# Short time existence and smoothness of the nonlocal mean curvature flow of
graphs

Anoumou Attiogbe  A. A. : African Institute for Mathematical Sciences in
Senegal, KM 2, Route de Joal, B.P. 14 18. Mbour, Senegal.
<EMAIL_ADDRESS>,  Mouhamed Moustapha Fall  M. M. F.:
African Institute for Mathematical Sciences in Senegal, KM 2, Route de Joal,
B.P. 14 18. Mbour, Senegal<EMAIL_ADDRESS>and  Tobias Weth
T. W.: Institut für Mathematik, Goethe-Universität, Frankfurt, Robert-Mayer-
Str. 10, D-60629 Frankfurt<EMAIL_ADDRESS>
###### Abstract.

We consider the geometric evolution problem of entire graphs moving by
fractional mean curvature. For this, we study the associated nonlocal
quasilinear evolution equation satisfied by the family of graph functions. We
establish, using an analytic semigroup approach, short time existence,
uniqueness and optimal Hölder regularity in time and space of classical
solutions of the nonlocal equation, depending on the regularity of the initial
graph. The method also yields $C^{\infty}-$smoothness estimates of the
evolving graphs for positive times.

Keywords: Nonlocal mean curvature flow; Quasilinear evolution equations;
Analytic semigroup theory; Fixed point theorem.

##  1\. Introduction

In the present paper, we study the geometric evolution problem of
hypersurfaces moving by fractional mean curvature. More precisely we consider
the problem of existence and uniqueness of a family of (sufficiently regular)
open subsets $\\{E(t)\\}_{t>0}$ of $\mathbb{R}^{N}$ satisfying

$\displaystyle\partial_{t}X_{t}\cdot\nu(X_{t}):=-H_{E(t)}^{\alpha}(X_{t}),\;\;\;\mbox{for
all $X_{t}\in\partial E(t)$ and }t\in[0,T],$ (1.1)

where, $\nu$ is the unit normal vector field on $\partial E(t)$ and
$H_{E(t)}^{\alpha}$ is the fractional mean curvature of order $\alpha\in(0,1)$
at $X_{t}\in\partial E(t)$. Recall that for a set $E\subset\mathbb{R}^{N}$ of
class $C^{1+\beta}$, with $\beta>\alpha$, the fractional (nonlocal) mean
curvature of $\partial E$ is well defined and it is given at a point
$x\in\partial E$ by

$\displaystyle
H_{E}^{\alpha}(x):=P.V.\int_{\mathbb{R}^{N}}\frac{\mathbbm{1}_{E^{c}}(y)-\mathbbm{1}_{E}(y)}{|x-y|^{N+\alpha}}dy=\lim_{\varepsilon\to
0}\int_{\mathbb{R}^{N}\setminus
B_{\varepsilon}(x)}\frac{\mathbbm{1}_{E^{c}}(y)-\mathbbm{1}_{E}(y)}{|x-y|^{N+\alpha}}dy,$
(1.2)

where $\mathbbm{1}_{E}$ denotes the characteristic function of the set $E$ and
${E^{c}}:=\mathbb{R}^{N}\setminus E$. We recall that the notion of fractional
(nonlocal) mean curvature of order $\alpha\in(0,1)$ appeared for the first
time in the work of Caffarelli and Souganidis in [8] around 2008. Since that
time, the nonlocal mean curvature has been studied extensively in various
settings, see e.g. [1, 5, 25, 16] and the references therein.

As a nonlocal conterpart of the classical mean curvature flow (see e.g. [18,
23] and the references therein), the nonlocal mean curvature flow has
attracted much interest in recent years, see e.g [14, 10, 11, 12, 13, 19, 20,
25, 3] and the survey paper [13]. The first paper dealing with the existence
and uniqueness of solutions of this flow is [19], where Imbert used the level
set formulation of the geometric flow to prove existence and uniqueness of
viscosity solutions while the recent papers [10] and [12] deal with regularity
results of viscosity solution of the flow starting from Lipschitz graphs. We
also mention the paper [25] where the autors proved a comparison principle for
the flow (1.1) and found bounds on the maximal existence time and uniqueness
of smooth solutions.

Despite the attention that the nonlocal mean curvature flow has already
received, the existence and regularity of classical solutions remained an open
problem until the recent work [20]. In this paper, the authors prove short
time existence and uniqueness of classical solution to the fractional mean
curvature flow, starting from a bounded $C^{1,1}$ initial set.

In the present paper, we consider the complementary case of short time
existence and uniqueness of classical solutions of problem (1.1) when the sets
$E(t)$ are given by subgraphs of functions $u(t,\cdot)\in
C^{1+\beta}_{loc}(\mathbb{R}^{N-1})$, where $\beta>\alpha$. We recall first
that if $E_{u}:=\\{(x,y)\in\mathbb{R}^{N-1}\times\mathbb{R}:y<u(x)\\}$ then,
see e.g. [16, Proposition 3.5], we have that

$\displaystyle
H(u)(x):=H^{\alpha}_{E_{u}}(x,u(x))=P.V.\int_{\mathbb{R}^{N-1}}\frac{{\mathcal{G}}(p_{u}({x},{y}))}{|{x}-{y}|^{N-1+\alpha}}d{y},$
(1.3)

where

$p_{u}({x},{y})=\frac{u({y})-u({x})}{|{x}-{y}|},\qquad{\mathcal{G}}(p):=-\int_{-p}^{p}\frac{d\tau}{(1+\tau^{2})^{\frac{N+\alpha}{2}}}.$
(1.4)

We point out that, due to the boundedness of the function ${\mathcal{G}}$, the
expression $H(u)(x)$ is well-defined if $u:\mathbb{R}^{N-1}\to\mathbb{R}$ is
measurable and of class $C^{1+\beta}$ for some $\beta>\alpha$ in a
neighborhood of $x$. With this expression of the fractional mean curvature, we
can derive the quasilinear evolution problem corresponding to (1.1) when
$E_{u}(t):=\\{(x(t),y(t))\in\mathbb{R}^{N-1}\times\mathbb{R}:y(t)<u(t,x(t))\\}$,
with $u:[0,T]\times\mathbb{R}^{N-1}\to\mathbb{R}$. Indeed, recall that the
unit interior normal vector field on $\partial E_{u}(t)$ is given by
$\nu(X_{t}):=\frac{(-\nabla u(t,x(t)),1)}{\sqrt{1+|\nabla u(t,x(t))|^{2}}}$.
Therefore for every $X_{t}\in\partial
E_{u}(t)=\\{(x(t),u(t,x(t))):x(t)\in\mathbb{R}^{N-1}\\}$, we have that

$\displaystyle\partial_{t}X_{t}=\Big{(}\dot{x}(t),\dot{x}(t)\cdot\nabla
u(t,x(t))+\partial_{t}u(t,x(t))\Big{)}.$

Hence

$\displaystyle\partial_{t}X_{t}\cdot\nu(X_{t})$
$\displaystyle=\frac{-\dot{x}(t)\cdot\nabla u(t,x(t))}{\sqrt{1+|\nabla
u(t,x(t))|^{2}}}+\frac{\dot{x}(t)\cdot\nabla u(t,x(t))}{\sqrt{1+|\nabla
u(t,x(t))|^{2}}}+\frac{\partial_{t}u(t,x(t))}{\sqrt{1+|\nabla
u(t,x(t))|^{2}}}$ $\displaystyle=\frac{\partial_{t}u(t,x(t))}{\sqrt{1+|\nabla
u(t,x(t))|^{2}}}.$ (1.5)

Therefore, from (1) and (1.1), the evolution of $u$ is given by the flow
associated to the quasilinear evolution equation

$\displaystyle\partial_{t}u=-{\sqrt{1+|\nabla
u|^{2}}}H(u),\;\;\;t\in(0,T],\;\;\;\;\;u(0)=u_{0}.$ (1.6)

Our first main result is the following.

######  Theorem 1.1.

Let $\nu>0$, $\beta\in(\alpha,1)$, $\rho\in(0,\frac{1}{1+\alpha})$ and
$\gamma_{\rho}:=\beta+\rho(1+\alpha)$. Then for all $u_{0}\in
C^{1+\gamma_{\rho}}_{loc}(\mathbb{R}^{N-1})$, with $\|\nabla
u_{0}\|_{C^{\gamma_{\rho}}(\mathbb{R}^{N-1})}\leq\nu$, there exist $T,C_{0}>0$
only depending on $\rho,\alpha,\beta,\gamma,N$ and $\nu$ such that the problem

$\displaystyle\left\\{\begin{array}[]{rll}\partial_{t}u+{\sqrt{1+|\nabla
u|^{2}}}H(u)&=0&\hskip 14.22636pt\mbox{in }\;[0,T]\times\mathbb{R}^{N-1}\\\
u(0)&=u_{0}&\hskip 14.22636pt\mbox{in }\;\mathbb{R}^{N-1}\end{array}\right.$
(1.9)

admits a unique solution $u\in
C^{\rho}([0,T],C^{1+\beta}_{loc}(\mathbb{R}^{N-1}))\cap
C^{1+\rho}([0,T],C^{\beta-\alpha}_{loc}(\mathbb{R}^{N-1}))$ satisfying

$\|u-u_{0}\|_{C^{\rho}([0,T],C^{1+\beta}(\mathbb{R}^{N-1}))\cap
C^{1+\rho}([0,T],C^{\beta-\alpha}(\mathbb{R}^{N-1}))}\leq C_{0}.$ (1.10)

If, in addition, $\nabla u_{0}\in C^{1+\gamma_{\rho}}(\mathbb{R}^{N-1})$ then
for all $\beta^{\prime}\in(\alpha,\beta)$ there exists $C>0$ only depending on
$\rho,\alpha,\beta,\gamma,N,\nu,T$ and $\beta^{\prime}$ such that

$\|\nabla u\|_{C^{\rho}([0,T],C^{1+\beta^{\prime}}(\mathbb{R}^{N-1}))}\leq
C\|\nabla u_{0}\|_{C^{1+\gamma_{\rho}}(\mathbb{R}^{N-1})}.$ (1.11)

The following result complements Theorem 1.1 by providing, under the same
assumptions, smoothness estimates for positive times.

######  Theorem 1.2.

Under the assumptions of Theorem 1.1, we have $u(t,\cdot)\in
C^{\infty}(\mathbb{R}^{N-1})$ for every $t\in(0,T]$. Moreover, for every
$\beta^{\prime}\in(\alpha,\beta)$, $\rho\in(0,\frac{\alpha}{1+\alpha}]$ and
for all $k\in\mathbb{N}\setminus\\{0\\}$, there exists $C_{k}>0$ only
depending on $\rho,\alpha,\beta,\gamma,N,\nu,\beta^{\prime},T$ and $k$ such
that

$\|t^{k}\nabla
u\|_{C^{\rho}([0,T],C^{k+\beta^{\prime}}(\mathbb{R}^{N-1}))}\leq C_{k}.$
(1.12)

Related to Theorem 1.2, we also mention [12], where the authors state that
solutions to the fractional mean flow of graph, starting from a Lipschitz
graph, are $C^{\infty}$-smooth for positive times provided $H(u_{0})\in
L^{\infty}(\mathbb{R}^{N-1})$. However [12] does not provide enough details
that confirm this statement. Indeed, the authors apply a priori Hölder
regularity estimates from [27, 26] to the linearization of the first equation
of (1.9). However the linearization of the weighted nonlocal mean curvature
operator $u\mapsto{\sqrt{1+|\nabla u|^{2}}}H(u)$ does not fall in the class of
nonlocal operators considered in [27, 26].

Our next result provides universal estimates of the Lipschitz norm and the
mean curvature of the evolving graph in terms of the initial data.

######  Theorem 1.3.

Under the assumptions of Theorem 1.1, we have

$\|\nabla u\|_{L^{\infty}((0,T)\times\mathbb{R}^{N-1})}\leq\|\nabla
u_{0}\|_{L^{\infty}(\mathbb{R}^{N-1})}$

and

$\|\partial_{t}u\|_{L^{\infty}((0,T)\times\mathbb{R}^{N-1})}\leq\|{\sqrt{1+|\nabla
u_{0}|^{2}}}H(u_{0})\|_{L^{\infty}(\mathbb{R}^{N-1})}.$

If moreover $u_{0}\in L^{\infty}(\mathbb{R}^{N-1})$, then
$\|u\|_{L^{\infty}((0,T)\times\mathbb{R}^{N-1})}\leq\|u_{0}\|_{L^{\infty}(\mathbb{R}^{N-1})}.$

As we shall see below, the existence in Theorem 1.1 is a consequence of a more
general existence result concerning problem (1.9). Indeed, we define first the
Banach space

${\mathcal{C}}^{\theta}_{0}(\mathbb{R}^{N-1})=\overline{C^{\infty}_{c}(\mathbb{R}^{N-1})}^{\|\cdot\|_{C^{\theta}(\mathbb{R}^{N-1})}}\qquad\textrm{
for $\theta\in\mathbb{R}_{+}\setminus\mathbb{N}$},$

endowed with the norm of $C^{\theta}(\mathbb{R}^{N-1})$. We then write (1.9)
as

$\displaystyle\left\\{\begin{array}[]{rll}\partial_{t}u+{\mathcal{L}}_{0}u&=F(u)&\hskip
14.22636pt\mbox{in }\;[0,T]\times\mathbb{R}^{N-1}\\\ u(0)&=u_{0}&\hskip
14.22636pt\mbox{in }\;\mathbb{R}^{N-1},\end{array}\right.$ (1.15)

where
${\mathcal{L}}_{0}:=D{\mathcal{H}}(u_{0}):{\mathcal{C}}^{1+\beta}_{0}(\mathbb{R}^{N-1})\to{\mathcal{C}}^{\beta-\alpha}_{0}(\mathbb{R}^{N-1})$
is the derivative of the weighted mean curvature operator

$\displaystyle u\mapsto{\mathcal{H}}(u):={\sqrt{1+|\nabla u|^{2}}}H(u)$ (1.16)

at the initial condition $u_{0}$ and the nonlinear map $u\mapsto
F(u)=-{\mathcal{H}}(u)+{\mathcal{L}}_{0}u$ has the property that
$F(u)-F(u_{0})$ has superlinear growth in $u-u_{0}$. Our strategy is now to
show that ${\mathcal{L}}_{0}$ generates a strongly continuous analytic
semigroup on ${\mathcal{C}}^{\beta-\alpha}_{0}(\mathbb{R}^{N-1})$ and to apply
a fixed point argument. However, we need precise estimates, as we want all the
regularity estimated by the Hölder norms of $\nabla u_{0}$.

The main result from which we partly derive Theorem 1.1 is the following

######  Theorem 1.4.

Let $\beta\in(\alpha,1)$, $\nu>0$, $\rho\in(0,\frac{1}{1+\alpha})$ and
$\gamma_{\rho}:=\beta+\rho(1+\alpha)$. Then, there exists $T>0$, depending
only on $\rho,\alpha,\beta,N$ and $\nu$, such that for all
$u_{0}\in{\mathcal{C}}^{1+\beta}_{0}(\mathbb{R}^{N-1})$, with

$\|\nabla u_{0}\|_{C^{\gamma_{\rho}}(\mathbb{R}^{N-1})}\leq\nu,$

the initial value problem

$\displaystyle\left\\{\begin{array}[]{rll}\partial_{t}u+{\sqrt{1+|\nabla
u|^{2}}}H(u)&=0&\hskip 14.22636pt\mbox{in }\;[0,T]\times\mathbb{R}^{N-1}\\\
u(0)&=u_{0}&\hskip 14.22636pt\mbox{in }\;\mathbb{R}^{N-1}\end{array}\right.$
(1.19)

admits a unique solution $u\in
C^{1+\rho}([0,T],{\mathcal{C}}^{\beta-\alpha}_{0}(\mathbb{R}^{N-1}))\cap
C^{\rho}([0,T],{\mathcal{C}}^{1+\beta}_{0}(\mathbb{R}^{N-1}))$. Moreover,
there exists $C_{0}>0$, depending only on $\rho,\alpha,\beta,N$ and $\nu$,
such that

$\|u-u_{0}\|_{C^{1+\rho}([0,T],C^{\beta-\alpha}(\mathbb{R}^{N-1}))\cap
C^{\rho}([0,T],C^{1+\beta}(\mathbb{R}^{N-1}))}\leq C_{0}.$ (1.20)

We note that, thanks to the regularity of $E_{u}(t)$ proved in Theorem 1.4,
the uniqueness of the flow $(E_{u}(t))_{t\in[0,T]}$ also follows from [25],
where the authors proved that (1.1) admits a unique classical solution. On the
other hand, the proof of our Theorem 1.4 yields existence and uniqueness
simultaneously.

Recall that, provided the linear operator
${\mathcal{L}}_{0}:=D{\mathcal{H}}(u_{0}):{\mathcal{C}}^{1+\beta}_{0}(\mathbb{R}^{N-1})\to{\mathcal{C}}^{\beta-\alpha}_{0}(\mathbb{R}^{N-1})$
generates an analytic semigroup, to get optimal
$C^{\rho}([0,T],{\mathcal{C}}^{1+\beta}_{0}(\mathbb{R}^{N-1}))$-regularity in
time for the problem (1.15), it is necessary and sufficient that $u_{0}$
satisfies
$F(u_{0})-{\mathcal{L}}_{0}u_{0}=-{\mathcal{H}}(u_{0})\in\mathcal{D}_{{\mathcal{L}}_{0}}(\rho,\infty)$.
Here $\mathcal{D}_{{\mathcal{L}}_{0}}(\rho,\infty)$ is a real interpolation
space between the space ${\mathcal{C}}^{\beta-\alpha}_{0}(\mathbb{R}^{N-1})$
and the domain ${\mathcal{C}}^{1+\beta}_{0}(\mathbb{R}^{N-1})$. We refer the
reader to Lunardi [22] and Sinestrari [28] on this type of optimal regularity.
We shall show here that
$\mathcal{D}_{{\mathcal{L}}_{0}}(\rho,\infty)=C^{\beta+\rho(1+\alpha)-\alpha}(\mathbb{R}^{N-1})\cap{\mathcal{C}}^{\beta-\alpha}_{0}(\mathbb{R}^{N-1})$
for $\rho\not=\frac{1+\alpha-\beta}{1+\alpha}$ (see Proposition 3.3 below). In
addition, we will prove that
${\mathcal{H}}(u_{0})\in\mathcal{D}_{{\mathcal{L}}_{0}}(\rho,\infty)$ if and
only if $\nabla u_{0}\in C^{\beta+\rho(1+\alpha)}(\mathbb{R}^{N-1})$.

The proof of Theorem 1.4 is given in Section 5 below. Main steps in this proof
consist in establishing suitable regularity properties of the fractional mean
curvature operator $H$ in Hölder spaces and in showing that the operator
$D{\mathcal{H}}(u_{0}):{\mathcal{C}}^{1+\beta}_{0}(\mathbb{R}^{N-1})\subset{\mathcal{C}}^{\beta-\alpha}_{0}(\mathbb{R}^{N-1})\to{\mathcal{C}}^{\beta-\alpha}_{0}(\mathbb{R}^{N-1})$
is the generator of a strongly continuous analytic semigroup. We shall derive
the generation of a strongly analytic semigroup by combining a result from [2]
for a class of non-smooth integro-differential operators defined
${\mathcal{C}}^{1+\beta}_{0}(\mathbb{R}^{N-1})$ with Hölder estimates for
integral operators with anisotropic kernels represented suitably in polar
coordinates. To obtain existence and uniqueness for the non homogeneous Cauchy
problem (1.15), we also need a characterization of the real interpolation
space $\mathcal{D}_{{\mathcal{L}}_{0}}(\rho,\infty)$, see Section 3.2. Once
Theorem 1.4 is established, we derive Theorem 1.1 from it by an approximation
argument. The uniqueness in Theorem 1.1 and the universal estimates in Theorem
1.3 are based on a suitable version of the comparison principle, which we also
prove. In fact, our comparison principle yields further interesting
qualitative properties of the flow such as preservation of positivity of
$u_{0}$ and $H(u_{0})$.

