# Minimizing Movements for Anisotropic and Inhomogeneous Mean Curvature Flows

Antonin Chambolle  CEREMADE department, Université Paris-Dauphine, Université
PSL, CNRS, pl. du Maréchal de Lattre de Tassigny, 75016 Paris, France  ,
Daniele De Gennaro  CEREMADE department, Université Paris-Dauphine, Université
PSL, CNRS, pl. du Maréchal de Lattre de Tassigny, 75016 Paris, France  and
Massimiliano Morini  Dipartimento di Scienze Matematiche, Fisiche e
Informatiche, Università di Parma, Parco Area delle Scienze 7/A, Parma, Italy

###### Abstract.

In this paper we address anisotropic and inhomogeneous mean curvature flows
with forcing and mobility, and show that the minimizing movements scheme
converges to level set/viscosity solutions and to distributional solutions à
la Luckhaus-Sturzenhecker to such flows, the latter result holding in low
dimension and conditionally to the convergence of the energies. By doing so we
generalize recent works concerning the evolution by mean curvature by removing
the hypothesis of translation invariance, which in the classical theory allows
one to simplify many arguments.

##  1\. Introduction

In this paper we deal with the anisotropic, inhomogeneous mean curvature flow
with forcing and mobility. By inhomogeneous we mean that the flow is driven by
surface tensions depending on the position in addition to the orientation of
the surface. The evolution of sets $t\mapsto E_{t}\subseteq\mathbb{R}^{N}$
considered is (formally) governed by the law

(1)
$V(x,t)=\psi(x,\nu_{E_{t}}(x))\left(-H^{\phi}_{E_{t}}(x)+f(x,t)\right),\quad
x\in\partial E_{t},\ t\in(0,T),$

where $V(x,t)$ is the (outer) normal velocity of the boundary $\partial E_{t}$
at $x$, $\phi(x,p)$ is a given anisotropy representing the surface tension,
$H^{\phi}$ is the anisotropic mean curvature of $\partial E_{t}$ associated to
$\phi$, $\psi(x,p)$ is an anisotropy evaluated at the outer unit normal
$\nu_{E_{t}}(x)$ to $\partial E_{t}$ which represents a velocity modifier
(also called the mobility term), and $f$ is the forcing term. We will be
mainly concerned with smooth anisotropies (and the regularity assumptions will
be made precise later on): in this case, the curvature $H^{\phi}$ is the first
variation of the anisotropic and inhomogeneous perimeter associated to the
anisotropy $\phi$ (in short, $\phi-$perimeter) defined as

(2)
$P_{\phi}(E):=\int_{\partial^{*}E}\phi(x,\nu_{E}(x))\,\textnormal{d}\mathcal{H}^{N-1}(x)$

for any set $E$ of finite perimeter (where $\partial^{*}E$ denotes the reduced
boundary of $E$) and, if $E$ is sufficiently smooth, it takes the form

$H^{\phi}_{E}(x)=\textnormal{div}(\nabla_{p}\phi(x,\nu_{E}(x))),$

where with $\nabla_{p}$ we denote the gradient made with respect to the second
variable. Note that evolution (1) can be red as the motion of sets in
$\mathbb{R}^{N}$, when the latter is endowed with the Finsler metric induced
by the anisotropy (see Remark 4.13). Equation (1) is relevant in Material
Sciences, Crystal Growth, Image Segmentation, Geometry Processing and other
fields see e.g. [1, 19, 28, 43, 44].

The mathematical literature for inhomogeneous mean curvature flows is not as
extensive as in the homogeneous case, mainly due to the difficulties arising
from the lack of translational invariance. Indeed, assuming that the evolution
is invariant under translations allows to simplify many arguments used in the
classical proofs of, for example, comparison results and estimates on the
speed of evolution. In the homogeneous case the well-posedness theory is
nowadays well established and quite satisfactory, both in the local and
nonlocal case, and even in the much more challenging crystalline case (that
is, when the anisotropy $\phi$ is piecewise affine) see [2, 3, 8, 12, 13, 14,
16, 27, 36, 38, 41] to cite a few. Concerning the inhomogeneous mean curvature
flow, we cite [30, 31] where the short time existence of smooth solutions on
manifolds is shown, and [26, 34], where the viscosity level set approach
(introduced for the homogeneous evolution in [16, 23]) is extended,
respectively, to the equation (1) and to the Riemannian setting.

In the present work we implement the minimizing movement approach à la
Almgren-Taylor-Wang (in short, ATW scheme) [3] to prove existence via
approximation of a level set solution to the generalized anisotropic and
inhomogeneous motion (1). To carry on this scheme (which has only been
sketched in [8], but lacks a formal proof) we gain insights from [14]. We also
show that, under the additional hypothesis of convergence of the energies (4)
and low dimension (LABEL:hp_dim)(which are nowadays classical for this
approach), the same approximate solutions provide in the limit a suitable
notion of “BV-solutions”, also termed distributional solutions, see [38, 41].

There are many more concepts of weak solution for the mean curvature flow. In
particular, we cite the diffuse-interface approximation provided by the Allen-
Cahn equation [22, 33, 29, 37] and the threshold dynamic scheme [40, 20] (see
also the relative entropy methods of [36]). Other recent results concern the
weak-strong uniqueness problem, which consists in proving that weak solutions
coincide with the smooth ones as long as the latter exist. After classical
works concerning viscosity solutions, a new definition of “BV-solution” (whose
existence is proved via the Allen-Cahn approximation scheme) allows the
authors in [29, 37] to prove weak-strong uniqueness for isotropic and
anisotropic mean curvature flows. This result is based upon the so-called
optimal dissipation inequality satisfied by their weak solution. In general,
it is very difficult to say if the ATW scheme could satisfy such a property,
mainly because of the “degeneracy” of the dissipation term in the incremental
problem defined via the distance function. Even if all these results concern
the translationally invariant case, a study of some of these properties in the
inhomogeneous setting seems very interesting and challenging.

Other remarks on possible research directions are the following. To begin
with, the new arguments which are used to compensate the lack of translation
invariance are based on the locality of the anisotropic curvature $H^{\phi}$
associated with a smooth anisotropy $\phi$. This implies that the proofs are
not straightforwardly adaptable to the so-called “variational curvatures”
considered in [14], which are non-local in nature. On the other hand, since
the crystalline curvatures are highly nonlocal and degenerate operators (see
e.g. [12, 10]), they do not fall in the theory constructed in the present
work. In principle, it would be possible to follow the same perturbative study
conducted in [12] in order to prove at least existence for an inhomogeneous
and crystalline mean curvature flow. However, a satisfactory characterization
of the limiting motion equation bearing a comparison principle is lacking so
far.

This work can be seen as a first step towards constructing a general theory of
motions driven by non-translationally invariant and possibly nonlocal
curvatures, in the spirit of [14].

###  1.1. Main results

Now briefly recall the minimizing movements procedure in order to state the
main results of the paper. Given an initial bounded set $E_{0}$ and a
parameter $h>0$, we define the discrete flow
$E_{t}^{(h)}:=T_{h,t-h}E_{t-h}^{(h)}$ for any $t\geq h$ and
$E_{t}^{(h)}=E_{0}$ for $t\in[0,h),$ where the functional $T_{h,t}$ is defined
for $t\geq 0$ as follows: for any bounded set $E$ we set $T_{h,t}E$ (or,
sometimes, $T_{h,t}^{-}E$) as the minimal solution to the problem

(3)
$\min\left\\{P_{\phi}(F)+\int_{F}\left(\dfrac{\textnormal{sd}^{\psi}_{E}(x)}{h}+\fint_{[\frac{t}{h}]h}^{[\frac{t}{h}]h+h}f(x,s)\,\textnormal{d}s\right)\,\textnormal{d}\mathcal{H}^{N-1}(x)\
:\ F\text{ is measurable}\right\\},$

where $\textnormal{sd}^{\psi}_{E}(x)$ is the signed geodesic distance between
$x$ and $E$ induced by the anisotropy $\psi$ (see (10) for the precise
definition) and $[s]=\max\\{n\leq s,\ n\in\mathbb{N}\cup\\{0\\}\\}$ denotes
the integer part of a non-negative real number $s\in[0,+\infty)$. We will then
define $T^{+}_{h,t}E$ as the maximal solution to the problem above. Any
$L^{1}-$limit point as $h\to 0$ of the family $\\{E_{t}^{(h)}\\}_{t\geq 0}$
will be called a flat flow. In the whole paper we will assume that

(H0) $\begin{split}&\phi\in\mathscr{E}\text{ (see Definition \ref{def reg ell
integr}) and }\psi\text{ is an anisotropy as in Definition \ref{def
anisotropy}, }\\\ &\forall t\in[0,+\infty)\text{ it holds }f(\cdot,t)\in
C^{0}(\mathbb{R}^{N}),\|f\|_{L^{\infty}(\mathbb{R}^{N}\times[0,+\infty))}<\infty.\end{split}$

With more effort one could weaken the hypothesis and require
$\int_{0}^{t}f(\cdot,s)\,\textnormal{d}s$ to be continuous (see [15]). For the
sake of simplicity we will require the global-in-time boundedness. We prove
existence and H ölder regularity for flat flows.

######  Theorem 1.1 (Existence of flat flows).

Let $E_{0}$ be a bounded set of finite perimeter and $\phi,\psi,f$ satisfy
(LABEL:standing_hp). Fix $T>0$. For any $h>0$, let
$\\{E_{t}^{(h)}\\}_{{t\in[0,T)}}$ be a discrete flow with initial datum
$E_{0}$. Then, there exists a family of sets of finite perimeter
$\\{E_{t}\\}_{t\in[0,T)}$ and a subsequence $h_{k}\searrow 0$ such that

$E_{t}^{(h)}\to E_{t}\quad\text{in }L^{1},$

for a.e. $t\in[0,T)$. Such flow satisfies the following regularity property:
there exists a constant $c$, depending on $T$, such that for every $0\leq
s\leq t<T$,

$\displaystyle|E_{s}\triangle E_{t}|$ $\displaystyle\leq c|t-s|^{1/2},$
$\displaystyle P_{\phi}(E_{t})$ $\displaystyle\leq P_{\phi}(E_{0})+c.$

Subsequently, we will show that flat flow s are distributional solutions, as
defined in [38]. We will require additional hypothesis: firstly, low dimension
(LABEL:hp_dim) (linked to the complete regularity of the $\phi-$perimeter
minimizer, compare [38, 41]), moreover

(H1) $\displaystyle\exists\,c_{\psi}>0\ \text{s.t. }|\psi(x,v)-\psi(y,v)|\leq
c_{\psi}|x-y|,\quad\forall x,y\in\mathbb{R}^{N},v\in{S^{N-1}},$ (H2)
$\displaystyle f\in C^{0}(\mathbb{R}^{N}\times[0,\infty)]).$

######  Theorem 1.2 (Existence of distributional solutions).

Assume (LABEL:standing_hp), (H1), (H2) and (LABEL:hp_dim). For any $T>0$, if

(4)
$\lim_{k\to\infty}\int_{0}^{T}P_{\phi}(E_{t}^{(h_{k})})=\int_{0}^{T}P_{\phi}(E_{t}),$

then $\\{E_{t}\\}_{{t\in[0,T]}}$ is a distributional solution (1) with initial
datum $E_{0}$ in the following sense:

  * (1)

for a.e. $t\in[0,T)$ he set $E_{t}$ has weak $\phi-$curvature
$H^{\phi}_{E_{t}}$ (see (19) for details) satisfying

$\int_{0}^{T}\int_{\partial^{*}E_{t}}|H^{\phi}_{E_{t}}|^{2}<\infty;$

  * (2)

there exist $v:\mathbb{R}^{N}\times(0,T)\to\mathbb{R}$ with
$\int_{0}^{T}\int_{\partial^{*}E_{t}}v^{2}\,\textnormal{d}\mathcal{H}^{N-1}\,\textnormal{d}t<\infty$
and $v(\cdot,t)\big{\lvert}_{\partial E_{t}}\in L^{2}(\partial E_{t})$ for
a.e. $t\in[0,T)$, such that

(5)
$\displaystyle-\int_{0}^{T}\int_{\partial^{*}E_{t}}v\eta\,\textnormal{d}\mathcal{H}^{N-1}\,\textnormal{d}t$
$\displaystyle=\int_{0}^{T}\int_{\partial^{*}E_{t}}\left(H^{\phi}_{E_{t}}-f\right)\eta\,\textnormal{d}\mathcal{H}^{N-1}\,\textnormal{d}t$
(6)
$\displaystyle\int_{0}^{T}\int_{E_{t}}\partial_{t}\eta\,\textnormal{d}x\,\textnormal{d}t+\int_{E_{0}}\eta(\cdot,0)\,\textnormal{d}x$
$\displaystyle=-\int_{0}^{T}\int_{\partial^{*}E_{t}}\psi(\cdot,\nu_{E_{t}})v\eta\,\textnormal{d}\mathcal{H}^{N-1}\,\textnormal{d}t,$

for every $\eta\in C^{1}_{c}(\mathbb{R}^{N}\times{[0,T)}).$

The definitions $1),2)$ extend to our case the definition of $BV$-solutions of
[38] and the distributional solutions of [41]. We recall that hypothesis (4)
ensures that the evolving sets avoid the so-called “fattening” phenomenon. It
is known that this hypothesis is satisfied in the case of evolution of convex
or mean-convex sets, see e.g. [10, 17, 24], but in general is not known under
which general hypothesis it is valid. We also remark that the proof of the
theorem above provides a detailed proof of [10, Theorem 3.2], which had only
been sketched. Moreover, we bypass the use of a Bernstein-type result (which
is usually employed) by a double blow-up technique.

