Short time existence of the classical solution to the fractional Mean curvature flow

TL;DR

This paper proves that smooth solutions to the fractional mean curvature flow exist for a short time when starting from bounded, regular initial sets, including volume-preserving cases.

Contribution

It establishes the short-time existence of solutions for the fractional mean curvature flow and volume-preserving variants, extending the understanding of these geometric flows.

Findings

Short-time existence of smooth solutions proven

Results apply to volume-preserving fractional mean curvature flow

Solutions exist for initial sets with C^{1,1} regularity

Abstract

We establish short-time existence of the smooth solution to the fractional mean curvature flow when the initial set is bounded and C^{1,1}-regular. We provide the same result also for the volume preserving fractional mean curvature flow.

Equations

$$V_t=-H_{E_t}^s\text{on }\partial E_t,$$

$$H_E^s(x):=\text{p.v.}\left({\int }_{E^c}\frac{dy}{|y-x{|}^{n+1+s}}-{\int }_E\frac{dy}{|x-y{|}^{n+1+s}}\right).$$

$$V_t=-(H_{E_t}^s-{\bar{H}}_{E_t}^s)\text{on }\partial E_t,$$

$${\partial }_{t}u={\Delta }^{\frac{1+s}{2}}u+Q(x,u,\nabla u)-H_{\Sigma }^s\text{on }\Sigma \times (0,T],$$

$$\nabla u(X)={\nabla }_{X}u=Xu,$$

$$\nabla F(Y_1,\dots ,Y_k,X)=({\nabla }_{X}F)(Y_1,\dots ,Y_k),$$

$$({\nabla }_{X}F)(Y_1,\dots ,Y_k)=XF(Y_1,\dots ,Y_k)-\sum _{i=1}^{k}F(Y_1,\dots ,{\nabla }_X{Y}_i,\dots ,Y_k).$$

$${\nabla }_{X_k}\cdots {\nabla }_{X_1}u={\nabla }^{k}u(X_1,\dots ,X_k).$$

$$\Vert u{\Vert }_{C^0(\Sigma )}:=\underset{x\in \Sigma }{\sup }|u(x)|$$

$$\Vert u{\Vert }_{C^{\alpha }(\Sigma )}:=\underset{\begin{array}{c}x,y\in \Sigma \\ x\ne y\end{array}}{\sup }\frac{|u(y)-u(x)|}{|y-x{|}^{\alpha }}+\Vert u{\Vert }_{C^0(\Sigma )}.$$

$$\Vert F{\Vert }_{C^{\alpha }(\Sigma )}:=\sup \{\Vert F(X_1,\dots ,X_k){\Vert }_{C^{\alpha }(\Sigma )}:X_i\in \mathcal{T}(\Sigma )\text{with}\Vert X_i{\Vert }_{C^1(\Sigma )}\le 1,i=1\dots ,k\}.$$

$$\Vert u{\Vert }_{C^{k+\alpha }(\Sigma )}:=\sum _{l=0}^k\Vert {\nabla }^{l}u{\Vert }_{C^{\alpha }(\Sigma )}.$$

$$\Vert u{\Vert }_{C^{1+\alpha }(\Sigma )}\le \delta \Vert u{\Vert }_{C^{1+s+\alpha }(\Sigma )}+C_{\delta }\Vert u{\Vert }_{C^0(\Sigma )}.$$

$${\nabla }^{k}u(X_1,\dots ,X_i,\dots ,X_j,\dots ,Y_k)={\nabla }^{k}u(X_1,\dots ,X_j,\dots ,X_i,\dots ,Y_k)+{\partial }^{k-1}u.$$

$$\Vert {\partial }^{l}u{\Vert }_{C^{\gamma }(\Sigma )}\le C_{l,\gamma }\Vert u{\Vert }_{C^{l+\gamma }(\Sigma )}$$

$${\nabla }_{X_k}\cdots {\nabla }_{X_1}u=X_k\cdots X_{1}u+{\partial }^{k-1}u,$$

$$\begin{array}{rl}\sup \{\Vert X_k\cdots X_{1}u{\Vert }_{C^{\gamma }(\Sigma )}& :X_i\in \mathcal{T}(\Sigma ),\Vert X_i{\Vert }_{C^{k+2}(\Sigma )}\le 1,i=1,\dots ,k\}\\ & \ge \frac{1}{C_k}\Vert u{\Vert }_{C^{k+\gamma }(\Sigma )}-C_k\Vert u{\Vert }_{C^{k-1+\gamma }(\Sigma )},\end{array}$$

$$D_{\tau }F(x)=D\tilde{F}(x)(I-\nu (x)\otimes \nu (x)),$$

$$v(u)=D_{\tau }u(x)\cdot p\text{for all }u\in C^{\infty }(\Sigma ),$$

$${\nabla }_{X}u=D_{\tau }u\cdot \tilde{X}\text{for all }u\in C^{\infty }(\Sigma ).$$

$$\frac{1}{C}\Vert u{\Vert }_{C^{1+\alpha }(\Sigma )}\le \Vert D_{\tau }u{\Vert }_{C^{\alpha }(\Sigma )}+\Vert u{\Vert }_{C^0(\Sigma )}\le C\Vert u{\Vert }_{C^{1+\alpha }(\Sigma )}.$$

$${\int }_{\Sigma }\mathrm{div}Xd{\mathcal{H}}^n=0.$$

$${\int }_{\Sigma }\mathrm{div}\tilde{X}d{\mathcal{H}}^n={\int }_{\Sigma }{H}_{\Sigma }\tilde{X}\cdot \nu d{\mathcal{H}}^n,$$

$${\Delta }^{\frac{s+1}{2}}u(x):=2{\int }_{\Sigma }\frac{u(y)-u(x)}{|y-x{|}^{n+1+s}}d{\mathcal{H}}_y^n.$$

$$L[u](x):={\Delta }^{\frac{s+1}{2}}u(x)+c_s^2(x)u(x),$$

$$c_s^2(x)={\int }_{\Sigma }\frac{|\nu (y)-\nu (x){|}^2}{|y-x{|}^{n+1+s}}d{\mathcal{H}}_y^n.$$

$$\{\begin{array}{ll}& {\partial }_{t}u={\Delta }^{\frac{s+1}{2}}u+f(x,t)+g(x)\text{on }\Sigma \times (0,T]\\ & u(x,0)=u_0(x)\text{for }x\in \Sigma .\end{array}$$

$$\sup_{0<t<T}\|u(\cdot,t)\|_{C^{1+s+\alpha}(\Sigma)}\leq C(1+T)\big{(}\|u_{0}\|_{C^{1+s+\alpha}(\Sigma)}+\sup_{0<t<T}\|f(\cdot,t)\|_{C^{\alpha}(\Sigma)}+T\|g\|_{C^{1+s+\alpha}(\Sigma)}\big{)}$$

$$\sup_{0<t<T}\|u(\cdot,t)\|_{C^{0}(\Sigma)}\leq\|u_{0}\|_{C^{0}(\Sigma)}+T\big{(}\sup_{0<t<T}\|f(\cdot,t)\|_{C^{0}(\Sigma)}+\|g\|_{C^{0}(\Sigma)}\big{)}$$

$$\partial E_0=\{x+h_0(x)\nu (x):x\in \Sigma \}$$

Full text

1. Introduction

In this paper we study the motion of sets by their fractional mean curvature from the classical point of view. In the problem we are given a bounded regular set $E_{0}\subset\mathbb{R}^{n+1}$ and we let it evolve to a family of smooth sets $(E_{t})_{t\in(0,T]}$ according to the law where the normal velocity at a point equals its fractional mean curvature. More precisely this can be written as

[TABLE]

where $V_{t}$ denotes the normal velocity and $H_{E}^{s}$ is the fractional mean curvature of $E\subset\mathbb{R}^{n+1}$ with $s\in(0,1)$, given by

[TABLE]

One motivation to study (1.1) is that it can be interpreted as the gradient flow of the fractional perimeter and is therefore the evolutionary counterpart to the problem of minimizing the fractional perimeter. The stationary problem has received a lot of attention since the work [5], where the authors study the regularity of sets which locally minimize the fractional perimeter. By [5] and [3] we know that the local perimeter minimizers are smooth up to a small singular set which precise dimension is not known (see [29] and [14] and the references therein). The global isoperimetric problem in the Euclidian space is also well understood. It is proven in [18] that the ball is the solution to the fractional isoperimetric problem, and we even have the sharp quantification of the isoperimetric inequality [17, 19]. A related result is the generalization of the Alexandroff theorem [4, 13], where the authors prove that the ball is the only smooth compact set with constant fractional mean curvature. Note that one does not need to assume the set to be connected, which is in contrast to the classical Alexandroff theorem.

Concerning the evolutionary problem, the first existence result is due to Imbert [24] who defines the viscosity solution of (1.1) and proves that it exists for all times and is unique. This means that the flow (1.1) has a well defined weak solution which is unique up to fattening. In [6] Caffarelli-Souganidis construct weak solution by using threshold dynamics, and in [9] Chambolle-Morini-Ponsiglione use the gradient flow structure to construct weak solution by using the minimizing movement scheme (see also [8]).

The issue with the weak solution is, as observed in [11] (see also [7]), that the flow may develop singularities even in the planar case. Cinti-Sinestrari-Valdinoci [12] avoid singularities by studying the volume preserving fractional mean curvature flow

[TABLE]

for convex initial sets, where $\bar{H}_{E_{t}}^{s}=\fint_{\partial E_{t}}H_{E_{t}}^{s}\,d\mathcal{H}^{n}$. By [10], the fractional mean curvature flow with a forcing term preserves convexity and thus one expects Huisken’s result [22] to hold also in the fractional case, i.e., the flow (1.3) remains convex, does not develop singularities and converges to the sphere. This is indeed the main result in [12] under an additional regularity assumption. We remark that also (1.1) preserves the convexity and therefore one may expect a result similar to [21] to hold also for (1.1), but to the best of our knowledge no result in this direction exists. Finally we refer to [28] for interesting analysis of the smooth solution of (1.1).

Here we are interested in the classical solution of (1.1) which means that the sets $(E_{t})_{t}$ are smooth and diffeomorphic to $E_{0}$. As we mentioned we may expect the classical solution to exist only for a short time interval $(0,T)$ as the flow will vanish in finite time or it may develop singularities before that. We prove the short time existence of the classical solution under the assumption that the initial set is $C^{1,1}$-regular, or $C^{1+s+\alpha}$-close to a $C^{1,1}$-regular set. The same result holds also for the volume preserving flow (1.3) and we choose to state the result for both cases simultaneously.

Main Theorem**.**

Let $E_{0}\subset\mathbb{R}^{n+1}$ be a bounded set such that $\partial E_{0}$ is a $C^{1,1}$-regular hypersurface and let $\alpha\in(0,1)$. There exists $T\in(0,1)$ such that the fractional mean curvature flow (1.1), and the volume preserving flow (1.3), has a unique classical solution $(E_{t})_{t\in(0,T]}$ starting from $E_{0}$. The flow becomes instantaneously smooth, i.e., each surface $\partial E_{t}$ with $t\in(0,T]$ is $C^{\infty}$-hypersurface. Moreover, there is $\varepsilon>0$ with the property that if the $C^{1+s+\alpha}$-distance between $E_{0}$ and $E$ is less than $\varepsilon$, then the flow (1.1) (and the flow (1.3)) starting from $E$ also exists for the time interval $(0,T]$.

We give a more quantitative statement of the main theorem in the last section in Theorem 5.1 and in Theorem 5.3. We expect the smooth solution of (1.1) to agree with the viscosity solution on the time interval $(0,T]$ but we do not prove this.