The paper is organized as follows. In Section 2, we introduce the function
spaces we will use in this paper and we prove some preliminary results.
Section 3 is devoted to the strongly continuous analytic semigroup generated
by a general class of nonlocal operators. Section4 deals with the regularity
properties satisfied by the weighted nonlocal mean curvature operator
${\mathcal{H}}(u)$ and its linearization. Finally, in Section 5, we prove our
main results.

Acknowledgments
This work is supported by the Alexander von Humboldt foundation and the German
Academic Exchange Service (DAAD). The authors would like to thank Sven Jarohs
and Esther Cabezas-Rivas for many fruitful discussions.

##  2\. Notation and preliminary estimates

For $m\in\mathbb{N}$, we denote by $C_{b}^{m}(\mathbb{R}^{N-1})$ the space of
$m$-times bounded continuously differentiable functions endowed with the norm

$\displaystyle\|u\|_{C_{b}^{m}}=\sum_{k=0}^{m}\|D^{k}u\|_{L^{\infty}}:=\sum_{k=0}^{m}\sup_{x\in\mathbb{R}^{N-1}}|D^{k}u(x)|.$

For $0<\gamma<1$, we consider the space of Hölder continuous functions

$\displaystyle C^{\gamma}(\mathbb{R}^{N-1})=\bigg{\\{}u\in
C_{b}(\mathbb{R}^{N-1}):\,[u]_{C^{\gamma}}:=\sup_{\stackrel{{\scriptstyle
x,y\in\mathbb{R}^{N-1}}}{{x\neq
y}}}\frac{|u(x)-u(y)|}{|x-y|^{\gamma}}<\infty\bigg{\\}}$ (2.1)

endowed with the norm

$\displaystyle
u\mapsto\|u\|_{C^{\gamma}}=\|u\|_{L^{\infty}}+[u]_{C^{\gamma}}.$

More generally, for $m\in\mathbb{N}$ and $0<\gamma<1$, we let

$C^{m+\gamma}(\mathbb{R}^{N-1})=\\{u\in C_{b}^{m}(\mathbb{R}^{N-1}):D^{m}u\in
C^{\gamma}(\mathbb{R}^{N-1})\\},$

endowed with the norm

$u\mapsto\|u\|_{C^{m+\gamma}}=\|u\|_{C_{b}^{m}}+[D^{m}u]_{C^{\gamma}}.$

For late use, we recall the interpolation inequality that, for all
$\gamma\in(0,1)$, $\gamma^{\prime}\in[0,\gamma)$ and for all $\varepsilon>0$,
there exists $C_{\varepsilon}>0$ such that

$\|\nabla v\|_{C^{\gamma^{\prime}}}\leq\varepsilon\|\nabla
v\|_{C^{\gamma}}+C_{\varepsilon}\|v\|_{L^{\infty}}\qquad\textrm{ for all $v\in
C^{1,\gamma}(\mathbb{R}^{N-1})$.}$ (2.2)

Next, we define

${\mathcal{C}}^{m+\gamma}_{0}(\mathbb{R}^{N-1}):=\overline{C^{\infty}_{c}(\mathbb{R}^{N-1})}^{\|\cdot\|_{C^{m+\gamma}}}$
(2.3)

endowed with the $C^{m+\gamma}$-norm. We then recall the following result.

######  Proposition 2.1.

[2, Proposition 2.1] Let $\beta,\gamma\in(0,1)$, with $\gamma>\beta$. Let
$u\in C^{\gamma}(\mathbb{R}^{N-1})$ be such that
$\lim_{R\to\infty}\|f\|_{C^{\beta}(\mathbb{R}^{N-1}\setminus B_{R})}=0$. Then
$f\in{\mathcal{C}}^{\beta}_{0}(\mathbb{R}^{N-1})$.

We need some more notation related to space of time dependent functions. Let
${\mathcal{X}}$ be a Banach space. For $T>0$, we consider the Banach space
$L^{\infty}([0,T],{\mathcal{X}})$, consisting of bounded functions
$u:[0,T]\rightarrow{\mathcal{X}}.$ The spaces
$L^{\infty}([0,T],{\mathcal{X}})$ is endowed with the norm

$\|u\|_{L^{\infty}([0,T],{\mathcal{X}})}=\sup_{t\in[0,T]}\|u(t)\|_{\mathcal{X}}.$

For $\mu\in(0,1)$, we define the Banach space

$C^{1+\mu}([0,T],{\mathcal{X}})=\left\\{u\in
C^{1}([0,T],{\mathcal{X}})\,:\,[\partial_{t}u]_{C^{\mu}([0,T],{\mathcal{X}})}=\sup_{0\leq
s<t\leq
T}\frac{\|\partial_{t}u(t)-\partial_{t}u(s)\|_{\mathcal{X}}}{|t-s|^{\mu}}<+\infty\right\\},$

endowed with the norm
$\|u\|_{C^{1+\mu}([0,T],{\mathcal{X}})}=\|u\|_{L^{\infty}([0,T],{\mathcal{X}})}+[\partial_{t}u]_{C^{\mu}([0,T],{\mathcal{X}})}.$

###  2.1. Estimates in polar coordinates

In our study of linearizations of the nonlocal mean curvature operator, we
will have to consider integral operators whose associated kernels have
anisotropic singularities which can be resolved partially in polar
coordinates. For this, the following basic estimates will be of key
importance.

######  Lemma 2.2.

Let $\gamma\in(\alpha,1)$, $v\in C^{1+\gamma}_{loc}(\mathbb{R}^{N-1})$ be such
that $\nabla v\in L^{\infty}(\mathbb{R}^{N-1})$. Let $\mu\in
C^{\gamma-\alpha}(\mathbb{R}^{N-1}\times[0,\infty)\times S^{N-2})$ and $\nu\in
C^{\gamma}(\mathbb{R}^{N-1}\times[0,\infty)\times S^{N-2})$ with
$\nu(\cdot,0,\cdot)=0$. We define

$I_{e}v(x):=\int_{S^{N-2}}\int_{0}^{\infty}\delta_{e}v(x,r,\theta)\mu(x,r,\theta)r^{-2-\alpha}drd\theta$

and

$I_{o}v(x):=\int_{S^{N-2}}\int_{0}^{\infty}\delta_{o}v(x,r,\theta)\nu(x,r,\theta)r^{-2-\alpha}drd\theta$

for $x\in\mathbb{R}^{N-1}$, where

$\delta_{e}v(x,r,\theta):=\frac{1}{2}(2v(x)-v(x+r\theta)-v(x-r\theta))\quad\textrm{
and }\quad\delta_{o}v(x,r,\theta):=v(x)-v(x-r\theta).$ (2.4)

Then the functions $I_{e}v$ and $I_{o}v$ satisfy the following estimates with
constants $C$ independent of $\mu,\nu$ and $v$.

  * (i)

We have

$\|I_{o}v\|_{C^{\gamma-\alpha}}\leq C\|\nabla
v\|_{L^{\infty}}\|\nu\|_{C^{\gamma}(\mathbb{R}^{N-1}\times[0,\infty)\times
S^{N-2})}$ (2.5)

and if $\nabla v\in C^{\gamma}(\mathbb{R}^{N-1})$, then

$\|I_{e}v\|_{C^{\gamma-\alpha}}\leq C\|\nabla
v\|_{C^{\gamma}}\|\mu\|_{C^{\gamma-\alpha}(\mathbb{R}^{N-1}\times[0,\infty)\times
S^{N-2})}.$ (2.6)

  * (ii)

If $\nabla v\in C^{\gamma}(\mathbb{R}^{N-1})$ and $\nabla v$ is compactly
supported, then

$\lim_{R\to\infty}\|I_{e}v\|_{C^{\gamma-\alpha}(\mathbb{R}^{N-1}\setminus
B_{R})}=\lim_{R\to\infty}\|I_{o}v\|_{C^{\gamma-\alpha}(\mathbb{R}^{N-1}\setminus
B_{R})}=0.$ (2.7)

  * (iii)

If $v\in{\mathcal{C}}^{1+\beta}_{0}(\mathbb{R}^{N-1})$, with
$\beta\in(\alpha,\gamma)$, then
$I_{e}v,I_{o}v\in{{\mathcal{C}}^{\beta-\alpha}_{0}}(\mathbb{R}^{N-1})$.

###### Proof.

We assume for simplicity that
$\|\nu\|_{C^{\gamma}(\mathbb{R}^{N-1}\times[0,\infty)\times S^{N-2})}\leq 1$.
We first observe that

$|\nu(x,r,\theta)|\leq\min(1,r^{\gamma}),\qquad|\delta_{o}v(x,r,\theta)|=r\left|\int_{0}^{1}\nabla
v(x-rt\theta)\cdot\theta dt\right|\leq r\|\nabla v\|_{L^{\infty}}$ (2.8)

and for $x,x^{\prime}\in\mathbb{R}^{N-1}$

$|\nu(x,r,\theta)-\nu(x^{\prime},r,\theta)|\leq\min(|x-x^{\prime}|^{\gamma},r^{\gamma}),\qquad$
(2.9)

and

$|\delta_{o}v(x,r,\theta)-\delta_{o}v(x^{\prime},r,\theta)|\leq 2\|\nabla
v\|_{L^{\infty}}\min(|x-x^{\prime}|,r).$ (2.10)

Then, by (2.8), we have

$\displaystyle\|I_{o}v\|_{L^{\infty}}$
$\displaystyle=\sup_{x\in\mathbb{R}^{N-1}}\int_{S^{N-2}}\int_{0}^{\infty}|\delta_{o}v(x,r,\theta)\nu(x,r,\theta)|r^{-2-\alpha}drd\theta$
$\displaystyle\leq C\|\nabla
v\|_{L^{\infty}}\int_{0}^{\infty}\min(1,r^{\gamma})r^{-1-\alpha}dr\leq
C\|\nabla v\|_{L^{\infty}}.$ (2.11)

We write

$\displaystyle I_{o}v(x)-I_{o}v(z)$
$\displaystyle=\int_{S^{N-2}}\int_{0}^{\infty}r^{-2-\alpha}\delta_{o}v(x,r,\theta)[\nu(x,r,\theta)-\nu(z,r,\theta)]\;drd\theta$
$\displaystyle+\int_{S^{N-2}}\int_{0}^{\infty}r^{-2-\alpha}[\delta_{o}v(x,r,\theta)-\delta_{o}v(z,r,\theta)]\nu(x,r,\theta)\;drd\theta.$

From this, (2.8), (2.9) and (2.10), we get

$\displaystyle|I_{o}v(x)-I_{o}v(z)|$ $\displaystyle\leq C\|\nabla
v\|_{L^{\infty}}\int_{0}^{\infty}r^{-1-\alpha}\min\\{r^{\gamma},|x-z|^{\gamma}\\}\;dr$
$\displaystyle+C\|\nabla
v\|_{L^{\infty}}\int_{0}^{\infty}r^{\gamma-2-\alpha}\min\\{|x-z|,r\\}\;dr\leq
C\|\nabla v\|_{L^{\infty}}|x-z|^{\gamma-\alpha}.$

Therefore

$\displaystyle[I_{o}v]_{C^{\gamma-\alpha}}\leq C\|\nabla v\|_{L^{\infty}}.$
(2.12)

The inequalities (2.11) and (2.12) imply (2.5).
For simplicity, we assume that
$\|\mu\|_{C^{\gamma}(\mathbb{R}^{N-1}\times[0,\infty)\times S^{N-2})}\leq 1$.
We write

$\delta_{e}v(x,r,\theta)=\frac{r}{2}\int_{0}^{1}[\nabla v(x-rt\theta)-\nabla
v(x+rt\theta)]dt.$ (2.13)

Hence, we have

$|\delta_{e}v(x,r,\theta)|\leq r\min(1,r^{\gamma})\|\nabla v\|_{C^{\gamma}}$
(2.14)

and for $x,x^{\prime}\in\mathbb{R}^{N-1}$

$|\mu(x,r,\theta)-\mu(x^{\prime},r,\theta)|\leq\min(|x-x^{\prime}|^{\gamma},1)$
(2.15)

and

$|\delta_{e}v(x,r,\theta)-\delta_{e}v(x^{\prime},r,\theta)|\leq r\|\nabla
v\|_{C^{\gamma}}\min(|x-x^{\prime}|^{\gamma},r^{\gamma}).$ (2.16)

We now estimate, using (2.14),

$\displaystyle\|I_{e}v\|_{L^{\infty}}$
$\displaystyle=\sup_{x\in\mathbb{R}^{N-1}}\int_{S^{N-2}}\int_{0}^{\infty}|\delta_{e}v(x,r,\theta)\mu(x,r,\theta)|r^{-2-\alpha}drd\theta$
$\displaystyle\leq C\|\nabla
v\|_{C^{\gamma}}\int_{0}^{\infty}\min(1,r^{\gamma})r^{-1-\alpha}dr\leq
C\|\nabla v\|_{C^{\gamma}}.$ (2.17)

We have

$\displaystyle I_{e}v(x)-I_{e}v(z)$
$\displaystyle=\int_{S^{N-2}}\int_{0}^{\infty}r^{-2-\alpha}\delta_{e}v(x,r,\theta)[\mu(x,r,\theta)-\mu(z,r,\theta)]\;drd\theta$
$\displaystyle+\int_{S^{N-2}}\int_{0}^{\infty}r^{-2-\alpha}[\delta_{e}v(x,r,\theta)-\delta_{e}v(z,r,\theta)]\mu(x,r,\theta)\;drd\theta.$

From this, (2.8), (2.9) and (2.10), we get

$\displaystyle|I_{e}v(x)-I_{e}v(z)|$ $\displaystyle\leq C\|\nabla
v\|_{C^{\gamma}}\min\\{1,|x-z|^{\gamma}\\}\int_{0}^{\infty}r^{-1-\alpha}\
\min\\{1,r^{\gamma}\\};dr$ $\displaystyle+C\|\nabla
v\|_{C^{\gamma}}\int_{0}^{\infty}r^{-1-\alpha}\min\\{|x-z|^{\gamma},r^{\gamma}\\}\;dr\leq
C\|\nabla v\|_{C^{\gamma}}|x-z|^{\gamma-\alpha}.$

This and (2.17) give (2.6). The proof of $(i)$ is thus complete.

For $(ii)$, we assume that Supp$\nabla v\subset B_{R^{\prime}}$, for some
$R^{\prime}>1$. We let $R>2R^{\prime}$ and we split

$\displaystyle I_{e}v(x)$
$\displaystyle=\int_{S^{N-2}}\int_{0}^{R}\delta_{e}v(x,r,\theta)\mu(x,r,\theta)r^{-2-\alpha}drd\theta+\int_{S^{N-2}}\int_{R}^{\infty}\delta_{e}v(x,r,\theta)\mu(x,r,\theta)r^{-2-\alpha}drd\theta$
$\displaystyle=:I_{e}^{1}(x)+I_{e}^{2}(x).$

By (2.13),

$I_{e}^{1}(x)=0\qquad\textrm{ for $|x|\geq R^{\prime}+R$.}$ (2.18)

In addition, for $|x|>R$, by (2.13),

$\displaystyle|I_{e}^{2}(x)|\leq\|\nabla
v\|_{L^{\infty}}\int_{0}^{1}\int_{R}^{\frac{|x|+R^{\prime}}{t}}r^{-1-\alpha}dr\,dt\leq
C\|\nabla
v\|_{L^{\infty}}\left(R^{-\alpha}+(|x|+R^{\prime})^{-\alpha}\right),$ (2.19)

so that

$\lim_{R\to\infty}\|I_{e}\|_{L^{\infty}}=0.$ (2.20)

Now, for $h\in\mathbb{R}^{N-1}$,

$\displaystyle|I_{e}^{2}(x+h)-I_{e}^{2}(x)|$
$\displaystyle\leq|h|^{\gamma-\alpha}\|\nabla
v\|_{C^{\gamma-\alpha}}\int_{0}^{1}\int_{R}^{\frac{|x|+|h|+R^{\prime}}{t}}r^{-1-\alpha}dr\,dt$
$\displaystyle\leq C|h|^{\gamma-\alpha}\|\nabla
v\|_{C^{\gamma-\alpha}}\left(R^{-\alpha}+(|x|+|h|+R)^{-\alpha}\right).$

Combining this with (2.20) and (2.18), we deduce that
$\lim_{R\to\infty}\|I_{e}v\|_{C^{\gamma-\alpha}(\mathbb{R}^{N-1}\setminus
B_{R})}=0$. By similar argument we have
$\lim_{R\to\infty}\|I_{o}v\|_{C^{\gamma-\alpha}(\mathbb{R}^{N-1}\setminus
B_{R})}=0$, so that (2.7) holds.

For $(iii)$, we pick $v\in{\mathcal{C}}^{1+\gamma}_{0}(\mathbb{R}^{N-1})$.
Then there exists $v_{n}\in C^{\infty}_{c}(\mathbb{R}^{N-1})$ such that
$v_{n}\to v$ in $C^{1+\gamma}(\mathbb{R}^{N-1})$. Thanks to $(i)$, we have
$I_{e}v_{n},I_{o}v_{n}\in{\mathcal{C}}^{\gamma-\alpha}_{0}(\mathbb{R}^{N-1})$.
Now, by $(ii)$ and Proposition 2.1, we get
$I_{o}v_{n},I_{e}v_{n}\in{\mathcal{C}}^{\beta-\alpha}_{0}(\mathbb{R}^{N-1})$.
Once again by $(i)$, we have $I_{e}v_{n}\to I_{e}v$ and $I_{o}v_{n}\to I_{o}v$
in $C^{\beta-\alpha}(\mathbb{R}^{N-1})$. This then yields the conclusion,
since ${\mathcal{C}}^{\beta-\alpha}_{0}(\mathbb{R}^{N-1})$ is closed in
$C^{\beta-\alpha}(\mathbb{R}^{N-1})$. ∎

###  2.2. A nonlocal comparison principle in the entire space

In this subsection we provide a comparison principle for a class of nonlocal
evolution equations relying on kernel assumptions in polar coordinates, which
will be used later on.