In the second part of the work we will focus on the level set approach.
Briefly, given an initial compact set $E_{0}$, we set $u_{0}$ such that
$\\{u_{0}\geq 0\\}=E_{0}$ and we look for a solution $u$ in the viscosity
sense (in a sense made precise in Definition 4.5) to

(7) $\begin{cases}\partial_{t}u+\psi(x,-\nabla
u)\left(\textnormal{div}\nabla_{p}\phi(x,\nabla u(x))-f(x,t)\right)=0\\\
u(\cdot,t)=u_{0}.\end{cases}$

Classical remarks ensure that any level set $\\{u\geq s\\}$ is evolving
following the mean curvature flow (1). To prove existence for (7) we use an
approximating procedure. For $h>0$ and $t\in(0,+\infty)$ we set iteratively
$u_{h}^{\pm}(\cdot,t)=u_{0}$ for $t\in[0,h)$ and for $t\geq h$

$\begin{split}u^{+}_{h}(x,t)&:=\sup\left\\{s\in\mathbb{R}\ :\ x\in
T_{h,t-h}^{+}\\{u_{h}^{+}(\cdot,t-h)\geq s\\}\right\\}\\\
u_{h}^{-}(x,t)&:=\sup\left\\{s\in\mathbb{R}\ :\ x\in
T_{h,t-h}^{-}\\{u_{h}^{-}(\cdot,t-h)>s\\}\right\\},\end{split}$

where the operator $T_{h,t}^{\pm}$ has been previously introduced. We remark
that these are maps piecewise constant in time, since
$T^{\pm}_{h,t}=T^{\pm}_{h,[t/h]h}$, which are only upper and lower
semicontinuous in space respectively. Then, we will pass to the limit $h\to 0$
on the families $\\{u_{h}^{\pm}\\}_{h}$ to find functions $u^{+},u^{-}$ which
are viscosity sub - and supersolution respectively of equation (7). Passing to
the limit as $h\to 0$ in our case is not straightforward. The main issue is
that we do not have an uniform estimate on the modulus of continuity of the
functions $u_{h}$ (compare [14]) and thus we can not pass to the (locally)
uniform limit of the sequence. (More precisely, our best estimate contained in
Lemma 4.8 decays too fast as $h\to 0$ to provide any useful information).
Nonetheless, motivated by [6, 5, 7] we can define the half-relaxed limits

(8) $\begin{split}u^{+}(x,t):=&\sup_{(x_{h},t_{h})\to(x,t)}\limsup_{h\to
0}u_{h}^{+}(x_{h},t_{h})\\\
u^{-}(x,t):=&\inf_{(x_{h},t_{h})\to(x,t)}\liminf_{h\to
0}u_{h}^{-}(x_{h},t_{h}),\end{split}$

and prove that the functions defined above are sub - and supersolutions,
respectively, to (7). The main difficulty in this regard is that we need to
work with just semicontinuous functions in space, as in the translationally
invariant setting one can easily prove the uniform equicontinuity of the
approximating sequence. We prove the following.

######  Theorem 1.3.

Assume (LABEL:standing_hp), (H1) and $f\in
C^{0}(\mathbb{R}^{n}\times[0,+\infty))$. The function $u^{+}$ (respectively
$u^{-}$) defined in (8) is a viscosity subsolution (respectively a viscosity
supersolution) of (7).

Thanks to the results of [16] we then prove that, under the additional
hypothesis

(H3) $\begin{split}&\nabla_{x}\nabla_{p}\phi(\cdot,p)\text{ and
}\nabla^{2}_{p}\phi(\cdot,p)\text{ are Lipschitz, uniformly for
}p\in{S^{N-1}}\\\ &\nabla_{p}^{2}\phi^{2}(x,p)\text{ is uniformly elliptic in
}p,\text{uniformly in }x\\\ &\psi(\cdot,p)\text{ Lipschitz continuous,
uniformly in }p\\\ &f(\cdot,t)\text{ Lipschitz continuous, uniformly in
}t,\end{split}$

the following uniqueness result holds.

######  Theorem 1.4.

Assume (LABEL:standing_hp) and (H3). If $u_{0}$ is a continuous function which
is spatially constant outside a compact set, equation (7) with initial
condition $u_{0}$ admits a unique continuous viscosity solution $u$ given by
(8). In particular, $u^{+}=u^{-}=u$ is the unique continuous viscosity
solution to (7) and $u_{h}^{\pm}\to u$ as $h\to 0$, locally uniformly.

The previous result yields a proof of consistency between the level set
approach and the minimizing movements one to study the evolution (1). We
recall that it has been established for the classical mean curvature flow in
[11], in the anisotropic but homogeneous case in [21] and in a very general
nonlocal setting in [14].

##  2\. Preliminaries

We start introducing some notations. We consider $0\in\mathbb{N}$. We will use
both $B_{r}(x)$ and $B(x,r)$ to denote the Euclidean ball in $\mathbb{R}^{N}$
centered in $x$ and of radius $r$; with $B^{N-1}_{r}(x)$ we denote the
Euclidean ball in $\mathbb{R}^{N-1}$ centered in $x$ and of radius $r$; with
${S^{N-1}}$ we denote the sphere $\partial B_{1}(0)\subseteq\mathbb{R}^{N}$;
with $Sym_{N}$ the symmetric real matrices of size $N\times N$. In the
following, we will always speak about measurable sets and refer to a set as
the union of all the points of density $1$ of that set i.e. $E=E^{(1)}.$ If
not otherwise stated, we implicitly assume that the function spaces considered
are defined on $\mathbb{R}^{N}$, e.g $L^{\infty}=L^{\infty}(\mathbb{R}^{N})$;
the space $C^{0}$ denotes the space of continuous functions. Moreover, we
often drop the measure with respect to which we are integrating, if clear from
the context.

######  Definition 2.1.

We define anisotropy (sometimes defined as an elliptic integrand) a function
$\psi$ with the following properties:
$\psi(x,p):\mathbb{R}^{N}\times\mathbb{R}^{N}\to[0,+\infty)$ is a continuous
function, which is convex and positively 1-homogeneous in the second variable,
such that

$\dfrac{1}{c_{\psi}}|p|\leq\psi(x,p)\leq c_{\psi}|p|$

for any point $x\in\mathbb{R}^{N}$ and vector $p\in\mathbb{R}^{N}$.

We remark that, as standard, we define a real function $f$ positively
1-homogeneous if for any $\lambda\geq 0,$ it holds $f(\lambda x)=\lambda
f(x)$. In particular, the anisotropies that we will consider are not
symmetric. In the following, we will always denote the gradient of an
anisotropy with respect to the first (respectively second) variable as
$\nabla_{x}\psi$ (respectively $\nabla_{p}\psi$). We then recall the
definition of some well-known quantities (see [8]). Define the polar function
of an anisotropy $\psi$, denoted with $\psi^{\circ}$, as

(9) $\psi^{\circ}(\cdot,\xi):=\sup_{p\in\mathbb{R}^{N}}\left\\{\xi\cdot p\ :\
\psi(\cdot,p)\leq 1\right\\}.$

Using the definition it is easy to see that for all $p,\xi\in\mathbb{R}^{N}$
it holds

$\psi(\cdot,p)\psi^{\circ}(\cdot,\xi)\geq
p\cdot\xi,\quad-\psi(\cdot,-p)\psi^{\circ}(\cdot,\xi)\leq p\cdot\xi.$

Furthermore, one can prove that (see [8]) for $p\neq 0$

$\psi^{\circ}(\nabla_{p}\psi)=1,\ \psi(\nabla_{p}\psi^{\circ})=1,\
(\psi^{\circ})^{\circ}=\psi.$

We define for any $x,y\in\mathbb{R}^{N}$ the geodesic distance induced by
$\psi,$ or $\psi-$distance in short, as

$\textnormal{dist}^{\psi}(x,y):=\inf\left\\{\int_{0}^{1}\psi^{\circ}(\gamma(t),\dot{\gamma}(t))\,\textnormal{d}t\
:\ \gamma\in W^{1,1}([0,1];\mathbb{R}^{N}),\gamma(0)=x,\gamma(1)=y\right\\}.$

We remark that this function is not symmetric in general. We define the signed
distance function from a closed set $E\subseteq\mathbb{R}^{N}$ as

(10) $\textnormal{sd}^{\psi}_{E}(x):=\inf_{y\in
E}\textnormal{dist}^{\psi}(y,x)-\inf_{y\notin
E}\textnormal{dist}^{\psi}(x,y),$

so that $\textnormal{sd}^{\psi}_{E}\geq 0$ on $E^{c}$ and
$\textnormal{sd}^{\psi}_{E}\leq 0$ in $E$. We remark that the bounds stated in
Definition 2.1 imply

(11) $\dfrac{1}{c_{\psi}}\textnormal{dist}\leq\textnormal{dist}^{\psi}\leq
c_{\psi}\textnormal{dist},$

where here and in the following we will denote with
$\textnormal{dist},\textnormal{sd}$ the Euclidean distance and signed distance
function respectively. We define the $\psi-$balls as the balls associated to
the $\psi-$distance, that is

$B^{\psi}_{\rho}(x):=\\{y\in\mathbb{R}^{N}:\text{dist}^{\psi}(y,x)<\rho\\},$

which in general are not convex nor symmetric.

######  Definition 2.2.

We say that an anisotropy $\phi$ is a regular elliptic integrand, and write
$\phi\in\mathscr{E}$, if there exists two constants $\lambda\geq 1,\,l\geq 0$
such that if $\phi(x,\cdot)\big{\lvert}_{{S^{N-1}}}\in C^{2,1}({S^{N-1}})$ and
for every $x,y,e\in\mathbb{R}^{N},\,\nu,\nu^{\prime}\in{S^{N-1}}$ one has:

$\displaystyle\dfrac{1}{\lambda}\leq\phi(x$ $\displaystyle,\nu)\leq\lambda,$
$\displaystyle|\phi(x,\nu)-\phi(y,\nu)|+|\nabla_{p}$
$\displaystyle\phi(x,\nu)-\nabla_{p}\phi(y,\nu)|\leq l|x-y|$
$\displaystyle|\nabla_{p}\phi(x,\nu)|+\|\nabla_{p}^{2}\phi(x,\nu)\|+$
$\displaystyle\dfrac{\|\nabla_{p}^{2}\phi(x,\nu)-\nabla_{p}^{2}\phi(x,\nu^{\prime})\|}{|\nu-\nu^{\prime}|}\leq\lambda$
$\displaystyle e\cdot\nabla_{p}^{2}\phi(x,\nu)[e]$
$\displaystyle\geq\dfrac{|e-(e\cdot\nu)\nu|^{2}}{\lambda}.$

Given any set of finite perimeter $E$, one can define the $\phi-$perimeter
$P_{\phi}$ as follows

$P_{\phi}(E):=\int_{\partial^{*}E}\phi(x,\nu_{E}(x))\,\textnormal{d}\mathcal{H}^{N-1}(x),$

where $\partial^{*}E$ is the reduced boundary of $E$ and $\nu_{E}$ is the
measure-theoretic outer normal, see [39] for further references on sets of
finite perimeter. The $\phi-$perimeter of a set of finite perimeter $E$ in an
open set $A$ is defined as

$P_{\phi}(E;A):=\int_{\partial^{*}E\cap
A}\phi(x,\nu_{E}(x))\,\textnormal{d}\mathcal{H}^{N-1}(x).$

We remark that, by definition of regular elliptic integrand, for any set $E$
of finite perimeter it holds

$\frac{1}{\lambda}P(E)\leq P_{\phi}(E)\leq\lambda P(E).$

Some additional remarks on this definition can be found in [18]. We just
recall the submodularity property of the $\phi-$perimeter, which can be proved
for instance by using the formulae for the reduced boundary and measure-
theoretic normal of union and intersection of sets of finite perimeter (see
[39]).

######  Proposition 2.3 (Submodularity property).

For any two sets $E,F\subseteq\mathbb{R}^{N}$ of finite perimeter, one has

(12) $P_{\phi}(E\cup F)+P_{\phi}(E\cap F)\leq P_{\phi}(E)+P_{\phi}(F).$

Moreover, by homogeneity, (9) and recalling that for any set $E$ of finite
perimeter it holds
$D\chi_{E}=-\nu_{E}\,\textnormal{d}\mathcal{H}^{N-1}{\big{\lvert}_{\partial^{*}E}}$
we have the following equivalent definitions

(13) $\displaystyle P_{\phi}(E)$
$\displaystyle=\sup\left\\{\int_{\mathbb{R}^{N}}-D\chi_{E}\cdot\xi\ :\ \xi\in
C^{1}_{c}(\mathbb{R}^{N};\mathbb{R}^{N}),\phi^{\circ}(\cdot,\xi)\leq
1\right\\}$
$\displaystyle=\sup\left\\{\int_{E}\textnormal{div}\,\xi\,\textnormal{d}\mathcal{H}^{N-1}\
:\ \xi\in C^{1}_{c}(\mathbb{R}^{N};\mathbb{R}^{N}),\phi^{\circ}(\cdot,\xi)\leq
1\right\\}.$

Concerning the regularity property of the $\phi-$perimeter minimizers, we
refer to [42]. We just recall the following results. Given two anisotropies
$\phi,\psi\in\mathscr{E}$, we define the “distance” between them as

$\displaystyle\textnormal{dist}_{\mathscr{E}}(\phi,\psi):=\sup\\{|\phi(x,p)-\psi(x,p)|$
$\displaystyle+|\nabla_{p}\phi(x,p)-\psi(x,p)|+|\nabla_{p}^{2}\phi(x,p)-\nabla_{p}^{2}\psi(x,p)|:x\in\mathbb{R}^{N},p\in{S^{N-1}}\\},$

where $|\cdot|$ denotes the Euclidian norm. Given $\phi\in\mathscr{E}$, we
recall that $E$ is a $0-$minimizer for the $\phi-$perimeter if for any
$x\in\mathbb{R}^{N},r>0$

$P_{\phi}(E;B_{r}(x))\leq P_{\phi}(F;B_{r}(x))$

for every $F\subset\mathbb{R}^{N}$ such that $F\triangle
E\subset\joinrel\subset B_{r}$. Then, some regularity properties of minimizers
of $\phi-$perimeter can be found in the theorems of part $II.7$ and $II.8$ in
[42], which are recalled below.