The proof of the main theorem is based on Schauder estimates on parabolic equations. As in [23] we first parametrize the flow (1.1) by using the ’height’ function over a smooth reference surface $\Sigma$, which is close to the initial surface. This leads us to solve the nonlinear nonlocal parabolic equation

[TABLE]

where $\Delta^{\frac{1+s}{2}}$ denotes the fractional Laplacian on $\Sigma$ and $Q$ is nonlinear. Following the idea in [16] we prove that the nonlinear term $Q$ in (1.4) remains small for a short time interval and the equation can thus be seen as a small perturbation of the fractional heat equation. We then use Schauder estimates and a standard fixed point argument to obtain the existence of a solution which is $C^{1+s+\alpha}$-regular in space and obtain the smoothness by differentiating the equation. The right hand side of (1.4) is essentially the parametrization of the fractional mean curvature. Similarly as in [3] and [14], the main difficulty in our analysis is due to the complicated structure of the nonlinear term $Q$, which makes it challenging to estimate its $C^{k+\alpha}$-norm in a quantitative way. In order to differentiate the equation (1.4) multiple times we need effective notation and basic tools from differential geometry. We also point out that the $C^{1,1}$-regularity of the intial set causes additional difficulties, because some of the constants in (1.4) depend on the $C^{\alpha}$-norm of the curvature of $\Sigma$, which we cannot bound uniformly if we want $\Sigma$ to be close to $\partial E_{0}$. Finally another technical issue, although a minor one, is that there is no comprehensive Schauder theory for the fractional heat equation on compact hypersurfaces in the literature, and therefore we have to prove the appropriate estimates ourselves (see Theorem 2.2).

The paper is organized as follows. After a short preliminary section (Section 2) we derive the equation (1.4) in Section 3 by using the parametrization by the height function (Proposition 3.2). In Section 4 we study the spatial regularity properties of the operator on the right-hand-side of (1.4) and give the proof of the main theorem in Section 5. The Appendix contains the Schauder estimates for the fractional heat equation with a forcing term on compact hypersurfaces, which might be of independent interest.

2. Preliminaries

Throughout the paper we assume that $\Sigma\subset\mathbb{R}^{n+1}$ is a smooth compact hypersurface, i.e., there is a smooth bounded set $G\subset\mathbb{R}^{n+1}$ such that $\partial G=\Sigma$. We choose $G$ as our reference set and define the classical solution of (1.1) (and (1.3)) as in the case of the classical mean curvature flow [26], i.e., we say that $(E_{t})_{t\in(0,T]}$ is a classical solution of (1.1) starting from $E_{0}$ if there exists a map $\Psi\in C(\mathbb{R}^{n+1}\times[0,T];\mathbb{R}^{n+1})\cap C^{\infty}(\mathbb{R}^{n+1}\times(0,T];\mathbb{R}^{n+1})$ such that $\Psi(\cdot,t)$ is a $C^{1+s}$-diffeomorphism for $t\in[0,T]$ and smooth diffeomorphism for $t\in(0,T]$, with $E_{t}=\Psi(G,t)$ for $t\in(0,T]$, $E_{0}=\Psi(G,0)$ and $E_{t}$ satisfies (1.1) (or (1.3) in the volume preserving case).

2.1. Geometric preliminaries

We recall some basic analysis related to Riemannian manifolds. For an introduction to the topic we refer to [25], from where we also adopt our notation.

Since $\Sigma$ is embedded in $\mathbb{R}^{n+1}$ it has a natural metric $g$ induced by the Euclidian metric. Then $(\Sigma,g)$ is a Riemannian manifold and we denote the inner product on each tangent space $X,Y\in T_{x}\Sigma$ by $\langle X,Y\rangle$. We extend the inner product in a natural way for tensors. We denote by $\mathscr{T}(\Sigma)$ the smooth vector fields on $\Sigma$ and recall that for $X\in\mathscr{T}(\Sigma)$ and $u\in C^{\infty}(\Sigma)$ the notation $Xu$ means the derivative of $u$ in direction of $X$. We emphasize that we assume every vector field to be smooth.

We denote the Riemannian connection on $\Sigma$ by $\nabla$ and recall that for a function $u\in C^{\infty}(\Sigma)$ the covariant derivative $\nabla u$ is a $1$-tensor field defined for $X\in\mathscr{T}(\Sigma)$ as

[TABLE]

i.e., the derivative of $u$ in the direction of $X$. The covariant derivative of a smooth $k$-tensor field $F\in\mathscr{T}^{k}(\Sigma)$, denoted by $\nabla F$, is a $(k+1)$-tensor field and is defined for $Y_{1},\dots,Y_{k},X\in\mathscr{T}(\Sigma)$ as

[TABLE]

where

[TABLE]

Here $\nabla_{X}Y$ is the covariant derivative of $Y$ in the direction of $X$ (see [25]) and since $\nabla$ is the Riemannian connection it holds $\nabla_{X}Y=\nabla_{Y}X+[X,Y]$ for every $X,Y\in\mathscr{T}(\Sigma)$.

We denote the $k$th order covariant derivative of a smooth function $u\in C^{\infty}(\Sigma)$ by $\nabla^{k}u$, which is a $k$-tensor field defined recursively as $\nabla^{k}u=\nabla(\nabla^{k-1}u)$. Let $X_{1},\dots,X_{k}\in\mathscr{T}(\Sigma)$ be vector fields on $\Sigma$. Then $\nabla^{k}u(X_{1},\dots,X_{k})$ denotes the covariant derivative applied to $X_{1},\dots,X_{k}$ and we often use the notation

[TABLE]

We use the fact that $\Sigma$ is embedded in $\mathbb{R}^{n+1}$ and define the sup-norm and the $\alpha$-Hölder norm, for $\alpha\in(0,1)$, of a function $u\in C(\Sigma)$ in a standard way,

[TABLE]

and

[TABLE]

Moreover, we set $\|u\|_{C^{k}(\Sigma)}:=\sum_{l=0}^{k}\|\nabla^{l}u\|_{C^{0}(\Sigma)}$ for all $k\in\mathbb{N}$. We define further the $\alpha$-Hölder norm of a $k$-tensor field $F\in\mathscr{T}^{k}(\Sigma)$ by

[TABLE]

It is straightforward to check that this agrees with the more standard definition via partition of unity. Using this we define the $C^{k+\alpha}$- norm of function $u$, with $k\in\mathbb{N}$ and $\alpha\in[0,1)$, as

[TABLE]

We use the notation $u\in C^{k+\alpha}(\Sigma)$ for a function $u$ with bounded $C^{k+\alpha}(\Sigma)$-norm when $\alpha\in(0,1)$ and $u\in C^{k,1}(\Sigma)$ when $\alpha=1$. We assume that $\Sigma=\partial G$ is uniformly $C^{1,1}$-regular surface and define its $C^{1,1}$-norm as the smallest number $R$ such that $G$ satisfies the interior and the exterior ball condition with radius $1/R$. Note that this norm also bounds $\|\nu\|_{C^{1}(\Sigma)}$. If a constant depends on the $C^{1,1}$-norm of $\Sigma$ we choose not to mention it and call such a constant uniform.

We recall the following interpolation inequality. The proof is essentially the same as [20, Lemma 6.32] (see also [2]).

Lemma 2.1**.**

Assume $\Sigma$ is a compact $C^{1,1}$-hypersurface and let $s\in(0,1)$ and $\alpha\in(0,1-s)$. For every $\delta\in(0,1)$ there is $C_{\delta}>0$ such that for $u\in C^{1+s+\alpha}(\Sigma)$ it holds

[TABLE]

Observe that $\nabla^{2}u$ is symmetric, i.e., $\nabla^{2}u(X,Y)=\nabla^{2}u(Y,X)$ for every vector fields $X$ and $Y$, while $\nabla^{k}u$ for $k\geq 3$ is not. From the definition we see that for $X_{1},\dots,X_{k}\in\mathscr{T}(\Sigma)$ with $\|X_{i}\|_{C^{k+2}(\Sigma)}\leq 1$, $i=1,\dots,k$ it holds

[TABLE]

Here the notation $\partial^{l}u$ stands for a function which satisfies

[TABLE]

for any $\gamma\in(0,2)$. Note also that in general $\nabla_{Y}\nabla_{X}u\neq\nabla_{Y}(\nabla_{X}u)$ for $X,Y\in\mathscr{T}(\Sigma)$ since $\nabla_{Y}(\nabla_{X}u)=Y(Xu)=YXu$ and $\nabla_{Y}\nabla_{X}u=YXu-(\nabla_{Y}X)u$. On the other hand if $X_{1},\dots,X_{k}$ are vector fields with $\|X_{i}\|_{C^{k+2}(\Sigma)}\leq 1$, $i=1,\dots,k$, then it holds

[TABLE]

where $\partial^{k-1}u$ denotes a function which satisfies (2.2). It is then straightforward to check that for any $\gamma\in(0,2)$ it holds

[TABLE]

where $C_{k}$ depends on $k$.

We may use the fact that $\Sigma$ is embedded in $\mathbb{R}^{n+1}$ to extend any function $F\in C^{1}(\Sigma;\mathbb{R}^{m})$ to $\tilde{F}\in C^{1}(\mathbb{R}^{n+1};\mathbb{R}^{m})$ such that $\tilde{F}=F$ on $\Sigma$. We define the tangential differential of $F$ by

[TABLE]

where $\nu$ denotes the unit outer normal of $\Sigma=\partial G$ (outer with respect to $G$). We denote $\nu_{\Sigma}$ if we want to be emphasize that the normal is related to $\Sigma$ and denote by $\nu_{E}(x)$ the normal of a generic set $E$. It is clear that $D_{\tau}F(x)$ does not depend on the chosen extension. With a slight abuse of notation we denote the tangential gradient of $u\in C^{\infty}(\Sigma)$ at $x$ also by $D_{\tau}u(x)$, even if it is a vector in $\mathbb{R}^{n+1}$.

We may use the embedding to associate the tangent space $T_{x}\Sigma$ with the linear subspace $\{p\in\mathbb{R}^{n+1}:p\cdot\nu(x)=0\}$ by the relation

[TABLE]

where $v\in T_{x}\Sigma$, i.e., a derivation at $x$, and $p\in\mathbb{R}^{n+1}$ with $p\cdot\nu(x)=0$. The components of the vector $p$ are then given by $p_{i}=v(x_{i})$. Indeed, by ’tangent space’ we usually mean the geometric tangent space, i.e., a linear subspace of $\mathbb{R}^{n+1}$, but for clarity we use ’$\cdot$’ for the standard inner product of two vectors in $\mathbb{R}^{n+1}$ while ’$\langle\cdot,\cdot\rangle$’ denotes the inner product on the tangent space. Similarly we may associate a smooth vector field $X\in\mathscr{T}(\Sigma)$ with the vector valued function $\tilde{X}\in C^{\infty}(\Sigma,\mathbb{R}^{n+1})$ which satisfies $\tilde{X}(x)\cdot\nu(x)=0$ for all $x\in\Sigma$ and

[TABLE]

Therefore, by a vector field $X$ we usually mean a vector valued function which values are on the (geometric) tangent space, $X\cdot\nu=0$, with the convention that $Xu$ denotes the derivative of $u$ in direction of $X$. It is also clear that the tangential gradient of $u\in C^{\infty}(\Sigma)$ is equivalent to its covariant derivative and for every $\alpha\in(0,1)$ it holds

[TABLE]

We denote the divergence of a vector field $X\in\mathscr{T}(\Sigma)$ by $\operatorname{div}X$ and the divergence theorem states

[TABLE]

For clarity we denote the divergence of a vector valued function $F\in C^{\infty}(\mathbb{R}^{n+1},\mathbb{R}^{n+1})$ in $\mathbb{R}^{n+1}$ by $\operatorname{div}_{\mathbb{R}^{n+1}}F$. We may extend the definition of divergence to vector valued functions $\tilde{X}\in C^{\infty}(\Sigma,\mathbb{R}^{n+1})$ by $\operatorname{div}\tilde{X}:=\text{Trace}(D_{\tau}\tilde{X})$. Then the divergence theorem generalizes to

[TABLE]

where $H_{\Sigma}$ denotes the mean curvature of $\Sigma$, which is the sum of the principal curvatures.