######  Proposition 2.3.

Let $\alpha\in(0,1)$, $\beta\in(\alpha,1)$, $T>0$, $V\in
C\left([0,T],C_{b}(\mathbb{R}^{N-1})\right)$ and let let $\mu\in
C\left([0,T],C^{\beta}(\mathbb{R}^{N-1}\times[0,\infty)\times S^{N-2})\right)$
with $\mu>0$ and $\mu(t,x,0,\theta)=\mu(t,x,0,-\theta)$ Moreover, let
$\beta>\alpha$, and let $u\in L^{\infty}([0,T],C^{\beta}(\mathbb{R}^{N-1}))$
satisfy

$u(\cdot,x)\in C^{1}([0,T])\quad\text{for every $x\in\mathbb{R}^{N-1}$,}\quad
u(t,\cdot)\in C^{1+\beta}_{loc}(\mathbb{R}^{N-1})\quad\text{for every
$t\in[0,T]$}$

and

$\partial_{t}u(t,x)+P.V.\int_{\mathbb{R}^{N-1}}\frac{u(t,x)-u(t,x+y)}{|y|^{N+\alpha}}\mu(t,x,|y|,y/|y|)\,dy+V(t,x)\cdot\nabla
u(t,x)\leq 0$ (2.21)

for $(t,x)\in[0,T]\times\mathbb{R}^{N-1}$. Then
$\displaystyle\sup_{[0,T]\times\mathbb{R}^{N-1}}u=\sup_{x\in\mathbb{R}^{N-1}}u(0,x).$

###### Proof.

We consider $\eta\in C^{\infty}_{c}(B_{2})$ with $0\leq\eta\leq 1$ on
$\mathbb{R}^{N-1}$ and $\eta=1$ on $B_{1}$. We define, for $\varepsilon,R>0$,

$v(t,x)=\eta_{R}(x)u(t,x)-\varepsilon t,$

where $\eta_{R}(x)=\eta(x/R)$. We also define, for $w\in
C^{1+\beta}_{loc}(\mathbb{R}^{N-1})\cap L^{\infty}(\mathbb{R}^{N-1})$ and
$x\in\mathbb{R}^{N-1}$,

$\overline{L}(t)w(x)=P.V.\int_{\mathbb{R}^{N-1}}\frac{w(x)-w(x+y)}{|y|^{N+\alpha}}\mu(t,x,|y|,y/|y|)\,dy.$

We then have

$\displaystyle\overline{L}(t)(\eta_{R}u(t))(x)=\eta_{R}(x)\overline{L}(t)(u(t))(x)$
$\displaystyle+u(t,x)P.V.\int_{\mathbb{R}^{N-1}}\frac{\eta_{R}(x)-\eta_{R}(x+y)}{|y|^{N+\alpha}}\mu(t,x,|y|,y/|y|)\,dy$
$\displaystyle+P.V.\int_{\mathbb{R}^{N-1}}\frac{\eta_{R}(x)-\eta_{R}(x+y)}{|y|^{N+\alpha}}(u(t,x+y)-u(t,x))\mu(t,x,|y|,y/|y|)\,dy.$

From this, we get

$\displaystyle\overline{L}(t)(\eta_{R}u(t))(x)-\eta_{R}(x)P.V.\int_{\mathbb{R}^{N-1}}\frac{u(t,x)-u(t,x+y)}{|y|^{N+\alpha}}\mu(t,x,|y|,y/|y|)\,dy$
$\displaystyle=\frac{u(t,x)}{2}P.V.\int_{\mathbb{R}^{N-1}}\frac{2\eta_{R}(x)-\eta_{R}(x-y)-\eta_{R}(x+y)}{|y|^{N+\alpha}}\mu(t,x,0,y/|y|)\,dy$
$\displaystyle+{u(t,x)}P.V.\int_{\mathbb{R}^{N-1}}\frac{\eta_{R}(x)-\eta_{R}(x+y)}{|y|^{N+\alpha}}[\mu(t,x,|y|,y/|y|)-\mu(t,x,0,y/|y|)]\,dy$
$\displaystyle+P.V.\int_{\mathbb{R}^{N-1}}\frac{\eta_{R}(x)-\eta_{R}(x+y)}{|y|^{N+\alpha}}(u(t,x+y)-u(t,x))\mu(t,x,|y|,y/|y|)\,dy.$

Hence, by Lemma 2.2 we get

$\|\overline{L}(t)(\eta_{R}u(t))-\eta_{R}\overline{L}(t)u(t)\|_{L^{\infty}(\mathbb{R}^{N-1})}\leq
C\|\nabla\eta_{R}\|_{C^{1}_{b}}\|u(t)\|_{C^{\beta}}\leq\frac{C}{R}\|u(t)\|_{C^{\beta}}$

for $t\in[0,T]$. From this and (2.21), we then obtain

$\partial_{t}v+\overline{L}(t)v+V\cdot\nabla
v\leq-\varepsilon+F_{R}\qquad\textrm{ in $[0,T]\times\mathbb{R}^{N-1}$,}$
(2.22)

with

$\|F_{R}\|_{L^{\infty}((0,T)\times\mathbb{R}^{N-1})}\leq\frac{C\|u\|_{L^{\infty}((0,T),C^{\beta})}}{R}(1+\|V\|_{L^{\infty}((0,T)\times\mathbb{R}^{N-1})}).$
(2.23)

We claim that

$\max_{[0,T]\times\mathbb{R}^{N-1}}v=\max_{x\in\mathbb{R}^{N-1}}v(0,x).$
(2.24)

Indeed, let $(t_{0},x_{0})\in[0,T]\times\mathbb{R}^{N-1}$ be such that
$v(t_{0},x_{0})=\max_{[0,T]\times\mathbb{R}^{N-1}}v$. Suppose that $t_{0}>0$.
The maximality property then implies that $\nabla_{x}v(t_{0},x_{0})=0$,
$\partial_{t}v(t_{0},x_{0})\geq 0$ and also $L(t_{0})v(x_{0})\geq 0$, since
$\mu\geq 0$ by assumption. By (2.22) we thus have

$0\leq\overline{L}(t_{0})v(x_{0})\leq-\varepsilon+\|F_{R}\|_{L^{\infty}((0,T)\times\mathbb{R}^{N-1})}$

which is not possible if $R$ is large enough, thanks to (2.23). Therefore
$t_{0}=0$ and thus (2.24) holds as claimed.
Letting now $R\to\infty$ and then $\varepsilon\to 0$ in (2.24), we get the
result. ∎

##  3\. Analytic semigroups, their generators, intermediate spaces and
associated semilinear evolution equations

We begin by introducing some notions regarding function spaces and analytic
semigroups (see e.g. [22, Chapter 2]). For normed vector spaces
${\mathcal{X}},{\mathcal{X}}^{\prime}$ we let
${\mathcal{L}}({\mathcal{X}},{\mathcal{X}}^{\prime})$ denote the space of
continuous linear operators ${\mathcal{X}}\to{\mathcal{X}}^{\prime}$, endowed
with the usual operator norm. As usual, we also set
${\mathcal{L}}({\mathcal{X}}):={\mathcal{L}}({\mathcal{X}},{\mathcal{X}})$.

A strongly continuous analytic semigroup on a Banach space ${\mathcal{X}}$ is
a family of operators $\\{T(t)\\}_{t\geq
0}\subset{\mathcal{L}}({\mathcal{X}})$ with the following properties:

  * (i)

$T(0)=\operatorname{id}$, $T(t+s)=T(t)T(s)$ for all $t,s\geq 0$.

  * (ii)

The function $(0,\infty)\to{\mathcal{L}}({\mathcal{X}})$, $t\mapsto T(t)$ is
analytic.

  * (iii)

The function $[0,\infty)\to{\mathcal{X}}$, $t\mapsto T(t)u$ is continuous for
every $u\in{\mathcal{X}}$.

The generator of such a semigroup is given as an (unbounded) linear operator
$B_{0}:{\mathcal{Y}}\subset{\mathcal{X}}\to{\mathcal{X}}$ by

$B_{0}u:=\lim_{t\to 0^{+}}\frac{T(t)u-u}{t},\qquad u\in{\mathcal{Y}},$

where the domain ${\mathcal{Y}}\subset{\mathcal{X}}$ of $B_{0}$ is given as
the subspace of $u\in{\mathcal{X}}$ for which this limit exists.

We recall that an operator
$B_{0}:{\mathcal{Y}}\subset{\mathcal{X}}\to{\mathcal{X}}$ generates a strongly
continuous analytic semigroup in this sense if and only if $B_{0}$ is
sectorial (see e.g. [22, Def. 2.0.1] for a definition) and its domain
${\mathcal{Y}}$ is dense in ${\mathcal{X}}$. In such a case, the operator
$B_{0}$ is also closed, which means that ${\mathcal{Y}}$ is a Banach space
with the graph norm
$u\mapsto\|u\|_{{\mathcal{X}}}+\|{\mathcal{B}}_{0}u\|_{{\mathcal{X}}}$ on
${\mathcal{Y}}$. As a consequence, by the open mapping theorem, the graph norm
on ${\mathcal{Y}}$ is equivalent to any other given norm
$\|\cdot\|_{{\mathcal{Y}}}$ on ${\mathcal{Y}}$ for which
$({\mathcal{Y}},\|\cdot\|_{{\mathcal{Y}}})$ is a Banach space.

In the following, if
${\mathcal{B}}_{0}:{\mathcal{Y}}\subset{\mathcal{X}}\to{\mathcal{X}}$
generates a strongly continuous analytic semigroup, we shall denote this
semigroup by

$t\mapsto e^{B_{0}t}\in{\mathcal{L}}({\mathcal{X}}),\qquad t\geq 0.$

As noted in [22, Proposition 2.1.1], for all $k\in\mathbb{N}$, there exists
$M_{k}>0$, such that

$\displaystyle\|t^{k}B_{0}^{k}e^{B_{0}t}\|_{\mathcal{L}(Y)}\leq M_{k},\hskip
14.22636pt\mbox{for all $t\in(0,1]$.}$ (3.1)

In order to obtain optimal regularity estimates in time, it is convenient to
introduce, for $\rho\in[0,1)$, the intermediate space

$\displaystyle\mathcal{D}_{B_{0}}(\rho,\infty)=\\{f\in{\mathcal{Y}}:[f]_{\mathcal{D}_{B_{0}}(\rho,\infty)}=\sup_{0<t\leq
1}\|t^{1-\rho}B_{0}e^{B_{0}t}f\|_{\mathcal{Y}}<\infty\\},$ (3.2)

endowed with the norm
$\|f\|_{\mathcal{D}_{B_{0}}(\rho,\infty)}=\|f\|_{\mathcal{Y}}+[f]_{\mathcal{D}_{B_{0}}(\rho,\infty)}.$

We then have the following result taken from [22, Theorem 4.3.1 (iii)].

######  Theorem 3.1.

Let $B_{0}:{\mathcal{Y}}\subset{\mathcal{X}}\to{\mathcal{X}}$ be the generator
of a strongly continuous analytic semigroup. Let $T>0$, $\rho\in(0,1)$,
$u_{0}\in{\mathcal{X}}$, and let $f\in C^{\rho}([0,T],{\mathcal{Y}})$ be such
that $f(0)-B_{0}u_{0}\in{\mathcal{D}_{B_{0}}(\rho,\infty)}$. Then the problem

$\left\\{\begin{array}[]{rll}u^{\prime}(t)+B_{0}u(t)&=&f(t),\hskip
28.45274ptt\in(0,T]\\\ u(0)&=&u_{0}\end{array}\right.$

admits a unique solution $u\in C^{\rho}([0,T],{\mathcal{X}})\cap
C^{1+\rho}([0,T],{\mathcal{Y}})$. Moreover, there exists
$C_{T}=C(T,\rho,M_{0},M_{1},M_{2})>0$ such that

$\displaystyle\|B_{0}(u-u_{0})\|_{C^{\rho}([0,T],{\mathcal{Y}})}+$
$\displaystyle\|u-u_{0}\|_{C^{1+\rho}([0,T],{\mathcal{Y}})}$
$\displaystyle\leq
C_{T}\Big{(}\|f-B_{0}u_{0}\|_{C^{\rho}([0,T],{\mathcal{Y}})}+\|f(0)-B_{0}u_{0}\|_{\mathcal{D}_{B_{0}}(\rho,\infty)}\Big{)}$
(3.3)

and $C_{T}\leq C_{T_{0}}$ for all $T\leq T_{0}$.

###### Proof.

The existence, uniqueness and (3.1) follow from [22, Theorem 4.3.1]. Moreover,
as explained in the beginning of Section 4.1 in [22], the constant $C_{T}$ is
increasing in $T$. ∎

######  Remark 3.2.

Let $B_{0}:{\mathcal{Y}}\subset{\mathcal{X}}\to{\mathcal{X}}$ be the generator
of a strongly continuous analytic semigroup and $\sigma\in(0,1)$. Then, as
noted in [22, Proposition 2.2.2], the intermediate space
$\mathcal{D}_{B_{0}}(\rho,\infty)$ does not depend on the operator $B_{0}$
itself, as it coincides with a real interpolation space between the spaces
${\mathcal{Y}}$ and ${\mathcal{X}}$ (with equivalence of respective norms). We
shall use this fact in the following where we consider a Hölder space setting.

###  3.1. Intermediate spaces in a Hölder space setting

In our application to the nonlocal mean curvature flow, we will need to
consider the special case where
${\mathcal{Y}}={\mathcal{C}}^{1+\beta}_{0}(\mathbb{R}^{N-1})$ and
${\mathcal{X}}={\mathcal{C}}^{\beta-\sigma}_{0}(\mathbb{R}^{N-1})$ for values
$\sigma\in(0,1)$, $\beta\in(\sigma,1)$. Our next result provides a
characterization of the intermediate space $\mathcal{D}_{B_{0}}(\rho,\infty)$
defined in (3.2) in this particular case. As noted in Remark 3.2, this space
does not depend on the particular choice of a generator
$B_{0}:{\mathcal{Y}}\subset{\mathcal{X}}\to{\mathcal{X}}$ of a strongly
continuous analytic semigroup.

######  Proposition 3.3.

Let $\sigma\in(-1,1)$, $\beta\in(\sigma,1+\sigma)$ with
$\beta\not\in\mathbb{N}$. Let
$B_{0}:{\mathcal{C}}^{1+\beta}_{0}(\mathbb{R}^{N-1})\subset{\mathcal{C}}^{\beta-\sigma}_{0}(\mathbb{R}^{N-1})\to{\mathcal{C}}^{\beta-\sigma}_{0}(\mathbb{R}^{N-1})$
be any infinitesimal generator of a strongly continuous analytic semigroup on
${\mathcal{C}}^{\beta-\sigma}_{0}(\mathbb{R}^{N-1})$. Let
$\rho\in(0,\min(1,\frac{1}{1+\sigma}))$ and put
$\gamma_{\rho}:=\beta+\rho(1+\sigma).$ Then
${\mathcal{C}}^{\beta-\sigma}_{0}(\mathbb{R}^{N-1})\cap
C^{\gamma_{\rho}-\sigma}({\mathbb{R}^{N-1}})\subset\mathcal{D}_{B_{0}}(\rho,\infty)$
and there exists $c=c(\sigma,\beta,\rho)>0$ such that

$\|f\|_{\mathcal{D}_{B_{0}}(\rho,\infty)}\leq
c\|f\|_{C^{\gamma_{\rho}-\sigma}}\qquad\textrm{for all
$f\in{\mathcal{C}}^{\beta-\sigma}_{0}(\mathbb{R}^{N-1})\cap
C^{\gamma_{\rho}-\sigma}({\mathbb{R}^{N-1}})$.}$ (3.4)

If moreover $\gamma_{\rho}-\sigma\not\in\mathbb{N}$, then
${\mathcal{C}}^{\beta-\sigma}_{0}(\mathbb{R}^{N-1})\cap
C^{\gamma_{\rho}-\sigma}({\mathbb{R}^{N-1}})=\mathcal{D}_{B_{0}}(\rho,\infty)$,
with equivalence of their respective norms.

###### Proof.

By [2, Corollary 2.17], for $\sigma\in(-1,1)$, the standard fractional
Laplacian
$(-\Delta)^{\frac{1+\sigma}{2}}:{\mathcal{C}}^{1+\beta}_{0}(\mathbb{R}^{N-1})\to{\mathcal{C}}^{\beta-\sigma}_{0}(\mathbb{R}^{N-1})$
is a generator of a strongly continuous analytic semigroup, and therefore

$\mathcal{D}_{B_{0}}(\rho,\infty)=\mathcal{D}_{(-\Delta)^{\frac{1+\sigma}{2}}}(\rho,\infty),$
(3.5)

with equivalence of their respective norms, see Remark 3.2, .

For simplicity, we write $\gamma=\gamma_{\rho}$. Letting
$f\in{\mathcal{C}}^{\beta-\sigma}_{0}(\mathbb{R}^{N-1})\cap
C^{\gamma-\sigma}({\mathbb{R}^{N-1}})$, we then have

$e^{t(-\Delta)^{\frac{1+\sigma}{2}}}f(x)=K(\cdot,t)\star
f(x)=\int_{\mathbb{R}^{N-1}}K(x-y,t)f(y)dy,$ (3.6)

where $K$ is the heat kernel of $(-\Delta)^{\frac{1+\sigma}{2}}$. It is known,
see e.g. [6], that

$K(x,t)=t^{-\frac{N-1}{1+\sigma}}P(t^{-\frac{1}{1+\sigma}}x),$

for some radially symmetric function $P\in C^{\infty}(\mathbb{R}^{N-1})$, with

$\qquad|D^{k}P(y)|\leq\frac{C(k,N,\sigma)}{1+|y|^{N+\sigma+k}}.$

From this, we can apply [17, Lemma 2.1] to get

$|(-\Delta)^{\frac{1+\sigma}{2}}P(z)|\leq\frac{C}{1+|z|^{N+\sigma}}.$ (3.7)

We have, using that
$\int_{\mathbb{R}^{N-1}}(-\Delta)^{\frac{1+\sigma}{2}}P(z)\,dz=0$ and a change
of variables,

$\displaystyle H_{t}(x)$
$\displaystyle:=t(-\Delta)^{\frac{1+\sigma}{2}}(e^{t(-\Delta)^{\frac{1+\sigma}{2}}}f)(x)=t\int_{\mathbb{R}^{N-1}}(-\Delta)^{\frac{1+\sigma}{2}}K(x-y,t)f(y)\,dy$
$\displaystyle=t^{-\frac{N-1}{1+\sigma}}\int_{\mathbb{R}^{N-1}}((-\Delta)^{\frac{1+\sigma}{2}}P)(t^{-\frac{1}{1+\sigma}}(x-y))f(y)\,dy$
$\displaystyle=\int_{\mathbb{R}^{N-1}}(-\Delta)^{\frac{1+\sigma}{2}}P(z)f(x-t^{\frac{1}{1+\sigma}}z)\,dz=\int_{\mathbb{R}^{N-1}}(-\Delta)^{\frac{1+\sigma}{2}}P(z)[f(x-t^{\frac{1}{1+\sigma}}z)-f(x)]\,dz.$

Now using that $(-\Delta)^{\frac{1+\sigma}{2}}P$ is even, we conclude that

$H_{t}(x)=\frac{1}{2}\int_{\mathbb{R}^{N-1}}(-\Delta)^{\frac{1+\sigma}{2}}P(z)[f(x-t^{\frac{1}{1+\sigma}}z)+f(x+t^{\frac{1}{1+\sigma}}z)-2f(x)]\,dz.$
(3.8)

Suppose that $\gamma-\beta\leq 1$. We then deduce, from that

$\displaystyle|H_{t}(x)|\leq
t^{\frac{\gamma-\beta}{1+\sigma}}C[f]_{C^{\gamma-\beta}}\int_{\mathbb{R}^{N-1}}|z|^{\gamma-\beta}|(-\Delta)^{\frac{1+\sigma}{2}}P(z)|\,dz.$
(3.9)

If also $\gamma-\beta>1$ then since

$\displaystyle|[f(x-t^{\frac{1}{1+\sigma}}z)$
$\displaystyle+f(x+t^{\frac{1}{1+\sigma}}z)-2f(x)]|\leq 2[\nabla
f]_{C^{\gamma-\beta-1}}t^{\frac{1}{1+\sigma}}|z|\min(1,(t^{\frac{1}{1+\sigma}}|z|)^{\gamma-\beta-1}),$

we still have (3.9). As a consequence

$\displaystyle\|H_{t}\|_{L^{\infty}}\leq
Ct^{\rho}\|f\|_{C^{\gamma-\sigma}}\qquad\textrm{for all
$\rho\in(0,\min(1,\frac{1}{1+\sigma})).$}$ (3.10)

To proceed, we start with the case $\gamma-\sigma\leq 1$. Then for
$x\not=x^{\prime}\in\mathbb{R}^{N-1}$, by (3.7) and the fact that
$0<\gamma-\beta=\rho(1+\sigma)<1+\sigma$, we have