######  Theorem 2.4.

Assume $\phi\in\mathscr{E}$. Then, for any 0-minimizer $E$ of the
$\phi-$perimeter, the reduced boundary $\partial^{*}E$ of the set $E$ is of
class $C^{1,1/2}$ and the singular set $\Sigma:=\partial
E\setminus\partial^{*}E$ satisfies

$\mathcal{H}^{N-3}(\Sigma)=0.$

######  Theorem 2.5.

Let $m>0,\alpha\in(0,1)$. Then, there exists
$\varepsilon=\varepsilon(m,\alpha)>0$ with the following property: let
$\phi=\phi(p)\in\mathscr{E}$, $\phi\in
C^{3,\alpha}(\mathbb{R}^{N}\setminus\\{0\\})$ with

$\|\phi|_{S^{N-1}}\|_{C^{3,\alpha}}\leq m\ \text{and
}\textnormal{dist}_{\mathscr{E}}(\phi,|\cdot|)\leq\varepsilon.$

Then, for any 0-minimizer $E$ of the $\phi-$perimeter, the reduced boundary
$\partial^{*}E$ of the set $E$ is of class $C^{1,1/2}$ and the singular set
$\Sigma:=\partial E\setminus\partial^{*}E$ satisfies

$\mathcal{H}^{N-7}(\Sigma)=0.$

We sum up these hypotheses that yield the complete regularity of minimizers of
parametric elliptic integrands:

(14) $\begin{split}&\text{either }\phi\in\mathscr{E}\text{ and }N\leq 3,\\\
&\text{or }N\leq 7\text{ and the hypotheses of Theorem \ref{teo reg ellipt
integr} are satisfied}.\end{split}$

###  2.1. The first variation of the $\phi-$perimeter

In this section we compute the first variation of the $\phi$-perimeter and
define some additional operators associated to it.

Assume $E$ is of class $C^{2}$. Let $X$ be a smooth and compactly supported
vector field and assume $\Psi(x,t)=:\Psi_{t}(x)$ is the associated flow. To
simplify the notation, we write

$\nu(x,t)=\nabla_{x}\textnormal{sd}_{\Psi(E,t)}(x).$

By classical formulae (see e.g. [9]) we can compute the following. For the
sake of brevity, we avoid writing the evaluation $\phi=\phi(x,\nu_{E}(x))$, if
not otherwise specified, and assume that all the integrals are made with
respect to the Hausdorff $(N-1)$-dimensional measure $\mathcal{H}^{N-1}.$

$\displaystyle\dfrac{\,\textnormal{d}}{\,\textnormal{d}t}\Big{\lvert}_{t=0}P_{\phi}(E_{t})=\dfrac{\,\textnormal{d}}{\,\textnormal{d}t}\Big{\lvert}_{t=0}\int_{\partial
E}\phi(\Psi_{t}(x),\nu(\Psi_{t}(x),t))J\Psi_{t}$ (15)
$\displaystyle=\int_{\partial E}\nabla_{x}\phi\cdot
X+\nabla_{p}\phi\cdot\left(-\nabla_{\tau}(X\cdot\nu)+D\nu[X]\right)+\phi\,\textnormal{div}_{\tau}X$
$\displaystyle=\int_{\partial E}\nabla_{x}\phi\cdot
X+\nabla_{p}\phi\cdot\left(-\nabla_{\tau}(X\cdot\nu)+D\nu[X]\right)+\textnormal{div}_{\tau}(\phi
X)-\nabla\phi\cdot X+(\nabla\phi\cdot\nu)(X\cdot\nu)$
$\displaystyle=\int_{\partial E}\nabla_{x}\phi\cdot
X+\nabla_{p}\phi\cdot\left(-\nabla_{\tau}(X\cdot\nu)+D\nu[X]\right)-\nabla_{x}\phi\cdot
X-D\nu[\nabla_{p}\phi]\cdot X$
$\displaystyle\quad+\textnormal{div}_{\tau}(\phi
X)+(\nabla\phi\cdot\nu)(X\cdot\nu)$ $\displaystyle=\int_{\partial
E}-\nabla_{p}\phi\cdot\nabla_{\tau}(X\cdot\nu)+(\nabla_{x}\phi\cdot\nu)(X\cdot\nu)+\left(D\nu[\nabla_{p}\phi]\cdot\nu\right)(X\cdot\nu)+\textnormal{div}_{\tau}(\phi
X)$ $\displaystyle=\int_{\partial
E}\textnormal{div}_{\tau}\left(\nabla_{p}\phi(X\cdot\nu)\right)-\nabla_{p}\phi\cdot\nabla_{\tau}(X\cdot\nu)+(X\cdot\nu)(\nabla_{x}\phi\cdot\nu)$
$\displaystyle=\int_{\partial
E}(\textnormal{div}_{\tau}\nabla_{p}\phi)(X\cdot\nu)+\nabla_{p}\phi\cdot\nabla_{\tau}(X\cdot\nu)-\nabla_{p}\phi\cdot\nabla_{\tau}(X\cdot\nu)+(\nabla_{x}\phi\cdot\nu)(X\cdot\nu)$
$\displaystyle=\int_{\partial
E}(X\cdot\nu)\left(\textnormal{div}_{\tau}\nabla_{p}\phi+\nabla_{x}\phi\cdot\nu\right)=\int_{\partial
E}(X\cdot\nu)\,\textnormal{div}\nabla_{p}\phi$

where the last equality follows from the definition of
$\textnormal{div}_{\tau}$ and the fact that $\phi$ is $1-$homogeneous with
respect to the $p$ variable, since

$\displaystyle\textnormal{div}\nabla_{p}\phi$
$\displaystyle=\textnormal{div}_{\tau}\nabla_{p}\phi+\sum_{i}\nu_{i}\left({\partial_{x_{i}}}\nabla_{p}\phi\right)[\nu]$
$\displaystyle=\textnormal{div}_{\tau}\nabla_{p}\phi+\sum_{i}\nu_{i}\nabla_{p}(\partial_{x_{i}}\phi)\cdot\nu+\nu\cdot\left(\nabla^{2}_{p}\phi
D\nu\right)[\nu]$
$\displaystyle=\textnormal{div}_{\tau}\nabla_{p}\phi+\nabla_{x}\phi\cdot\nu.$

Therefore, we define the first variation of a $C^{2}-$regular set $E$, induced
by the vector field $X$, as

(16) $\delta P_{\phi}(E)[X\cdot\nu]:=\int_{\partial
E}(X(x)\cdot\nu(x))\,\textnormal{div}\nabla_{p}\phi(x,\nu(x))\,\textnormal{d}\mathcal{H}^{N-1}(x)$

and the $\phi-$curvature of the set $E$ as

(17) $H^{\phi}_{E}(x):=\textnormal{div}\nabla_{p}\phi(x,\nu(x)).$

If we now consider equation (15), we develop the tangential gradient to find

$\displaystyle\nabla_{p}\phi\cdot(-\nabla_{\tau}(X\cdot\nu)+D\nu[X])=\nabla_{p}\phi\cdot(-\nabla_{\tau}X[\nu]-D\nu[X]+D\nu[X])=0.$

This shows that for any set $E$ of class $C^{2}$ it holds

$\delta P_{\phi}(E)[X\cdot\nu]:=\int_{\partial E}\left(\nabla_{x}\phi\cdot
X+\phi\,\textnormal{div}_{\tau}X\right)\,\textnormal{d}\mathcal{H}^{N-1},$

where we dropped the evaluation of $\phi$ at $(x,\nu_{E}(x))$. We remark that
the expression on the right hand side makes sense even if the set $E$ is just
of finite perimeter. Defining the $\phi-$divergence operator
$\textnormal{div}_{\phi}$ as

(18) $\displaystyle\textnormal{div}_{\phi}X:=\nabla_{x}\phi\cdot
X+\phi\,\textnormal{div}_{\tau}X,$

we are led to define the distributional $\phi-$curvature of a set $E$ of
finite perimeter as an operator $H^{\phi}_{E}\in L^{1}(\partial E)$ (if it
exists) such that the following representation formula holds

(19) $\int_{\partial
E}\textnormal{div}_{\phi}X\,\textnormal{d}\mathcal{H}^{N-1}=\int_{\partial
E}H^{\phi}_{E}\,\nu_{E}\cdot X\,\textnormal{d}\mathcal{H}^{N-1},\qquad\forall
X\in C^{\infty}_{c}(\mathbb{R}^{N};\mathbb{R}^{N}).$

The previous computations allow to say that the distributional
$\phi-$curvature can be expressed as (17) if the set is of class $C^{2}$.
Finally, since $\phi$ is a regular elliptic integrand, one can prove the
following monotonicity result.

######  Lemma 2.6.

Let $E,F$ be two $C^{2}$ sets of finite $\phi-$perimeter with $E\subseteq F$,
and assume that $x\in\partial F\cap\partial E$: then $H^{\phi}_{F}(x)\leq
H^{\phi}_{E}(x).$

###### Proof.

Since the anisotropy is smooth, we can expand the curvature formula (17) as

(20)
$H^{\phi}=\text{tr}\left(\nabla_{x}\nabla_{p}\phi(x,\nu)+\nabla^{2}_{p}\phi(x,\nu)D\nu\right)$

and compare $H^{\phi}_{E}$ with $H^{\phi}_{F}$. We consider separately the two
terms appearing in (20). The first one depends on $\nu$ just by the value it
has at the point $x$. Therefore, since $\nu_{E}(x)=\nu_{F}(x)$ we have the
equality. The second one falls in the classical framework of smooth
anisotropies that do not depend on the space variable. Since $D\nu_{F}\leq
D\nu_{E}$ (as matrices) one concludes the proof. ∎

##  3\. The minimizing movements approach

In this section we follow the work of [41] (see also [3, 38]) to prove the
existence for the mean curvature flow via the minimizing movements approach.
We recall that in the whole paper we will assume the hypothesis
(LABEL:standing_hp).

###  3.1. The discrete scheme

In this subsection we will define the discrete scheme approximating the weak
solution of the mean curvature flow, and we shall study some of its
properties.

We define the following iterative scheme. Given $h>0,f\in
L^{\infty}(\mathbb{R}^{N}\times[0,\infty))$ and $t\geq h$, and given a bounded
set of finite perimeter $F$, we minimize the energy functional

(21)
$\mathscr{F}_{h,t}^{F}(E)=P_{\phi}(E)+\dfrac{1}{h}\int_{E}\textnormal{sd}^{\psi}_{F}(x)\,\textnormal{d}x-\int_{E}F_{h}(x,t)\,\textnormal{d}x$

in the class of all measurable sets $E\subseteq\mathbb{R}^{N}$, and where we
have set

$F_{h}(x,t):=\fint_{t}^{t+h}f(x,s)\,\textnormal{d}s.$

Equivalently, we could define the energy functional as

$\mathscr{F}_{h,t}^{F}(E)=P_{\phi}(E)+\dfrac{1}{h}\int_{E\triangle
F}|\textnormal{sd}^{\psi}_{F}|-\int_{E}F_{h}(x,t)\,\textnormal{d}x,$

which agrees with (21) up to a constant. Then, we denote

$T_{h,t}F=E\in\text{argmin}\ \mathscr{F}_{h,t}^{F}.$

We will refer to this minimizing procedure as the incremental problem. It is
well-known (compare (16) and [39, Proposition 17.8]) that a minimimum of (21)
of class $C^{2}$ satisfies the Euler-Lagrange equation

(22)
$\int_{\partial{E}}H^{\phi}_{{E}}X\cdot\nu_{{E}}\,\textnormal{d}\mathcal{H}^{N-1}=-\int_{\partial{E}}\left(\frac{1}{h}\textnormal{sd}^{\psi}_{{F}}(x)-F_{h}(x,t)\right)X(x)\cdot\nu_{{E}}(x)\,\textnormal{d}\mathcal{H}^{N-1}(x)$

for all $X\in C^{\infty}_{c}(\mathbb{R}^{N};\mathbb{R}^{N})$. We can then
define the discrete flow, which can be seen as a discrete-in-time
approximation of the mean curvature flow starting from the initial set
$E_{0}$. We define iteratively the discrete flow by setting
$E_{t}^{(h)}=E_{0}$ for $t\in[0,h)$ and

(23)
$E_{t}^{(h)}=T_{h,t-h}E_{t-h}^{(h)}=T_{h,([\frac{t}{h}]-1)h}E_{t-h}^{(h)},\qquad
t\in[h,+\infty),$

where $[\cdot]$ denotes the integer part of a real number. This section is
devoted to recall and prove some estimates on the discrete flow. The first one
is a well-known existence result.

######  Lemma.

For any measurable function $g:\mathbb{R}^{N}\to\mathbb{R}$ such that
$\min\\{g,0\\}\in L^{1}_{loc},$ the problem

$\min\left\\{P(E)+\int_{E}g\ :\ E\text{ is of finite perimeter}\right\\}$

admits a solution.

Consider now $F$ as a bounded set of finite perimeter. Then, the function
$g=\textnormal{sd}^{\psi}_{F}/h-F_{h}$ is coercive, thus $\min\\{g,0\\}\in
L^{1}$. Therefore, by the previous result and by classical arguments see [14,
Proposition 6.1] for a proof, one can prove the following result.

######  Lemma 3.1.

For any given set $F$ of finite perimeter, the problem (21) admits a solution
$E$, which satisfies the discrete dissipation inequality

$P_{\phi}(E)+\dfrac{1}{h}\int_{E\triangle F}|\textnormal{sd}^{\psi}_{F}|\leq
P_{\phi}(F)+\int_{E\setminus F}F_{h}(x,t)\,\textnormal{d}x-\int_{F\setminus
E}F_{h}(x,t)\,\textnormal{d}x.$

Moreover, the problem (21) admits a minimal and a maximal solution.

We define $T_{h,t}^{+}F$ (respectively $T_{h,t}^{-}F$) as the maximal
(respectively minimal) solution to (21) having as initial datum $F$. In the
following, whenever no confusion is possible, we shall write $T_{h,t}$ instead
of $T_{h,t}^{-}$.

A comparison result holds. We will consider just bounded sets as datum for the
problem (21), but the same result holds in general for unbounded sets (see
also Section 4.1 for the case of unbounded sets with bounded boundary). The
proof of this result is classical (see e.g. [14]) and it is based on the
submodularity of the perimeter (12). We will omit it.