2.2. Fractional Laplacian

We define the fractional Laplacian on $\Sigma$ as

[TABLE]

This should be understood in principal valued sense, but from now on we assume this without further mention. It is not difficult to see, and it actually follows from Proposition 4.9, that if $u\in C^{\infty}(\Sigma)$ then $\Delta^{\frac{s+1}{2}}u$ is a smooth function on $\Sigma$. It is well known [14, 17] that by linearizing the fractional mean curvature at $\Sigma$ one obtains the following Jacobi operator

[TABLE]

where

[TABLE]

We note that since $\Sigma$ is a smooth surface, $c_{s}^{2}(\cdot)$ defines a smooth function on $\Sigma$. Again this is not difficult to see and it follows from our analysis in Section 4. Moreover, since we assume $\Sigma$ is uniformly $C^{1,1}$-regular, the $\alpha$-Hölder norm of $c_{s}^{2}$, for small $\alpha$, is uniformly bounded (see Lemma 4.3).

As we mentioned in the introduction, the proof of the main theorem is based on regularity estimates for nonlinear nonlocal parabolic equation. To this aim we need standard Schauder estimates for the fractional heat equation with a forcing term

[TABLE]

We prove the following Schauder estimate. We give the proof in the Appendix.

Theorem 2.2**.**

Assume that $f:\Sigma\times[0,T]\to\mathbb{R}$ and $u_{0},g:\Sigma\to\mathbb{R}$ are smooth and fix $\alpha\in(0,1-s)$. Then (2.6) has a unique smooth solution and it holds

[TABLE]

and

[TABLE]

The second statement is in fact a simple consequence of the maximum principle.

3. Parametrization of the flow (1.1)

In this section we follow [23] (see also [26]) and parametrize the equation (1.1) by using the height function over a smooth reference surface. Note first that since $\partial E_{0}$ is a compact $C^{1,1}$-hypersurface we find for any $\varepsilon>0$ a smooth compact hypersurface $\Sigma$ such that we may write $\partial E_{0}$ as a graph over $\Sigma$,

[TABLE]

with $\|h_{0}\|_{C^{0}(\Sigma)}<\varepsilon$ and $\|h_{0}\|_{C^{2}(\Sigma)}\leq C$. Indeed, we may first fix a smooth surface $\Sigma_{0}$ such that

[TABLE]

where $\tilde{h}\in C^{1,1}(\Sigma_{0})$ (note that $\|\tilde{h}\|_{C^{0}(\Sigma_{0})}$ is not necessarily small). By standard mollification argument we find $\tilde{h}_{\varepsilon}\in C^{\infty}(\Sigma_{0})$ with

[TABLE]

Thus we may define $\Sigma=\{x+\tilde{h}_{\varepsilon}(x)\nu_{\Sigma_{0}}(x):x\in\Sigma_{0}\}$.

From now on we assume that $\alpha$ is a positive number such that $\alpha<(1-s)/2$. We note that because $\partial E_{0}$ is only $C^{1,1}$-regular we have $\|\nu_{\Sigma}\|_{C^{1}(\Sigma)}\leq C$ but $\|\nu_{\Sigma}\|_{C^{1+s+\alpha}(\Sigma)}\leq C\varepsilon^{-s-\alpha}$. This means that we have to be careful in our analysis whenever we have terms which depend on the norm $\|\nu_{\Sigma}\|_{C^{1+s+\alpha}(\Sigma)}$, because we cannot bound it uniformly if we want $\Sigma$ to be close to $\partial E_{0}$. Note that

[TABLE]

and therefore even if $\|\nu_{\Sigma}\|_{C^{1+s+\alpha}(\Sigma)}$ is large it still holds

[TABLE]

for $\alpha<(1-s)/2$. Therefore for any $\delta>0$ we may choose $\varepsilon$ small such that

[TABLE]

In particular, this implies $\|h_{0}\,\nu_{\Sigma}\|_{C^{1+s+\alpha}(\Sigma)}\leq C\delta$.

Our goal is to write the family of sets $(E_{t})_{t\in(0,T]}$, which is a solution of (1.1), as a graph over the reference surface $\Sigma$. To be more precise, we look for a function $h\in C(\Sigma\times[0,T])\cap C^{\infty}(\Sigma\times(0,T])$ such that the family of sets $E_{t}$ given by

[TABLE]

is a solution of (1.1). In this section we provide the calculations which show that this leads to the equation

[TABLE]

where $H_{\Sigma}^{s}$ is the fractional mean curvature of the reference surface $\Sigma$ and $L[\cdot]$ is the linear operator defined in (2.5). The precise formula for the remainder term $P$ is given in Proposition 3.2. Our goal in the next section is then to show that for $\delta>0$ small the function $x\mapsto P(x,u,\nabla u)$ satisfies

[TABLE]

when $\|u\|_{C^{1+s+\alpha}(\Sigma)}\leq\delta$. This means that we may treat (3.3) as a small perturbation of the fractional heat equation, i.e., (2.6) with $f=0$ and $g=0$.

In order to calculate (3.3) we define the class of sets $\mathfrak{h}_{\delta}(\Sigma)$ such that $E\in\mathfrak{h}_{\delta}(\Sigma)$ if its boundary can be written as

[TABLE]

In particular, if $E\in\mathfrak{h}_{\delta}(\Sigma)$ then its boundary is a compact $C^{1+s+\alpha}$-hypersurface.

We begin with a standard calculation.

Lemma 3.1**.**

Let $E\subset\mathbb{R}^{n+1}$ be a smooth bounded set, let $\Phi_{\tau}$ be a family of diffeomorphisms such that $\Phi_{0}(x)=x$, denote the velocity field by X(x)=\frac{d}{d\tau}\big{|}_{\tau=0}\Phi_{\tau}(x) and suppose $X\in C^{1+s+\alpha}(\Sigma)$. Then it holds

[TABLE]

Proof.

Let us denote $E_{\tau}=\Phi_{\tau}(E)$ and $k_{\varepsilon}(z)=(|z|^{2}+\varepsilon)^{-\frac{n+s+1}{2}}$. It is enough to show that

[TABLE]

Indeed, by repeating the same calculations for the second term in (1.2) and letting $\varepsilon\to 0$ yields the result. We split the above term as

[TABLE]

Let us denote the Jacobian of $\Phi_{\tau}$ by $J_{\Phi_{\tau},\mathbb{R}^{n+1}}$. Since \frac{d}{d\tau}\big{|}_{\tau=0}J_{\Phi_{\tau},\mathbb{R}^{n+1}}=\operatorname{div}_{\mathbb{R}^{n+1}}X, we may write the first term by change of variables as

[TABLE]

By symmetry it holds $D_{x}k_{\varepsilon}(x-y)=-D_{y}k_{\varepsilon}(x-y)$. Therefore we have for the second term

[TABLE]

∎

We may use Lemma 3.1 to write the fractional mean curvature $H_{E}^{s}$ over the reference surface $\Sigma$. Let $E\in\mathfrak{h}_{\delta}(\Sigma)$ with $\partial E=\{x+h(x)\nu(x):x\in\Sigma\}$. We define the sets $E_{t^{\prime}}$ as $\partial E_{t^{\prime}}:=\{x+t^{\prime}h(x)\nu(x):x\in\Sigma\}$, with $t^{\prime}\in[0,1]$, and family of diffeomorphisms $\Phi_{t^{\prime}h}:\Sigma\to\partial E_{t^{\prime}}$ as

[TABLE]

Then for $x\in\Sigma$ we have

[TABLE]

We denote the tangential Jacobian of $\Phi_{t^{\prime}h}$ by $J_{\Phi_{t^{\prime}h}}$ (see [1] for details) and define $\Phi_{\tau}(x):=\Phi_{(t^{\prime}+\tau)h}(\Phi_{t^{\prime}h}^{-1}(x))$. Note that $\Phi_{\tau}:\partial E_{t^{\prime}}\to\partial E_{t^{\prime}+\tau}$ is a diffeomorphism and

[TABLE]

We apply Lemma 3.1 and change of variables to deduce

[TABLE]

We may write the normal $\nu_{E_{t^{\prime}}}$ (see [26, Section 1.5]) as

[TABLE]

where $Q_{i}$ are smooth functions which depend on the second fundamental form of $\Sigma$ and $Q_{i}(y,0,0)=0$, for $i=1,2$ for all $y\in\Sigma$. Moreover, $Q_{2}$ takes values on the tangent space, i.e., $Q_{2}(y,\cdot,\cdot)\cdot\nu(y)=0$. We may thus write

[TABLE]

We write $2(\nu(y)-\nu(x))\cdot\nu(y)=|\nu(y)-\nu(x)|^{2}$ and obtain by (3.6)

[TABLE]

To shorten the notation we denote the kernel $K_{u}:\Sigma\times\Sigma\to[0,\infty]$ generated by $u\in C^{1+s+\alpha}(\Sigma)$ as

[TABLE]

Recall that $Q_{1}(y,0,0)=0$. We may thus write

[TABLE]

We may finally write the fractional mean curvature of $E\in\mathfrak{h}_{\delta}(\Sigma)$ by recalling the linear operator $L[\cdot]$ in (2.5), by (3.5) and by the previous calculations

[TABLE]

The remainder terms $R_{1,u}$ and $R_{2,u}$ are defined for a generic function $u\in C^{1+s+\alpha}(\Sigma)$ with $\|u\|_{C^{1+s+\alpha}(\Sigma)}\leq\delta$ as

[TABLE]

and

[TABLE]

where the kernel $K_{u}$ is defined in (3.7), and $Q_{1},Q_{2}$ are smooth functions which satisfy $Q_{1}(y,0,0)=Q_{2}(y,0,0)=0$ for all $y\in\Sigma$.

In order to write the flow (1.1) as an equation we recall from [26] that the normal velocity of the flow $(E_{t})_{t}$ given by $E_{t}=\Phi_{t}(E)$, where $\Phi_{t}(x)=x+h(x,t)\nu(x)$ on $\Sigma$, is

[TABLE]

By choosing in $t^{\prime}=1$ in (3.6) we have

[TABLE]

and the Jacobian can be written as $J_{\Phi_{t}}=1+Q_{3}(y,h,\nabla h)$. Here $Q_{1}$ and $Q_{3}$ are smooth functions with $Q_{1}(x,0,0)=Q_{3}(x,0,0)=0$ for all $x\in\Sigma$. Thus when $E_{t}\in\mathfrak{h}_{\delta}(\Sigma)$ for small enough $\delta$ we may write

[TABLE]

where $Q$ is a smooth function with $Q(x,0,0)=0$ for all $x\in\Sigma$. We may finally write the equation for $h$ by combing (3.8) and (3.11). We state this in the following proposition.

Proposition 3.2**.**

Assume that the flow $(E_{t})_{t\in(0,T]}$, with $E_{t}\in\mathfrak{h}_{\delta}(\Sigma)$ for $t\in(0,T]$, is a classical solution of (1.1) starting from $E_{0}$ with $\partial E_{0}=\{x+h_{0}(x)\nu(x):x\in\Sigma\}$ and assume $\delta$ is small. Then the function $h\in C(\Sigma\times[0,T])\cap C^{\infty}(\Sigma\times(0,T])$ with $\partial E_{t}=\{x+h(x,t)\nu(x):x\in\Sigma\}$ is a solution of the equation

[TABLE]

with $h(x,0)=h_{0}$. Here $L$ is the linear operator defined in (2.5) and $H_{\Sigma}^{s}$ is the fractional mean curvature of the reference surface $\Sigma$. The remainder terms $R_{1,h(\cdot,t)}$ and $R_{2,h(\cdot,t)}$ are defined in (3.9) and (3.10) respectively and $Q$ is a smooth function with $Q(x,0,0)=0$ for all $x\in\Sigma$.

Conversely, if $h\in C(\Sigma\times[0,T])\cap C^{\infty}(\Sigma\times(0,T])$ is a solution of (3.12) with

[TABLE]

then $\partial E_{t}=\{x+h(x,t)\nu(x):x\in\Sigma\}$ defines a family of sets which is a solution of (1.1) starting from $E_{0}$.