$\displaystyle|H_{t}(x)-H_{t}(x^{\prime})|\leq
C[f]_{C^{\gamma-\sigma}}\int_{\mathbb{R}^{N-1}}|(-\Delta)^{\frac{1+\sigma}{2}}P(z)|\min(|x-x^{\prime}|,t^{\frac{1}{1+\sigma}}|z|)^{\gamma-\sigma}\,dz$
$\displaystyle\leq
C[f]_{C^{\gamma-\sigma}}t^{\frac{\gamma-\sigma}{1+\sigma}}\int_{|z|\leq
t^{-\frac{1}{1+\sigma}}|x-x^{\prime}|}|z|^{\gamma-\sigma}|(-\Delta)^{\frac{1+\sigma}{2}}P(z)|\,dz$
$\displaystyle+C[f]_{C^{\gamma-\sigma}}|x-x^{\prime}|^{\gamma-\sigma}\int_{|z|\geq
t^{\frac{-1}{1+\sigma}}|x-x^{\prime}|}|(-\Delta)^{\frac{1+\sigma}{2}}P(z)|\,dz$
$\displaystyle\leq
C[f]_{C^{\gamma-\sigma}}t^{\frac{\gamma-\sigma}{1+\sigma}}t^{-\frac{\beta-\sigma}{1+\sigma}}|x-x^{\prime}|^{\beta-\sigma}\int_{\mathbb{R}^{N-1}}|z|^{\gamma-\beta}|(-\Delta)^{\frac{1+\sigma}{2}}P(z)|\,dz$
$\displaystyle+C[f]_{C^{\gamma-\sigma}}|x-x^{\prime}|^{\gamma-\sigma}\int_{|z|\geq
t^{\frac{-1}{1+\sigma}}|x-x^{\prime}|}\frac{1}{1+|z|^{N-1+(\gamma-\beta)}}\,dz$
$\displaystyle\leq
C[f]_{C^{\gamma-\sigma}}\left(t^{\frac{\gamma-\beta}{1+\sigma}}|x-x^{\prime}|^{\beta-\sigma}+|x-x^{\prime}|^{\gamma-\sigma}(t^{\frac{-1}{1+\alpha}}|x-x^{\prime}|)^{-(\gamma-\beta)}\right)$
$\displaystyle\leq
C[f]_{C^{\gamma-\sigma}}t^{\frac{\gamma-\beta}{1+\sigma}}|x-x^{\prime}|^{\beta-\sigma}.$

This clearly implies that

$\sup_{t\in(0,1)}t^{-\rho}[H_{t}]_{C^{\beta-\sigma}}\leq
C[f]_{C^{\gamma-\sigma}(\mathbb{R}^{N-1})}.$

As a consequence, using also (3.10), we have

$\sup_{t\in(0,1)}t^{-\rho}\|H_{t}\|_{C^{\beta-\sigma}}\leq
C\|f\|_{C^{\gamma-\sigma}}\qquad\textrm{ for $\gamma-\sigma\leq 1.$}$ (3.11)

We now assume that $\gamma-\sigma>1$ and we observe that from the upper bounds
of $\rho$ and $\beta$ we have $\gamma-\sigma\leq 2$. Therefore (recalling
(3.8)) using that

$\displaystyle|[f(x-t^{\frac{1}{1+\sigma}}z)$
$\displaystyle+f(x+t^{\frac{1}{1+\sigma}}z)-2f(x)]-[f(x^{\prime}-t^{\frac{1}{1+\sigma}}z)+f(x^{\prime}+t^{\frac{1}{1+\sigma}}z)-2f(x^{\prime})]|$
$\displaystyle\leq 2[\nabla
f]_{C^{\gamma-\sigma-1}}t^{\frac{1}{1+\sigma}}|z|\min(|x-x^{\prime}|,(t^{\frac{1}{1+\sigma}}|z|))^{\gamma-\sigma-1}$

and the same argument as above, we get

$\sup_{t\in(0,1)}t^{-\rho}[H_{t}]_{C^{\beta-\sigma}}\leq C[\nabla
f]_{C^{\gamma-\sigma-1}}.$

From this, (3.10) and (3.11), we conclude that for all
$\rho\in(0,\min(1,\frac{1}{1+\sigma}))$,

$\|f\|_{\mathcal{D}_{(-\Delta)^{\frac{1+\sigma}{2}}}(\rho,\infty)}=\|f\|_{C^{\beta-\sigma}}+\sup_{t\in(0,1)}t^{-\rho}\|H_{t}\|_{C^{\beta-\sigma}}\leq
C\|f\|_{C^{\gamma-\sigma}}.$

Thanks to (3.5) and (3.9) we obtain

$\|f\|_{\mathcal{D}_{B_{0}}({\rho},\infty)}\leq C\|f\|_{C^{\gamma-\sigma}}.$

Therefore ${\mathcal{C}}^{\beta-\sigma}_{0}(\mathbb{R}^{N-1})\cap
C^{\gamma_{\rho}-\sigma}({\mathbb{R}^{N-1}})\subset\mathcal{D}_{B_{0}}(\rho,\infty)$
and (3.4) holds.
Next, by [22, Proposition 2.2.2] and [22, Corollary 1.2.18] we have that
$\mathcal{D}_{B_{0}}(\rho,\infty)$ is continuously embedded in
$C^{\gamma_{\rho}-\sigma}(\mathbb{R}^{N-1})$, provided
$\gamma_{\rho}-\sigma\not\in\mathbb{N}$. Since, by definition
$\mathcal{D}_{B_{0}}(\rho,\infty)\subset{\mathcal{C}}^{\beta-\sigma}_{0}(\mathbb{R}^{N-1})$,
we get the desired result.

∎

######  Remark 3.4.

We point out that Proposition 3.3 still holds when $\beta\in\\{0,1\\}$ and
$\gamma_{\rho}-\sigma=1$, provided that, for $k\in\\{1,2\\}$, the spaces
$C^{k}(\mathbb{R}^{N-1})$ and ${\mathcal{C}}^{k}_{0}(\mathbb{R}^{N-1})$ are,
respectively, replaced with the Hölder-Zygmund space
${\mathcal{C}}^{k}(\mathbb{R}^{N-1})$ and the space
${\mathcal{C}}^{k}_{0}(\mathbb{R}^{N-1})=\overline{C^{\infty}_{c}(\mathbb{R}^{N-1})}^{\|\cdot\|_{{\mathcal{C}}^{k}}}$.
Recall that the space ${\mathcal{C}}^{1}(\mathbb{R}^{N-1})$ is defined by

$\displaystyle{\mathcal{C}}^{1}(\mathbb{R}^{N-1}):=\bigg{\\{}u\in
C_{b}(\mathbb{R}^{N-1}):\,[u]_{{\mathcal{C}}^{1}}:=\sup_{\stackrel{{\scriptstyle
x,y\in\mathbb{R}^{N-1}}}{{x\neq
y}}}\frac{|u(x)-2u(\frac{x+y}{2})+u(y)|}{|x-y|}<\infty\bigg{\\}}$ (3.12)

and ${\mathcal{C}}^{2}(\mathbb{R}^{N-1})$ is given by the set of
$u\in{\mathcal{C}}^{1}(\mathbb{R}^{N-1})$ such that
$\partial_{i}u\in{\mathcal{C}}^{1}(\mathbb{R}^{N-1})$ for $i=1,\dots,N-1$.

###  3.2. A class of nonlocal operators generating strongly continous analytic
semigroups

In this section, we consider a class of nonlocal operators which we prove to
generate strongly continuous analytic semigroups.
For fixed $\gamma\in(\alpha,1)$, we consider linear nonlocal operator
$\mathbb{L}_{K}:C^{1+\gamma}(\mathbb{R}^{N-1})\to
C^{\gamma-\alpha}(\mathbb{R}^{N-1})$ given by

$\mathbb{L}_{K}u(x)=P.V.\int_{\mathbb{R}^{N-1}}\frac{u(x)-u(y)}{|x-y|^{N+\alpha}}K(x,y)\,dy,$
(3.13)

where
$K:\big{(}\mathbb{R}^{N-1}\times\mathbb{R}^{N-1}\big{)}\setminus\\{(x,x):x\in\mathbb{R}^{N-1}\\}\rightarrow\mathbb{R}$
is a measurable function satisfying the following assumptions.

######  Assumptions 3.5.

  1. (i)

$K(x,y)=K(y,x)$ for all $x,y\in\mathbb{R}^{N-1}$, $x\neq y$.

  2. (ii)

$\frac{1}{\kappa}\leq K(x,y)\leq\kappa$, for some $\kappa>1$.

  3. (iii)

there exists $A_{K}\in C^{\gamma}(\mathbb{R}^{N-1}\times[0,\infty)\times
S^{N-2})$, with $A_{K}(x,0,\theta)=A_{K}(x,0,-\theta)$ such that
$A_{K}(x,r,\theta)=K(x,x-r\theta)$ for all $r>0$ and
$\|A_{K}\|_{C^{\gamma}(\mathbb{R}^{N-1}\times[0,\infty)\times
S^{N-2})}\leq\kappa$.

In order to apply the estimates in Lemma 2.2, it will be useful to decompose
$\mathbb{L}_{K}$ in two parts, writing

$\mathbb{L}_{K}u(x)=\int_{S^{N-2}}\int_{0}^{\infty}\frac{\delta_{e}u(x,r,\theta)}{r^{2+\alpha}}{A}_{K}^{e}(x,r,\theta)drd\theta+\int_{S^{N-2}}\int_{0}^{\infty}\frac{\delta_{o}u(x,r,\theta)}{r^{2+\alpha}}{A}_{K}^{o}(x,r,\theta)drd\theta,$
(3.14)

for $u\in C^{1+\gamma}(\mathbb{R}^{N-1})$ with

${A}_{K}^{e}(x,r,\theta)=A_{K}(x,r,\theta)+A_{K}(x,r,-\theta),\qquad{A}_{K}^{o}(x,r,\theta)=A_{K}(x,r,\theta)-A_{K}(x,r,-\theta)$

and $\delta_{e}u$, $\delta_{o}u$ defined as in (2.4). We now define the set

$\mathcal{O}_{\nu}^{\gamma}:=\\{u\in
C^{1+\gamma}_{loc}(\mathbb{R}^{N-1})\,:\,\|\nabla u\|_{C^{\gamma}}\leq\nu\\},$
(3.15)

for $\nu>0$, and we state the following estimates.

######  Lemma 3.6.

  1. (i)

Let $\gamma>\alpha$. Then there exist $C,C^{\prime}>0$ depending only on
$N,\alpha,\gamma,\kappa$ such that

$\|\mathbb{L}_{K}u\|_{C^{\gamma-\alpha}}\leq C^{\prime}\|\nabla
u\|_{C^{\gamma}}\qquad\text{for all $u\in\mathcal{O}_{\nu}^{\gamma}$}$ (3.16)

and

$\|\nabla u\|_{C^{\gamma}}\leq
C(\|\mathbb{L}_{K}u\|_{C^{\gamma-\alpha}}+\|\nabla
u\|_{L^{\infty}})\qquad\text{for all $u\in\mathcal{O}_{\nu}^{\gamma}$.}$
(3.17)

  2. (ii)

If $\beta\in(\alpha,\gamma)$, then we have
$\mathbb{L}_{K}u\in{\mathcal{C}}^{\beta-\alpha}_{0}(\mathbb{R}^{N-1})$ for all
$u\in{\mathcal{C}}^{1+\beta}_{0}(\mathbb{R}^{N-1})\cap\mathcal{O}_{\nu}^{\gamma}$.

###### Proof.

(i) It is clear from Lemma 2.2 and (3.14) that (3.16) holds.
Now we let $f(x)=\mathbb{L}_{K}u(x)$. Since $K(\cdot+x_{0},\cdot+x_{0})$
satisfies also Assumptions 3.5 for all $x_{0}\in\mathbb{R}^{N-1}$, by [15,
Theorem 1.3$(ii)$, Theorem 1.4$(iii)$], we get

$\displaystyle\|\nabla u\|_{C^{\gamma}(B_{1}(x_{0}))}$
$\displaystyle=\|\nabla(u-u(x_{0}))\|_{C^{\gamma}(B_{1}(x_{0}))}$
$\displaystyle\leq
C\left(\|f\|_{C^{\gamma-\alpha}(B_{2}(x_{0}))}+\|u-u(x_{0})\|_{L^{\infty}(B_{2}(x_{0}))}+\int_{\mathbb{R}^{N-1}}\frac{|u(x)-u(x_{0})|}{1+|x-x_{0}|^{N+\alpha}}\,dy\right)$
$\displaystyle\leq C\left(\|f\|_{C^{\gamma-\alpha}}+\|\nabla
u\|_{L^{\infty}}+\int_{|x-x_{0}|\geq
2}\frac{|u(x)-u(x_{0})|}{|x-x_{0}|^{N+\alpha}}\,dy\right)$ $\displaystyle\leq
C\left(\|f\|_{C^{\gamma-\alpha}}+\|\nabla u\|_{L^{\infty}}+\|\nabla
u\|_{L^{\infty}}\int_{|x-x_{0}|\geq
2}\frac{1}{|x-x_{0}|^{N-1+\alpha}}\,dy\right).$

From this, we then get

$\|\nabla u\|_{C^{\gamma}(B_{1}(x_{0}))}\leq
C\left(\|f\|_{C^{\gamma-\alpha}}+\|\nabla u\|_{L^{\infty}}\right),$

where $C$ may change value from one line to an other. Since $x_{0}$ is
arbitrary, (3.17) follows.
(ii) By Lemma 2.2 and (3.14) if
$u\in{\mathcal{C}}^{1+\beta}_{0}(\mathbb{R}^{N-1})$, then
$\mathbb{L}_{K}u\in{\mathcal{C}}^{\beta-\alpha}_{0}(\mathbb{R}^{N-1})$. ∎

Combining Lemma 3.6 and a result from [2], we now exhibit a useful class of
operators generating strongly continuous analytic semigroups in Hölder spaces.

######  Proposition 3.7.

Let $\gamma\in(\beta,1)$ and
$b\in{{\mathcal{C}}^{\beta-\alpha}_{0}}(\mathbb{R}^{N-1})$ be such that

$\frac{1}{\kappa}\leq|b(x)|\leq\kappa\qquad\textrm{ for all
$x\in\mathbb{R}^{N-1}$}.$

Let
$L:{{\mathcal{C}}^{1+\beta}_{0}}(\mathbb{R}^{N-1})\to{{\mathcal{C}}^{\beta-\alpha}_{0}}(\mathbb{R}^{N-1})$
be a bounded linear operator satisfying for all $\varepsilon>0$, there exists
$c_{\varepsilon}$ such that

$\displaystyle\|Lu\|_{C^{\beta-\alpha}}\leq\varepsilon\|\mathbb{L}_{K}u\|_{C^{\beta-\alpha}}+c_{\varepsilon}\|u\|_{C^{\beta-\alpha}}\;\;\;\mbox{
for all }u\in{\mathcal{C}}^{1+\beta}_{0}(\mathbb{R}^{N-1}).$ (3.18)

Suppose that for all $k\in\mathbb{N}$,

$\sup_{\theta\in
S^{N-2},}\|\nabla^{k}_{\theta}A_{K}^{e}(\cdot,0,\theta)\|_{C^{\gamma}}\leq
C_{k}.$ (3.19)

Then, the operator

$b(x)\mathbb{L}_{K}+L:{\mathcal{C}}^{1+\beta}_{0}(\mathbb{R}^{N-1})\to{\mathcal{C}}^{\beta-\alpha}_{0}(\mathbb{R}^{N-1}),$
(3.20)

with domain ${{\mathcal{C}}^{1+\beta}_{0}}(\mathbb{R}^{N-1})$, generates a
strongly continuous analytic semigroup on
${{\mathcal{C}}^{\beta-\alpha}_{0}}(\mathbb{R}^{N-1})$.

###### Proof.

Recalling (3.14), we write

$b(x)\mathbb{L}_{K}:=L_{1}+L_{2},$ (3.21)

where

$\displaystyle L_{1}u(x)$
$\displaystyle=b(x)\Bigl{(}P.V.\displaystyle\int_{\mathbb{R}^{N-1}}\frac{(u(x)-u(x+y))}{|y|^{N+\alpha}}A_{K}^{e}(x,0,y/|y|)\;dy\Bigr{)}$
$\displaystyle=\frac{b(x)}{2}\int_{\mathbb{R}^{N-1}}\frac{(2u(x)-u(x+y)-u(x-y))}{|y|^{N+\alpha}}A_{K}^{e}(x,0,y/|y|)\;dy$

and

$L_{2}u(x)=\displaystyle\int_{\mathbb{R}^{N-1}}\frac{(u(x)-u(x+y))}{|y|^{N+\alpha}}b(x)[A_{K}^{o}(x,|y|,y/|y|)+A_{K}^{e}(x,|y|,y/|y|)-A_{K}^{e}(x,0,y/|y|)]\;dy.$

By Lemma 2.2, the operator
$L_{1}:{{\mathcal{C}}^{1+\beta}_{0}}(\mathbb{R}^{N-1})\to{{\mathcal{C}}^{\beta-\alpha}_{0}}(\mathbb{R}^{N-1})$
is bounded.

Letting $k(x,y):=b(x)A_{K}^{e}(x,0,y/|y|)|y|^{-N-\alpha}$ and $\chi\in
C^{\infty}_{c}(\mathbb{R}^{N-1})$ be nonnegative, radially symmetric and
satisfy $\chi=1$ for $|y|\leq 2$. One can split $k$ as $k=k_{1}+k_{2}$, where
$k_{1}(x,y)=k(x,y)\chi(y)$ and $k_{2}(x,y)=k(x,y)(1-\chi(y))$. Then, it is
easy to see that $k_{1}=0$ for $|y|\geq 2$ and $k_{2}=0$ for $|y|\leq 1$. In
addition, using also (3.19), for all $n\in\mathbb{N}^{N-1}$ with $|n|\leq N$,
there exists $C>0,c>0$ only depending on $N,\alpha,\gamma,n$ and $\kappa$ such
that

$\left\\{\begin{aligned} &\|\partial_{y}^{n}k_{1}(\cdot,y)\|_{C^{\gamma}}\leq
C|y|^{-N-\alpha-|n|},\qquad 0<|y|\leq 2,\\\ &k_{1}(x,y)\leq
C|y|^{-N-\alpha},\qquad\;0<|y|\leq 1,\;x\in\mathbb{R}^{N-1},\\\
&\|k_{2}(\cdot,y)\|_{C^{\gamma}}\leq C|y|^{-N-\alpha^{\prime}},\qquad
0<|y|\leq 1,\;\;\mbox{ for }\;\;0\leq\alpha^{\prime}<\alpha<1\\\
&\displaystyle\int_{|y|\geq 1}\|k_{2}(\cdot,y)\|_{C^{\gamma}}\;dy<\infty,\\\
&\lim\limits_{|y|\rightarrow\infty}\|k_{2}(\cdot,y)\|_{C^{\gamma}}=0.\end{aligned}\right.$
(3.22)

In view of this, we can apply [2, Corollary 2.17], to deduce that, for
$\beta\in(\alpha,\gamma)$, the operator
$L_{1}:{{\mathcal{C}}^{1+\beta}_{0}}(\mathbb{R}^{N-1})\to{{\mathcal{C}}^{\beta-\alpha}_{0}}(\mathbb{R}^{N-1})$
generates a strongly continuous analytic semigroup.
By Lemma 2.2 and (2.2), for all $\varepsilon>0$, there exists
$C_{2}(\varepsilon)>0$ such that

$\displaystyle\|L_{2}u\|_{C^{\beta-\alpha}}\leq C\|\nabla u\|_{C_{b}}\leq
C\left(\varepsilon\|u\|_{C^{1+\beta}}+C_{2}(\varepsilon)\|u\|_{L^{\infty}}\right).$

Combining this information with Lemma 3.6 (i), we get

$\displaystyle\|L_{2}u\|_{C^{\beta-\alpha}}\leq
C\left(\varepsilon\|(L_{1}+L_{2})u\|_{C^{\beta-\alpha}}+C_{2}(\varepsilon)\|u\|_{L^{\infty}}\right).$

with a possibly different constant $C$ not depending on $\varepsilon$. The
above two estimates give

$\displaystyle\|L_{2}u\|_{C^{\beta-\alpha}}\leq
C\left(\varepsilon\|L_{1}u\|_{C^{\beta-\alpha}}+C_{2}(\varepsilon)\|u\|_{L^{\infty}}\right).$

From this and (3.18), we have

$\displaystyle\|(L_{2}+L)u\|_{C^{\beta-\alpha}}\leq\varepsilon\|L_{1}u\|_{C^{\beta-\alpha}}+C(\varepsilon)\|u\|_{C^{\beta-\alpha}}\;\;\;\mbox{
for all }u\in{{\mathcal{C}}^{1+\beta}_{0}(\mathbb{R}^{N-1})}.$

Now applying [24, Theorem 2.1], we deduce that
$b(x)\mathbb{L}_{K}+L=L_{1}+L_{2}+L$ is a generator of a strongly continuous
analytic semigroup. ∎

We shall now derive the following existence and uniqueness result from Theorem
3.1, Proposition 3.3 and Proposition 3.7.