######  Lemma 3.2 (Weak comparison principle).

Assume that $F_{1},F_{2}$ are bounded sets with
$F_{1}{\subset\joinrel\subset}F_{2}$ and consider $g_{1},g_{2}\in L^{\infty}$
with $g_{1}\geq g_{2}$. Then, for any two solutions $E_{i}$, $i=1,2$ of the
problems

$\min\left\\{P_{\phi}(E)+\int_{E}\dfrac{\textnormal{sd}^{\psi}_{F_{i}}}{h}+g_{i}:E\textnormal{
is of finite perimeter}\right\\},$

we have $E_{1}\subseteq E_{2}$. If, instead, $F_{1}\subseteq F_{2}$, then we
have that the minimal (respectively maximal) solution to (21) for $i=1$ is
contained in the minimal (respectively maximal) solution to (21) for $i=2$.

We now prove the volume-density estimates for minimizers of problem (21). This
result is based on the minimality properties of almost-minimizers for
perimeters induced by regular elliptic integrands (see [18, Remark 1.9] for
further results). These estimates have the disadvantage that the smallness
condition on the radius depends on the parameter $h$. Subsequently, we will
recall a finer result in the spirit of [38], where we can drop this dependence
by making some restrictions on the balls considered.

######  Lemma 3.3.

Let $g\in L^{\infty}$ and assume $E$ minimizes the functional

$\mathscr{F}(F)=P_{\phi}(F)+\int_{F}g$

among all measurable subsets of $\mathbb{R}^{N}$. Then the density estimate

$\displaystyle\sigma\rho^{N}$ $\displaystyle\leq|B_{\rho}(x)\cap
E|\leq(1-\sigma)\rho^{N}$ (24) $\displaystyle\sigma\rho^{N-1}$
$\displaystyle\leq P_{\phi}(E;B_{\rho}(x))\leq(1-\sigma)\rho^{N-1}$

holds for all $x\in\partial^{*}E,$
$0<\rho<(2\lambda\|g\|_{\infty})^{-1}:=\rho_{0},$ for a suitable
$\sigma=\sigma(N,c_{\psi},\lambda)$.

###### Proof.

By minimality,

$P_{\phi}(E)\leq P_{\phi}(F)+\|g\|_{\infty}|E\triangle F|\qquad\forall
F\subseteq\mathbb{R}^{N},$

thus [18, Lemma 2.8] implies the thesis. ∎

######  Remark 3.4.

We remark that the previous result allows us to choose the minimal solution to
(21) to be an open set, and the maximal one to be a closed set. This follows
from the fact that the density estimates imply that the boundary of any
minimizer has zero measure.

We now recall [12, Lemma 3.7], which is an anisotropic version of [38, Remark
1.4]. It provides volume-density estimates for minimizers of (21) starting
from $E$, uniform in $\psi$ and $h$, holding in the exterior of $E$. We remark
that, even if in the reference the anisotropy $\phi$ considered did not depend
on $x$, all the arguments hold with minor modifications also in our case. We
recall the proof of this result, as similar techniques will be used later on.

######  Lemma 3.5.

Let $E$ be a bounded, closed set, $h>0$ , and $g\in
L^{\infty}(\mathbb{R}^{N})$. Let $E^{\prime}$ be a minimizer of

$P_{\phi}(F)+\int_{F}\dfrac{\textnormal{sd}^{\psi}_{E}}{h}+g.$

Then, there exists $\sigma>0$, depending on $\lambda$, and $r_{0}\in(0,1)$,
depending only on $N,\lambda,G:=\|g\|_{L^{\infty}(F)}$, with the following
property: if $\bar{x}$ is such that $|E^{\prime}\cap B_{s}(\bar{x})|>0$ for
all $s>0$ and $B_{r}(\bar{x})\cap E=\emptyset$ with $r\leq r_{0}$, then

(25) $|E^{\prime}\cap B_{r}(\bar{x})|\geq\sigma r^{N}.$

Analogously, if $\bar{x}$ is such that $|B_{s}(\bar{x})\setminus
E^{\prime}|>0$ for all $s>0$ and $B_{r}(\bar{x})\subseteq E$ with $r\leq
r_{0}$, then

$|B_{r}(\bar{x})\setminus E^{\prime}|\geq\sigma r^{N}.$

###### Proof.

For all $s\in(0,r)$, set $E^{\prime}(s):=E^{\prime}\setminus B_{s}(\bar{x})$.
Note that, for a.e. $s$ we have

$P_{\phi}(E^{\prime}(s))=P_{\phi}(E^{\prime})-P_{\phi}(E^{\prime}\cap
B_{s}(\bar{x}))+\int_{E^{\prime}\cap\partial
B_{s}(\bar{x})}\left(\phi(x,\nu(x))+\phi(x,-\nu(x))\right)\,\textnormal{d}\mathcal{H}^{N-1}(x),$

where $\nu$ denotes the outer normal vector of the set $E^{\prime}\cap\partial
B_{s}(\bar{x})$. Since $E^{\prime}\cap B_{s}(\bar{x})\subset E^{c}$ and
$\textnormal{sd}^{\psi}_{E}\geq 0$ in $E^{c}$, one has $\int_{E^{\prime}\cap
B_{s}(\bar{x})}\textnormal{sd}^{\psi}_{E}\geq 0,$ and therefore the minimality
of $E^{\prime}$ implies

$P_{\phi}(E^{\prime}\cap B_{s}(\bar{x}))+\int_{E^{\prime}\cap
B_{s}(\bar{x})}g\leq\int_{E^{\prime}\cap\partial
B_{s}(\bar{x})}\left(\phi(x,\nu(x))+\phi(x,-\nu(x))\right)\,\textnormal{d}\mathcal{H}^{N-1}(x).$

By the bound on the $\phi-$perimeter and using the classical isoperimetric
inequality (whose constant is denoted $C_{N}$) we obtain

$\displaystyle 2\lambda\mathcal{H}^{N-1}(E^{\prime}\cap\partial
B_{s}(\bar{x}))$ $\displaystyle\geq\frac{1}{\lambda}P(E^{\prime}\cap
B_{s}(\bar{x}))+\int_{E^{\prime}\cap B_{s}(\bar{x})}g$
$\displaystyle\geq\frac{1}{\lambda}C_{N}|E^{\prime}\cap
B_{s}(\bar{x})|^{\frac{N-1}{N}}-\|g\|_{\infty}|E^{\prime}\cap
B_{s}(\bar{x})|\geq\dfrac{C_{N}}{2\lambda}|E^{\prime}\cap
B_{s}(\bar{x})|^{\frac{N-1}{N}},$

provided $|E^{\prime}\cap B_{s}(\bar{x})|^{1/N}\leq
C_{N}/(2\lambda\|g\|_{\infty})$, which is true if $r_{0}$ is small enough.
Since the rhs is positive for every $s$, we conclude

(26) $\dfrac{\,\textnormal{d}}{\,\textnormal{d}s}|E^{\prime}\cap
B_{s}(\bar{x})|^{\frac{1}{N}}\geq\dfrac{C_{N}}{4\lambda^{2}N}\quad\text{for
a.e. }s\in(0,r).$

The thesis follows by integrating the above differential inequality. The other
case is analogous. ∎

######  Remark 3.6.

Requiring that the anisotropy $\psi$ is bounded uniformly from above and below
ensures that the results of the previous Lemmas 3.3 and 3.5 can be read in
terms of the $\psi-$balls. For example, for any $r\geq 0$ and
$x\in\mathbb{R}^{N}$, equation (25) could be read as $|E^{\prime}\cap
B^{\psi}_{r}(\bar{x})|\geq\sigma c_{\psi}^{-N}r^{N},$ provided $\bar{x}$ is
such that $|E^{\prime}\cap B^{\psi}_{s}(\bar{x})|>0$ for all $s>0$ and
$B_{r}^{\psi}(\bar{x})\cap E=\emptyset$, and holds for all $r\leq
r_{0}/c_{\psi}$. Here, $\sigma$ is as in Lemma 3.5 and depends only on
$\lambda$. Analogous statements holds for Lemma 3.9.

We now provide some estimates on the evolution of balls under the discrete
flow. We start by a simple remark concerning the boundedness of the evolving
sets.

######  Remark 3.7.

A simple estimate on the energies implies that the minimizers of (21) are
bounded whenever $F$ is bounded. Indeed, assume $F\subseteq B_{R}$ and
consider $B_{\rho}(x)\cap(E\setminus B_{R})\neq\emptyset$: testing the
minimality of $E$ against $F$ we easily deduce

$\dfrac{R}{2h}|B_{\rho}(x)\cap E|\leq\int_{E\cap
B_{\rho}(x)}\frac{\textnormal{sd}^{\psi}_{F}}{h}\leq
P_{\phi}(F)+\|F_{h}(\cdot,t)\|_{\infty}|E\triangle F|\leq
P_{\phi}(F)+\|f\|_{\infty}(|F|+|E|).$

Employing the density estimates of Lemma 3.5 and sending $R\to\infty$, we get
a contradiction, as the isoperimetric inequality implies that $|E|$ is bounded
since $\mathscr{F}_{h,t}^{F}(F)<\infty.$

We now want to prove finer estimates on the speed of evolution of balls. These
estimates are classically a crucial step in order to prove existence of the
flow. In the case under study, the main difficulties come from the
inhomogeneity of the functionals considered, as in the homogeneous case
convexity arguments easily yield the boundedness result, for example. We will
use a “variational” approach in the spirit of [14] (but see also [41, Lemma
3.8] for a different proof relying more on the smoothness of the evolving
set).

######  Lemma 3.8.

For every $R_{0}>0$ there exist $h_{0}(R_{0})>0$ and $C(R_{0},\phi,\psi,f)>0$
with the following property: For all $R\geq R_{0}$, $h\in(0,h_{0})$, $t>0$ and
$x\in\mathbb{R}^{N}$ one has

(27) $T_{h,t}(B_{R}(x))\supset B_{R-Ch}(x).$

###### Proof.

We divide the proof into three steps. In the following, the constants
$\sigma,r_{0}$ are those of Lemma 3.5. We will assume $x=0$ for simplicity. We
fix $R\geq R_{0}$ and denote $E:=T_{h,t}B_{R}$.

Step 1. We prove that, given $a\in(0,\sigma),\varepsilon\in(0,1)$, we can
ensure $|B_{R(1-\varepsilon)}\setminus E|<a\,R^{N}(1-\varepsilon)^{N}$ for $h$
small enough. Indeed, assume by contradiction $|B_{R(1-\varepsilon)}\setminus
E|\geq a\,R^{N}(1-\varepsilon)^{N}$. Testing the minimality of $E$ against
$B_{R}$, we obtain

$\int_{(B_{R(1-\varepsilon)}\setminus E)\cup(E\setminus
B_{R})}\dfrac{|\textnormal{sd}^{\psi}_{B_{R}}|}{h}\leq\dfrac{1}{h}\int_{B_{R}\triangle
E}|\textnormal{sd}^{\psi}_{B_{R}}|\leq P_{\phi}(B_{R}){-\int_{B_{R}\setminus
E}F_{h}+\int_{E\setminus B_{R}}F_{h}},$

and estimating $|\textnormal{sd}^{\psi}_{B_{R}}|\geq{R}\varepsilon/c_{\psi}$
on $B_{R(1-\varepsilon)}\setminus E$, we get

$\displaystyle\dfrac{{R}\varepsilon}{hc_{\psi}}|B_{R(1-\varepsilon)}\setminus
E|\leq
P_{\phi}(B_{R})+\|f\|_{\infty}\left(\omega_{N}R^{N}+|B_{R(1+\varepsilon)}\setminus
B_{R}|\right)+\int_{E\setminus
B_{R(1+\varepsilon)}}\left(F_{h}-\dfrac{|\textnormal{sd}^{\psi}_{B_{R}}|}{h}\right).$

Taking $h\leq\varepsilon/(c_{\psi}\|f\|_{\infty})$, the last term on the rhs
is negative, thus

$\dfrac{{R}\varepsilon}{hc_{\psi}}|B_{R(1-\varepsilon)}\setminus E|\leq
P_{\phi}(B_{R})+\|f\|_{\infty}R^{N}(\omega_{N}+2^{N+1}\varepsilon).$

We employ the hypothesis to obtain

$\dfrac{a}{hc_{\psi}}\,\varepsilon(1-\varepsilon)^{N}R^{N+1}\leq
c_{\psi}N\omega_{N}R^{N-1}+cR^{N},$

a contradiction for $h\leq
ca\,\varepsilon\,(1-\varepsilon)^{N}\min\\{1,{R^{2}}\\},$ where $c$ is a
constant depending on $N,\phi,\psi,\|f\|_{\infty}.$

Step 2. Using Step 1, we prove that $B_{R/2}\subset E$ for $h$ small. Assume
that $R\leq r_{0}$: by following the second part of the proof of Lemma 3.5 we
obtain equation (26), which reads

$\dfrac{\,\textnormal{d}}{\,\textnormal{d}s}|B_{s}\setminus
E|^{1/N}\geq\dfrac{C_{N}}{4\lambda^{2}N}=\sigma^{1/N}\quad\text{for a.e
}s\in(0,R).$

Applying the previous step with $\varepsilon=1/4,a=\sigma/3^{N}$, it holds
$|B_{3R/4}\setminus E|\leq\sigma R^{N}/4^{N}$ for all $h\leq
c(N,\phi,\psi,f)R$. Therefore, one deduces the existence of a positive
extinction radius

(28) $R^{*}=\dfrac{3R}{4}-\dfrac{|B_{3R/4}\setminus
E|^{1/N}}{\sigma^{1/N}}\geq\frac{R}{2}$

such that $|B_{R^{*}}\setminus E|=0$, which proves the claim. Clearly, taking
$h\leq cR_{0}$ the smallness assumption on $h$ is uniform for $R\geq R_{0}$.