4. Regularity estimates for the nonlocal operators

In this section we study the spatial regularity issues related to the equation (3.12) and, in particular, the remainder terms $R_{1,u}$ and $R_{2,u}$ defined in (3.9) and (3.10). As we mentioned in the previous section, our goal is to prove that if $\|u\|_{C^{1+s+\alpha}}$ is small then $R_{1,u}$ and $R_{2,u}$ are small in the $C^{\alpha}$-sense, which then implies that we may regard the equation (3.12) as a linear equation with a small perturbation. We study also the higher order regularity of $R_{1,u}$ and $R_{2,u}$ in order to prove that the solution of (3.12) becomes instantaneously smooth. The complicated structure of $R_{1,u}$ and $R_{2,u}$ makes this section challenging.

Throughout this section $K$ denotes a generic kernel, if not otherwise mentioned, while $K_{u}$ is the kernel defined in (3.7). Next we define the class of kernels which we will use throughout the section.

Definition 4.1**.**

Let $\kappa>0$ and $K:\Sigma\times\Sigma\to\mathbb{R}\cup\{\pm\infty\}$. We say that $K\in\mathcal{S}_{\kappa}$ if the following three conditions hold:

• (i)

$K$ is continuous at every $y,x\in\Sigma$, $x\neq y$, and it holds

[TABLE]

• (ii)

The function $x\mapsto K(y,x)$ is differentiable at every $x,y\in\Sigma$, $x\neq y$, and

[TABLE]

• (iii)

The function

[TABLE]

is Hölder continuous with $\|\psi\|_{C^{\alpha}(\Sigma)}\leq\kappa$.

Remark 4.2**.**

Throughout the paper we assume that $\Sigma$ is a compact hypersurface, but Definition 4.1 can be extended to the case $\Sigma=\mathbb{R}^{n}$. For instance the autonomous kernel $|y-x|^{-n-1-s}$ in $\mathbb{R}^{n}$ trivially satisfies the conditions (i)-(iii) for $\kappa=n+1+s$.

The first two conditions in Definition 4.1 state that the kernel $K$ behaves similarly as the model case $|y-x|^{-n-1-s}$, while the third condition is somewhat more involved. Indeed, it is not trivial to prove the condition (iii) for $K_{u}$ defined in (3.7), since $\Sigma$ is not flat and thus there is no cancellation due to symmetry as in the case $\Sigma=\mathbb{R}^{n}$. (We will prove this in Lemma 4.5.) However, it is important first to notice that there are cases when we do not need the condition (iii) to prove Hölder continuity estimates. This is stated in the following useful auxiliary lemma.

Lemma 4.3**.**

Assume that $K:\Sigma\times\Sigma\to\mathbb{R}\cup\{\pm\infty\}$ satisfies the conditions (i) and (ii) in Definition 4.1 with constant $\kappa>0$. Moreover, assume that $F\in C(\Sigma\times\Sigma)$ satisfies the following:

• (1)

For all $x,y\in\Sigma$ it holds

[TABLE]

• (2)

For all $x,y,z\in\Sigma$ with $|y-x|\geq 2|z-x|$ it holds

[TABLE]

Then the function

[TABLE]

is Hölder continuous and $\|\psi\|_{C^{\alpha}(\Sigma)}\leq C\kappa_{0}\kappa$.

Proof.

By the condition (i) in Definition 4.1 and by the assumption (1) we immediately obtain $\|\psi\|_{C^{0}(\Sigma)}\leq C\kappa_{0}\kappa$, because the function $y\mapsto F(y,x)K(y,x)$ is integrable over $\Sigma$ for every $x$. To show the Hölder continuity we may assume that $0\in\Sigma$ and we need to show $|\psi(z)-\psi(0)|\leq C\kappa_{0}\kappa|z|^{\alpha}$ for $z\in\Sigma$ close to [math]. We divide the set $\Sigma$ into $\Sigma_{-}=\Sigma\cap\{|y|\leq 2|z|\}$ and $\Sigma_{+}=\Sigma\cap\{|y|>2|z|\}$. For all $y\in\Sigma_{-}$ it holds by the condition (i) in Definition 4.1 and by the assumption (1) that

[TABLE]

Similarly it holds

[TABLE]

On the other hand it follows from the condition (ii) in Definition 4.1 that for all $y\in\Sigma_{+}$, i.e., $|y|\geq 2|z|$, it holds

[TABLE]

This together with the condition (i) and with the assumptions (1) and (2) yield

[TABLE]

These imply $|\psi(z)-\psi(0)|\leq C\kappa_{1}\kappa_{2}|z|^{\alpha}$. ∎

We proceed by stating first the crucial regularity estimates we need repeatedly in the paper, and then prove that $K_{u}\in\mathcal{S}_{\kappa}$ (see Definition 4.1) for bounded $\kappa$.

Lemma 4.4**.**

Let $K\in\mathcal{S}_{\kappa}$ (see Definition 4.1) and assume $v_{1}\in C^{1+s+\alpha}(\Sigma)$, $v_{2}\in C^{s+\alpha}(\Sigma)$ and $v_{3}\in C^{\alpha}(\Sigma)$. Then the function

[TABLE]

is Hölder continuous and

[TABLE]

Proof.

We write $\psi$ as

[TABLE]

Note that $\|\psi\|_{C^{\alpha}(\Sigma)}\leq\|v_{3}\|_{C^{\alpha}(\Sigma)}\|\psi_{1}\|_{C^{\alpha}(\Sigma)}+\|v_{3}\|_{C^{\alpha}(\Sigma)}\|v_{2}\|_{C^{\alpha}(\Sigma)}\|\psi_{2}\|_{C^{\alpha}(\Sigma)}$. Therefore it is enough to estimate $\|\psi_{1}\|_{C^{\alpha}(\Sigma)}$ and $\|\psi_{2}\|_{C^{\alpha}(\Sigma)}$. We define

[TABLE]

It is straightforward to check that $F_{1}$ satisfies the assumptions of Lemma 4.3 with $\kappa_{0}\leq C\,\|v_{1}\|_{C^{1}(\Sigma)}\|v_{2}\|_{C^{s+\alpha}(\Sigma)}$. Therefore Lemma 4.3 yields $\|\psi_{1}\|_{C^{\alpha}(\Sigma)}\leq C\kappa\,\|v_{1}\|_{C^{1}(\Sigma)}\|v_{2}\|_{C^{s+\alpha}(\Sigma)}$. We need thus to show that

[TABLE]

To this aim we write $\psi_{2}$ as

[TABLE]

It follows immediately from the condition (iii) in Definition 4.1 that the second term on the right-hand-side is Hölder-continuous with $C^{\alpha}$-norm bounded by $C\kappa\|v_{1}\|_{C^{1+\alpha}(\Sigma)}$. We need thus to prove the Hölder-continuity of $\psi_{3}$. We notice that for every $x,y\in\Sigma$ it holds

[TABLE]

Therefore the function

[TABLE]

satisfies the assumption (1) of Lemma 4.3 with $\kappa_{0}\leq\|v_{1}\|_{C^{1+s+\alpha}(\Sigma)}$. Moreover for every $x,y,z\in\Sigma$ with $|y-x|\geq 2|x-z|$ it holds

[TABLE]

Therefore $F_{2}$ satisfies the assumption (2) of Lemma 4.3 with $\kappa_{0}\leq 2\|v_{1}\|_{C^{1+s+\alpha}(\Sigma)}$, and we conclude by Lemma 4.3 that

[TABLE]

∎

Let us now prove that the kernel $K_{u}$ defined in (3.7) belongs to the class $\mathcal{S}_{\kappa}$(see Definition 4.1) for bounded $\kappa$, when the norm $\|u\,\nu_{\Sigma}\|_{C^{1+s+\alpha}(\Sigma)}$ is small. Recall that this is a reasonable assumption by (3.2). We denote

[TABLE]

and write $K_{u}$ defined in (3.7) as

[TABLE]

We study also the linearization of $K_{u}$, which means that for a given $w\in C^{1+s+\alpha}(\Sigma)$ we consider

[TABLE]

Lemma 4.5**.**

Assume that $u\in C^{1+s+\alpha}(\Sigma)$ is such that $\|u\|_{C^{1+s+\alpha}(\Sigma)}+\|u\,\nu_{\Sigma}\|_{C^{1+s+\alpha}(\Sigma)}\leq\delta$ and $w\in C^{1+s+\alpha}(\Sigma)$. Then the following hold.

• (a)

When $\delta$ is small enough the kernel $K_{u}$ defined in (3.7) belongs to the class $\mathcal{S}_{\kappa_{1}}$, with $\kappa_{1}\leq C$.

• (b)

When $\delta$ is small enough the kernel \frac{d}{d\xi}\Big{|}_{\xi=0}K_{u+\xi w} belongs to the class $\mathcal{S}_{\kappa_{2}}$, with

[TABLE]

Proof.

Claim (a): We denote $\Phi_{u}(x):=x+u(x)\nu(x)$ and recall that

[TABLE]

It follows from the assumption $\|u\,\nu_{\Sigma}\|_{C^{1+s+\alpha}(\Sigma)}\leq\delta$ that

[TABLE]

when $\delta$ is small. Therefore it is clear that $K_{u}$ satisfies the conditions (i) and (ii) in Definition 4.1 of $\mathcal{S}_{\kappa_{1}}$ with $\kappa_{1}\leq C$.

The condition (iii) in Definition 4.1 is technically more involved to verify. We note that in principle we should regularize the kernel $K_{u}$ for the forthcoming calculations as in the proof of Lemma 3.1, but we ignore this since it can be done with obvious changes. Recall that we need to show that the function

[TABLE]

is Hölder continuous. The idea is to use integration parts in order to write $\psi$ as a nonsingular integral. To be more precise, we prove the following equality

[TABLE]

where $\tilde{Q}$ is a smooth function with $\tilde{Q}(x,0)=0$ for all $x\in\Sigma$, and

[TABLE]

Recall that we already know that $K_{u}$ satisfies the conditions (i) and (ii) in Definition 4.1. The idea is then to show that $F_{\Phi_{u}}$ defined in (4.7) satisfies the assumptions of Lemma 4.3, which then implies that the RHS of (4.6) defines a Hölder continuous function.

In order to show (4.6) we shorten the notation by $\Phi(x)=\Phi_{u}(x)$ and notice that the tangential gradient of $y\mapsto|\Phi(y)-\Phi(x)|^{-n+1-s}$ is

[TABLE]

By the divergence theorem it holds

[TABLE]

Therefore the two previous equalities yield

[TABLE]

We write the term on the left-hand-side as

[TABLE]

The equality (4.6) then follows from (4.8), (4.9) and from the fact that

[TABLE]

where $\tilde{Q}(x,0)=0$ for all $x\in\Sigma$.

When $\|u\,\nu_{\Sigma}\|_{C^{1}}\leq\delta$, with $\delta$ small enough, the matrix $I+\tilde{Q}(x,D_{\tau}(u\,\nu_{\Sigma}))$ is invertible. Therefore in order to show that

[TABLE]

is Hölder continuous it is enough to show that the RHS in (4.6) is Hölder continuous. As we mentioned, we will do this by showing that $F_{\Phi_{u}}$ defined in (4.7) satisfies the assumptions of Lemma 4.3 with $\kappa_{0}\leq C$.

First, by using (4.4) and $\|u\,\nu_{\Sigma}\|_{C^{1+s+\alpha}(\Sigma)}\leq\delta$ it is straightforward to check that

[TABLE]

satisfies the assumptions of Lemma 4.3 with $\kappa_{0}\leq C$. Moreover we have that

[TABLE]

Therefore, arguing as with (4.2) we deduce that

[TABLE]

also satisfies the assumptions of Lemma 4.3 with $\kappa_{0}\leq C$.

We need yet to treat the term

[TABLE]

Since $\Sigma$ is uniformly $C^{1,1}$-regular hypersurface, there is a constant $C$ such that $|(y-x)\cdot\nu(x)|\leq C|y-x|^{2}$ for every $x,y\in\Sigma$. Therefore $F_{3}$ satisfies the assumption (1) of Lemma 4.3. The assumption (2) is straightforward to verify but we do this for the reader’s convenience. We have

[TABLE]

Note that $|y-x|\geq 2|z-x|$ implies $|y-z|\leq 2|y-x|$. Therefore $F_{3}$ satisfies the assumption (2) of Lemma 4.3 and the RHS of (4.6) is Hölder continuous. This proves the claim (a).

