $B_{s}(p)\times(-s^{2},0)$. This is true provided

[TABLE]

In particular, it is true if $|p|>\frac{5\sqrt{2}}{3}R$ and $s=|p|/10$. From (22), we get

[TABLE]

The last factor of the right hand side can be estimated using equation (20) to obtain

[TABLE]

Using the notation $\xi=p/\sqrt{2}$, we have

[TABLE]

Similarly, we get an estimate on the second derivative

[TABLE]

Therefore

[TABLE]

Since $R<2$, the estimate (23) is valid for $|\xi|\geq 4$ in particular.

Step 2: Estimates up to the boundary.

Consider the equation $\mathcal{L}U_{1}=0\textrm{ in the annulus }A_{R,4}=B_{4}\setminus\bar{B}_{R}.$ The domain $A_{R,4}$ is bounded so global regularity results for elliptic equations from [3] imply that

[TABLE]

where $\psi$ is a function in $W^{4,2}(A_{R,4})$ such that $\psi-U_{1}\in W^{1,2}_{0}(A_{R,4})$.

Note that we can bound the first four derivatives of $U_{1}$ for $|\xi|=4$ by an argument similar to the one in step 1,

[TABLE]

It is then clear that we can find a constant $C$ independent of $F$ and $U_{1}$, and a function $\psi$ for which

[TABLE]

With this choice of $\psi$, equation (24) gives us

[TABLE]

Using Sobolev’s inequality, (25), (26) and (20), we get

[TABLE]

Combining (23) and (27), we obtain the desired result. ∎

4. Finding sub and supersolutions

This section is devoted to the construction of a subsolution $u_{-}$ and a supersolution $u_{+}$ to the problem (6)-(4), or equivalently, to

[TABLE]

with small $\varepsilon$. Let us discuss the strategy for finding a supersolution first. Since the boundary data is of order $\varepsilon$, we write $u=\varepsilon U$,

[TABLE]

and decompose $U$ into the two terms $U=U_{1}+\varepsilon^{2}U_{2}$, where

[TABLE]

We discussed the existence of such a function $U_{1}$ in section 3.6. Choosing an appropriate function $U_{2}$ is done below.

4.1. Preliminary computations for finding a supersolution

In order to get a supersolution, we want $U_{2}$ to satisfy $\mathcal{L}U_{2}-\frac{D_{i}UD_{j}UD_{ij}U}{1+\varepsilon^{2}|DU|^{2}}\leq 0.$ Roughly estimating $U$ by $U_{1}$ and bounding the derivatives of $U_{1}$ using Lemma 9, we are looking for a $U_{2}$ so that

[TABLE]

where $C_{1}=K_{1}\|F\|_{C^{4}[0,2\pi]}$. Define $v_{a}=a\left(r-\frac{R^{2}}{r}\right).$ A simple computation gives

[TABLE]

This makes $v_{a}$ a perfect candidate for $U_{2}$.

Remark 10*.*

The term $a(1-2R^{2})/r$ has to counterbalance the contribution $C_{1}^{3}/r$ from the nonlinear term. Therefore, we need $(1-2R^{2})$ to be negative. This justifies the imposed lower bound $R>1/\sqrt{2}$.

4.2. Existence of a supersolution and a subsolution

Let $\varepsilon>0$ be a small constant. Consider the function

[TABLE]

where $a$ is a constant to be chosen later.

[TABLE]

We have $|D_{i}v_{a}|\leq 2a$ and $|D_{ij}v_{a}|\leq\frac{6a}{r}$, which combined with (21), give us

[TABLE]

Here we used the notation $C_{1}=K_{1}\|F\|_{C^{4}}$. We replace $a$ by $a=kC_{1}^{3}$, with $k$ a constant to be chosen later, to get

[TABLE]

If $\varepsilon$ is small enough so that $\eta:=\varepsilon^{2}kC_{1}^{2}\leq 1,$ the equation (32) becomes

[TABLE]

Hence, to have a supersolution, it suffices to find $k$ and $\varepsilon$ that satisfy

[TABLE]

If we take $R_{0}$ so that $R\geq R_{0}>1/\sqrt{2}$, both inequalities are true for $k=\frac{2}{2R_{0}^{2}-1}$ and $\varepsilon^{2}C_{1}^{2}\leq\frac{1}{62k}=\frac{2R_{0}^{2}-1}{124}$. Recall that $C_{1}=K_{1}\|F\|_{C^{4}}$, therefore, for $\varepsilon$ and $k$ so that

[TABLE]

the function $u_{+}=\varepsilon(U_{1}+\varepsilon^{2}kC_{1}^{3}(r-\frac{R^{2}}{r}))$ is a supersolution $\mathcal{E}u_{+}\leq 0$ with boundary condition $u_{+}|_{{\partial}\Omega}=\varepsilon F$. A similar argument shows that $u_{-}=\varepsilon(U_{1}-\varepsilon^{2}kC_{1}^{3}(r-\frac{R^{2}}{r}))$ is a subsolution with boundary condition $u_{-}|_{{\partial}\Omega}=\varepsilon F$. If we denote by $f=\varepsilon F$ and $u_{1}=\varepsilon U_{1}$, we just proved the following result.