If $R\geq r_{0}$ one simply uses a covering argument. For any $x\in
B_{R-r_{0}}$, applying the previous result to the ball $B_{r_{0}}(x)$ and
using the comparison principle of Lemma 3.2, we conclude that $\forall h\leq
c\,r_{0}$ it holds

$\bigcup_{x\in B_{R-r_{0}}}B_{r_{0}/2}(x){\subset\joinrel\subset}E.$

Step 3. We conclude the proof. By the previous two steps and Remark 3.7,
taking $h$ small enough, we see that

$\rho:=\sup\\{r>0\ :\ |B_{r}\setminus E|=0\\}\in{(R/2,+\infty)}.$

We can assume $\rho\leq R,$ otherwise the result of the lemma is trivial.
Consider the vector field $\nabla_{p}\phi\left(x,\frac{x}{|x|}\right)\in
C^{1}(\mathbb{R}^{N},\mathbb{R}^{N})$. Then, recalling (13), we get
$P_{\phi}({G})\geq-\int_{\mathbb{R}^{N}}D\chi_{G}\cdot\nabla_{p}\phi(x,x/|x|)$
for all $G$ set of finite perimeter and

$P_{\phi}((1+\varepsilon)B_{\rho})=\int_{\mathbb{R}^{N}}D\chi_{(1+\varepsilon)B_{\rho}}\cdot\left(-\nabla_{p}\phi\left(x,\frac{x}{|x|}\right)\right).$

Setting $W_{\varepsilon}=(1+\varepsilon)B_{\rho}\setminus E$, by submodularity
on $(1+\varepsilon)B_{\rho},E$ and exploiting the minimality of $E$, we obtain

$\displaystyle\int_{\mathbb{R}^{N}}\nabla_{p}\phi\left(x,\dfrac{x}{|x|}\right)\cdot
D\chi_{W_{\varepsilon}}$
$\displaystyle=\int_{\mathbb{R}^{N}}\nabla_{p}\phi\left(x,\frac{x}{|x|}\right)\cdot\left(D\chi_{(1+\varepsilon)B_{\rho}}-D\chi_{(1+\varepsilon)B_{\rho}\cap
E}\right)$ $\displaystyle\leq P_{\phi}((1+\varepsilon)B_{\rho}\cap
E)-P_{\phi}((1+\varepsilon)B_{\rho})$ $\displaystyle\leq
P_{\phi}(E)-P_{\phi}((1+\varepsilon)B_{\rho}\cup E)$
$\displaystyle\leq\dfrac{1}{h}\int_{W^{\varepsilon}}\textnormal{sd}^{\psi}_{B_{R}}{-}\int_{W_{\varepsilon}}F_{h}(x,t)\,\textnormal{d}x.$

We conclude, using the divergence theorem ,

$\int_{W^{\varepsilon}}-\textnormal{div}\nabla_{p}\phi\left(x,\frac{x}{|x|}\right)\leq\dfrac{1}{h}\int_{W^{\varepsilon}}\textnormal{sd}^{\psi}_{B_{R}}+\|f\|_{\infty}|W_{\varepsilon}|.$

Dividing by $|W^{\varepsilon}|$ and sending $\varepsilon\to 0$ we obtain

$\fint_{\partial B_{\rho}\cap
E}-\textnormal{div}\nabla_{p}\phi\left(x,\frac{x}{|x|}\right){\,\textnormal{d}\mathcal{H}^{N-1}}\leq\frac{1}{c_{\psi}}\dfrac{\rho-R}{h}+\|f\|_{\infty}.$

Exploiting the regularity assumptions on $\phi$, we remark that

$|\textnormal{div}\nabla_{p}\phi|=|\text{tr}\left(\nabla_{x}\nabla_{p}\phi+\nabla^{2}_{p}\phi\nabla(x/|x|)\right)|\leq
C\left(1+\frac{1}{|x|}\right).$

Thus, we obtain

$-C\left(1+\dfrac{1}{\rho}\right)\leq\dfrac{\rho-R}{h},$

which implies that $\rho\in(0,\rho_{1})\cup(\rho_{2},R)$ for
$\rho_{1,2}=\left(R-Ch\mp\sqrt{(R-Ch)^{2}-4Ch}\right)/2,$ as long as $h\leq
R_{0}^{2}/(4C)$. Since the choice $\rho\leq\rho_{1}<R/2$ is not admissible, we
conclude the proof by estimating

$\rho_{2}=R-Ch+\dfrac{R-Ch}{2}\left(\sqrt{1-\dfrac{4Ch}{(R-Ch)^{2}}}-1\right)\geq
R-Ch-\dfrac{Ch}{R-Ch},$

from which the thesis follows. ∎

The proof of the previous result can be employed to prove an estimate from
above of the evolution speed of the flow, as the following result shows. Since
the proof follows the same lines and is easier in this case, we only sketch
it.

######  Lemma 3.9.

Fix $T>0$ and $R_{0}>0$. Then, there exist positive constants
$C=C(\phi,\psi,f,R_{0})$ and $h_{0}=h_{0}(R_{0})$ such that, for every $R\geq
R_{0}$ and $h\leq h_{0}$, if $E_{0}\subseteq B_{R}$, then
$E_{t}^{(h)}\subseteq B_{R+CT}$ for all $t\in(0,T)$.

###### Proof.

Choose $h$ small as in the previous result and set

$\rho=\inf\\{r>0\ :\ |E\setminus B_{r}|=0\\}\in{(R/2,+\infty)}.$

We can assume $\rho\geq R,$ otherwise the result is trivial. Defining
$W_{\varepsilon}=E\setminus(1-\varepsilon)B_{\rho}$ and reasoning as before we
obtain

$\displaystyle\int_{\mathbb{R}^{N}}\nabla_{p}\phi\left(x,\dfrac{x}{|x|}\right)\cdot
D\chi_{W^{\varepsilon}}$
$\displaystyle=\int_{\mathbb{R}^{N}}\nabla_{p}\phi\left(x,\dfrac{x}{|x|}\right)\cdot\left(D\chi_{(1-\varepsilon)B_{\rho}\cup
E}-D\chi_{(1-\varepsilon)B_{\rho}}\right)$ $\displaystyle\geq-
P_{\phi}((1-\varepsilon)B_{\rho}\cup E)+P_{\phi}((1-\varepsilon)B_{\rho})$
$\displaystyle\geq-P_{\phi}(E)+P_{\phi}((1-\varepsilon)B_{\rho}\cap E)$
$\displaystyle\geq\dfrac{1}{h}\int_{W^{\varepsilon}}\textnormal{sd}^{\psi}_{B_{R}}{-}\int_{W_{\varepsilon}}F_{h}(x,t)\,\textnormal{d}x.$

As in the previous proof, we arrive at

$\dfrac{\rho-R}{h}\leq C\left(1+\dfrac{1}{\rho}\right),$

which implies that
$\rho\leq\rho_{2}=\left(R+Ch+\sqrt{(R+Ch)^{2}+4Ch}\right)/2\leq R+Ch,$ up to
changing $C$. ∎

###  3.2. Existence of flat flows

In the following, we will prove that the discrete flow (defined in (23))
defines a discrete-in-time approximation of a weak solution to the mean
curvature flow, which is usually known as a “flat” flow (because the
approximating surfaces $\partial^{*}E_{t}^{(h)}$ converge in the “flat”
distance of Whitney to the limit $\partial^{*}E_{t}$, see [3]).

We start by proving uniform bounds on the distance between two consecutive
sets of the discrete flow and on the symmetric difference between them. We
introduce the time-discrete normal velocity: for all $t\geq 0$ and
$x\in\mathbb{R}^{N}$, we set

$v_{h}(x,t):=\begin{cases}\frac{1}{h}\textnormal{sd}^{\psi}_{E_{t-h}^{(h)}}(x)\quad&\text{for
}t\in[h,+\infty)\\\ 0&\text{for }t\in[0,h).\end{cases}$

The following result provides a bound on the $L^{\infty}-$norm of the discrete
velocity. Since the proof is essentially the same of [38, Lemma 2.1], we will
omit it. The only difference is that we use the upper and lower bounds of (11)
to work with Euclidean balls.

######  Lemma 3.10.

There exists a positive constant $c_{\infty}$ depending only on $N,\psi$ with
the following property. Let $E_{0}$ be a bounded set of finite perimeter and
let $\\{E_{t}^{(h)}\\}_{t\in(0,T)}$ be a discrete flow starting from $E_{0}$.
Then,

$\sup_{E_{t}^{(h)}\triangle E_{t-h}^{(h)}}|v_{h}(\cdot,t)|\leq
c_{\infty}h^{-1/2}$

for all $h$ sufficiently small.

The following result can be found in [41, Proposition 3.4] (see also [25,
Lemma 2.2]): it provides an estimate on the volume of the symmetric difference
of two consecutive sets of the discrete flow. The proof is analogous to the
one in the reference.

######  Lemma 3.11.

There exists a constant $C$ such that for every $t\geq h$ the discrete flow
$E_{t}^{(h)}$ satisfies for all $h$ sufficiently small

(29) $|E_{t+h}^{(h)}\triangle E_{t}^{(h)}|\leq
C\left(lP_{\phi}(E_{t}^{(h)})+\dfrac{1}{l}\int_{E_{t}^{(h)}\triangle
E_{t+h}^{(h)}}|\textnormal{sd}^{\psi}_{E_{t}^{(h)}}|\right)\quad\forall l\leq
c\sqrt{h},$

where $c$ is a positive constant depending on $N,\psi$.

We are now able to prove an uniform bound on the perimeter of the evolving
sets. The proof follows [25, Proposition 2.3].

######  Lemma 3.12.

For any initial bounded set $E_{0}$ of finite $\phi-$perimeter and $h$ small
enough, the discrete flow $\\{E_{t}^{(h)}\\}$ satisfies

$P_{\phi}(E_{t}^{(h)})\leq C_{T}\quad\forall t\in(0,T),$

for a suitable constant $C_{T}=C_{T}(T,E_{0},f,\phi,\psi)$.

###### Proof.

By testing the minimality of $E^{(h)}_{t}$ against $E^{(h)}_{t-h}$ we obtain
$\forall t\in[h,T)$

(30) $P_{\phi}(E^{(h)}_{t})+{\frac{1}{h}}\int_{E_{t}^{(h)}\triangle
E_{t-h}^{(h)}}|\textnormal{sd}^{\psi}_{E_{t-h}^{(h)}}|\leq
P_{\phi}(E^{(h)}_{t-h})+\|f\|_{\infty}|E_{t}^{(h)}\triangle E_{t-h}^{(h)}|.$

Combining this estimate with (29) for $l=2Ch\|f\|_{\infty}\ll\sqrt{h}$, where
$C$ is the constant appearing in equation (29), we obtain for $h$ sufficiently
small

(31) $P_{\phi}(E^{(h)}_{t})+\dfrac{1}{{2h}}\int_{E_{t}^{(h)}\triangle
E_{t-h}^{(h)}}|\textnormal{sd}^{\psi}_{E_{t-h}^{(h)}}|\leq\left(1+{2C^{2}}h\|f\|_{\infty}^{2}\right)P_{\phi}(E^{(h)}_{t-h})$

Iterating the previous estimate, we find

$\displaystyle P_{\phi}(E^{(h)}_{t})$
$\displaystyle\leq(1+{2C^{2}\|f\|_{\infty}}h)^{\left[\frac{t}{h}\right]-1}P_{\phi}(E_{h}^{(h)}).$

In order to estimate $P_{\phi}(E^{(h)}_{h})$ we start by observing that Remark
3.7, for $h=h(E_{0})$ small enough, implies $E_{h}^{(h)}\subseteq B_{2r}$,
where $E_{0}\subseteq B_{r}$. Therefore, by (30) for $t=h$ we obtain
$P_{\phi}(E_{h}^{(h)})\leq P_{\phi}(E_{0})+c$ and we conclude
$P_{\phi}(E_{t}^{(h)})\leq C_{T}(P_{\phi}(E_{0})+1)$. ∎

We then present a sketch of the proof of the local Hölder continuity in time
of the discrete flow, uniformly in $h$, which can be deduced as in [25,
Proposition 2.3]. We highlight the main differences.

######  Proposition 3.13.

Let $E_{0}$ be an initial bounded set of finite $\phi-$perimeter and $T>0$.
Then, for $h$ small enough, for a discrete flow $\\{E^{(h)}_{t}\\}$ starting
from $E_{0}$ it holds

$|E_{t}^{(h)}\triangle E_{s}^{(h)}|\leq C_{T}|t-s|^{1/2}\quad\forall h\leq
t\leq s<T,$

for a suitable constant $C_{T}=C_{T}(T,E_{0},f,\phi,\psi)$.

###### Proof.

Following the previous proof, employing again (31) we find

$\displaystyle P_{\phi}({E_{2h}^{(h)}})$
$\displaystyle+{\frac{1}{2}}\int_{E_{{2h}}^{(h)}\triangle
E_{h}^{(h)}}|v_{h}{(\cdot,2h)}|+{\frac{1}{2}}\int_{E_{h}^{(h)}\triangle
E_{0}^{(h)}}|v_{h}{(\cdot,h)}|$
$\displaystyle\leq(1+ch)P_{\phi}(E_{h}^{(h)})+{\frac{1}{2}}\int_{E_{h}^{(h)}\triangle
E_{0}^{(h)}}|v_{h}{(\cdot,h)}|$
$\displaystyle\leq(1+ch)\left(P_{\phi}(E_{h}^{(h)})+\int_{E_{h}^{(h)}\triangle
E_{0}^{(h)}}|v_{h}|{(\cdot,h)}\right)\leq(1+ch)^{2}P_{\phi}(E_{0}).$

Iterating, we conclude as before

(32) $\sum_{k=1}^{[T/h]}\int_{E_{kh}^{(h)}\triangle
E_{(k-1)h}^{(h)}}|v_{h}{(\cdot,kh)}|\leq C_{T}(P_{\phi}(E_{0})+1).$

Therefore, combining the previous results and applying (29) with
$l=h\ll\sqrt{h}$, we obtain

(33) $\int_{h}^{T}|E_{t}^{(h)}\triangle E_{t-h}^{(h)}|\leq
c\sum_{k=1}^{[T/h]}\left(hP_{\phi}(E_{kh}^{(h)})+\int_{E_{kh}^{(h)}\triangle
E_{(k-1)h}^{(h)}}|v_{h}{(\cdot,kh)}|\right)\leq
C_{T}\left(P_{\phi}(E_{0})+1\right).$

The proof then follows the one of [25, Proposition 2.3], from equation $(2.5)$
onward. ∎

We finally prove the main result of this section, the existence of flat flows.