Claim (b):

We denote \partial_{w}K_{u}=\frac{d}{d\xi}\Big{|}_{\xi=0}K_{u+\xi w} for short. Note that from (4.3) and (4.4) it follows

[TABLE]

for every $x,y\in\Sigma,x\neq y$. Therefore $\partial_{w}K_{u}$ satisfies the condition (i) in Definition 4.1 with $\kappa\leq C\|w\,\nu_{\Sigma}\|_{C^{1}(\Sigma)}$. It is straightforward to check that $\partial_{w}K_{u}$ satisfies also the condition (ii) in Definition 4.1 with $\kappa\leq C\|w\|_{C^{1}(\Sigma)}$.

We need thus to verify the last condition, i.e., we show that

[TABLE]

satisfies $\|\tilde{\psi}\|_{C^{\alpha}(\Sigma)}\leq C\|w\,\nu_{\Sigma}\|_{C^{1+s+\alpha}(\Sigma)}$. To this aim we recall that by (4.6) for small $\xi$ it holds

[TABLE]

where $F_{\Phi_{u+\xi w}}$ is defined in (4.7).

Let us denote the RHS of (4.11) by

[TABLE]

By differentiating we have

[TABLE]

Recall that $\partial_{w}K_{u}$ satisfies the conditions (i) and (ii) in Definition 4.1 with $\kappa\leq C\|w\,\nu_{\Sigma}\|_{C^{1}(\Sigma)}$ and we already proved that $F_{\Phi_{u}}$ satisfies the assumptions (1) and (2) of Lemma 4.3 with $\kappa_{0}\leq C$. Lemma 4.3 then implies that

[TABLE]

To treat $\varphi_{\xi,1}$ we recall that we already proved that $K_{u}\in\mathcal{S}_{\kappa}$ for $\kappa\leq C$. We simplify the expression (4.7) by writing it as

[TABLE]

We denote \Phi_{u}^{\prime}(x)=\frac{\partial}{\partial\xi}\big{|}_{\xi=0}\Phi_{u+\xi w}(x)=w(x)\nu(x), differentiate the above equality and obtain

[TABLE]

Since $\|\Phi_{u}^{\prime}\|_{C^{1+s+\alpha}(\Sigma)},\|D_{\tau}\Phi_{u}^{\prime}\|_{C^{s+\alpha}(\Sigma)}\leq C\|w\,\nu_{\Sigma}\|_{C^{1+s+\alpha}(\Sigma)}$ we may use Lemma 4.4 to deduce

[TABLE]

The Hölder continuity of $\tilde{\psi}$, defined in (4.10), then follows from (4.11) and from the fact that

[TABLE]

Hence we have

[TABLE]

This proves the claim (b). ∎

Remark 4.6**.**

It is clear that the results of Lemma 4.4 and Lemma 4.5 hold also in the case $\Sigma=\mathbb{R}^{n}$ if we assume that the functions $v_{1},v_{2},v_{3},u$ and $w$ are in the corresponding Hölder spaces globally, i.e., $v_{i}\in C^{1+s+\alpha}(\mathbb{R}^{n})$ for $i=1,2,3$ and $\|u\|_{C^{1+s+\alpha}(\mathbb{R}^{n})}\leq\delta$.

We may use the previous results to prove that when $\|u\|_{C^{1+s+\alpha}}$ is small then the remainder terms $R_{1,u}$ and $R_{2,u}$ defined in (3.9) and (3.10) are small. This is stated more precisely in the following proposition. Recall that we may ignore the dependence on $C^{1}$-norm of $\nu_{\Sigma}$, but we do however need to keep track on the dependence on higher norm of $\nu_{\Sigma}$ for later purpose.

Proposition 4.7**.**

Assume $u\in C^{1+s+\alpha}(\Sigma)$ is such that $\|u\|_{C^{1+s+\alpha}(\Sigma)}+\|u\,\nu_{\Sigma}\|_{C^{1+s+\alpha}(\Sigma)}\leq\delta$, and let $R_{1,u}$ and $R_{2,u}$ be the functions defined in (3.9) and in (3.10) respectively. Then for $\delta>0$ small enough it holds

[TABLE]

Proof.

Estimate for $R_{1,u}$:   Recall that

[TABLE]

For later purpose we prove a slightly more general claim. Assume that $u$ is as in the assumption, $v\in C^{1+s+\alpha}(\Sigma)$ and define

[TABLE]

We claim that it holds

[TABLE]

for all $\xi\in[0,1]$. The estimate for $R_{1,u}$ then follows from (4.13) by choosing $v=u$.

In order to prove (4.13) we write (4.12) as

[TABLE]

When $\delta$ is small Lemma 4.5 yields $K_{\xi u}\in\mathcal{S}_{\kappa_{1}}$ with $\kappa_{1}\leq C$ and $\frac{d}{d\xi}K_{\xi u}\in\mathcal{S}_{\kappa_{2}}$ with $\kappa_{2}\leq C\|u\,\nu_{\Sigma}\|_{C^{1+s+\alpha}(\Sigma)}\leq C\delta$ for all $\xi\in[0,1]$. We note that the functions $Q_{1}$ and $Q_{2}$ in (3.6) depend on the second fundamental form of $\Sigma$ such that $\|Q_{i}(x,u,\nabla u)\|_{C^{\beta}(\Sigma)}\leq C(\|u\|_{C^{1+\beta}(\Sigma)}+\|u\nu_{\Sigma}\|_{C^{1+\beta}(\Sigma)})$ for $\beta\in(0,1)$. Therefore it holds

[TABLE]

and we also have

[TABLE]

for all $\xi\in[0,1]$. Hence, the estimate (4.13) follows from Lemma 4.4.

Estimate for $R_{2,u}$:   Recall that

[TABLE]

We use $|\nu(y)-\nu(x)|^{2}=-2\nu(x)\cdot(\nu(y)-\nu(x))$ and write the first term as

[TABLE]

The function inside the integral is of type (4.12) with $v=\nu$ and therefore (4.13) implies $\|\psi_{1}\|_{C^{\alpha}(\Sigma)}\leq C\delta\|\nu_{\Sigma}\|_{C^{1+s+\alpha}(\Sigma)}$.

Let us fix $t^{\prime}\in[0,1]$. We need yet to prove that the function

[TABLE]

satisfies $\|\psi_{2}\|_{C^{\alpha}(\Sigma)}\leq C\delta$. To this aim we recall that we may estimate

[TABLE]

Since $K_{t^{\prime}u}\in\mathcal{S}_{\kappa_{1}}$ with $\kappa_{1}\leq C$, Lemma 4.4 implies $\|\psi_{2}\|_{C^{\alpha}(\Sigma)}\leq C\delta\|\nu_{\Sigma}\|_{C^{1+s+\alpha}(\Sigma)}$. ∎

We need similar estimate as Proposition 4.7 for the linearization of the remainder terms $R_{1,u}$ and $R_{2,u}$.

Proposition 4.8**.**

Assume $u,w\in C^{1+s+\alpha}(\Sigma)$ and $\|u\|_{C^{1+s+\alpha}(\Sigma)}+\|u\,\nu_{\Sigma}\|_{C^{1+s+\alpha}(\Sigma)}\leq\delta$. Let $R_{1,u+\xi w}$ and $R_{2,u+\xi w}$ be functions defined in (3.9) and in (3.10) respectively and define

[TABLE]

Then for $\delta$ small it holds

[TABLE]

and

[TABLE]

Here $C$ is a uniform constant while $C_{\Sigma}$ depends on $\|\nu_{\Sigma}\|_{C^{1+s+\alpha}(\Sigma)}$.

Proof.

This time we prove the claim only for $\partial_{w}R_{1,u}$ since the argument for $\partial_{w}R_{2,u}$ is similar. We differentiate $R_{1,u+\eta w}$, defined in (3.9), and obtain

[TABLE]

Note that the first term is of type (4.12) with $v=w$ and therefore (4.13) implies that it is Hölder continuous and its $C^{\alpha}$-norm is bounded by $C\delta\|w\|_{C^{1+s+\alpha}(\Sigma)}$. Concerning the two last terms in (4.16), note first that Lemma 4.5 (b) yields \frac{d}{d\eta}\Big{|}_{\eta=0}K_{t^{\prime}u+t^{\prime}\eta w}\in\mathcal{S}_{\kappa_{2}} with $\kappa_{2}\leq C\|w\,\nu_{\Sigma}\|_{C^{1+s+\alpha}(\Sigma)}$. By the interpolation inequality in Lemma 2.1 we may estimate

[TABLE]

Moreover, we have by (4.14)

[TABLE]

and as in (4.15) we have

[TABLE]

Thus we deduce by Lemma 4.4 that the two last terms in (4.16) are Hölder continuous with $C^{\alpha}$-norms bounded by $C\delta\|w\|_{C^{1+s+\alpha}(\Sigma)}+C_{\Sigma}\|w\|_{C^{0}(\Sigma)}$. Hence, we have

[TABLE]

∎

At the end of the section we study how to control the higher order norms of $R_{1,u}$ and $R_{2,u}$ in order to differentiate the equation (3.12) with respect to $x$. Moreover, even if the fractional Laplacian is linear it is not obvious how to bound its higher order covariant derivatives. Before that we remark on how we write the derivative of the function

[TABLE]

with respect to a vector field $X\in\mathscr{T}(\Sigma)$. First it holds

[TABLE]

where $\nabla_{X(x)}G(y,x)$ denotes the derivative of $G(y,\cdot)$ with respect to $X$. On the other hand it holds

[TABLE]

Therefore we have by the divergence theorem that

[TABLE]

where

[TABLE]

The following proposition gives us a formula to commute differentiation and the fractional Laplacian. In the following, and in the rest of the paper, $C_{k}$ denotes a constant which depends on $k$ and on $\Sigma$ in an unspecified way while $C$ denotes a uniform constant.

Proposition 4.9**.**

Let $X_{1},\dots,X_{k}\in\mathscr{T}(\Sigma)$ be vector fields such that $\|X_{i}\|_{C^{k+2}(\Sigma)}\leq 1$ for $i=1,\dots,k$ and assume $u\in C^{\infty}(\Sigma)$. Then

[TABLE]

where $\partial^{k+s}u$ denotes a function which satisfies $\|\partial^{k+s}u\|_{C^{\alpha}(\Sigma)}\leq C_{k}\|u\|_{C^{k+s+\alpha}(\Sigma)}$. Moreover, it holds

[TABLE]

for every $k\in\mathbb{N}$.

Proof.

Let us denote $K(y,x)=|y-x|^{-n-1-s}$ and let $X_{1},\dots,X_{k}\in\mathscr{T}(\Sigma)$ be as in the assumption. We define $\partial_{1}K(y,x):=\nabla_{X_{1}(y)}K(y,x)+\nabla_{X_{1}(x)}K(y,x)$ and $\partial_{j}K(y,x)$, for $2\leq j\leq k$, recursively as

[TABLE]

We begin by claiming that $\partial_{k}K(y,x)$ satisfies the conditions (i)-(iii) in Definition 4.1 with $\kappa\leq C_{k}$, i.e., $\partial_{k}K(y,x)\in\mathcal{S}_{C_{k}}$. Note that the constant does not depend on the chosen vector fields $X_{1},\dots,X_{k}$ once they satisfy $\|X_{i}\|_{C^{k+2}(\Sigma)}\leq 1$.

It is straightforward to check that $\partial_{k}K(y,x)$ satisfies the conditions (i) and (ii) in the Definition 4.1 with $\kappa\leq C_{k}$ for some $C_{k}$. We need thus to prove the condition (iii). We prove this by induction and fix $X\in\mathscr{T}(\Sigma)$ such that $\|X\|_{C^{3}(\Sigma)}\leq 1$. We need first to show that the function

[TABLE]

is $\alpha$-Hölder continuous.