######  Theorem 3.8.

Let $0<\alpha<\beta<1$ and
$B_{0}:=b(x)\mathbb{L}_{K}+L:{\mathcal{C}}^{1+\beta}_{0}(\mathbb{R}^{N-1})\to{\mathcal{C}}^{\beta-\alpha}_{0}(\mathbb{R}^{N-1})$
be given by (3.20). Let $T>0$ and $\rho\in(0,\frac{1}{1+\alpha})$. Let
$u_{0}\in{\mathcal{C}}^{1+\beta}_{0}(\mathbb{R}^{N-1})$ and $f\in
C^{\rho}([0,T],{{\mathcal{C}}^{\beta-\alpha}_{0}(\mathbb{R}^{N-1})})$ be such
that $f(0)-B_{0}u_{0}\in{C^{\gamma_{\rho}-\alpha}(\mathbb{R}^{N-1})}$ with
$\gamma_{\rho}=\beta+\rho(1+\alpha)$. Then the problem

$\left\\{\begin{array}[]{rll}u^{\prime}(t)+B_{0}u(t)&=&f(t),\hskip
28.45274ptt\in(0,T]\\\ u(0)&=&u_{0}\end{array}\right.$ (3.23)

admits a unique solution $u\in C^{\rho}([0,T],{\mathcal{C}}^{1+\beta}_{0})\cap
C^{1+\rho}([0,T],{\mathcal{C}}^{\beta-\alpha}_{0})$. Moreover, there exists
$C_{T}=C(T,\rho,\kappa,N,\alpha,\gamma,\beta)>0$ such that

$\displaystyle\|u-u_{0}\|_{C^{\rho}([0,T],C^{1+\beta})}$
$\displaystyle+\|u-u_{0}\|_{C^{1+\rho}([0,T],C^{\beta-\alpha})}$
$\displaystyle\leq
C_{T}\Big{(}\|f-B_{0}u_{0}\|_{C^{\rho}([0,T],C^{\beta-\alpha})}+\|f(0)-B_{0}u_{0}\|_{C^{\gamma_{\rho}-\alpha}}\Big{)}.$

In addition, $C_{T}\leq C_{T_{0}}$ for all $T\leq T_{0}$.

###### Proof.

From Theorem 3.1 and Proposition 3.7, we get a unique solution $u\in
C^{\rho}([0,T],{\mathcal{C}}^{1+\beta}_{0})\cap
C^{1+\rho}([0,T],{\mathcal{C}}^{\beta-\alpha}_{0})$ to (3.23). Moreover,

$\displaystyle\|B_{0}(u-u_{0})\|_{C^{\rho}([0,T],C^{\beta-\alpha})}+\|u-u_{0}\|_{C^{1+\rho}([0,T],C^{\beta-\alpha})}$
$\displaystyle\leq
C_{T}\Big{(}\|f-B_{0}u_{0}\|_{C^{\rho}([0,T],C^{\beta-\alpha}))}+\|f(0)-B_{0}u_{0}\|_{\mathcal{D}_{B_{0}}(\rho,\infty)}\Big{)}.$
(3.24)

By Proposition 3.3 we have
$\|f(0)-B_{0}u_{0}\|_{\mathcal{D}_{B_{0}}(\rho,\infty)}\leq
C^{\prime}\|f(0)-B_{0}u_{0}\|_{C^{\gamma_{\rho}-\alpha}}$. On the other hand,
since $B_{0}$ is closed, we obtain

$\|B_{0}v\|_{C^{\beta-\alpha}}+\|v\|_{C^{\beta-\alpha}}\geq
C\left(\|v\|_{C^{1+\beta}}+\|v\|_{C^{\beta-\alpha}}\right),\quad\forall v\in
C^{1+\beta}(\mathbb{R}^{N-1}).$

From this and (3.2), we get the result. ∎

##  4\. Regularity property of the nonlocal mean curvature operator

By changes of variables and the fact that
${\mathcal{G}}(p)=-\displaystyle\int_{-p}^{p}\frac{d\tau}{(1+\tau^{2})^{\frac{N+\alpha}{2}}}$
is odd, for $w\in C^{1,\gamma}_{loc}(\mathbb{R}^{N-1})$, with $\gamma>\alpha$,
we have

$\displaystyle H(w)(x)$
$\displaystyle=P.V.\int_{\mathbb{R}^{N-1}}\frac{{\mathcal{G}}(p_{w}({x},{y}))}{|{x}-{y}|^{N-1+\alpha}}d{y}$
$\displaystyle=\frac{P.V.}{2}\int_{\mathbb{R}^{N-1}}\frac{{\mathcal{G}}(p_{w}({x},{x-z}))+{\mathcal{G}}(p_{w}({x},{x+z}))}{|{z}|^{N-1+\alpha}}d{z}$
$\displaystyle=\frac{P.V.}{2}\int_{\mathbb{R}^{N-1}}\frac{{\mathcal{G}}(p_{w}({x},{x-z}))-{\mathcal{G}}(-p_{w}({x},{x+z}))}{|{z}|^{N-1+\alpha}}d{z}.$

Therefore, by the fundamental theorem of calculus and the fact that
${\mathcal{G}}^{\prime}(p)=-2(1+p^{2})^{\frac{-N-\alpha}{2}}$ is even, we get

$\displaystyle
H(w)(x)=\int_{\mathbb{R}^{N-1}}\frac{2w(x)-w(x-z)-w(x+z)}{|z|^{N+\alpha}}{\mathcal{K}}_{w}(x,z)dz,$
(4.1)

where

$\displaystyle{\mathcal{K}}_{w}(x,z)=\int_{0}^{1}\bigg{(}1+\left(\tau
p_{w}({x},{x-z})-(1-\tau)p_{w}({x},{x+z})\right)^{2}\bigg{)}^{-\frac{N+\alpha}{2}}d\tau.$
(4.2)

For the remainder of this section, it will be convenient to write

$\displaystyle
H(w)(x)=\int_{S^{N-2}}\int_{0}^{\infty}\frac{\delta_{e}w(x,r,\theta)}{r^{2+\alpha}}\mathcal{A}_{w}(x,r,\theta)drd\theta,$
(4.3)

where $\delta_{e}w(x,r,\theta)=2w(x)-w(x-r\theta)-w(x+r\theta)$, for all
$(x,r,\theta)\in\mathbb{R}^{N-1}\times[0,\infty)\times S^{N-2}$ and

$\displaystyle\left.\begin{array}[]{rll}\mathcal{A}_{w}&:\mathbb{R}^{N-1}\times[0,\infty)\times
S^{N-2}\rightarrow\mathbb{R}\\\
&(x,r,\theta)\mapsto\mathcal{A}_{w}(x,r,\theta)={\mathcal{K}}_{w}(x,r\theta),\;\;\;{\mathcal{A}}_{w}(x,0,\theta)=\bigg{(}1+(\nabla
w(x)\cdot\theta)^{2}\bigg{)}^{-\frac{N+\alpha}{2}}.\end{array}\right.$ (4.6)

######  Lemma 4.1.

Let $k\in\mathbb{N}$ and $0<\gamma<1$. Then there exists
$C=C(k,N,\alpha,\gamma)>0$ such that for all $u,w_{1},\dots,w_{k}\in
C^{1+\gamma}(\mathbb{R}^{N-1})$ , we have

$\|\partial^{k}_{u}\mathcal{A}_{u}[w_{1},\dots,w_{k}]\|_{C^{\gamma}(\mathbb{R}^{N-1}\times[0,\infty)\times
S^{N-2})}\leq C\left(1+\|\nabla
u\|_{C^{\gamma}}\right)^{C}\prod_{i=1}^{k}\|\nabla w_{i}\|_{C^{\gamma}}.$

###### Proof.

This follows from the definition of ${\mathcal{A}}$. ∎

We have the following lemma.

######  Lemma 4.2.

Let $\gamma\in(\alpha,1)$. Then the map $H:C^{1+\gamma}(\mathbb{R}^{N-1})\to
C^{\gamma-\alpha}(\mathbb{R}^{N-1})$ is of class $C^{\infty}$ and for all
$k\in\mathbb{N},$ $u,w_{1},\dots,w_{k}\in C^{1+\gamma}(\mathbb{R}^{N-1}),$

$\|D^{k}H(u)[w_{1},\dots,w_{k}]\|_{C^{\gamma-\alpha}}\leq C\left(1+\|\nabla
u\|_{C^{\gamma}}\right)^{C}\prod_{i=1}^{k}\|\nabla w_{i}\|_{C^{\gamma}}.$
(4.7)

with $C>0$ only depending on $k,N,\alpha,\gamma$. Moreover
$H:{\mathcal{C}}^{1+\gamma}_{0}(\mathbb{R}^{N-1})\to{{\mathcal{C}}^{\gamma-\alpha}_{0}}(\mathbb{R}^{N-1})$
is of class $C^{\infty}$ and (4.7) holds for all
$u,w_{1},\dots,w_{k}\in{\mathcal{C}}^{1+\gamma}_{0}(\mathbb{R}^{N-1})$.

###### Proof.

For all $k\in\mathbb{N}$, $w_{0},w_{1},\dots,w_{k}\in
C^{1+\gamma}(\mathbb{R}^{N-1}),$ we define

$\mathcal{B}_{k}^{e}(u)[w_{0};w_{1},\dots,w_{k}](x)=\int_{S^{N-2}}\int_{0}^{\infty}\delta_{e}w_{0}(x,r,\theta)\partial^{k}_{u}\mathcal{A}_{u}^{e}[w_{1},\dots,w_{k}](x,r,\theta)r^{-2-\alpha}drd\theta.$

We now prove that $H$ is of class $C^{\infty}$ and for all $k\in\mathbb{N}$,

$\displaystyle D^{k}H(u)[w_{1}\dots,w_{k}]$
$\displaystyle=\mathcal{B}^{e}_{k}(u)[u;w_{1},\dots,w_{k}]+\sum_{j=1}^{k}\mathcal{B}^{e}_{k-1}(u)[w_{j};w_{1},\dots,w_{j-1},w_{j+1},\dots,w_{k}].$
(4.8)

For this, it is enough to prove that
$u\mapsto\mathcal{B}^{e}_{k}(u)[u;w_{1},\dots,w_{k}]$ is differentiable. By
the linearity of the map $u\mapsto\delta_{e}u$ it is also enough to prove the
differentiability of $u\mapsto\mathcal{B}^{e}_{k}(u)[w_{0};w_{1},\dots,w_{k}]$
with

$\displaystyle\|D_{u}\mathcal{B}^{e}_{k}(u)[w_{0};w_{1},\dots,w_{k}]\|_{C^{\gamma-\alpha}}\leq
C\left(1+\|\nabla u\|_{C^{\gamma}}\right)^{C}\prod_{i=0}^{k}\|\nabla
w_{i}\|_{C^{\gamma}}.$ (4.9)

Let $v\in C^{1+\gamma}(\mathbb{R}^{N-1})$, with $\|\nabla v\|_{C^{\gamma}}\leq
1$. We have

$\displaystyle\mathcal{B}^{e}_{k}(u+v)[w_{0};w_{1},\dots,w_{k}](x)-\mathcal{B}^{e}_{k}(u)[w_{0};w_{1},\dots,w_{k}](x)-$
$\displaystyle\int_{S^{N-2}}\int_{0}^{\infty}\delta_{e}w_{0}(x,r,\theta)\partial^{k+1}_{u}\mathcal{A}_{u}^{e}[w_{1},\dots,w_{k},v](x,r,\theta)r^{-2-\alpha}\,drd\theta$
$\displaystyle=\int_{S^{N-2}}\int_{0}^{\infty}r^{-2-\alpha}\delta_{e}w_{0}(x,r,\theta)$
$\displaystyle\times\int_{0}^{1}\left(\partial^{k+1}_{u}\mathcal{A}_{u+\rho
v}^{e}[w_{1},\dots,w_{k},v](x,r,\theta)-\partial^{k+1}_{u}\mathcal{A}_{u}^{e}[w_{1},\dots,w_{k},v](x,r,\theta)\right)d\rho\,drd\theta$
$\displaystyle=\int_{S^{N-2}}\int_{0}^{\infty}r^{-2-\alpha}\delta_{e}w_{0}(x,r,\theta)\int_{0}^{1}\rho\int_{0}^{1}\partial^{k+2}_{u}\mathcal{A}_{u+\rho\overline{\rho}v}^{e}[w_{1},\dots,w_{k},v,v](x,r,\theta)d\overline{\rho}d\rho\,drd\theta$

Now by Lemma 2.2 and Lemma 4.1, we get

$\displaystyle\Big{\|}\mathcal{B}^{e}_{k}(u+v)[w_{0},w_{1},\dots,w_{k}]-\mathcal{B}^{e}_{k}(u)[w_{0},w_{1},\dots,w_{k}]-$
$\displaystyle\int_{S^{N-2}}\int_{0}^{\infty}r^{-2-\alpha}\delta_{e}w_{0}(\cdot,r,\theta)\partial^{k+1}_{u}\mathcal{A}_{u}^{e}[w_{1},\dots,w_{k},v](\cdot,r,\theta)\,drd\theta\Big{\|}_{C^{\gamma-\alpha}}$
$\displaystyle\leq C\left(1+\|\nabla
u\|_{C^{\gamma-\alpha}}\right)^{C}\prod_{i=0}^{k}\|\nabla
w_{i}\|_{C^{\gamma}}\|\nabla v\|_{C^{\gamma-\alpha}}^{2}.$

From this, we deduce that
$u\mapsto\mathcal{B}^{e}_{k}(u)[w_{0};w_{1},\dots,w_{k}]$ is differentiable
and (4.9) holds.
Now the fact that $H:C^{1+\gamma}(\mathbb{R}^{N-1})\to
C^{\gamma-\alpha}(\mathbb{R}^{N-1})$ is of class $C^{\infty}$, follows easily
by induction, thanks to (4.8) and the estimates on $\mathcal{B}^{e}_{k}$.
Moreover the estimate on the derivative $H$ is an immediate consequence of
those of $\mathcal{B}^{e}_{k}$.

In view of (4.8), Lemma 2.2 and Lemma 4.1, the fact that
$H:{\mathcal{C}}^{1+\gamma}_{0}(\mathbb{R}^{N-1})\to{\mathcal{C}}^{\gamma-\alpha}_{0}(\mathbb{R}^{N-1})$
is of class $C^{\infty}$ follows similarly as above. In fact one can simply
replace in the above argument $C^{1+\gamma}(\mathbb{R}^{N-1})$ with
${\mathcal{C}}^{1+\gamma}_{0}(\mathbb{R}^{N-1})$ and
$C^{\gamma-\alpha}(\mathbb{R}^{N-1})$ with
${{\mathcal{C}}^{\gamma-\alpha}_{0}}(\mathbb{R}^{N-1})$. ∎

We compute next the explicit expression of $DH$.

######  Lemma 4.3.

Let $u\in C^{1+\gamma}_{loc}(\mathbb{R}^{N-1})$, for some
$\gamma\in(\alpha,1)$, with $\nabla u\in C^{\gamma}(\mathbb{R}^{N-1})$. Then
for all $w\in C^{1+\gamma}(\mathbb{R}^{N-1})$

$DH(u)[w](x)=-P.V.\int_{\mathbb{R}^{N-1}}\frac{w(x)-w(y)}{|x-y|^{N+\alpha}}{\mathcal{G}}^{\prime}(p_{u}(x,y))\,dy$
(4.10)

and

$\|DH(u)[w]\|_{C^{\gamma-\alpha}}\leq C(1+\|\nabla u\|_{C^{\gamma}})\|\nabla
w\|_{C^{\gamma}},$

where ${\mathcal{G}}^{\prime}(p)=-2(1+p^{2})^{-\frac{N+\alpha}{2}}.$ Here
$C>0$, depends only on $N,\alpha,\beta$ and $\gamma$. If moreover
$\gamma>\beta$ and $w\in{\mathcal{C}}^{1+\beta}_{0}(\mathbb{R}^{N-1})$, then
$DH(u)[w]\in{{\mathcal{C}}^{\beta-\alpha}_{0}}(\mathbb{R}^{N-1})$.

###### Proof.

We consider the linear operator $M$ defined as

$Mw(x):=-P.V.\int_{\mathbb{R}^{N-1}}\frac{w(x)-w(y)}{|x-y|^{N+\alpha}}{\mathcal{G}}^{\prime}(p_{u}(x,y))\,dy.$

We then have

$\displaystyle H(u+w)(x)-H(u)(x)-Mw(x)$
$\displaystyle=P.V.\int_{\mathbb{R}^{N-1}}\int_{0}^{1}\frac{[{\mathcal{G}}^{\prime}(p_{u+\tau
w}(x,y))-{\mathcal{G}}^{\prime}(p_{u}(x,y))]p_{w}(x,y)}{|x-y|^{N-1+\alpha}}d\tau\,dx.$

We then get

$\displaystyle\Gamma(x):=H(u+w)(x)-H(u)(x)-Mw(x)=P.V.\int_{\mathbb{R}^{N-1}}\frac{(p_{w}(x,x-y))^{2}\mu(x,|y|,y/|y|)}{|y|^{N-1+\alpha}}\,dx,$
(4.11)

where

$\mu(x,r,\theta):=2\int_{0}^{1}\tau\int_{0}^{1}{\mathcal{G}}^{\prime\prime}(p_{u+\tau\varrho
w}(x,x-r\theta))d\tau d\varrho$

and

$\mu(x,0,\theta):=2\int_{0}^{1}\tau\int_{0}^{1}{\mathcal{G}}^{\prime\prime}(\nabla
u(x)\cdot\theta+\tau\varrho\nabla w(x)\cdot\theta)d\tau d\varrho.$

Using polar coordinates and recalling (2.4), we can write

$\displaystyle\Gamma(x):=P.V.\int_{S^{N-2}}\int_{0}^{\infty}r^{-2-\alpha}(w(x)-w(x-r\theta))\int_{0}^{1}\nabla
w(x-r\tau\theta)\cdot\theta\,d\tau\mu(x,r,\theta)\,drd\theta$
$\displaystyle=\int_{S^{N-2}}\int_{0}^{\infty}r^{-2-\alpha}\delta_{o}w(x,r,\theta)\int_{0}^{1}(\nabla
w(x-r\tau\theta)-\nabla w(x))\cdot\theta\,d\tau\mu(x,r,\theta)\,drd\theta$
$\displaystyle+\int_{S^{N-2}}\int_{0}^{\infty}r^{-2-\alpha}\delta_{o}w(x,r,\theta)\nabla
w(x)\cdot\theta(\mu(x,r,\theta)-\mu(x,0,\theta))\,drd\theta$
$\displaystyle+\int_{S^{N-2}}\int_{0}^{\infty}r^{-2-\alpha}\delta_{e}w(x,r,\theta)\nabla
w(x)\cdot\theta\mu(x,0,\theta)\,drd\theta,$

where we used that $\mu(x,0,-\theta)=-\mu(x,0,\theta)$, from the oddness of
${\mathcal{G}}^{\prime\prime}$. Since

$\|\mu\|_{C^{\gamma}(\mathbb{R}^{N-1}\times[0,\infty)\times S^{N-2})}\leq
C(1+\|\nabla w\|_{C^{\gamma}}+\|\nabla u\|_{C^{\gamma}}),$

we can thus apply Lemma 2.2$(i)$ to deduce that

$\|\Gamma\|_{C^{\gamma-\alpha}}\leq C(1+\|\nabla w\|_{C^{\gamma}}+\|\nabla
u\|_{C^{\gamma}})\|\nabla w\|_{C^{\gamma}}^{2}.$

Recalling (4.11), this completes the proof of the first statement of the
lemma. The second statement follows from Lemma 2.2$(iii)$. ∎

We deduce the following important result for the weighted fractional mean
curvature operator ${\mathcal{H}}$ defined in (1.16).