###### Proof of Theorem 1.1.

The proof is classical and we only sketch it. By the uniform equicontinuity of
the approximating sequence of Proposition 3.13 and compactness of sets of
finite perimeter (by Lemma 3.9 and 3.12) we can use the Ascoli-Arzelà theorem
to prove that the sequence $(E_{t}^{(h_{k})})_{k\in\mathbb{N}}$ converges in
$L^{1}$ to sets $E_{t}$ for all times $t\geq 0$ and that the family
$\\{E_{t}\\}_{t\geq 0}$ satisfies the $1/2-$Hölder continuity property,
locally uniformly in time. The other property is then easily deduced. ∎

###  3.3. Existence of distributional solutions

From Theorem 1.1 we deduce the existence of a subsequence $(h_{k})_{k\geq 0}$
such that

(34)
$D\chi_{E_{t}^{(h_{k})}}\overset{\ast}{\rightharpoonup}D\chi_{E_{t}}\qquad\forall
t\geq 0.$

We will also assume (4), remarking that it implies

(35)
$\lim_{k\to\infty}P_{\phi}(E_{t}^{(h_{k})})=P_{\phi}(E_{t})\qquad\text{for
a.e. }t\in[0,+\infty).$

Our aim is to derive (5) and (6) from the Euler-Lagrange equation (22) and
passing to the limit $h\to 0$. To achieve this, we will prove that the
discrete velocity is a good approximation (up to multiplicative factors) of
the discrete evolution speed of the sets. Notice that (5) is a weak
formulation of (1), while (6) establishes the link between $v$ and the
velocity of the boundaries of $E_{t}$. Indeed, law (1) can be interpreted as
looking for a family $\\{E_{t}\\}_{t\geq 0}$ of sets, whose normal vector
$\nu_{E_{t}}$ and $\phi-$curvature $H^{\phi}_{E_{t}}$ are well-defined objects
and a function $v\ :\ [0,\infty)\times\mathbb{R}^{N}\to\mathbb{R}$ such that
for every $t\in[0,+\infty)$ and $x\in\partial E_{t}$

(36) $\begin{cases}v(x,t)&=-H^{\phi}_{E_{t}}(x)+f(x,t)\\\
V(x,t)&=\psi(x,\nu_{E_{t}}(x))v(x,t),\end{cases}$

where $V$ represents the normal velocity of evolution, obtained as the limit
as $h\to 0$ (in a suitable sense) of the ratio

$\frac{\chi_{E_{t}}-\chi_{E_{t-h}}}{h}.$

In this whole section we will assume that hypothesis (LABEL:hp_dim) holds. In
particular, the sets defining the discrete flow are smooth hypersurfaces in
$\mathbb{R}^{N}$. Moreover, we require hypotheses (H1) to hold.

We start by estimating in time the $L^{2}-$norm of the discrete velocity. The
proof is the same as the one presented in [41, Lemma 3.6], up to using the
density estimates on the $\phi-$perimeter of Lemma 3.3 and considering the
$\psi-$balls instead of the Euclidean one.

######  Proposition 3.14.

Let $\\{E_{t}^{(h)}\\}_{t\geq 0}$ be a discrete flow starting from an initial
bounded set $E_{0}$ of finite $\phi-$perimeter. Then, for any $T>0$ and for
$h$ small enough, it holds

$\int_{0}^{T}\int_{\partial
E_{t}^{(h)}}v_{h}^{2}\,\textnormal{d}\mathcal{H}^{N-1}\,\textnormal{d}t\leq
C_{T},$

for a suitable constant $C_{T}=C_{T}(T,E_{0},\phi,\psi,f)$.

Recalling now the Euler-Lagrange equation (22) and Lemma 3.12 we conclude

(37) $\int_{0}^{T}\int_{\partial
E_{t}^{(h)}}\left(H^{\phi}_{E_{t}^{(h)}}\right)^{2}=\int_{0}^{T}\int_{\partial
E_{t}^{(h)}}\left(v_{h}-F_{h}\right)^{2}\leq C_{T},$

We now prove an estimate on the error between the discrete velocity
$\psi(\cdot,\nu_{E_{t}})v_{h}(\cdot,t)$ and the discrete time derivative of
$\chi_{h}$. The proof of this result is based on a double blow-up argument,
and the smoothness of sets (locally) minimizing the $\phi-$perimeter is
essential. We will split the proof in various lemmas: the first one concerns
the composition of blow-ups, and is a well-known result to the experts. We
present a simple proof since we could not find a reference.

######  Lemma 3.15 (Composition of blow-ups).

Consider $0<\beta<\beta^{\prime}<1$. Assume that $(A_{h})_{h\in[0,1]}$ is a
family of measurable sets such that the following blow-ups converge as $h\to
0$

$\displaystyle\dfrac{{A_{h}-x_{h}}}{h^{\beta}}\to A_{1}$ in $L^{1}_{loc}$
$\displaystyle h^{-(\beta^{\prime}-\beta)}A_{1}\to A_{2}$
$\displaystyle\quad\text{in $L^{1}_{loc}$},$

where $x_{h}\in\partial A_{h}$ for all $h\in[0,1]$. Then, if the composition
of the blow-ups $h^{-\beta^{\prime}}{(A_{h}-x_{h})}$ converges in
$L^{1}_{loc}$, the limit coincides with $A_{2}$.

###### Proof.

We can assume wlog ${x_{h}}=0$. Denote with $A_{3}=L^{1}_{loc}-\lim_{h\to
0}h^{-\beta^{\prime}}{A_{h}}$. We fix a ball $B_{M}$ and $\varepsilon>0$.
There exists $h^{*}$ such that $\forall h\leq h^{*}$ it holds

$|((h^{-\beta^{\prime}}{A_{h}})\triangle A_{3})\cap
B_{M}|\leq\varepsilon,\quad|((h^{-\beta^{\prime}+\beta}A_{1})\triangle
A_{2})\cap B_{M}|\leq\varepsilon.$

We fix $h$ and wlog assume $Mh^{\beta^{\prime}-\beta}\leq 1$. Taking
$\tilde{h}<h$ suitably small (depending on $h,\varepsilon$), we can ensure

$|((\tilde{h}^{-\beta}{A_{h}})\triangle A_{1})\cap B_{1}|\leq\varepsilon
h^{N(\beta^{\prime}-\beta)}.$

Since $\tilde{h}^{-\beta}h^{-(\beta^{\prime}-\beta)}>h^{-\beta^{\prime}}$,
there exists $\bar{h}<h$ such that
$\bar{h}^{-\beta^{\prime}}=\tilde{h}^{-\beta}h^{-(\beta^{\prime}-\beta)}.$ We
can then estimate

$\displaystyle|(A_{3}\triangle A_{2})\cap B_{M}|$
$\displaystyle\leq|(A_{3}\triangle\bar{h}^{-\beta^{\prime}}{A_{h}})\cap
B_{M}|+|((h^{-\beta^{\prime}+\beta})A_{1}\triangle(\bar{h}^{-\beta^{\prime}}{A_{h}}))\cap
B_{M}|$ $\displaystyle\ +|((h^{-\beta^{\prime}+\beta}A_{1})\triangle
A_{2})\cap B_{M}|$ $\displaystyle\leq
2\varepsilon+h^{-N(\beta^{\prime}-\beta)}|(A_{1}\triangle(\tilde{h}^{-\beta}{A_{h}}))\cap
B_{Mh^{\beta^{\prime}-\beta}}|$ $\displaystyle\leq
2\varepsilon+h^{-N(\beta^{\prime}-\beta)}|(A_{1}\triangle(\tilde{h}^{-\beta}{A_{h}}))\cap
B_{1}|\leq 3\varepsilon.$

∎

We now compute some estimates on the normal vector on the boundary of the
evolving sets, following the proof of [41, Lemma 4.2] (see also [38,
Proposition 2.2]). We fix $c_{\infty}$ as the constant appearing in Lemma
3.10.

In the sequel, we will denote by $\omega(h)$ a modulus of continuity, that is
a continuous increasing function $\omega:[0,1]\to\mathbb{R}$ with
$\omega(0)=0$, which can possibly change from statement to statement and line
to line to absorb constants independent of $h$.

######  Lemma 3.16.

Assume (LABEL:standing_hp) and (H1). For given constants
$1/2<\beta^{\prime}<\alpha<1$ and $T>2$, there exists a modulus of continuity
$\omega$ with the following property. Consider $t\in[2h,T]$ and
${x_{h}}\in\partial E_{t}^{(h)}$ such that

(38) $|v_{h}(t,y)|\leq h^{\alpha-1}\quad\forall y\in
B_{c_{\infty}\sqrt{h}}({x_{h}})\cap(E_{t}^{(h)}\triangle E_{t-h}^{(h)}).$

Then, there exists $\nu\in{S^{N-1}}$ such that

$\displaystyle|\nu_{E_{t}^{(h)}}(\cdot)-\nu|$
$\displaystyle\leq\omega(h)\quad\text{in
}B_{h^{\beta^{\prime}}}({x_{h}})\cap\partial E_{t}^{(h)}$ (39)
$\displaystyle|\nu_{E_{t-h}^{(h)}}(\cdot)-\nu|$
$\displaystyle\leq\omega(h)\quad\text{in
}B_{h^{\beta^{\prime}}}({x_{h}})\cap\partial E_{t-h}^{(h)}.$

###### Proof.

We fix $\frac{1}{2}<\beta<\beta^{\prime}<\alpha$ and
$0<R<h^{\frac{1}{2}-\beta}/c_{\psi}$. Testing the minimality of
$E^{(h)}_{s},s=t,t-h$, we find

(40) $P_{\phi}(E_{s}^{(h)},B_{Rh^{\beta}}({x_{h}}))\leq
P_{\phi}(G,B_{Rh^{\beta}}({x_{h}}))+\dfrac{1}{h}\int_{G\triangle
E_{s}^{(h)}}|\textnormal{sd}^{\psi}_{E_{s-h}^{(h)}}|+\int_{G\triangle
E_{s}^{(h)}}|F_{h}|,$

for any set $G$ of finite perimeter such that $G\triangle
E_{s}^{(h)}{\subset\joinrel\subset}B_{Rh^{\beta}}({x_{h}})$. Using Lemma 3.10,
the $1-$Lipschitz regularity of $\textnormal{sd}^{\psi}$ and (38), we deduce
$|v_{h}(s,y)|\leq
c_{\psi}Rh^{\beta-1}+c_{\infty}h^{-1/2}\leq(1+c_{\infty})h^{-1/2}$ for any
$y\in B_{Rh^{\beta}}({x_{h}})\cap(E^{(h)}_{s}\triangle F)$. Plugging this
inequality in (40), we find

(41) $P_{\phi}(E_{s}^{(h)},B_{Rh^{\beta}}({x_{h}}))\leq
P_{\phi}(G,B_{Rh^{\beta}}({x_{h}}))+\dfrac{1+c}{\sqrt{h}}|F\triangle
E_{s}^{(h)}|+\|f\|_{\infty}|G\triangle E_{s}^{(h)}|.$

We then introduce the blown-up sets for $s=t,t-h$, defined as

$\displaystyle E_{s}^{(h),\beta}:=h^{-\beta}\left(E^{(h)}_{s}-{x_{h}}\right).$

Rescaling equation (41), we easily find that $E_{s}^{(h),\beta}$ is a
$(\Lambda_{h},r_{h})-$minimizer of the
$\phi({x_{h}}+h^{\beta}\cdot,\cdot)-$perimeter, with
$\Lambda_{h}=(1+c)h^{\beta-1/2},\ r_{h}=h^{1/2-\beta}$. Moreover, scaling the
density estimates (24) we have a uniform bound on the perimeters of the sets
$E_{s}^{(h),\beta}$ in each ball $B_{R}$. By compactness, there exist two sets
$E_{1}^{\beta},E_{2}^{\beta}$ such that

$E_{t}^{(h),\beta}\to E_{1}^{\beta},\ E_{t-h}^{(h),\beta}\to
E_{2}^{\beta}\qquad\text{in }L^{1}_{loc}.$

Then, by scaling and (38) we find

$|\textnormal{sd}^{\psi}_{E_{t-h}^{(h),\beta}}(\cdot)|\leq{c_{\infty}}h^{\alpha-\beta}\qquad\text{on
}B_{h^{1/2-\beta}}(0)\cap(E_{t}^{(h),\beta}\triangle E_{t-h}^{(h),\beta}),$

thus we easily conclude that $E^{\beta}:=E_{1}^{\beta}=E^{\beta}_{2}.$ By
Lemma 3.9 we can assume that $x_{h}\to x_{0}$ as $h\to 0$, up to subsequences.
Moreover, by closeness of $\Lambda_{h}-$minimizers under
$L^{1}_{loc}-$convergence (see e.g. [18, Theorem 2.9]), one can see that
$E^{\beta}$ is a 0-minimizer for the $\phi(x_{0},\cdot)-$perimeter. Thus, by
complete regularity, it is a smooth $C^{2}$ set. We can then employ the
classic blow-up theorem to deduce that, for a fixed
$\beta^{\prime}\in(\beta,\alpha)$, the blow-up
$h^{-(\beta^{\prime}-\beta)}E^{\beta}$ converges to a half-space
$\mathbb{H}=\\{x\cdot\nu\leq 0\\}$ as $h\to 0$. Finally, the blow-ups

$E_{s}^{(h),\beta^{\prime}}:=\dfrac{E^{(h)}_{s}-{x_{h}}}{h^{\beta^{\prime}}}$

admit a converging subsequence by compactness of sets of finite perimeter and
by rescaling equation (41). Thus, the previous Lemma 3.15 implies

$E_{s}^{(h),\beta^{\prime}}\to\mathbb{H}\quad\text{in }L^{1}_{loc}$

as $h\to 0$. To conclude, the $\varepsilon-$regularity Theorem for
$\Lambda-$minimizers (see e.g. [18, Theorem 3.1]) ensures that
$E_{s}^{(h),\beta^{\prime}}$ are uniformly $C^{1,\frac{1}{2}}$ sets in
$B_{1}(0)$ for $s=t,t-h$ as $h\to 0$. ∎

We recall here an approximation result proved in [38] (see also [41] for a
more detailed proof). We remark that the proof of this result is purely
geometric and does not rely on the variational problem satisfied by the sets
$E_{t}^{(h)},E_{t-h}^{(h)}$.