The formula (4.6) in the case $u=0$ reads as

[TABLE]

where

[TABLE]

We differentiate (4.19) with respect to $X(x)$ and obtain by (4.17)

[TABLE]

where

[TABLE]

First, recall that in the proof of Lemma 4.5 we already verified that $F$ satisfies the assumptions of Lemma 4.3. Since $\partial_{1}K(y,x)$ satisfies the conditions (i) and (ii) in Definition 4.1 with $\kappa\leq C_{1}$, Lemma 4.3 yields $\|\tilde{\psi}_{1}\|_{C^{\alpha}(\Sigma)}\leq C_{1}$. Second, we may write $F_{1}$ as

[TABLE]

where $v_{j,i}$ are such that $\|v_{j,i}\|_{C^{2}(\Sigma)}\leq C_{1}$. Moreover, by Lemma 4.5 it holds $K(y,x)\in\mathcal{S}_{\kappa}$ with $\kappa\leq C$ and we may thus use Lemma 4.4 to conclude that $\|\varphi_{1}\|_{C^{\alpha}(\Sigma)}\leq C_{1}$. Hence

[TABLE]

and therefore $\partial_{1}K(y,x)\in\mathcal{S}_{\kappa}$ with $\kappa\leq C_{1}$.

We argue by induction and assume that $\partial_{k-1}K(y,x)\in\mathcal{S}_{\kappa}$ with $\kappa\leq C_{k-1}$. Note that this holds for any vector fields $X_{1},\dots,X_{k-1}$ with $\|X_{i}\|_{C^{k+1}(\Sigma)}\leq 1$. Let us fix $X_{1},\dots,X_{k}$ as in the assumption. We differentiate (4.19) with respect to $X_{1},\dots,X_{k}$ and obtain by (4.17)

[TABLE]

Here $\tilde{F}_{i}$ can be written as

[TABLE]

where $v_{j,i}$ are such that $\|v_{j,i}\|_{C^{2}(\Sigma)}\leq C_{k}$. Again we recall that $F$ satisfies the assumptions of Lemma 4.3 and $\partial_{k}K(y,x)$ satisfies the conditions (i) and (ii) in Definition 4.1 for $\kappa\leq C_{k}$. Lemma 4.3 then yields $\|\tilde{\psi}_{k}\|_{C^{\alpha}(\Sigma)}\leq C_{k}$. To prove the Hölder continuity of $\varphi_{k}$ we recall that by induction assumption $\partial_{j}K(y,x)\in\mathcal{S}_{\kappa}$ with $\kappa\leq C_{k-1}$ for every $j\leq k-1$. We may thus use Lemma 4.4 to deduce $\|\varphi_{k}\|_{C^{\alpha}(\Sigma)}\leq C_{k}$. Therefore we conclude that

[TABLE]

is Hölder continuous with $\|\psi_{k}\|_{C^{\alpha}(\Sigma)}\leq C_{k}$ and thus $\partial_{k}K(y,x)$ satisfies the condition (iii) in Definition 4.1 with $\kappa\leq C_{k}$.

We prove the claim by first choosing $\tilde{X}_{1},\dots,\tilde{X}_{l}$, with $1\leq l\leq k$, such that $\|\tilde{X}_{i}\|_{C^{k+2}(\Sigma)}\leq 1$ for $1\leq i\leq l$. Recall that the function $\tilde{X}_{l}\cdots\tilde{X}_{1}u$ is defined recursively as $\tilde{X}_{1}u=\nabla_{\tilde{X}_{1}}u$ and $\tilde{X}_{j+1}\tilde{X}_{j}\cdots\tilde{X}_{1}u=\nabla_{\tilde{X}_{j+1}}(\tilde{X}_{j}\cdots\tilde{X}_{1}u)$ for $j\geq 1$. We apply (4.17) $l$ times for $\tilde{X}_{1},\dots,\tilde{X}_{l}$ and obtain

[TABLE]

where $\partial^{l-1-j}u$ denotes a function which satisfies $\|\partial^{l-1-j}u\|_{C^{1+s+\alpha}(\Sigma)}\leq C_{l-1}\|u\|_{C^{l-j+s+\alpha}(\Sigma)}$ and $v_{j}$ are such that $\|v_{j}\|_{C^{2}(\Sigma)}\leq C_{l}$. By using Lemma 4.4 and $\partial_{j}K(y,x)\in\mathcal{S}_{\kappa}$ with $\kappa\leq C_{j}$ we deduce

[TABLE]

where $\partial^{l+s}u$ denotes a function which satisfies $\|\partial^{l+s}u\|_{C^{\alpha}(\Sigma)}\leq C_{l}\|u\|_{C^{l+s+\alpha}(\Sigma)}$. Hence, we deduce by (2.4), Lemma 4.4 and (4.20) that

[TABLE]

for every $l\leq k$. Note that Lemma 4.4 implies $\|\Delta^{\frac{1+s}{2}}u\|_{C^{\alpha}(\Sigma)}\leq C\|u\|_{C^{1+s+\alpha}(\Sigma)}$. Therefore by iterating the above inequality for $l=1,\dots,k$ we obtain

[TABLE]

This implies the second statement.

Let $X_{1},\dots,X_{k}$ be as in the assumption. We deduce from (4.20) that

[TABLE]

where $\partial^{k+s}u$ denotes a function which satisfies $\|\partial^{k+s}u\|_{C^{\alpha}(\Sigma)}\leq C_{k}\|u\|_{C^{k+s+\alpha}(\Sigma)}$. By (2.3) we have

[TABLE]

where $\partial^{k-1}(\Delta^{\frac{1+s}{2}}u)$ denotes a function which satisfies

[TABLE]

The estimate (4.21) applied to $(k-1)$ yields

[TABLE]

and the claim follows. ∎

Similar result holds for the remainder terms $R_{1}$ and $R_{2}$ .

Lemma 4.10**.**

Assume $u\in C^{\infty}(\Sigma)$ with $\|u\|_{C^{1+s+\alpha}(\Sigma)}+\|u\,\nu_{\Sigma}\|_{C^{1+s+\alpha}(\Sigma)}\leq\delta$ and let $R_{1,u}$ and $R_{2,u}$ be the functions defined in (3.9) and in (3.10) respectively. Let $X_{1},\dots,X_{k}\in\mathscr{T}(\Sigma)$ be vector fields with $\|X_{i}\|_{C^{k+2}(\Sigma)}\leq 1$ for $i=1,\dots,k$ . There is a constant $C_{k}$, which depends on $k$ and $\Sigma$, such that for $\delta>0$ small enough it holds

[TABLE]

and

[TABLE]

In particular, it holds

[TABLE]

for every $k\in\mathbb{N}$.

Proof.

Since the proof is similar to the proof of Proposition 4.9 we only sketch it. In addition we only prove the claim for $R_{1,u}$ as the estimate for $R_{2,u}$ follows from a similar argument. Let $u\in C^{\infty}(\Sigma)$ be as in the assumption and let $K_{u}(y,x)$ be as defined in (3.7). As in the proof of the previous proposition we define $\partial_{j}K_{u}(y,x)$, for $1\leq j\leq k$, by $\partial_{1}K_{u}(y,x)=\nabla_{X_{1}(y)}K_{u}(y,x)+\nabla_{X_{1}(x)}K_{u}(y,x)$ and for $j\geq 2$ recursively as

[TABLE]

We claim that $\partial_{k}K_{u}(y,x)$ satisfies the conditions (i)-(iii) in Definition 4.1 with

[TABLE]

for some $C_{k}$ and $C$, where the latter is independent of $k$. Moreover, the constants in (4.22) do not depend on the chosen vector fields $X_{1},\dots,X_{k}$. The argument for (4.22) is similar to the one in the beginning of Proposition 4.9 and thus we omit it.

To prove the claim we recall the definition of $R_{1,u}$ in (3.9). As in Proposition 4.9 we first choose $\tilde{X}_{1},\dots,\tilde{X}_{l}$, with $1\leq l\leq k$, such that $\|\tilde{X}_{i}\|_{C^{l+2}(\Sigma)}\leq 1$ for $1\leq i\leq l$. We apply (4.17) $l$ times for $\tilde{X}_{1},\dots,\tilde{X}_{l}$ and obtain

[TABLE]

where $\partial^{j}w$ denotes a function which satisfies $\|\partial^{j}w\|_{C^{1+s+\alpha}(\Sigma)}\leq C\|w\|_{C^{j+1+s+\alpha}(\Sigma)}$ and $v_{j}$ are such that $\|v_{j}\|_{C^{2}(\Sigma)}\leq C_{l}$. Here $\varphi_{1}$ denotes the fuction on the first row in (4.23), $\varphi_{2}$ the function on the second row etc. The function $\varphi_{1}$ is of type (4.12), with $v=\tilde{X}_{l}\cdots\tilde{X}_{1}u$ and therefore (4.13) implies

[TABLE]

On the other hand Lemma 4.4, the assumption $\|u\|_{C^{1+s+\alpha}(\Sigma)}\leq\delta$, (2.1) and (2.3) imply

[TABLE]

Since $\partial_{l}K_{t^{\prime}u}(y,x)$ belongs to the class $\mathcal{S}_{\kappa_{l}}$, with $\kappa_{l}$ given by (4.22), we conclude by Lemma 4.4 that

[TABLE]

Similarly we deduce that $\|f\|_{C^{\alpha}(\Sigma)}\leq C_{l}(1+\|u\|_{C^{l+s+\alpha}(\Sigma)}^{l})$. Combining the previous inequalities with (4.23) yields

[TABLE]

We deduce by (2.4) and (4.24) that

[TABLE]

for every $l\leq k$. Recall that by Proposition 4.7 we have $\|R_{1,u}u\|_{C^{\alpha}(\Sigma)}\leq C\delta\|u\|_{C^{1+s+\alpha}(\Sigma)}$. Therefore by using the above inequality $k$ times for $l=1,\dots,k$ we obtain

[TABLE]

This proves the second claim.

Let $X_{1},\dots,X_{k}$ be as in the assumption. We deduce from (4.24) that

[TABLE]

Then (2.3) implies

[TABLE]

The claim follows from (4.25) with $(k-1)$.

∎

5. Proof of the Main Theorem

We will first prove the main theorem for the flow (1.1) and explain at the end of the section how the proof can be applied to deal with the volume preserving case (1.3). By Proposition 3.2 we need to prove that the equation (3.12) has a unique solution $h\in C(\Sigma\times[0,T])\cap C^{\infty}(\Sigma\times(0,T])$ with $h(x,0)=h_{0}(x)$ for $x\in\Sigma$.

Suppose that $\delta$ is small such that the results in Section 4 hold. Recall that by the discussion at the beginning of Section 3 we may choose $\Sigma$ in such a way that we have

[TABLE]

where $\varepsilon\in(0,\delta)$ and $C$ will be chosen later. Here is the statement of the main theorem for (1.1).

Theorem 5.1** (Main Theorem).**

Let $0<\alpha<(1-s)/2$. Assume $\Sigma\subset\mathbb{R}^{n+1}$ is a smooth compact hypersurface and $h_{0}:\Sigma\to\mathbb{R}$ is such that (5.1) holds. For $\delta$ and $\varepsilon$ small enough, there is $T\in(0,1)$, depending on $\delta$ and $\varepsilon$, such that the equation (3.12) has a unique classical solution $h\in C(\Sigma\times[0,T])\cap C^{\infty}(\Sigma\times(0,T])$ with initial value $h(x,0)=h_{0}(x)$ for all $x\in\Sigma$. Moreover, it holds

[TABLE]

and for every $k\in\mathbb{N}$ there is a constant $\Lambda_{k}$ such that

[TABLE]

Note that Theorem 5.1 implies that the solution of (3.12) exists as long as its $C^{1+s+\alpha}$-norm stays small. This means that the fractional mean curvature flow has a smooth solution as long as it stays $C^{1+s+\alpha}$-close to the initial set. We also remark that the exponent $k!$ in the final statement is not optimal and we expect the optimal exponent to be linear in $k$. However, the most important consequence of the last inequality is that it quantifies the smoothness of $h(\cdot,t)$ for every $t\in(0,T]$.