######  Corollary 4.4.

The map
${\mathcal{H}}:{\mathcal{C}}^{1+\beta}_{0}(\mathbb{R}^{N-1})\to{\mathcal{C}}^{\beta-\alpha}_{0}(\mathbb{R}^{N-1})$
is of class $C^{\infty}$. Moreover, letting $u_{0}\in
C^{1+\gamma}_{loc}(\mathbb{R}^{N-1})$ for some $\gamma\in(\beta,1)$, with
$\nabla u_{0}\in C^{\gamma}(\mathbb{R}^{N-1})$, then the map

$F:{\mathcal{C}}^{1+\beta}_{0}(\mathbb{R}^{N-1})\to{\mathcal{C}}^{\beta-\alpha}_{0}(\mathbb{R}^{N-1}),\qquad
F(u)=D{\mathcal{H}}(u_{0})[u]-{\mathcal{H}}(u)$

is of class $C^{\infty}$ and for all $k\in\mathbb{N}$,

$\|D^{k}F(u)\|_{C^{\beta-\alpha}}\leq C\left(1+\|\nabla
u\|_{C^{\beta}}+\|\nabla u_{0}\|_{C^{\beta}}\right)^{C},$

with $C=C(k,N,\alpha,\beta)>1$.

###### Proof.

Recall that ${\mathcal{H}}(u)=Q(u)H(u)$ with $Q(u)=\sqrt{1+|\nabla u|^{2}}$
and we observe that $Q\in
C^{\infty}({\mathcal{C}}^{1+\beta}_{0}(\mathbb{R}^{N-1}),{\mathcal{C}}^{\beta-\alpha}_{0}(\mathbb{R}^{N-1}))$
and $DQ(u_{0})\in
C^{\infty}({\mathcal{C}}^{1+\beta}_{0}(\mathbb{R}^{N-1}),{\mathcal{C}}^{\beta-\alpha}_{0}(\mathbb{R}^{N-1}))$.
The conclusion then follows from Lemma 4.2 and Lemma 4.3. ∎

###  4.1. Regularity properties and linearization of the weighted fractional
mean curvature operator

We recall that

${\mathcal{H}}(u):=Q(u)H(u),\qquad Q(u):=\sqrt{1+|\nabla u(x)|^{2}}.$

Our aim is to prove that, provided $u_{0}\in
C^{1+\gamma}_{loc}(\mathbb{R}^{N-1})$ with $\gamma\in(\beta,1)$ and $\nabla
u_{0}\in C^{\gamma}_{loc}(\mathbb{R}^{N-1})$, we have that
$D{\mathcal{H}}(u_{0}):{\mathcal{C}}^{1+\beta}_{0}(\mathbb{R}^{N-1})\to{\mathcal{C}}^{\beta-\alpha}_{0}(\mathbb{R}^{N-1})$
generates a strongly continuous analytic semigroup. We observe that for all
$w\in C^{1+\beta}(\mathbb{R}^{N-1})$,

$D{\mathcal{H}}(u_{0})[w]:=Q(u_{0})\widetilde{L}_{1}w+\widetilde{L}_{2}w,$
(4.12)

where

$\displaystyle\widetilde{L}_{1}w(x)$
$\displaystyle:=-P.V.\int_{\mathbb{R}^{N-1}}\frac{w(x)-w(y)}{|x-y|^{N+\alpha}}{\mathcal{G}}^{\prime}(p_{u_{0}}(x,y))\,dy\
$
$\displaystyle=P.V.\int_{\mathbb{R}^{N-1}}\frac{w(x)-w(y)}{|x-y|^{N+\alpha}}(1+(p_{u_{0}}(x,y))^{2})^{\frac{N+\alpha}{2}}\,dy$
(4.13)

and

$\widetilde{L}_{2}w(x):=H(u_{0})(x)DQ(u_{0})[w](x)=H(u_{0})(x)\frac{\nabla
u_{0}(x)\cdot\nabla w(x)}{Q(u_{0})(x)}.$

For the following, we recall the set

$\mathcal{O}_{\nu}^{\gamma}:=\\{u\in
C^{1+\gamma}_{loc}(\mathbb{R}^{N-1})\,:\,\|\nabla u\|_{C^{\gamma}}\leq\nu\\},$
(4.14)

for $\nu>0$. We have the following result.

######  Lemma 4.5.

Let $\gamma\in(\alpha,1)$. Then the nonlocal mean curvature operator $H$
satisfies the following properties.

  1. (i)

There exists $C^{\prime}=C^{\prime}(N,\alpha,\gamma)>0$ such that for all
$u_{0}\in\mathcal{O}_{\nu}^{\gamma}$,

$\|{\mathcal{H}}(u_{0})\|_{C^{\gamma-\alpha}}+\|H(u_{0})\|_{C^{\gamma-\alpha}}\leq
C^{\prime}\|\nabla u_{0}\|_{C^{\gamma}}.$ (4.15)

  2. $(ii)$

If $u\in{\mathcal{C}}^{1+\gamma}_{0}(\mathbb{R}^{N-1})$ then
${\mathcal{H}}(u),H(u)\in{\mathcal{C}}^{\gamma-\alpha}_{0}(\mathbb{R}^{N-1})$.

  3. (iii)

There exists $C=C(N,\alpha,\gamma,\nu)$ such that if
$u\in\mathcal{O}_{\nu}^{\gamma-\alpha}\cap
C^{1+\beta}_{loc}(\mathbb{R}^{N-1})\cap L^{\infty}(\mathbb{R}^{N-1})$, for
some $\beta>\alpha$, and satisfies ${\mathcal{H}}(u)\in
C^{\gamma-\alpha}(\mathbb{R}^{N-1})$, then $u\in
C^{1+\gamma}(\mathbb{R}^{N-1})$ and

$C\|\nabla
u\|_{C^{\gamma}}\leq\|{\mathcal{H}}(u)\|_{C^{\gamma-\alpha}}+\|u\|_{L^{\infty}}.$
(4.16)

###### Proof.

In view of (4.3) and since ${\mathcal{H}}(u_{0})=\sqrt{1+|\nabla
u_{0}|^{2}}H(u_{0})$, we can apply Lemma 2.2 to get $(i)$ and $(ii)$.
For $(iii)$, we let $f:={\mathcal{H}}(u)$ and recalling (4.1), we then have

$\int_{\mathbb{R}^{N-1}}\frac{2u(x)-u(x-y)-u(x+y)}{|y|^{N+\alpha}}\widetilde{\mathcal{K}}_{u}(x,y)dy=f(x)\qquad\textrm{
for all $x\in\mathbb{R}^{N-1}$,}$

where $\widetilde{\mathcal{K}}_{u}(x,y)=\sqrt{1+|\nabla
u(x)|^{2}}{\mathcal{K}}_{u}(x,y)$. Since
$u\in{\mathcal{O}}_{\nu}^{\gamma-\alpha}$, we have

$\sup_{x,y\in\mathbb{R}^{N-1}}|\widetilde{\mathcal{K}}_{u}(x+h,y)-\widetilde{\mathcal{K}}_{u}(x,y)|\leq
C(\nu,N,\alpha,\gamma)|h|^{\gamma-\alpha}\qquad\textrm{ for all
$h\in\mathbb{R}^{N-1}$}.$

In addition
$(1+4\nu^{2})^{\frac{-N-\alpha}{2}}\leq\widetilde{\mathcal{K}}_{u}(x,y)\leq
1$, for all $x,y\in\mathbb{R}^{N-1}$. Therefore, applying [4, Theorem 1.2], we
obtain

$C\|\nabla u\|_{C^{\gamma}}\leq\|f\|_{C^{\gamma-\alpha}}+\|u\|_{L^{\infty}}.$

and the proof is complete. ∎

We now have the following result.

######  Lemma 4.6.

Let $w,v\in\mathcal{O}_{\nu}^{\gamma}$, for some $\gamma\in(\alpha,1)$. Then
the linear operator

$B[w,v]:C^{1+\gamma}(\mathbb{R}^{N-1})\mapsto
C^{\gamma-\alpha}(\mathbb{R}^{N-1}),\qquad
B[w,v]u:=\int_{0}^{1}D\mathcal{H}(\varrho w+(1-\varrho)v)[u]d\varrho,$ (4.17)

satisfies, for all $u\in C^{1+\gamma}_{loc}(\mathbb{R}^{N-1})$, with $\nabla
u\in C^{\gamma}(\mathbb{R}^{N-1})$,

$\|B[w,v]u\|_{C^{\gamma-\alpha}}\leq C\|\nabla u\|_{C^{\gamma}},$ (4.18)

and

$C^{\prime}\|\nabla
u\|_{C^{\gamma}}\leq\|B[w,v]u\|_{C^{\gamma-\alpha}}+\|\nabla
u\|_{L^{\infty}},$ (4.19)

for some constants $C,C^{\prime}$ depending only on $N,\gamma,\alpha$ and
$\nu$.
Moreover for all $\beta\in(\alpha,\gamma)$, the operator $B:=B[w,v]$, with
domain ${\mathcal{C}}^{1+\beta}_{0}(\mathbb{R}^{N-1})$, is an infinitesimal
generator of a strongly continuous analytic semigroup $\\{e^{tB}:t\geq 0\\}$
on ${\mathcal{C}}^{\beta-\alpha}_{0}(\mathbb{R}^{N-1})$.
In addition, there exists $m=m(N,\alpha,\beta,\gamma,\nu)\in\mathbb{R}$ such
that for $k\in\mathbb{N}$, there exists
$M_{k}=M_{k}(N,\alpha,\beta,\gamma,\nu,m)>0$, such that

$\displaystyle\|t^{k}B^{k}e^{tB}\|_{\mathcal{L}({\mathcal{C}}^{\beta-\alpha}_{0}(\mathbb{R}^{N-1}))}\leq
M_{k}e^{mt},\hskip 14.22636pt\mbox{for all $t>0$.}$ (4.20)

###### Proof.

Thanks to (4.12), we can write

$B[w,v]=Q(w)B_{1}+B_{2},$ (4.21)

where $Q(w)=\sqrt{1+|\nabla w|^{2}}$ and

$B_{1}u=-\int_{\mathbb{R}^{N-1}}\frac{u(x)-u(y)}{|x-y|^{N+\alpha}}\int_{-1}^{1}{\mathcal{G}}^{\prime}(\varrho
p_{w}(x,y)+(1-\varrho)p_{v}(x,y))d\varrho dy,$
$B_{2}u:=H(v)\int_{0}^{1}DQ(\varrho w+(1-\varrho)v)[u]d\varrho.$

Recall that ${\mathcal{G}}^{\prime}(p)=-(1+p^{2})^{-\frac{N+\alpha}{2}}$ and

$p_{U}(x,y)=-\frac{U(x)-U(y)}{|x-y|}=-\int_{0}^{1}\nabla
U(\tau(x-y)+y)\cdot\frac{x-y}{|x-y|}\,d\tau.$

Therefore since $p_{U}(x,x-r\theta)=-\int_{0}^{1}\nabla
U(x-r\tau\theta)\cdot\theta\,d\tau$, we see that $B_{1}$ satisfies the
properties in Assumptions 3.5, with
$K(x,y):=-\int_{-1}^{1}{\mathcal{G}}^{\prime}(\varrho
p_{w}(x,y)+(1-\varrho)p_{v}(x,y))d\varrho$ and $\kappa$, depending only on
$N,\alpha,\gamma,\beta$ and $\nu$. Applying Lemma 2.2, we obtain (4.18). On
the other hand by Lemma 3.6, we have

$C^{\prime}\|\nabla
u\|_{C^{\gamma}}\leq\|B_{1}u\|_{C^{\gamma-\alpha}}+\|\nabla u\|_{L^{\infty}}.$
(4.22)

From this and (2.2), we get

$\|B_{2}u\|_{C^{\gamma-\alpha}}\leq\varepsilon\|B_{1}u\|_{C^{\gamma-\alpha}}+c_{\varepsilon}\|\nabla
u\|_{L^{\infty}}.$ (4.23)

Combining this with (4.22), we obtain (4.19).

We now let $\beta\in(\alpha,\gamma)$ and
$u\in{\mathcal{C}}^{1+\beta}_{0}(\mathbb{R}^{N-1})$ so that
$B_{2}u\in{\mathcal{C}}^{\beta-\alpha}_{0}(\mathbb{R}^{N-1})$. Then (2.2) and
(4.23) imply that

$\|B_{2}u\|_{C^{\beta-\alpha}}\leq\varepsilon\|B_{1}u\|_{C^{\beta-\alpha}}+c_{\varepsilon}\|u\|_{C^{\beta-\alpha}}.$

Moreover,

$A_{K}^{e}(x,0,\theta)=-2\int_{-1}^{1}(1+(\varrho\nabla
w(x)\cdot\theta+(1-\varrho)\nabla
v(x)\cdot\theta)^{2})^{-\frac{N+\alpha}{2}}d\varrho$

and clearly satisfies $\sup_{\theta\in
S^{N-2}}\|\nabla^{k}_{\theta}A_{K}^{e}(\cdot,0,\theta)\|_{C^{\gamma}}\leq
C_{k}(N,\alpha,\gamma,\nu).$ We can thus apply Lemma 3.7, to deduce that
$B[w,v]:{\mathcal{C}}^{1+\beta}_{0}(\mathbb{R}^{N-1})\to{\mathcal{C}}^{\beta-\alpha}_{0}(\mathbb{R}^{N-1})$
generates a strongly continuous analytic semigroup. Since $B[w,v]$ is
sectorial, (4.20) follows from [22, Proposition 2.1.1].

∎

As a consequence of the above lemma, we the following result.

######  Corollary 4.7.

Let $\beta\in(\alpha,1)$ and $u_{0}\in\mathcal{O}_{\nu}^{\gamma}$, for some
$\gamma\in(\beta,1)$. Then the linear operator

$\mathcal{L}_{0}:{\mathcal{C}}^{1+\beta}_{0}(\mathbb{R}^{N-1})\mapsto{\mathcal{C}}^{\beta-\alpha}_{0}(\mathbb{R}^{N-1}),\qquad\mathcal{L}_{0}u:=D\mathcal{H}(u_{0})[u],$
(4.24)

is an infinitesimal generator of a strongly continuous analytic semigroup
$\\{e^{t{\mathcal{L}}_{0}}:t\geq 0\\}$ on
${\mathcal{C}}^{\beta-\alpha}_{0}(\mathbb{R}^{N-1})$.
Moreover, for $T>0$, $\rho\in(0,\frac{1}{1+\alpha})$,
$u_{0}\in{\mathcal{C}}^{1+\beta}_{0}(\mathbb{R}^{N-1})$ and $f\in
C^{\rho}([0,T],{{\mathcal{C}}^{\beta-\alpha}_{0}(\mathbb{R}^{N-1})})$ such
that
$f(0)-{\mathcal{L}}_{0}u_{0}\in{C^{\gamma_{\rho}-\alpha}(\mathbb{R}^{N-1})},$
with $\gamma_{\rho}=\beta+\rho(1+\alpha)$, the initial value problem

$\left\\{\begin{array}[]{rll}u^{\prime}(t)+{\mathcal{L}}_{0}u(t)&=&f(t),\hskip
28.45274ptt\in(0,T]\\\ u(0)&=&u_{0}\end{array}\right.$

admits a unique solution $u\in
C^{\rho}([0,T],{\mathcal{C}}^{1+\beta}_{0}(\mathbb{R}^{N-1}))\cap
C^{1+\rho}([0,T],{\mathcal{C}}^{\beta-\alpha}_{0}(\mathbb{R}^{N-1}))$. In
addition, there exists $C_{T}=C(T,\rho,\nu,N,\alpha,\gamma,\beta)>0$ such that

$\displaystyle\|u-u_{0}\|_{C^{\rho}([0,T],C^{1+\beta})}$
$\displaystyle+\|u-u_{0}\|_{C^{1+\rho}([0,T],C^{\beta-\alpha})}$
$\displaystyle\leq
C_{T}\Big{(}\|f-{\mathcal{L}}_{0}u_{0}\|_{C^{\rho}([0,T],C^{\beta-\alpha})}+\|f(0)-{\mathcal{L}}_{0}u_{0}\|_{C^{\gamma_{\rho}-\alpha}}\Big{)},$

with, $C_{T}\leq C_{T_{0}}$ for all $T\leq T_{0}$.

###### Proof.

Since $B[u_{0},u_{0}]=D\mathcal{H}(u_{0})$, the result follows from Lemma 4.6,
(4.21) and Theorem 3.8. ∎

######  Remark 4.8.

Let $\rho\in(0,\frac{1}{1+\alpha})$ and define
$\gamma_{\rho}=\beta+\rho(1+\alpha)$ and suppose that
$\gamma_{\rho}-\alpha\not=1$. Then by Proposition 3.3, if $B:=B[w,v]$ is given
by (4.17) then thanks to (4.18), $Bu\in\mathcal{D}_{B}(\rho,\infty)$ as soon
as $\nabla u\in
C^{\gamma_{\rho}}(\mathbb{R}^{N-1})\cap{\mathcal{C}}^{\beta-\alpha}_{0}(\mathbb{R}^{N-1})$.
As a consequence, if $u_{0}\in{\mathcal{C}}^{1+\beta}_{0}(\mathbb{R}^{N-1})$,
for some $\beta>\alpha$, then by (4.16),
$\mathcal{H}(u_{0})\in\mathcal{D}_{\mathcal{L}_{0}}(\rho,\infty)$ if and only
if $\nabla u_{0}\in
C^{\gamma_{\rho}}(\mathbb{R}^{N-1})\cap{\mathcal{C}}^{\beta-\alpha}_{0}(\mathbb{R}^{N-1})$,
where $\mathcal{L}_{0}:=D\mathcal{H}(u_{0})$.

Our next result shows that the operator $B[w,v]$ (defined in (4.17)) satisfies
the maximum principle.

######  Lemma 4.9.

Let $\gamma>\alpha$ and $w,v\in C([0,T],{\mathcal{O}}^{\gamma}_{\nu})$. Then
for all $u\in{\mathcal{O}}^{\gamma}_{\nu}$,

$\displaystyle
B[w(t),v(t)]u(x)=P.V.\int_{\mathbb{R}^{N-1}}\frac{u(x)-u(x+y)}{|y|^{N+\alpha}}\mu(t,x,|y|,y/|y|)\,dy+V(t,x)\cdot\nabla
u(x),$

where, for $(x,r,\theta)\in\mathbb{R}^{N-1}\times[0,\infty)\times S^{N-2}$ we
have $p_{w}(x,x-r\theta)=-\int_{0}^{1}\nabla w(x-r\tau\theta)\cdot\theta
d\tau$,

$\mu(t,x,r,\theta):=-\int_{-1}^{1}{\mathcal{G}}^{\prime}(\varrho
p_{w(t)}(x,x-r\theta)+(1-\varrho)p_{v(t)}(x,x-r\theta))Q(\varrho
w(t)+(1-\varrho)v(t))(x)d\varrho,$

and

$V(t,x):=\int_{-1}^{1}\frac{\varrho\nabla w(t,x)+(1-\varrho)\nabla
v(t,x)}{Q(\varrho w(t)+(1-\varrho)v(t))(x)}H(\varrho
w(t)+(1-\varrho)v(t))(x)d\varrho.$

Moreover $\mu(t,x,0,\theta)=\mu(t,x,0,-\theta)$ and $\mu\in
C([0,T],C^{\gamma}(\mathbb{R}^{N-1}\times[0,\infty)\times S^{N-2}))$ and $V\in
C_{b}([0,T]\times\mathbb{R}^{N-1}).$

###### Proof.

By the fundamental theorem of calculus, we can write
$p_{w}(x,x-r\theta)=\frac{w(x-r\theta)-w(x)}{r}=-\int_{0}^{1}\nabla
w(x-r\tau\theta)\cdot\theta d\tau$. Using (4.21), we get the expression of
$B[w(t),v(t)]$. Moreover from the evenness of ${\mathcal{G}}^{\prime}$ we have
$\mu(t,x,0,\theta)=\mu(t,x,0,-\theta)$. The regularity of $V$ is a consequence
of (4.15). ∎

As a consequence, of Lemma 4.9 and Proposition 2.3, we have the following
result.