Lemma 11**.**

Assume $R\geq R_{0}>1/\sqrt{2}$ and denote by $\Omega=\mathbf{R}^{2}\setminus\bar{B}_{R}$, the plane with a hole of radius $R$ at the origin. Let $f$ be a function in $C^{4}[0,2\pi]$ with the symmetries (2), $u_{1}$ a solution to the linear equation $\mathcal{L}u_{1}=0$, $u_{1}|_{{\partial}\Omega}=f$ and $K_{1}$ a constant (independent of $f$) so that

[TABLE]

If $\varepsilon=\|f\|_{C^{4}[0,2\pi]}$ satisfies

[TABLE]

then the function $u_{+}=u_{1}+K_{1}^{3}\varepsilon^{3}k(r-\frac{R^{2}}{r})$, where $k=\frac{2}{2R_{0}^{2}-1}$, satisfies

[TABLE]

and the function $u_{-}=u_{1}-K_{1}^{3}\varepsilon^{3}k(r-\frac{R^{2}}{r})$ satisfies

[TABLE]

From the bounds on the derivatives of $U_{1}$ and the definition of $u_{+}$ and $u_{-}$, we have

Corollary 12**.**

Let $u_{+}$ and $u_{-}$ be defined as in the lemma above. Then there exists a constant $K_{2}$ depending only on $\|f\|_{C^{4}}$ and $R_{0}$ so that

[TABLE]

5. The Parabolic Equation ${\partial}_{\tau}u=\mathcal{E}u$

In this section, we prove that there exists a solution to the parabolic equation ${\partial}_{\tau}u=\mathcal{E}u$ for all time $\tau\in[0,\infty)$ for well chosen initial and boundary conditions $u_{0}$ and $f$ respectively.

To avoid confusion, let us fix notations: we denote by $D_{i}$ the ordinary derivative with respect to $i$-th coordinate $x_{i}$ or $\xi_{i}$ of a spatial variable and by $Dv$ the gradient of the function $v$ with respect to spatial variables. The notation ${\partial}_{\tau}$ or ${\partial}_{t}$ is reserved for the derivative with respect to time.

5.1. Initial condition and short time existence

For the initial condition to the parabolic equation, we choose a $C^{3}$ function $u_{0}$ in $\Omega$ that stays between the sub and supersolutions constructed in Lemma 11 and has bounded derivatives

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Moreover, we assume that $u_{0}$ satisfies the following symmetries and compatibility conditions

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Let us define the domains $\mathcal{O}_{t}=\{x\in\mathbf{R}^{2}\mid|x|>\sqrt{2(1-t)}\ R\}$ and $\mathcal{Q}_{T}=\{(x,t)\mid 0<t<T,|x|>\sqrt{2(1-t)}\ R\}$. We consider the following corresponding problem for the mean curvature flow

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for some initial condition $v_{0}(x)$. Standard parabolic theory assures the existence of a smooth short time solution $v(x,t)$ to the boundary value problem (35) with initial condition $v_{0}(x)=\sqrt{2}u_{0}(\frac{x}{\sqrt{2}})$. Moreover, if we denote the maximal time interval in which $v$ exists by $[0,t_{0})$, we have

[TABLE]

For more details, we refer to [2].

In order to show that the solution $v$ exists for all time $t$, we argue by contradiction and assume that $t_{0}<\infty$ then show that $M_{t}$ is uniformly bounded for all $t\in[0,t_{0})$. Standard parabolic theory implies that the solution $v$ can then be continuated past $t_{0}$, which contradicts the maximality of $[0,t_{0})$. The rest of this section is devoted to the proof of a uniform bound on $M_{t}$.

The function $u$ defined by

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satisfies

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where we used the change of variables $\xi=\frac{x}{\sqrt{2(1-t)}},\tau=-\frac{1}{2}\ln(1-t)$ and $\tau_{0}=-\frac{1}{2}\ln(1-t_{0})$.

Lemma 13**.**

Let $u$ be a smooth solution of (38a) in $\Omega\times[0,\tau_{0})$ with boundary condition (38b), then

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Proof.

To each solution $u$ to (38) corresponds a solution $v$ to the problem (35) via the formula (37). This lemma is therefore a consequence of (36). ∎

5.2. Bounds on the function $u$

Here, we show that a solution $u$ to the problem (38) stays between the subsolution $u_{-}$ and the supersolution $u_{+}$:

Lemma 14**.**

Let $u(\xi,\tau)$ be a $C^{2}$ solution to the problem (38), then

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Proof.

The proof is a slight modification of the proof of a Maximum Principle for parabolic equations. Let $K_{2}$ be as in Corollary 12. For small $\varepsilon>0$, consider the function $W=\psi(u_{-}-u)-\varepsilon$, where $\psi(\xi,\tau):=e^{-\varepsilon|\xi|-(1+\delta)\tau}$ and $\delta$ is to be chosen later. Note that $W$ becomes negative as $|\xi|$ tends to infinity. We have ${\mathrm{\partial }}_{\tau }\psi =-(1+\delta )\psi ,D_i\psi =-\epsilon \frac{{\xi }_i}{}$