Proof.

**Step 1: ** (Set-up and basic estimates.)

Let us write the equation (3.12) as

[TABLE]

where the remainder term is defined for a generic function $u\in C^{\infty}(\Sigma)$ as

[TABLE]

Recall that $Q$ is a smooth function with $Q(x,0,0)=0$ for all $x\in\Sigma$, and $R_{1,u}$ and $R_{2,u}$ are defined in (3.9) and (3.10) respectively.

Let us first fix a small $\delta>0$ for which the results in Section 4 hold. Let us assume that

[TABLE]

and prove that this implies

[TABLE]

when $\varepsilon$ is small enough. Here $C_{\delta}$ depends on $\delta$.

First, we have by the assumption (5.1) that $\|\nu_{\Sigma}\|_{C^{1+s+\alpha}(\Sigma)}\leq C\varepsilon^{-s-\alpha}$. Therefore $\|u\|_{C^{0}(\Sigma)}\leq\varepsilon$ and (3.1) applied to $u$ imply

[TABLE]

when $\varepsilon$ is small. In particular, these imply $\|u\,\nu_{\Sigma}\|_{C^{1+s+\alpha}(\Sigma)}\leq C\delta$. It follows from Proposition 4.7 that

[TABLE]

The latter inequality and (5.5) yield

[TABLE]

Similarly it follows from Lemma 4.4 that $\|L[u]\|_{C^{\alpha}(\Sigma)}\leq C\delta$. Moreover, we may estimate as with (4.14) that

[TABLE]

By the interpolation inequality in Lemma 2.1 we estimate

[TABLE]

Hence, we have (5.4) by (5.5) and by the fact that $\|H_{\Sigma}^{s}\|_{C^{\alpha}(\Sigma)}$ is uniformly bounded for $\alpha<\frac{1-s}{2}$.

We will also ’linearize’ the equation (5.3). To this aim we prove that if $v_{1},v_{2}\in C^{1+s+\alpha}(\Sigma)$ are such that $\|v_{i}\|_{C^{1+s+\alpha}(\Sigma)}\leq\delta$ and $\|v_{i}\|_{C^{0}(\Sigma)}\leq\varepsilon$, for $i=1,2$, then it holds

[TABLE]

when $\varepsilon$ is small enough. Here $C_{\Sigma,\delta}$ depends on $\delta$ and on $\|\nu_{\Sigma}\|_{C^{1+s+\alpha}(\Sigma)}$.

Indeed, we denote $w=v_{2}-v_{1}$ and write

[TABLE]

Denote further $v_{\xi}=v_{1}+\xi w$ and recall that (5.5) holds also for $v_{\xi}$. In particular, it holds $\|v_{\xi}\,\nu_{\Sigma}\|_{C^{1+s+\alpha}(\Sigma)}\leq C\delta$. By recalling the definition of $P$ in (5.3) we obtain by differentiating

[TABLE]

It follows from Proposition 4.7 that

[TABLE]

for all $\xi\in(0,1)$. Moreover, we have by Proposition 4.8 that

[TABLE]

and

[TABLE]

Note that the latter inequality and (5.5) yield

[TABLE]

Lemma 4.4 implies

[TABLE]

and

[TABLE]

Finally we have by (5.5) and (5.6) that

[TABLE]

and since $Q$ is smooth we have

[TABLE]

By combining the previous estimates we obtain

[TABLE]

The inequality (5.7) then follows from the interpolation inequality in Lemma 2.1.

**Step 2: ** (Existence and Uniqueness of the strong solution.)

We define $X$ as the space of function $u\in C(\Sigma\times[0,T])$ such that $u\in X$ if

[TABLE]

and $h(x,0)=h_{0}(x)$ for all $x\in\Sigma$. We choose $\delta>0$ so small that the results in Section 4 hold and $\varepsilon>0$ even smaller if necessary and assume that $h_{0}$ satisfies (5.1). Finally $C$ is a large constant which we choose later.

We define a map $\mathcal{L}:X\to X$ such that for a given $h\in X$ the value $\mathcal{L}[h]:=u$ is the solution of the following linear equation with a forcing term

[TABLE]

Recall that by definition (2.5) $L[u]=\Delta^{\frac{1+s}{2}}u+c_{s}^{2}(x)u$. Therefore a fixed point of $\mathcal{L}:X\to X$ is a strong solution of (5.2). By a strong solution we mean that $h:\Sigma\times[0,T]\to\mathbb{R}$ is Lipschitz continuous in time, $C^{1+s+\alpha}$-regular in space and satisfies the equation (5.2) for almost every $t\in(0,T]$.

Let us first show that $\mathcal{L}:X\to X$ is well defined, i.e., $u$ defined by (5.8) belongs to $X$. By a standard approximation argument we may assume that $h_{0}$ and $h\in X$ are smooth. By Theorem 2.2 the solution $u$ is smooth and it holds

[TABLE]

Since we assume $\sup_{0\leq t<T}\|h(\cdot,t)\|_{C^{1+s+\alpha}(\Sigma)}\leq\delta$ and $\sup_{0\leq t<T}\|h(\cdot,t)\|_{C^{0}(\Sigma)}\leq\varepsilon$ we have by (5.4) that

[TABLE]

for every $t\in(0,T]$. Therefore since $\|h_{0}\|_{C^{0}(\Sigma)}\leq\varepsilon/2$ we find

[TABLE]

when $T$ is small. Hence we have the second condition in the definition of $X$.

In order to prove the first condition we recall that it holds

[TABLE]

for every $t\in(0,T]$. We use again Theorem 2.2, (5.9) and $\|h_{0}\|_{C^{1+s+\alpha}(\Sigma)}\leq\frac{\delta}{2C}$ and find

[TABLE]

where $C$ is a uniform constant and $C_{\delta}$ depends on $\delta$. By choosing first $\delta$, then $\varepsilon$ and finally $T$ small we find

[TABLE]

and the first condition follows.

Finally the bound $\sup_{0\leq t<T}\|\partial_{t}u(\cdot,t)\|_{C^{\alpha}(\Sigma)}\leq C$ follows from the equation (5.8) and from (5.4) as

[TABLE]

Hence we conclude that $\mathcal{L}:X\to X$ is well defined.

Let us next show that $\mathcal{L}:X\to X$ is a contraction with respect to the following norm

[TABLE]

where $\Lambda_{0}$ is a large constant which will be chosen later. Let us fix $h_{1},h_{2}\in X$ and denote $u_{1}=\mathcal{L}[h_{1}]$ and $u_{2}=\mathcal{L}[h_{2}]$. The function $v=u_{2}-u_{1}$ is a solution of the equation

[TABLE]

with $v(x,0)=0$ for all $x\in\Sigma$.

We denote $w=h_{2}-h_{1}$ and use (5.7) for $v_{1}(x)=h_{1}(x,t)$ and $v_{2}(x)=h_{2}(x,t)$ to conclude that

[TABLE]

Therefore the equation (5.11) and Theorem 2.2 yield

[TABLE]

and

[TABLE]

where the last inequality follows from the interpolation inequality in Lemma 2.1. We choose $\Lambda_{0}\geq\delta^{-1}C_{\Sigma,\delta}$ and $T\leq\Lambda_{0}^{-1}$ and have by the two above inequalities

[TABLE]

In other words

[TABLE]

when $\delta$ is small. Hence, $\mathcal{L}:X\to X$ is a contraction and by a standard fixed point argument we conclude that the equation (3.12) has a unique strong solution in $X$.

Step 3: (Higher order regularity.)

We prove the last statement of the theorem. In fact, we prove slightly stronger estimate, i.e., for every $k\in\mathbb{N}$ there is $\Lambda_{k}$ such that

[TABLE]

In particular, this implies that $h(\cdot,t)$ is smooth for $t>0$. One may then use the equation (3.12) to deduce that $h\in C^{\infty}(\Sigma\times(0,T])$.

Since $T<1$ we know by Step 2 that (5.12) holds for $k=0$. We argue by induction and assume that (5.12) holds for $k\in\mathbb{N}$ and prove that it holds also for $k+1$ with some large constant $\Lambda_{k+1}\geq\Lambda_{k}$. To this aim we fix vector fields $X_{1},\dots,X_{k}\in\mathscr{T}(\Sigma)$ with $\|X_{i}\|_{C^{k+2}(\Sigma)}\leq 1$, $i=1,\dots,k$, and define the function space $Y_{k+1}\subset X$ such that $u\in Y_{k+1}$ if $u\in X$ (defined in the beginning of Step 2) and

[TABLE]

where $\Lambda_{k}$ is the constant given by the induction assumption and $\tilde{\Lambda}_{k+1}$ is a constant which we will fix later. We note first that $Y_{k+1}$ is non-empty since at least the solution of the heat equation

[TABLE]

belongs to $Y_{k+1}$ when $\tilde{\Lambda}_{k+1}$ is chosen large enough.

Let $\mathcal{L}:X\to X$ be the map defined by (5.8). The goal is to show that for $h\in Y_{k+1}$ it holds $u=\mathcal{L}(h)\in Y_{k+1}$, i.e., $\mathcal{L}(Y_{k+1})\subset Y_{k+1}$. Therefore since the solution constructed in Step 2 is unique in $X$ we deduce that the solution belongs also to $Y_{k+1}$. In other words the solution of (3.12) satisfies

[TABLE]

Therefore it follows from $T<1$ and from the fact that (5.12) holds for $k$ that

[TABLE]

which proves (5.12). We need thus to prove that $u$ satisfies the second inequality in (5.13).

Let $u$ be the solution of (5.8) where $h_{0}$ and $h$ are smooth function such that $h\in Y_{k+1}$, i.e., $h$ satisfies (5.13). We denote

[TABLE]

We claim that $u_{k}$ is a solution of the equation

[TABLE]

where the function $P_{k}$ satisfies

[TABLE]

for all $t\in(0,T]$. Here $C_{k,\delta}$ is a constant which depends on $k,\delta$ and $\Sigma$, while $C$ is a uniform constant.

Indeed, we first note that since $h$ is smooth then the equation (5.8) and Theorem 2.2 imply that $u$ is smooth. We may thus differentiate (5.8) and obtain by Proposition 4.9 that

[TABLE]

where $\partial^{k}u(x,t)$ denotes a function which satisfies $\|\partial^{k}u(x,t)\|_{C^{\alpha}(\Sigma)}\leq C_{k}\|u(\cdot,t)\|_{C^{k+s+\alpha}(\Sigma)}$ for every $t\in(0,T]$. Recall that the function $P$ is defined in (5.3). We use Leibniz rule and Proposition 4.9 to deduce

[TABLE]

where $f_{k}$ is a function which satisfies

[TABLE]

We have

[TABLE]

and Proposition 4.9 yields

[TABLE]

Moreover by Lemma 4.10 we have

[TABLE]

Therefore it holds

[TABLE]

for all $t\in(0,T]$.

We use (2.1) to conclude that

[TABLE]

for all $t\in(0,T]$. By the interpolation inequality in Lemma 2.1 and by Young’s inequality we have

[TABLE]

Recall that by (5.5) the assumption $h\in X$ implies $\|\nu_{\Sigma}\|_{C^{1+s+\alpha}(\Sigma)}\|h(\cdot,t)\|_{C^{\alpha}(\Sigma)}\leq\delta$ for every $t\in(0,T)$ when $\varepsilon$ is small. Then Lemma 4.10 yields

[TABLE]

Finally, by (5.6) and by Lemma 4.4 we have

[TABLE]

The equation (5.14) and the estimate (5.15) then follows from the previous inequalities, from (5.16) (5.17) and (5.18) and from

[TABLE]

Let us then prove that $u\in Y_{k+1}$. We define $v(x,t)=t^{(k+1)!}u_{k}(x,t)$. Since $u_{k}$ is a solution of (5.14) then $v$ is a solution of

[TABLE]

with $v(x,0)=0$. Theorem 2.2 and (5.15) imply (recall that $T<1$)

[TABLE]

Recall that we assume $\sup_{0<t<T}t^{k!}\|u(\cdot,t)\|_{C^{k+s+\alpha}(\Sigma)}\leq\Lambda_{k}$ and $\sup_{0<t<T}t^{k!}\|h(\cdot,t)\|_{C^{k+s+\alpha}(\Sigma)}\leq\Lambda_{k}$. In particular, the latter implies

[TABLE]

Therefore we have

[TABLE]

Since we assume that the second inequality in (5.13) holds for $h_{k}=\nabla_{X_{k}}\cdots\nabla_{X_{1}}h$, the above inequality yields

[TABLE]

Let us first choose $\delta$ such that $C\delta\leq\frac{1}{4}$ and then $\tilde{\Lambda}_{k+1}=4C_{k,\delta}(1+\Lambda_{k}+\Lambda_{k}^{k})$. This gives us

[TABLE]

Therefore $u$ satisfies the second inequality in (5.13) and we conclude that $u\in Y_{k+1}$.