######  Corollary 4.10.

Under the assumptions of Lemma 4.9, let $u\in
L^{\infty}([0,T],C^{\beta}(\mathbb{R}^{N-1}))$, with $u(\cdot,x)\in
C^{1}([0,T])$ and $u(t,\cdot)\in C^{1+\beta}_{loc}(\mathbb{R}^{N-1})$, satisfy

$\partial_{t}u+B[w(t),v(t)]u\leq 0\qquad\textrm{ in
$[0,T]\times\mathbb{R}^{N-1}$.}$

Then $\sup_{[0,T]\times\mathbb{R}^{N-1}}u=\sup_{x\in\mathbb{R}^{N-1}}u(0,x)$.

##  5\. Proof of the main results

For $0<\rho<\frac{1}{1+\alpha}$, $\beta\in(\alpha,1)$ and $T>0$, we define for
the following Banach space

$E_{T}=C^{\rho}([0,T],{\mathcal{C}}^{1+\beta}_{0}(\mathbb{R}^{N-1}))\cap
C^{1+\rho}([0,T],{\mathcal{C}}^{\beta-\alpha}_{0}(\mathbb{R}^{N-1}))$ (5.1)

endowed with the norm
$\|\cdot\|_{E_{T}}=\|\cdot\|_{C^{\rho}([0,T],C^{1+\beta})}+\|\cdot\|_{C^{1+\rho}([0,T],C^{\beta-\alpha})}$.

######  Lemma 5.1.

Let $u_{0}\in
C^{1+\gamma}_{loc}(\mathbb{R}^{N-1})\cap{\mathcal{C}}^{1+\beta}_{0}(\mathbb{R}^{N-1})$,
for some $\gamma>\beta$, with $\nabla u_{0}\in C^{\gamma}(\mathbb{R}^{N-1})$.
For $R>0$, we define

$E_{T,R}:=\left\\{u\in E_{T}\,:\,u(0)=u_{0}\,,\,\|\nabla u-\nabla
u_{0}\|_{C^{\rho}([0,T],C^{\beta}(\mathbb{R}^{N-1}))}\leq R\right\\}.$ (5.2)

Then, there exists $C=C(N,\alpha,\beta)>1$ with the property that for every
$u,v\in E_{T,R}$ we have

$\displaystyle\|F(u)-F(v)\|_{C^{\rho}([0,T],{C^{\beta-\alpha}})}\leq
CR(1+R+\|\nabla u_{0}\|_{C^{\beta}})^{C}T^{\rho}(1+T^{\rho})\|u-v\|_{E_{T}},$

where
$F:{\mathcal{C}}^{1+\beta}_{0}(\mathbb{R}^{N-1})\to{\mathcal{C}}^{\beta-\alpha}_{0}(\mathbb{R}^{N-1})$
is given by $F(u)=-{\mathcal{H}}(u)+D{\mathcal{H}}(u_{0})u$.

###### Proof.

Let $T,R>0$, $u,v\in E_{T,R}$ and $w=u-v$. Then we have

$\displaystyle F(u)-F(v)=-\int_{0}^{1}(D{\mathcal{H}}(\lambda
u+(1-\lambda)v)-D{\mathcal{H}}(u_{0}))[w]\,d\lambda$
$\displaystyle=-\int_{0}^{1}\lambda\int_{0}^{1}D^{2}{\mathcal{H}}(\tau(\lambda
u+(1-\lambda)v)+(1-\tau)u_{0})[\lambda(u-u_{0})+(1-\lambda)(v-u_{0}),w]\,d\tau
d\lambda.$

Next, for $t\in[0,T]$, $\lambda,\tau\in[0,1]$ we define
$L_{\lambda,\tau}(t)\in{\mathcal{C}}^{\beta-\alpha}_{0}(\mathbb{R}^{N-1})$ by

$L_{\lambda,\tau}(t):=D^{2}{\mathcal{H}}(\tau(\lambda
u(t)+(1-\lambda)v(t))+(1-\tau)u_{0}).$

We observe that, by Corollary 4.4, for all $s,t\in[0,T]$,

$\|L_{\lambda,\tau}(t)-L_{\lambda,\tau}(s)\|_{C^{\beta-\alpha}}\leq
CR(1+R+\|\nabla u_{0}\|_{C^{\beta}})^{C}|s-t|^{\rho}$ (5.3)

and

$\|L_{\lambda,\tau}(t)\|_{C^{\beta-\alpha}}\leq CR(1+R+\|\nabla
u_{0}\|_{C^{\beta}})^{C}.$ (5.4)

We write

$\displaystyle[F(u)-F(v)](t)-[F(u)-F(v)](s)$
$\displaystyle=\int_{0}^{1}\lambda\int_{0}^{1}\Bigl{(}L_{\lambda,\tau}(s)[\lambda(u(s)-u_{0})+(1-\lambda)(v(s)-u_{0}),w(s)]$
$\displaystyle\qquad\qquad\;\>-L_{\lambda,\tau}(t)[\lambda(u(t)-u_{0})+(1-\lambda)(v(t)-u_{0}),w(t)]\Bigr{)}d\tau
d\lambda$
$\displaystyle=\int_{0}^{1}\lambda\int_{0}^{1}\Bigl{(}[L_{\lambda,\tau}(s)-L_{\lambda,\tau}(t)][\lambda(u(s)-u_{0})+(1-\lambda)(v(s)-u_{0}),w(s)]\,$
$\displaystyle\qquad\qquad\;\>+L_{\lambda,\tau}(t)[\lambda(u(s)-u_{0})+(1-\lambda)(v(s)-u_{0}),w(s)-w(t)]$
$\displaystyle\qquad\qquad\;\>+L_{\lambda,\tau}(t)[\lambda(u(s)-u(t))+(1-\lambda)(v(s)-v(t)),w(t)]\Bigr{)}d\tau\,d\lambda.$

Using (5.3), (5.4) and the fact that $u,v\in E_{T,R}$, we then get

$\displaystyle\|[F(u)-F(v)](t)-[F(u)-F(v)](s)\|_{C^{\beta-\alpha}}$
$\displaystyle\qquad\qquad\qquad\leq CR(1+R+\|\nabla
u_{0}\|_{C^{\beta}})^{C}(t^{\rho}+s^{\rho})|t-s|^{\rho}\|w\|_{E_{T}}.$ (5.5)

Using the fact that $[F(u)-F(v)](0)=0$, $w\in E_{T}$ and taking $s=0$ in
(5.5), we get

$\|F(u)-F(v)\|_{L^{\infty}([0,T],{{\mathcal{C}}^{\beta-\alpha}_{0}})}\leq
CR(1+R+\|\nabla u_{0}\|_{C^{\beta}})^{C}T^{2\rho}\|w\|_{E_{T}}.$ (5.6)

The result follows from (5.5) and (5.6).

∎

###### Proof of Theorem 1.4.

Let
$u_{0}\in{{\mathcal{C}}^{1+\beta}_{0}(\mathbb{R}^{N-1})}\cap{C^{\beta+\rho(1+\alpha)}_{loc}(\mathbb{R}^{N-1})}$
with $\|\nabla u_{0}\|_{C^{\beta+\rho(1+\alpha)}}\leq\nu$. Then by Corollary
4.7, the operator ${\mathcal{L}}_{0}:=D{\mathcal{H}}(u_{0})$ is a generator of
strongly continuous analytic semigroup. Now, to solve problem (1.19), we
rewrite it as

$\displaystyle\left\\{\begin{array}[]{rll}\partial_{t}u+{\mathcal{L}}_{0}u&=F(u)&\hskip
14.22636pt\mbox{in }\;[0,T]\times\mathbb{R}^{N-1}\\\ u(0)&=u_{0}&\hskip
14.22636pt\mbox{in }\;\mathbb{R}^{N-1}\end{array}\right.$ (5.9)

and use a fixed point argument, with
$F(u):=-{\mathcal{H}}(u)+D{\mathcal{H}}(u_{0})[u]=-{\mathcal{H}}(u)+{\mathcal{L}}_{0}u$.

Let $u\in E_{T}$ and recall (5.1) for the definition of $E_{T}$. Note that by
(4.18) and Lemma4.5,
$F(u)(0)-{\mathcal{L}}_{0}u(0)=-{\mathcal{H}}(u_{0})\in{\mathcal{C}}^{\beta-\alpha}_{0}(\mathbb{R}^{N-1})\cap
C^{\beta+\rho(1+\alpha)-\alpha}(\mathbb{R}^{N-1})$. Hence by Corollary 4.7,
there exists a unique function $\Phi(u)\in E_{T}$ satisfying

$\displaystyle\left\\{\begin{array}[]{rll}\partial_{t}\Phi(u)+{\mathcal{L}}_{0}\Phi(u)&=F(u)&\hskip
14.22636pt\mbox{in }\;[0,T]\times\mathbb{R}^{N-1}\\\
\Phi(u)(0)&=u_{0}&\hphantom{.5cm}\mbox{in
}\;\mathbb{R}^{N-1}.\end{array}\right.$ (5.12)

We define

$\mathcal{E}_{T,R}:=\left\\{u\in
E_{T}\,:\,u(0)=u_{0}\,,\,\|u-u_{0}\|_{E_{T}}\leq R\right\\}.$

Clearly a fixed point of the map $\Phi:E_{T}\to E_{T}$ in the set
$\mathcal{E}_{T,R}$ will be a solution to (1.19).

We claim that, provided $\|\nabla
u_{0}\|_{C^{\beta+\rho(1+\alpha)}(\mathbb{R}^{N-1})}\leq\nu$, there exist a
large constant $R>0$ and a small constant $T>0$, both depending only on
$N,\alpha,\beta,\rho,\gamma$ and $\nu$, such that

  1. $(i)$

$\Phi(\mathcal{E}_{T,R})\subset\mathcal{E}_{T,R}$ .

  2. $(ii)$

$\Phi$ is a contraction on $\mathcal{E}_{T,R}.$

To prove $(i)$, we use Corollary 4.7, (4.15) and Lemma 5.1, to get
$C=C(\rho,N,\alpha,\beta,\gamma,\nu)>1$ such that for all $T\in(0,1)$,

$\displaystyle\|\Phi(u)-u_{0}\|_{E_{T}}$ $\displaystyle\leq
C\left(\|F(u)-{\mathcal{L}}_{0}u_{0}\|_{C^{\rho}([0,T],{C^{\beta-\alpha}})}+\|{\mathcal{H}}(u_{0})\|_{C^{\beta+\rho(1+\alpha)-\alpha}}\right)$
$\displaystyle\leq
C\left(\|F(u)-F(u_{0})\|_{C^{\rho}([0,T],{C^{\beta-\alpha}})}+\|\nabla
u_{0}\|_{C^{\beta+\rho(1+\alpha)}}\right)$ $\displaystyle\leq
C\left(T^{\rho}(1+T^{\rho})R(1+R+\|\nabla u_{0}\|_{C^{\beta}})^{C}+\|\nabla
u_{0}\|_{C^{\beta+\rho(1+\alpha)}}\right).$

We can thus let $R=2\nu C$ and choose $T>0$ small, so that
$\Phi(u)\in\mathcal{E}_{T,R}$.

To prove $(ii)$, we let $u,v\in\mathcal{E}_{T,R}$. Then $W:=\Phi(u)-\Phi(v)$
satisfies

$\left\\{\begin{array}[]{rll}\partial_{t}W+{\mathcal{L}}_{0}W&=F(u)-F(v)&\hskip
14.22636pt\mbox{in }\;[0,T]\times\mathbb{R}^{N-1}\\\
W(0)&=0&\hphantom{.5cm}\mbox{in }\;\mathbb{R}^{N-1}.\end{array}\right.$

Hence by Corollary 4.7 and Lemma 5.1, there exists
$C=C(\rho,N,\alpha,\beta,\gamma,\nu)>1$ such that for $T\in(0,1)$,

$\displaystyle\|\Phi(u)-\Phi(v)\|_{E_{T}}$ $\displaystyle\leq
C\|F(u)-F(v)\|_{C^{\rho}([0,T],{C^{\beta-\alpha}})}\leq
CT^{\rho}R(1+R+\|\nabla u_{0}\|_{C^{\beta}})^{C}\|u-v\|_{E_{T}}.$

Now decreasing $T$, if necessary, we obtain
$\|\Phi(u)-\Phi(v)\|_{E_{T}}\leq\frac{1}{2}\|u-v\|_{E_{T}}$, which is $(ii)$.
This ends the proof of the claim.
We can thus apply the Banach fixed point theorem on $\mathcal{E}_{T,R}$ to get
a unique fixed point $u\in\mathcal{E}_{T,R}$ of $\Phi$. The proof of the
theorem is thus finished.

∎

Theorem 1.1 and Theorem 1.2 are consequences of the following result which
deals with bounded functions.

######  Theorem 5.2.

Under the assumptions of Theorem 1.4, we have the following properties.

  1. (i)

For all $\beta^{\prime}\in(\alpha,\beta)$,
$\rho\in(0,\frac{\alpha}{1+\alpha}]$, $k\in\mathbb{N}$ and $t\in(0,T]$, we
have $u(t)\in{\mathcal{C}}^{k+1+\beta^{\prime}}_{0}(\mathbb{R}^{N-1})$ and
there exists $C_{k}>0$ only depending on
$k,\rho,\alpha,\beta,\gamma,N,\nu,\beta^{\prime}$ and $T$ such that

$\|t^{k}\nabla u\|_{C^{\rho}([0,T],C^{k+\beta^{\prime}})}\leq C_{k}.$ (5.13)

  2. (ii)

If, in addition, $\nabla u_{0}\in C^{1+\gamma_{\rho}}(\mathbb{R}^{N-1})$ for
$\rho\in(0,\frac{1}{1+\alpha})$, then there exists $C>0$ only depending on
$\rho,\alpha,\beta,\gamma,N,\nu,T$ and $\beta^{\prime}$ such that

$\|\nabla u\|_{C^{\rho}([0,T],C^{1+\beta^{\prime}})}\leq C\|\nabla
u_{0}\|_{C^{1+\gamma_{\rho}}},$ (5.14)

where $\gamma_{\rho}=\beta+\rho(1+\alpha)$.

###### Proof.

We define $\tau_{h}u(t,x)=\frac{u(t,x+h)-u(t,x)}{|h|}$, for $0<|h|<1$. From
(1.19), we deduce that

$\left\\{\begin{array}[]{rll}\partial_{t}\tau_{h}u+L_{h}(t)\tau_{h}u&=0&\hskip
14.22636pt\mbox{in }\;[0,T]\times\mathbb{R}^{N-1}\\\
\tau_{h}u(0)&=\tau_{h}u_{0}&\hskip 14.22636pt\mbox{in
}\;\mathbb{R}^{N-1},\end{array}\right.$ (5.15)

where for all $a\in\mathbb{R}^{N-1}$, we define

$L_{a}(t)v:=\int_{0}^{1}D\mathcal{H}(\varrho
u(t,\cdot+a)+(1-\varrho)u(t))[v]\,d\varrho.$ (5.16)

We observe, from Lemma 4.9, that

$\displaystyle L_{a}(t)u(x)$
$\displaystyle=P.V.\int_{\mathbb{R}^{N-1}}\frac{u(t,x)-u(t,x+y)}{|y|^{N+\alpha}}\mu(t,x,|y|,y/|y|)\,dy+V(t,x)\cdot\nabla
u(t,x)$

where, for $(x,r,\theta)\in\mathbb{R}^{N-1}\times[0,\infty)\times S^{N-2}$ and
$a\in\mathbb{R}^{N-1}$,

$\mu(t,x,r,\theta):=-\int_{-1}^{1}{\mathcal{G}}^{\prime}(\varrho
p_{u(t,\cdot+a)}(x,x-r\theta)+(1-\varrho)p_{u}(x,x-r\theta))Q(\varrho
u(\cdot+a)+(1-\varrho)u)(x)d\varrho$

and

$V(t,x):=\int_{-1}^{1}\frac{\varrho\nabla u(t,x+a)+(1-\varrho)\nabla
u(t,x)}{Q(\varrho u(t,\cdot+a)+(1-\varrho)u(t,\cdot))(x)}H(\varrho
u(t,\cdot+a)+(1-\varrho)u(t))(x)d\varrho.$

We recall that ${\mathcal{G}}^{\prime}(p)=-(1+p^{2})^{-\frac{N+\alpha}{2}}$.
We now consider $(i)$. Clearly, by Lemma 4.6, for all $v\in
C^{1+\beta}(\mathbb{R}^{N-1})$,

$\|L_{a}(t)v\|_{C^{\beta-\alpha}}\leq C\|\nabla v\|_{C^{\beta}},$ (5.17)

and

$\|[L_{a}(t)-L_{a}(s)]v\|_{C^{\beta-\alpha}}\leq C|t-s|^{\rho}\|\nabla
v\|_{C^{\beta}},$ (5.18)

with $C$ independent on $a$. Again by Lemma 4.6, for all
$\beta^{\prime}\in(\alpha,\beta)$ and $a\in\mathbb{R}^{N-1}$,

$L_{a}(t):{{\mathcal{C}}^{1+\beta^{\prime}}_{0}(\mathbb{R}^{N-1})}\to{{\mathcal{C}}^{\beta^{\prime}-\alpha}_{0}(\mathbb{R}^{N-1})}$

is a generator of a strongly continuous analytic semigroup on
${{\mathcal{C}}^{\beta^{\prime}-\alpha}_{0}(\mathbb{R}^{N-1})}$, uniformly in
$t\in[0,T]$ and $a$.

We also observe, from Proposition 3.3, that for all $a\in\mathbb{R}^{N}$,

$C^{\beta^{\prime}+\rho(1+\alpha)-\alpha}(\mathbb{R}^{N-1})\cap{{\mathcal{C}}^{\beta^{\prime}-\alpha}_{0}(\mathbb{R}^{N-1})}\subset\mathcal{D}_{L_{a}(0)}(\rho,\infty)\qquad\textrm{
for all $\rho\in(0,\frac{1}{1+\alpha})$.}$ (5.19)

We now prove (5.13) by induction. Indeed, since
$\rho\leq\frac{\alpha}{1+\alpha}$, we have
$\beta^{\prime}+\rho(1+\alpha)-\alpha\leq\beta^{\prime}$.
We show by induction that for all $k\in\mathbb{N}$ and $t\in(0,T]$,

$u(t)\in{\mathcal{C}}^{k+1+\beta^{\prime}}_{0}(\mathbb{R}^{N-1})\qquad\textrm{
and }\qquad\|t^{k}\nabla u(t)\|_{C^{\rho}([0,T],C^{k+\beta^{\prime}})}\leq
C_{k},$ (5.20)

for some $C_{k}=C_{k}(N,\alpha,\beta,\beta^{\prime},\rho,\nu)>0$.
Clearly (5.20) holds for $k=0$ by the statement of the lemma. We assume that
(5.20) holds up to order $k\geq 1$ and we prove the result for $k+1$.