######  Corollary (Corollary 4.3 in [41]).

Under the hypotheses of Lemma 3.16, fix $0<\beta<\alpha$ and let
$\textbf{C}_{h^{\beta}}$ be the open cylinder defined as

$\textbf{C}_{h^{\beta}}(x_{h},\nu):=\left\\{x\in\mathbb{R}^{N}:|(x-x_{h})\cdot\nu|<\frac{h^{\beta}}{2},\
\bigg{|}(x-x_{h})-\left((x-x_{h})\cdot\nu\right)\nu\bigg{|}<\frac{h^{\beta}}{2}\right\\}.$

Then, it holds

$\displaystyle\Big{|}\int_{\textbf{C}_{h^{\beta}/2}(x_{h},\nu)}(\chi_{E_{t}^{(h)}}-\chi_{E_{t-h}^{(h)}})$
$\displaystyle\,\textnormal{d}x-\int_{\partial
E_{t}^{(h)}\cap\textbf{C}_{h^{\beta}/2}(x_{h},\nu)}\textnormal{sd}_{E^{(h)}_{t-h}}\,\textnormal{d}\mathcal{H}^{N-1}\Big{|}$
$\displaystyle\leq\omega(h)\int_{\textbf{C}_{h^{\beta}/2}(x_{h},\nu)}|\chi_{E_{t}^{(h)}}-\chi_{E_{t-h}^{(h)}}|.$

Carefully inspecting the proof, one indeed proves that there exists a
geometric constant $C$ such that for any $y\in B^{N-1}_{h^{\beta}/2}(x_{h})$

(42) $|\textnormal{sd}_{E_{t-h}^{(h)}}(y,f_{t}^{(h)}(y))\sqrt{1+|\nabla
f_{t}^{(h)}(y)|^{2}}-\left(f_{t}^{(h)}(y)-f_{t-h}^{(h)}(y)\right)|\leq\omega(h)|f_{t}^{(h)}(y)-f_{t-h}^{(h)}(y)|,$

where we set

$\partial
E_{s}^{(h)}\cap\textbf{C}=\\{(y,f_{s}^{(h)}(y))\in\mathbb{R}^{N-1}\times\mathbb{R},\
|y|\leq h^{\beta}/2\\},$

for $s=t,t-h$.

We briefly recall some classical results. Consider an anisotropy $\psi$,
independent of the position. It is well-known that, for any closed set
$G\subseteq\mathbb{R}^{N}$, setting $\textnormal{sd}^{\psi}_{G}$ as the
distance from $G$ induced by $\psi^{\circ}$, then the gradient of
$\textnormal{sd}^{\psi}_{G}$ exists almost everywhere and satisfies the
eikonal equation (for a proof see for instance [10, Remark 2.2])

(43) $\psi(\nabla\textnormal{sd}^{\psi}_{G})=1$

almost everywhere. Moreover, in this particular case, in the definition of
$\text{dist}^{\psi}$ we can consider just straight lines as follows from a
simple application of Jensen’s inequality: for any curve $\gamma$ as in the
definition of $\textnormal{dist}^{\psi},$ we have

$\int_{0}^{1}\psi^{\circ}(\dot{\gamma}(t))\,\textnormal{d}t\geq\psi^{\circ}\left(\int_{0}^{1}\dot{\gamma}\right)=\psi^{\circ}(y-x).$

######  Proposition 3.17 (Estimate on almost flat sets).

Under the hypotheses of Lemma 3.16 and with the same notation, fix
$\beta\in(0,\alpha)$ and let $\textbf{C}_{h^{\beta}}$ be the open cylinder
defined as

$\textbf{C}_{h^{\beta}}(x_{h},\nu):=\left\\{x\in\mathbb{R}^{N}:|(x-x_{h})\cdot\nu|<\frac{h^{\beta}}{2},\
\bigg{|}(x-x_{h})-\left((x-x_{h})\cdot\nu\right)\nu\bigg{|}<\frac{h^{\beta}}{2}\right\\}.$

Then, it holds

$\displaystyle\Big{|}\int_{\textbf{C}_{h^{\beta}/2}(x_{h},\nu)}(\chi_{E_{t}^{(h)}}-\chi_{E_{t-h}^{(h)}})$
$\displaystyle\,\textnormal{d}x-\int_{\partial
E_{t}^{(h)}\cap\textbf{C}_{h^{\beta}/2}(x_{h},\nu)}\psi(x,\nu_{E_{t}^{(h)}})\,\textnormal{sd}^{\psi}_{E^{(h)}_{t-h}}\,\textnormal{d}\mathcal{H}^{N-1}\Big{|}$
$\displaystyle\leq\omega(h)\int_{\textbf{C}_{h^{\beta}/2}(x_{h},\nu)}|\chi_{E_{t}^{(h)}}-\chi_{E_{t-h}^{(h)}}|.$

###### Proof.

We recall that the modulus of continuity $\omega$ may change from line to line
to absorb constants independent of $h$.

From the previous Lemma 3.16 we know that, for $h$ suitably small, both
$\partial E_{t}^{(h)}$ and $\partial E_{t-h}^{(h)}$ in
$\textbf{C}_{h^{\beta}/2}(x_{h},\nu)$ can be written as graphs of functions of
class $C^{1,\frac{1}{2}}$. Up to a change of coordinates, we can assume wlog
that $x_{h}=0,\nu=e_{N}$. For simplicity, we set $\textbf{
C}=\textbf{C}_{h^{\beta}/2}(0,e_{N}).$ We thus find

$\partial
E_{s}^{(h)}\cap\textbf{C}=\\{(y,f_{s}^{(h)}(y))\in\mathbb{R}^{N-1}\times\mathbb{R},\
|y|\leq h^{\beta}/2\\}$

for $s=t,t-h$, where $f_{s}^{(h)}:B_{h^{\beta}/2}^{N-1}\to\mathbb{R}$ are
$C^{1,\frac{1}{2}}$ functions with

$\|\nabla f_{s}^{(h)}\|_{L^{\infty}(B_{h^{\beta}/2})}\leq\omega(h).$

We want to prove the following slightly stronger pointwise inequality: namely,
that for any point $x=(y,f_{t}^{(h)}(y))\in\partial
E_{t}^{(h)}\cap\textbf{C}$, it holds

(44)
$\left\lvert\textnormal{sd}^{\psi}_{E_{t-h}^{(h)}}(x)\,\psi(x,\nu_{E_{t}^{(h)}}(x))\sqrt{1+|\nabla
f_{t}^{(h)}(y)|}-\left(f_{t}^{(h)}(y)-f_{t-h}^{(h)}(y)\right)\right\rvert\leq\omega(h)|f_{t}^{(h)}(y)-f_{t-h}^{(h)}(y)|.$

Integrating the previous inequality over C yields the thesis. Clearly, it is
enough to prove (44) at each point $x$ such that
$|\textnormal{sd}^{\psi}_{E_{t-h}^{(h)}}(x)|>0$. We thus fix
$x=(y,f_{t}^{(h)}(y))\in\partial E_{t}^{(h)}\cap\textbf{C}$ and denote by
$x^{\prime}:=(y,f_{t-h}^{(h)}(y))$. We remark that these points depend on $h$,
but we drop the subscript to ease notation. It can be assumed without loss of
generality that $x\notin E_{t-h}^{(h)}$, as the other case is analogous.
Step 1 We now prove that, with the notation previously introduced, it holds

(45)
$|\textnormal{sd}^{\prime}_{E_{t-h}^{(h)}}(x)-\textnormal{sd}^{\psi}_{E_{t-h}^{(h)}}(x)|\leq\omega(h)|f_{t}^{(h)}(y)-f_{t-h}^{(h)}(y)|,$

where $\textnormal{sd}^{\prime}$ denotes the signed distance function induced
by the anisotropy $\psi(x^{\prime},\cdot)$. Let $\gamma$ be a smooth curve,
with $\gamma(0)=x,\gamma(1)\in\partial E_{t-h}^{(h)}$ to be used in the
definition of the geodesic distance $\textnormal{sd}^{\psi}_{E^{(h)}_{t-h}}$.
Firstly, we remark that one could assume

(46) $\gamma([0,1])\subseteq
B(x,2c_{\psi}^{2}|f_{t}^{(h)}(y)-f_{t-h}^{(h)}(y)|)$

Indeed, if it were not the case, the lower bounds contained in (11) and (42)
allow us to estimate

(47)
$\displaystyle\int_{0}^{1}\psi^{\circ}(\gamma,\dot{\gamma})\,\textnormal{d}t\geq\dfrac{1}{c_{\psi}}\int_{0}^{1}|\dot{\gamma}|\,\textnormal{d}t\geq
2c_{\psi}|f_{t}^{(h)}(y)-f_{t-h}^{(h)}(y)|\geq
2c_{\psi}\,\textnormal{sd}_{E_{t-h}^{(h)}}(x)\geq
2\,\textnormal{sd}^{\psi}_{E_{t-h}^{(h)}}(x),$

a contradiction for $h$ small. We can reason analogously for
$\textnormal{sd}^{\prime}_{E^{(h)}_{t-h}}$. In particular, we can consider
just curves having length $\int_{0}^{1}|\dot{\gamma}|\leq
c|f_{t}^{(h)}(y)-f_{t-h}^{(h)}(y)|$. Therefore, we obtain (by homogeneity)

$\displaystyle\textnormal{sd}^{\psi}_{E_{t-h}^{(h)}}(x)$
$\displaystyle\leq\int_{0}^{1}\psi^{\circ}(\gamma,\dot{\gamma})\,\textnormal{d}t\leq\int_{0}^{1}\psi^{\circ}(x^{\prime},\dot{\gamma})\,\textnormal{d}t+\sup_{\nu\in{S^{N-1}},\,t\in[0,1]}|\psi(\gamma(t),\nu)-\psi(x^{\prime},\nu)|\int_{0}^{1}|\dot{\gamma}|$
$\displaystyle\leq\int_{0}^{1}\psi^{\circ}(x^{\prime},\dot{\gamma})\,\textnormal{d}t+c\,\omega(h)|f_{t}^{(h)}(y)-f_{t-h}^{(h)}(h)|,$

and, taking the $\inf_{\gamma}$, we obtain
$\textnormal{sd}^{\psi}_{E_{t-h}^{(h)}}(x)\leq\textnormal{sd}^{\prime}_{E_{t-h}^{(h)}}(x)+\omega(h)|f_{t}^{(h)}(y)-f_{t-h}^{(h)}(y)|$.
The converse inequality can be proved analogously, yielding (45).

Therefore, in what follows we will consider always the anisotropy frozen in
$x^{\prime}$, and use $\textnormal{sd}^{\prime}$ instead of
$\textnormal{sd}^{\psi}$. Finally, let $p\in\partial E_{t-h}^{(h)}$ a
minimizer for the definition of $\textnormal{sd}^{\prime}_{E^{(h)}_{t-h}}(x)$.
In the following, with $\Pi^{v}_{\mathbb{H}}z$, $\Pi_{\mathbb{H}}z$ we denote
respectively the projection on the hyperplane $\mathbb{H}$ of $z$ along the
direction $v$ and the orthogonal projection of $z$ on $\mathbb{H}$.

Step 2. In this step we assume that $E_{t-h}^{(h)}\cap\textbf{C}$ coincides
with the halfspace $\mathbb{H}=p+\\{z\cdot\nu\leq 0\\}$ intersected with the
same cylinder and prove claim (44).
To this aim, we start noticing that by translation we may assume $p=0$ and
that
$\textnormal{sd}^{\prime}_{\mathbb{H}}(z+\xi)=\textnormal{sd}^{\prime}_{\mathbb{H}}(z)$
for all $z\in\mathbb{R}^{N}$ and for all $\xi$ orthogonal to $\nu$. Hence, in
fact,

(48)
$\textnormal{sd}^{\prime}_{\mathbb{H}}(z)=\textnormal{sd}^{\prime}_{\mathbb{H}}((z\cdot\nu)\nu)=(z\cdot\nu)\textnormal{sd}^{\prime}_{\mathbb{H}}(\nu)\,.$

Therefore, $\textnormal{sd}^{\prime}_{\mathbb{H}}$ is differentiable
everywhere, with
$\nabla\textnormal{sd}^{\prime}_{\mathbb{H}}=\textnormal{sd}^{\prime}_{\mathbb{H}}(\nu)\nu$.
Recalling the eikonal equation (43), it must hold
$\textnormal{sd}^{\prime}_{\mathbb{H}}(\nu)=1/\psi(x^{\prime},\nu)$ and in
turn, from (48), and choosing $z=x$, we have

(49)
$\textnormal{sd}^{\prime}_{\mathbb{H}}(x)\psi(x^{\prime},\nu)=x\cdot\nu=\textnormal{sd}_{\mathbb{H}}(x).$

We remark that
$\textnormal{sd}^{\prime}_{\mathbb{H}}(x)=\textnormal{sd}^{\prime}_{E_{t-h}^{(h)}}(x)$
by (38), thus we conclude (44) by combining (49) with (42).