∎

We conclude this section by showing how to modify the previous proof to obtain a result analogous to Theorem 5.1 for the volume preserving fractional mean curvature flow (1.3). We use the same parametrization as in Section 3 to describe the motion of $E_{0}\in\mathfrak{h}_{\delta}(\Sigma)$ given by (1.3). If $E\in\mathfrak{h}_{\delta}(\Sigma)$ with $\partial E=\{x+h(x)\nu(x):x\in\Sigma\}$ then by (3.8) and by change of variables we have

[TABLE]

where $J_{\Phi}$ denotes the tangential Jacobian of $\Phi(x)=x+h(x)\nu(x)$. As we mentioned in Section 3 the tangential Jacobian can be written as $J_{\Phi}(x)=1+Q_{3}(x,h,\nabla h)$, where $Q_{3}$ is a smooth function such that $Q_{3}(x,0,0)=0$ for all $x\in\Sigma$. Recall that $L[h]$ is defined in (2.5) and notice that for $h\in C^{1+s+\alpha}(\Sigma)$ it holds

[TABLE]

Let us then define the number

[TABLE]

That is $\bar{R}_{3,u}=\bar{H}_{E}^{s}$.

We obtain immediately the following result which is analogous to Proposition 3.2.

Proposition 5.2**.**

Assume that $E_{0}\in\mathfrak{h}_{\delta}(\Sigma)$ for $\delta$ small. There exists a flow $(E_{t})_{t\in(0,T]}$ with $E_{t}\in\mathfrak{h}_{\delta}(\Sigma)$, for all $t\in(0,T]$, starting from $E_{0}$ which is a classical solution of (1.3) if and only if there exists a classical solution $h\in C(\Sigma\times[0,T])\cap C^{\infty}(\Sigma\times(0,T])$ of

[TABLE]

with $\sup_{0<t<T}\|h(\cdot,t)\|_{C^{1+s+\alpha}(\Sigma)}\leq\delta$. Here $\tilde{P}$ is defined for a generic function $u\in C^{\infty}(\Sigma)$ as

[TABLE]

where $P(x,u,\nabla u)$ is given by (5.3), $\bar{R}_{3,u}$ is defined in (5.19) and $Q$ is a smooth function such that $Q(x,0,0)=0$ for all $x\in\Sigma$.

Arguing as with (5.4) we deduce that if $u$ is such that $\|u\|_{C^{1+s+\alpha}(\Sigma)}\leq\delta$ and $\|u\|_{C^{0}(\Sigma)}\leq\varepsilon$ with $\varepsilon$ small enough it holds

[TABLE]

Similarly, we argue as with (5.7) and obtain for $v_{1},v_{2}$ with $\|v_{i}\|_{C^{1+s+\alpha}(\Sigma)}\leq\delta$ and $\|v_{i}\|_{C^{\alpha}(\Sigma)}\leq\varepsilon$ , $i=1,2$, that

[TABLE]

Therefore we obtain by (5.4) that

[TABLE]

and by (5.7) that

[TABLE]

We may thus use the argument in Step 2 in the proof of Theorem 5.1 to obtain the unique strong solution of (5.20). The smoothness of the strong solution follows immediately from Step 3 since $\bar{R}_{3,h(\cdot,t)}$ does not depend on $x$. We have thus the following result.

Theorem 5.3**.**

Let $0<\alpha(1-s)/2$. Assume $\Sigma\subset\mathbb{R}^{n+1}$ is a smooth compact hypersurface and $h_{0}:\Sigma\to\mathbb{R}$ is such that (5.1) holds. For $\delta$ and $\varepsilon$ small enough, there is $T\in(0,1)$, depending on $\delta$ and $\varepsilon$, such that (5.20) has a unique classical solution $h\in C(\Sigma\times[0,T])\cap C^{\infty}(\Sigma\times(0,T])$ with

[TABLE]

Moreover, for every $k\in\mathbb{N}$ there is a constant $\Lambda_{k}$ such that

[TABLE]

Thereom 5.3 together with Proposition 5.2 proves the main theorem for the volume preserving flow (1.3).

Appendix A

Here we give the proof of Theorem 2.2. We first recall the result in the case $\Sigma=\mathbb{R}^{n}$ and then use a perturbation argument to prove it for a compact and smooth hypersurface. The following result can be found in [27].

Theorem A.1**.**

Assume that $f:\mathbb{R}^{n}\times[0,T]\to\mathbb{R}$ is smooth and $|f(x,t)|\leq C(1+|x|)^{-n-1-s}$ for all $(x,t)\in\Sigma\times[0,T]$. Assume that $u$ with $\text{supp}\,u(\cdot,t)\subset B_{1}$ for all $t\in[0,T]$ is the solution of

[TABLE]

Then it holds

[TABLE]

Proof of Theorem 2.2.

The existence and uniqueness of the weak solution follows from Galerkin method and the smoothness follows by differentiating the equation with respect to time. Since the argument is standard we omit it and simply refer to [15].

Let us first prove the second inequality. It is clear that we may assume $g=0$. We write $u=v+w$ where

[TABLE]

By maximum principle it holds $|v(x,t)|\leq|u_{0}(x)|$ for all $(x,t)\in\Sigma\times(0,T]$. Let us then prove

[TABLE]

To this aim define $\tilde{w}(x,t)=t^{\varepsilon-1}w(x,t)$ for $\varepsilon>0$. Then $\tilde{w}$ is continuous on $\Sigma\times[0,T]$, $\tilde{w}(x,0)=0$ and assume it attains its maximum at $(\hat{x},\hat{t})\in\Sigma\times(0,T]$. By maximum principle it holds

[TABLE]

and

[TABLE]

Then the equation for $w$ implies

[TABLE]

Therefore, because $(\hat{x},\hat{t})$ is the maximum point, it holds for all $(x,t)\in\Sigma\times(0,T]$

[TABLE]

The estimate follows by letting $\varepsilon\to 0$. By repeating the argument for $-w$ we obtain the second inequality in Theorem 2.2.

Let us prove the first inequality in Theorem 2.2. We may assume that $g=u_{0}=0$, since the general case follows by considering the function $v(x,t)=u(x,t)-tg(x)-u_{0}(x)$.

Let us fix $x_{0}\in\Sigma$ and without loss of generality we may assume that $x_{0}=0$ and $\nu(0)=e_{n+1}$. Since $\Sigma$ is smooth and uniformly $C^{1,1}$-regular we may write it locally as a graph of a smooth function, i.e., there exists a smooth function $\phi:B_{2r}\subset\mathbb{R}^{n}\to\mathbb{R}$ such that

[TABLE]

where $C_{r}$ denotes the cylinder

[TABLE]

Note that the assumption $x_{0}=0$ and $\nu(0)=e_{n+1}$ implies $\phi(0)=0$, $D\phi(0)=0$ and

[TABLE]

Let $\zeta:\mathbb{R}^{+}\rightarrow\mathbb{R}$ be a smooth cut-off function such that $\zeta(\rho)=1$ for $\rho\in[0,r/2]$ and $\zeta(\rho)=0$ for $\rho\geq r$.

Let us denote $\Sigma_{r}=\Sigma\cap C_{r}$. The above notation in mind we may write the equation in (2.6) for $x\in\Sigma_{r}$ as

[TABLE]

where

[TABLE]

Since the function $1-\zeta(|y_{n+1}|)\zeta(|y^{\prime}|)$ vanishes on $\Sigma\cap C_{r/2}$ the above intergal is non-singular on $\Sigma\cap C_{r/4}$ and we have

[TABLE]

We may write every $x\in\Sigma_{r}$ as $x=(x^{\prime},\phi(x^{\prime}))$, where $x^{\prime}\in B_{r}\subset\mathbb{R}^{n}$. We denote, by slight abuse of notations, $u((x^{\prime},\phi(x^{\prime})),t)=u(x^{\prime},t)$ and similarly $G_{1}(x^{\prime},t)$, $f(x^{\prime},t)$ and $g(x^{\prime})$ for every point $x=(x^{\prime},\phi(x^{\prime}))$ on $\Sigma_{r}$. By change of variables we have

[TABLE]

We define $Q(z):=\sqrt{1+|z|^{2}}-1$ and

[TABLE]

Note that $Q$ is a smooth function with $Q(0)=0$ and $K_{\phi}(y^{\prime},x^{\prime})$ agrees with (3.7) when we choose $\Sigma=\mathbb{R}^{n}\times\{0\}\simeq\mathbb{R}^{n}$ and $u=\phi$. Note also that by (A.1) $\phi$ satisfies

[TABLE]

when $r$ is small enough. Using this notation we may write

[TABLE]

Let us define $w(x^{\prime},t)=\zeta(4|x^{\prime}|)u(x^{\prime},t)$ and extend $\phi$ to $\mathbb{R}^{n}$ such that (A.4) holds in $\mathbb{R}^{n}$. Then we have by (A.2) and by the above calculations

[TABLE]

We write

[TABLE]

We recall that $Q(0)=0$ and write

[TABLE]

By combining (A.5), (A.6) and (A.7) we obtain

[TABLE]

where

[TABLE]

[TABLE]

and

[TABLE]

We need to estimate the $C^{\alpha}$-norms of $F$, $G_{2}$ and $G_{3}$. Note first that trivially $\|w(\cdot,t)\|_{C^{\gamma}(\mathbb{R}^{n})}\leq C_{\gamma}\|u(\cdot,t)\|_{C^{\gamma}(\Sigma)}$ for all $\gamma\in(0,1)$. Since $(1-\zeta(|y^{\prime}|))$ vanishes for $|y^{\prime}|\leq r/2$ and $w(y^{\prime},t)$ vanishes for $|y^{\prime}|\geq r/4$ we have, similarly as with (A.3), that

[TABLE]

Next we recall that by Remark 4.6 and by (A.4), Lemma 4.4 and Lemma 4.5 hold also for $\Sigma=\mathbb{R}^{n}$ and $K_{\phi}$. Hence, we conclude by Lemma 4.4 that

[TABLE]

Similarly we observe that the term $F$ is of type (4.12) with $v=w(\cdot,t)$ and $u=\phi$. Therefore (4.13) and (A.4) yield

[TABLE]

for every $t\in(0,T]$.

We conclude by (A.3), (A.8), (A.9), (A.10), (A.11) and by Theorem A.1 that

[TABLE]

When $\delta$ is small we have

[TABLE]

Note that $\|u(\cdot,t)\|_{C^{1+s+\alpha}(\Sigma\cap C_{r/8})}\leq C\|w(\cdot,t)\|_{C^{1+s+\alpha}(\mathbb{R}^{n})}$ for every $t\in(0,T]$. Therefore since $\Sigma$ is compact we obtain by standard covering argument

[TABLE]

By the interpolation inequality in Lemma 2.1 and by the second inequality in Theorem 2.2 (recall that $u_{0}=g=0$) we have for all $t\in(0,T]$

[TABLE]

The claim then follows from (A.12) by choosing $\delta$ small. ∎

Acknowledgments

The first author was supported by the Academy of Finland grant 314227.