Consider $L_{a}(t)$ given by (5.16), which we can be written as (recall (2.4)
and (4.12))

$\displaystyle L_{a}(t)u(x)$
$\displaystyle=\frac{1}{2}\int_{0}^{\infty}\int_{S^{N-2}}r^{-2-\alpha}\delta_{e}u(x,r,\theta)(\mu(t,x,r,\theta)+\mu(t,x,r,-\theta))\,drd\theta$
$\displaystyle+\int_{0}^{\infty}\int_{S^{N-2}}r^{-2-\alpha}\delta_{o}u(x,r,\theta)(\mu(t,x,r,\theta)-\mu(t,x,r,-\theta))\,drd\theta+V(t,x)\cdot\nabla
u(t,x),$

with $\mu(t,x,0,\theta)=\mu(t,x,0,-\theta)$. From this and Lemma 2.2, we can
differentiate (5.15) $k$ times, so that, letting
$U=\frac{\partial^{k}}{\partial_{x_{1}\dots x_{k}}}(\tau_{h}u)$, we then get

$\begin{array}[]{rll}\partial_{t}U+L_{h}(t)U&=f(t,x)&\hskip 14.22636pt\mbox{in
}\;\mathbb{R}^{N-1}\times(0,T],\end{array}$ (5.21)

where

$\displaystyle f(t,x)$
$\displaystyle:=\sum_{S\in{\mathcal{S}}_{k-1}}P.V.\int_{\mathbb{R}^{N-1}}\frac{\partial^{|S|}}{\prod_{i\in
S}\partial
x_{i}}\frac{(\tau_{h}u)(x)-(\tau_{h}u)(x-y)}{|y|^{N+\alpha}}\frac{\partial^{k-|S|}}{\prod_{i\not\in
S}\partial x_{i}}\mu(t,x,|y|,y/|y|)\,dy$
$\displaystyle+\sum_{S\in{\mathcal{S}}_{k-1}}\frac{\partial^{|S|}}{\prod_{i\in
S}\partial x_{i}}(\tau_{h}\nabla
u)(x)\cdot\frac{\partial^{k-|S|}}{\prod_{i\not\in S}\partial x_{i}}V(t,x)$

and ${\mathcal{S}}_{k-1}$ is the set of subsets of $\\{1,\dots,k-1\\}$. By
(5.20) and the smoothness of ${\mathcal{G}}^{\prime}$, for all
$S\in{\mathcal{S}}_{k-1}$,

$\left\|t^{|S|}\frac{\partial^{k-|S|}\mu}{\prod_{i\not\in S}\partial
x_{i}}\right\|_{C^{\rho}([0,T],C^{\beta^{\prime}})}\leq C_{k-|S|}$

and in addition by Lemma 4.2,

$\left\|t^{|S|+1}\frac{\partial^{|S|}(\tau_{h}\nabla u)}{\prod_{i\in
S}\partial x_{i}}\right\|_{C^{\rho}([0,T],C^{\beta^{\prime}})}\leq
C_{|S|+1},\qquad\left\|t^{k-|S|}\frac{\partial^{k-|S|}}{\prod_{i\not\in
S}\partial x_{i}}V\right\|_{C^{\rho}([0,T],C^{\beta^{\prime}-\alpha})}\leq
C_{k-|S|}.$

It then follows, from (5.20) and Lemma 2.2, that
$f(t)\in{{\mathcal{C}}^{\beta^{\prime}-\alpha}_{0}}(\mathbb{R}^{N-1})$ and

$\left\|t^{k+1}f\right\|_{C^{\rho}([0,T],C^{\beta^{\prime}-\alpha})}\leq
CC_{k},\qquad t^{k+1}f\big{|}_{t=0}=0.$ (5.22)

We then define $v=t^{k+1}\frac{\partial^{k}(\tau_{h}u)}{\partial_{x_{1}\dots
x_{k}}}=t^{k+1}U\in C^{\rho}([0,T],{{\mathcal{C}}^{1+\beta^{\prime}}_{0}})$,
so that, by (5.21),

$\begin{array}[]{rll}\partial_{t}v+L_{h}(t)v&=(k+1)t^{k}\frac{\partial^{k}(\tau_{h}u)}{\partial_{x_{1}\dots
x_{k}}}(t,x)+t^{k+1}f(t,x)&\qquad\mbox{in }\;\mathbb{R}^{N-1}\times(0,T]\\\
v(0)&=0&\qquad\mbox{in }\;\mathbb{R}^{N-1}\times\\{0\\}.\end{array}$ (5.23)

Letting $g^{k}:=t^{k}\frac{\partial^{k}(\tau_{h}u)}{\partial_{x_{1}\dots
x_{k}}}$, then (5.20) implies that $g^{k}\in
C^{\rho}([0,T],{\mathcal{C}}^{\beta^{\prime}}_{0})$ and in addition, by (5.19)
and the choice of $\rho\leq\frac{\alpha}{1+\alpha}$, we find that

$\|g^{k}\|_{C^{\rho}([0,T],C^{\beta^{\prime}-\alpha})}+\|g^{k}(0)\|_{\mathcal{D}_{L_{0}(0)}(\rho,\infty)}\leq\|g^{k}\|_{C^{\rho}([0,T],C^{\beta^{\prime}})}+C\|g^{k}(0)\|_{C^{\beta^{\prime}}}\leq
C_{k}.$ (5.24)

In view of (5.17) and (5.18), we can thus apply [22, Proposition 6.1.3] to the
equation (5.23) and use (5.24) together with (5.22) and deduce that

$\displaystyle\|v\|_{C^{\rho}([0,T],C^{1+\beta^{\prime}})}$ $\displaystyle\leq
C\left(\|g^{k}\|_{C^{\rho}([0,T],C^{\beta^{\prime}-\alpha})}+\|g^{k}(0)\|_{\mathcal{D}_{L_{0}(0)}(\rho,\infty)}+\left\|t^{k+1}f\right\|_{C^{\rho}([0,T],C^{\beta^{\prime}-\alpha})}\right)$
$\displaystyle\leq C_{k+1}.$

Letting now $h\to 0$, we finally get

$\|t^{k+1}\nabla u\|_{C^{\rho}([0,T],C^{k+1+\beta^{\prime}})}\leq C_{k+1}.$

This completes the proof of $(i)$.
Finally for $(ii)$, we use (5.17) to obtain

$\|L_{h}(0)\tau_{h}u_{0}\|_{\mathcal{D}_{L_{a}(0)}(\rho,\infty)}\leq
C\|\nabla\tau_{h}u_{0}\|_{C^{\beta^{\prime}+\rho(1+\alpha)}}\leq
C\|\nabla\tau_{h}u_{0}\|_{C^{\gamma_{\rho}}}.$

We can thus apply [22, Proposition 6.1.3] to (5.15) to get

$\|\tau_{h}u\|_{C^{\rho}([0,T],C^{1+\beta^{\prime}})}\leq
C\left(\|\tau_{h}u_{0}\|_{C^{1+\beta}}+\|\nabla\tau_{h}u_{0}\|_{C^{\gamma_{\rho}}}\right).$

Letting $|h|\to 0$ in the above estimate, we find that

$\|\nabla u\|_{C^{\rho}([0,T],C^{1+\beta^{\prime}})}\leq C\|\nabla
u_{0}\|_{C^{1+{\gamma_{\rho}}}}.$

That is (5.14). ∎

###### Proof of Theorem 1.1 and Theorem 1.2 .

For $n\in\mathbb{N}$, we let $\eta_{n}\in C^{\infty}_{c}(\mathbb{R}^{N-1})$
such that $\eta\equiv 1$ for $|x|\leq n$, $\eta\equiv 0$ for $|x|\geq 2n$ and
$|D^{k}\eta_{n}|\leq\frac{1}{n^{k}}$ on $\mathbb{R}^{N-1}$. It then follows
that $\eta_{n}u_{0}\in{\mathcal{C}}^{1+\beta}_{0}(\mathbb{R}^{N-1})$ thanks to
Proposition 2.1 and that

$\|\nabla(\eta_{n}u_{0})\|_{C^{\gamma_{\rho}}}\leq 2\|\nabla
u_{0}\|_{C^{\gamma_{\rho}}}\leq 2\nu,$ (5.25)

where $\gamma_{\rho}=\beta+\rho(1+\alpha)$. By Theorem 1.4 there exists a
unique function $u^{n}\in E_{T}$ solving

$\displaystyle\left\\{\begin{array}[]{rll}\partial_{t}u^{n}+{\mathcal{H}}(u^{n})&=0&\hskip
14.22636pt\mbox{in }\;[0,T]\times\mathbb{R}^{N-1}\\\
u^{n}(0)&=\eta_{n}u_{0}&\hskip 14.22636pt\mbox{in
}\;\mathbb{R}^{N-1}\end{array}\right.$ (5.28)

and satisfying

$\|u^{n}-\eta_{n}u_{0}\|_{E_{T}}\leq C_{0}(N,\alpha,\beta,\gamma,\rho,\nu),$
(5.29)

Since, by (5.25), the sequence $(\nabla(\eta_{n}u_{0}))_{n}$ is bounded in
$C^{\beta}(\mathbb{R}^{N-1})$, it then follows, from (5.29) and the Arzelà-
Ascoli theorem that, for $\rho^{\prime}<\rho$ and $\beta^{\prime}<\beta$, up
to a subsequence, the sequence $(u^{n})_{n}$ converges in
$C^{\rho^{\prime}}([0,T],C^{1+\beta^{\prime}}_{loc}(\mathbb{R}^{N-1}))\cap
C^{1+\rho^{\prime}}([0,T],C^{\beta^{\prime}-\alpha}_{loc}(\mathbb{R}^{N-1}))$
to some function $u\in C^{\rho}([0,T],C^{1+\beta}_{loc}(\mathbb{R}^{N-1}))\cap
C^{1+\rho}([0,T],C^{\beta-\alpha}_{loc}(\mathbb{R}^{N-1}))$. In particular
${\mathcal{H}}(u^{n})\to{\mathcal{H}}(u)$ pointwise locally uniformly. Hence
passing to the limit in (5.28), we get

$\displaystyle\left\\{\begin{array}[]{rll}\partial_{t}u+{\sqrt{1+|\nabla
u|^{2}}}H(u)&=0&\hskip 14.22636pt\mbox{in }\;[0,T]\times\mathbb{R}^{N-1}\\\
u(0)&=u_{0}&\hskip 14.22636pt\mbox{in }\;\mathbb{R}^{N-1}.\end{array}\right.$
(5.32)

Letting $n\to\infty$ in (5.29), we then see that
$\|u-u_{0}\|_{C^{\rho}([0,T],C^{\beta})\cap
C^{1+\rho}([0,T],C^{\beta-\alpha})}\leq C_{0}^{\prime\prime}.$

For the uniqueness, we consider $v\in
C^{\rho}([0,T],C^{1+\beta}_{loc}(\mathbb{R}^{N-1}))\cap
C^{1+\rho}([0,T],C^{\beta-\alpha}_{loc}(\mathbb{R}^{N-1}))$ satisfying (5.32)
and $v-u_{0}\in{E_{T}}$. Then $w=u-v\in E_{T}$ satisfies

$\displaystyle\left\\{\begin{array}[]{rll}\partial_{t}w+B[u(t),v(t)]w&=0&\hskip
14.22636pt\mbox{in }\;[0,T]\times\mathbb{R}^{N-1}\\\ w(0)&=0&\hskip
14.22636pt\mbox{in }\;\mathbb{R}^{N-1},\end{array}\right.$ (5.35)

where, $B[u,v]w=\int_{0}^{1}D{\mathcal{H}}(\tau u+(1-\tau)v)[w]\,d\tau$. Hence
by Corollary 4.10, we deduce that $w=0$.
Now (1.11) is a consequence of (5.14), by compactness and thus the proof of
Theorem 1.1 is thus complete.
Using (5.13), we obtain (1.12) thanks to the Arzelà-Ascoli theorem, which is
Theorem 1.2.

∎

Our next result shows that the (global) sign of the initial condition $u_{0}$
and the one of the the initial fractional mean curvature $H(u_{0})$ are
preserved. It also completes the proof of Theorem 1.3.

######  Lemma 5.3.

Under the assumptions of Theorem 1.1, we have the following results.

  1. (i)

$\|\nabla u\|_{L^{\infty}((0,T)\times\mathbb{R}^{N-1})}\leq\|\nabla
u_{0}\|_{L^{\infty}}$,

  2. (ii)

$\displaystyle\sup_{[0,T]\times\mathbb{R}^{N-1}}\partial_{t}u=-\inf_{\mathbb{R}^{N-1}}{\mathcal{H}}(u_{0})\qquad$
and
$\qquad\displaystyle\inf_{[0,T]\times\mathbb{R}^{N-1}}\partial_{t}u=-\sup_{\mathbb{R}^{N-1}}{\mathcal{H}}(u_{0})$.

  3. (iii)

If moreover $u_{0}\in L^{\infty}(\mathbb{R}^{N-1})$ then

$\displaystyle\sup_{[0,T]\times\mathbb{R}^{N-1}}u=\sup_{\mathbb{R}^{N-1}}u_{0}\qquad\textrm{
and
}\qquad\displaystyle\inf_{[0,T]\times\mathbb{R}^{N-1}}u=\inf_{\mathbb{R}^{N-1}}u_{0}$

###### Proof.

We start with $(iii)$. If $u_{0}\in L^{\infty}(\mathbb{R}^{N-1})$ then by
(1.10) we have $u\in C^{\rho}([0,T],C^{1+\beta})\cap
C^{1+\rho}([0,T],C^{\beta-\alpha})$. Now, in view of (4.1), we can apply
Proposition 2.3 to $u$ and $-u$ to get $(iii)$.
Next to prove $(i)$, we define $\tau_{h}u(t,x)=\frac{u(t,x+h)-u(t,x)}{|h|}\in
C([0,T],C^{1+\beta})$, for $0<|h|<1$. We then have

$\partial_{t}\tau_{h}u+B[u(t,\cdot+h),u(t)]\tau_{h}u(t)=0\qquad\textrm{ in
$[0,T]\times\mathbb{R}^{N-1}$}.$

It thus follows from Corollary 4.10 that
$\|\tau_{h}u\|_{L^{\infty}((0,T)\times\mathbb{R}^{N-1})}\leq\|\tau_{h}u_{0}\|_{L^{\infty}}$.
Hence, letting $h\to 0$, we get $(i)$.
Finally, by Theorem 1.2 and Lemma 4.2 we have
$\partial_{t}u=-{\mathcal{H}}(u)\in
C^{\rho}([\varepsilon,T],C^{k+\beta^{\prime}-\alpha})$ for all
$\varepsilon\in(0,T)$ and $k\geq 0$. Letting $\tau\in(\varepsilon,T)$, we
define $u^{\tau}(t,x)=\frac{u(t+\tau,x)-u(t,x)}{\tau}$, for $\varepsilon<t\leq
T-\tau$. We thus get $u^{\tau}\in
C^{\rho}([\varepsilon,T-\tau],C^{1+\beta})\cap
C^{1+\rho}([\varepsilon,T-\tau],C^{\beta-\alpha})$ and

$\partial_{t}u^{\tau}+B[u(t+\tau),u(t)]u^{\tau}(t)=0\qquad\textrm{ on
$[\varepsilon,T-\tau]\times\mathbb{R}^{N-1}$}.$

Applying Corollary 4.10, we obtain

$\sup_{[\varepsilon,T]\times\mathbb{R}^{N-1}}u^{\tau}=\sup_{x\in\mathbb{R}^{N-1}}u^{\tau}(\varepsilon,x)\qquad\textrm{
and
}\qquad\inf_{[\varepsilon,T]\times\mathbb{R}^{N-1}}u^{\tau}=\inf_{x\in\mathbb{R}^{N-1}}u^{\tau}(\varepsilon,x).$

Since $\partial_{t}u=-{\mathcal{H}}(u)\in C^{\rho}([0,T],L^{\infty})$, letting
first $\tau\to 0$ and then $\varepsilon\to 0$ in the above identities, we get
$(ii)$. ∎

###### Proof of Theorem 1.3 (completed).

It suffices to apply Lemma 5.3.

∎

## References

  * [1] N. Abatangelo & E. Valdinoci, (2014). A notion of nonlocal curvature. Numerical Functional Analysis and Optimization, 35(7-9), 793-815.
  * [2] H. Abels & M. Kassmann (2009). The Cauchy problem and the martingale problem for integro-differential operators with non-smooth kernels. Osaka Journal of Mathematics, 46(3), 661-683.
  * [3] A. Attiogbe, M. M. Fall & E. H. A. Thiam (2022). Nonlocal diffusion of smooth sets. Math. Eng. 4 , no. 2, Paper No. 009, 22 pp. 35R11.
  * [4] R. F. Bass. (2009). Regularity results for stable-like operators. Journal of Functional Analysis, 257(8), 2693-2722.
  * [5] C. Bucur & E. Valdinoci, (2016). Nonlocal diffusion and applications (Vol. 20, pp. xii-155). Cham: Springer.
  * [6] R. M. Blumenthal and R. K. Getoor, (1960). Some theorems on stable processes, Transactions of the American Mathematical Society, 95(2), 263-273.
  * [7] L. Caffarelli, J. M. Roquejoffre & O. Savin (2010). Nonlocal minimal surfaces. Communications on Pure and Applied Mathematics, 63(9), 1111-1144.
  * [8] L. A. Caffarelli & P. E. Souganidis, (2010). Convergence of nonlocal threshold dynamics approximations to front propagation. Archive for rational mechanics and analysis, 195(1), 1-23.
  * [9] A. Chambolle, M. Novaga, and B. Ruffini, Some results on anisotropic fractional mean curvature flows, Interfaces Free Bound. 19 (2017), 393-415.
  * [10] S. Cameron, (2019). Eventual Regularization of Fractional Mean Curvature Flow. arXiv preprint arXiv:1905.09184.
  * [11] A. Cesaroni & M. Novaga, (2020). Symmetric self-shrinkers for the fractional mean curvature flow. The Journal of Geometric Analysis, 30(4), 3698-3715.
  * [12] A. Cesaroni & M. Novaga. (2021). Fractional mean curvature flow of Lipschitz graphs. arXiv:2103.11346.
  * [13] E. Cinti, The fractional mean curvature flow. Bruno Pini Mathematical Analysis Seminar, 11(1), 18-43. https://doi.org/10.6092/issn.2240-2829/10576.
  * [14] E. Cinti, C. Sinestrari & E. Valdinoci, Convex sets evolving by volume preserving fractional mean curvature flows. Preprint (2018).
  * [15] M. M. Fall, (2020). Regularity results for nonlocal equations and applications. Calculus of Variations and Partial Differential Equations, 59(5), 1-53.
  * [16] M. M. Fall, (2018). Constant nonlocal mean curvatures surfaces and related problems. In Proceedings of the International Congress of Mathematicians (ICM 2018) (In 4 Volumes) Proceedings of the International Congress of Mathematicians 2018 (pp. 1613-1637).
  * [17] M.M. Fall, T. Weth, (2016). Liouville theorems for a general class of nonlocal operators. Potential Anal. 45 , no. 1, 187-200.
  * [18] G. Huisken, Flow by mean curvature of convex surfaces into spheres, J. Differential Geom. 20 (1984), no. 1, 237-266.
  * [19] C. Imbert, (2009). Level set approach for fractional mean curvature flows. Interfaces and Free Boundaries, 11(1), 153-176.
  * [20] V. Julin & D. A. La Manna. Short time existence of the classical solution to the fractional mean curvature flow. In Annales de l’Institut Henri Poincaré, Analyse non linéaire (Vol. 37, No. 4, pp. 983-1016). Elsevier Masson.
  * [21] J. LeCrone, (2012). Elliptic operators and maximal regularity on periodic little-Hölder spaces. Journal of Evolution Equations, 12(2), 295-325.
  * [22] A. Lunardi, (2012). Analytic semigroups and optimal regularity in parabolic problems. Springer Science & Business Media.
  * [23] C. Mantegazza, (2011), Lecture Notes on Mean Curvature Flow, Progress in Mathematics, 290, Basel: Birkhäuser/Springer, doi:10.1007/978-3-0348-0145-4.
  * [24] A. Pazy, (1983). Semigroups of Linear Operators and Applications to Partial Differential Equations. Applied Mathematical Sciences, 44.
  * [25] M. Sáez & E. Valdinoci, (2019). On the evolution by fractional mean curvature. Communications in Analysis and Geometry, 27(1), 211-249.
  * [26] R. W. Schwab and L. Silvestre, (2016) Regularity for parabolic integro-differential equations with very irregular kernels, Anal. PDE 9 (2016), no. 3, 727-772.
  * [27] Joaquim Serra, Regularity for fully nonlinear nonlocal parabolic equations with rough kernels, Calc. Var. Partial Differential Equations 54 (2015), no. 1, 615-629.
  * [28] E. Sinestrari, (1985). On the abstract Cauchy problem of parabolic type in spaces of continuous functions. Journal of mathematical analysis and applications, 107(1), 16-66.