Step 3. We now conclude in the general case. With the notation introduced at
the end of Step 1, set $\nu=\nu_{E_{t-h}^{(h)}}{(p)}$, and consider the half-
space $\mathbb{H}=p+\\{z\cdot\nu\leq 0\\}$ and
$w:=x^{\prime}-\Pi_{\mathbb{H}}(x^{\prime})$ as depicted in Figure 1. We shall
prove that

$|w|\leq\omega(h)|f_{t}^{(h)}(y)-f_{t-h}^{(h)}(y)|.$

To see this, we start by remarking that (39) implies

$|e_{N}-e_{N}(e_{N}\cdot\nu_{E_{t-h}^{(h)}})|\leq\omega(h)\qquad\text{in
}\partial E_{t-h}^{(h)}\cap\textbf{C},$

implying $e_{N}\cdot\nu_{E_{t-h}^{(h)}}\geq 1-\omega(h)$, and thus, for any
versor $v$ tangent to $\partial E_{t-h}^{(h)}\cap\textbf{C}$ one has $|v\cdot
e_{N}|\leq\omega(h)$. Therefore, we have $(x^{\prime}-p)\cdot
e_{N}\leq\omega(h)|x^{\prime}-p|$ and also

$\displaystyle\dfrac{x^{\prime}-p}{|x^{\prime}-p|}\cdot\nu$
$\displaystyle=\dfrac{x^{\prime}-p}{|x^{\prime}-p|}\cdot\left(e_{N}(\nu\cdot
e_{N})+\nu-e_{N}(\nu\cdot e_{N})\right)$ $\displaystyle\leq\omega(h)+|\nu-
e_{N}(\nu\cdot e_{N})|=\omega(h)+\left(1-|\nu\cdot e_{N}|^{2}\right)^{1/2}$
$\displaystyle\leq 3\sqrt{\omega(h)},$

$\mathbb{H}$$x$$x^{\prime}$$E_{t-h}^{(h)}$$E_{t}^{(h)}$$p$$\Pi_{\mathbb{H}}^{\nu}x$$w$$\Pi_{\mathbb{H}}^{e_{N}}x$$\gamma$$\nu$
Figure 1. The situation in the proof of the lemma.

by choosing $h$ small. Up to defining $\sqrt{\omega}$ as $\omega$, using the
previous estimate and the bounds (46) we see that

(50)
$|w|=|x^{\prime}-p|\left(\dfrac{x^{\prime}-p}{|x^{\prime}-p|}\cdot\nu\right)\leq\omega(h)|x^{\prime}-p|\leq\omega(h)|f_{t}^{(h)}(y)-f_{t-h}^{(h)}(y)|.$

We now remark that
$\textnormal{sd}^{\prime}_{E_{t-h}^{(h)}}(x)=\textnormal{sd}^{\prime}_{\mathbb{H}}(x)$
(by convexity of the anisotropy $\psi(x^{\prime},\cdot)$) and so, applying the
previous step to $\mathbb{H}$ and using also (45), we get

$\left\lvert\textnormal{sd}^{\psi}_{E_{t-h}^{(h)}}(x)\,\psi(x,\nu_{E_{t}^{(h)}}(x))\sqrt{1+|\nabla
f_{t}^{(h)}(y)|}-|x-\Pi^{e_{N}}_{\mathbb{H}}x|\right\rvert\leq\omega(h)|x-\Pi_{\mathbb{H}}^{e_{N}}x|.$

We conclude (44) by estimating

$\big{\lvert}|x-\Pi_{\mathbb{H}}^{e_{N}}x|-|x-x^{\prime}|\Big{\rvert}\leq|x^{\prime}-\Pi_{\mathbb{H}}^{e_{N}}x|=|w|/|\nu\cdot
e_{N}|\leq\dfrac{\omega(h)}{1-\omega(h)}|f_{t}^{(h)}(y)-f_{t-h}^{(h)}(y)|,$

where we used (50). We conclude the proof by a simple change of coordinates
and using (44) to find

$\displaystyle\bigg{\lvert}$ $\displaystyle\int_{\partial
E_{t}^{(h)}\cap\textbf{C}}\psi(x,\nu_{E_{t}^{(h)}}(x))\,\textnormal{sd}^{\psi}_{E_{t-h}^{(h)}}(x)\,\textnormal{d}\mathcal{H}^{N-1}-\int_{B_{h^{\beta}/2}}f_{t}^{(h)}(y)-f_{t-h}^{(h)}(y)\,\textnormal{d}y\bigg{\rvert}$
$\displaystyle=\bigg{\lvert}\int_{B_{{h^{\beta}}/2}}\psi((y,f_{t}^{(h)}(y)),\nu_{E_{t}^{(h)}}(y,f_{t}^{(h)}(y)))\,\textnormal{sd}^{\psi}_{E_{t-h}^{(h)}}(y,f_{t}^{(h)}(y))\sqrt{1+|\nabla
f_{t}^{(h)}(y)|^{2}}-(f_{t}^{(h)}(y)-f_{t-h}^{(h)}(y))\,\textnormal{d}y\bigg{\lvert}$
$\displaystyle\leq\omega(h)\int_{B_{h^{\beta}/2}}|f_{t}^{(h)}-f_{t-h}^{(h)}|\,\textnormal{d}y.$

∎

Finally, we are able to prove that the error generated by approximating the
discrete velocity with $v_{h}$ goes to zero as $h\to 0$. We follow the lines
of [38, Proposition 2.2].

######  Proposition 3.18 (Error estimate).

Under the hypothesis of Lemma 3.16, the error in the discrete curvature
equation vanishes in the limit $h\to 0$, namely

(51) $\lim_{h\to
0}\,\left\lvert\frac{1}{h}\int_{0}^{T}\left(\int_{E_{t}^{(h)}}\eta\,\textnormal{d}x-\int_{E_{t-h}^{(h)}}\eta\,\textnormal{d}x\right)\,\textnormal{d}t-\int_{0}^{T}\int_{\partial
E_{t}^{(h)}}\psi(x,\nu_{E_{t}^{(h)}})v_{h}\eta\,\textnormal{d}\mathcal{H}^{N-1}(x)\,\textnormal{d}t\right\rvert=0$

for all $\eta\in C^{1}_{c}(\mathbb{R}^{N}\times[0,T))$.

###### Proof.

We fix $t\in[2h,\infty)$ and $\alpha\in(\frac{1}{2},\frac{N+2}{2N+2})$. For
any point $x_{h}\in\partial E_{t}^{(h)}$ we define the open set $A_{x_{h}}$
defined as follows:

  * (i)

if (38) holds, we set $A_{x_{h}}=\textbf{C}_{h^{\beta}/2}(x_{h},\nu)$, with
the notations of Corollary 3.17;

  * (ii)

otherwise we set $A_{x_{h}}=B(x_{h},c_{\infty}\sqrt{h})$, where $c_{\infty}$
is the constant of Lemma 3.10.

By Lemma 3.10, the family $\\{A_{x_{h}}:x_{h}\in\partial E_{t}^{(h)}\\}$ is a
covering of $E_{t}^{(h)}\triangle E_{t-h}^{(h)}.$ By a simple application of
Besicovitch’s theorem (see e.g. [39]), we find a finite collection of points
$I\subseteq\partial E_{t}^{(h)}$ such that $\\{A_{x_{h}}\\}_{x_{h}\in I}$ is a
covering of $E_{t}^{(h)}\triangle E_{t-h}^{(h)}$ with the finite intersection
property. We proceed to estimate (51) on each $A_{x_{h}}$ belonging to this
family.
Estimate in case (i) We use Proposition 3.17 to deduce

$\displaystyle\bigg{\lvert}$
$\displaystyle\int_{A_{x_{h}}}(\chi_{E_{t}^{(h)}}-\chi_{E_{t-h}^{(h)}})\eta\,\textnormal{d}x-\int_{\partial
E_{t}^{(h)}\cap
A_{x_{h}}}\psi(x,\nu_{E_{t}^{(h)}})\textnormal{sd}^{\psi}_{E_{t-h}^{(h)}}\eta\,\textnormal{d}\mathcal{H}^{N-1}\bigg{\rvert}$
$\displaystyle\leq|\eta(x_{h},t)|\bigg{\lvert}\int_{A_{x_{h}}}(\chi_{E_{t}^{(h)}}-\chi_{E_{t-h}^{(h)}})-\int_{\partial
E_{t}^{(h)}\cap
A_{x_{h}}}\psi(x,\nu_{E_{t}^{(h)}})\textnormal{sd}^{\psi}_{E_{t-h}^{(h)}}\,\textnormal{d}\mathcal{H}^{N-1}\bigg{\rvert}$
$\displaystyle+\
\bigg{\lvert}\int_{A_{x_{h}}}(\chi_{E_{t}^{(h)}}-\chi_{E_{t-h}^{(h)}})(\eta-\eta(x_{h},t))-\int_{\partial
E_{t}^{(h)}\cap
A_{x_{h}}}(\eta-\eta(x_{h},t))\,\psi(x,\nu_{E_{t}^{(h)}})\,\textnormal{sd}^{\psi}_{E_{t-h}^{(h)}}\,\textnormal{d}\mathcal{H}^{N-1}\bigg{\rvert}$
(52) $\displaystyle\leq
C(\omega(h)\|\eta\|_{\infty}+h^{\beta}\|\nabla\eta\|_{\infty})\int_{A_{x_{h}}}|\chi_{E_{t}^{(h)}}-\chi_{E_{t-h}^{(h)}}|\,\textnormal{d}\mathcal{H}^{N-1}+ch^{\beta}\|\nabla\eta\|_{\infty}P(E_{t}^{(h)},A_{x_{h}}).$

Estimate in case (ii) By assumption $\exists y\in
B_{c_{\infty}\sqrt{h}}(x_{h})\cap(E_{t}^{(h)}\triangle E_{t-h}^{(h)})$ such
that $|v_{h}(t,y)|>h^{\alpha-1}$. We can assume wlog $y\in E_{t}^{(h)}$. We
then have $B(y,h^{\alpha}/(2c_{\psi}))\subseteq\mathbb{R}^{N}\setminus
E_{t-h}^{(h)}$ and
$\textnormal{sd}^{\psi}_{E_{t-h}^{(h)}}>h^{\alpha}/(2c_{\psi}^{2})$ on
$B(y,h^{\alpha}/(2c_{\psi}))$. Since $h^{\alpha}<<h^{1/2}$, we can use the
density estimates of Lemma 3.3 to deduce

$ch^{(N+1)\alpha-1}\leq\int_{B(y,h^{\alpha}/(2c_{\psi}))\cap(E_{t}^{(h)}\triangle
E_{t-h}^{(h)})}|v_{h}|\,\textnormal{d}x.$

Analogously, recalling also Lemma 3.10, we deduce

$\int_{B(x_{h},c_{\infty}\sqrt{h})\cap\partial
E_{t}^{(h)}}|\psi(x,\nu_{E_{t-h}^{(h)}})\,\textnormal{sd}^{\psi}_{E_{t-h}^{(h)}}|\,\textnormal{d}\mathcal{H}^{N-1}(x)\leq
ch^{\frac{N}{2}}.$

Combining the two previous equations and $B(y,h^{\alpha}/(2c_{\psi}))\subseteq
B(y,c\sqrt{h})$, we infer

$\displaystyle\int_{A_{x_{h}}}|\chi_{E_{t}^{(h)}}$
$\displaystyle-\chi_{E_{t-h}^{(h)}}|+\int_{A_{x_{h}}\cap\partial
E_{t}^{(h)}}|\psi(x,\nu_{E_{t-h}^{(h)}})\,\textnormal{sd}^{\psi}_{E_{t-h}^{(h)}}|\,\textnormal{d}\mathcal{H}^{N-1}$
(53) $\displaystyle\leq
ch^{\frac{N}{2}-(N+1)\alpha+1}\int_{A_{x_{h}}\cap(E_{t}^{(h)}\triangle
E_{t-h}^{(h)})}|\psi(x,\nu_{E_{t-h}^{(h)}})v_{h}|.$

Summing over $x_{h}\in I$ both (52) and (53), and using the local finiteness
of the covering, we get

$\displaystyle\bigg{\lvert}$
$\displaystyle\int(\chi_{E_{t}^{(h)}}-\chi_{E_{t-h}^{(h)}})\eta\,\textnormal{d}x-\int_{\partial
E_{t}^{(h)}}\psi(x,\nu_{E_{t}^{(h)}})\textnormal{sd}^{\psi}_{E_{t-h}^{(h)}}\eta\,\textnormal{d}\mathcal{H}^{N-1}\bigg{\rvert}$
$\displaystyle\leq\sum_{x_{h}\in
I}\bigg{\lvert}\int_{A_{x_{h}}}(\chi_{E_{t}^{(h)}}-\chi_{E_{t-h}^{(h)}})\eta\,\textnormal{d}x-\int_{\partial
E_{t}^{(h)}\cap
A_{x_{h}}}\psi(x,\nu_{E_{t}^{(h)}})\textnormal{sd}^{\psi}_{E_{t-h}^{(h)}}\eta\,\textnormal{d}\mathcal{H}^{N-1}\bigg{\rvert}$
$\displaystyle\leq
c\left(\omega(h)\|\eta\|_{\infty}+h^{\beta}\|\nabla\eta\|_{\infty}+h^{\frac{N}{2}-(n+1)\alpha+1}\|\eta\|_{\infty}\right)\cdot$
$\displaystyle\quad\cdot\left(P(E_{t}^{(h)})+|E_{t}^{(h)}\triangle
E_{t-h}^{(h)}|+\int_{E_{t}^{(h)}\triangle E_{t-h}^{(h)}}|v_{h}|\right)$

where the last constant $c$ depends on $N,\psi$. We then use Lemma 3.12, (32)
and (33) to conclude

$\displaystyle\bigg{\lvert}\int_{2h}^{T}$
$\displaystyle\frac{1}{h}\left(\int_{E_{t}^{(h)}}\eta\,\textnormal{d}x-\int_{E_{t-h}^{(h)}}\eta\,\textnormal{d}x\right)-\int_{h}^{T}\int_{\partial
E_{t}^{(h)}}\psi(x,\nu_{E_{t}^{(h)}})v_{h}\eta\,\textnormal{d}\mathcal{H}^{N-1}\bigg{\rvert}$
$\displaystyle\leq
c\left(\omega(h)\|\eta\|_{\infty}+h^{\beta}\|\nabla\eta\|_{\infty}+h^{\frac{N}{2}-(n+1)\alpha+1}\|\eta\|_{\infty}\right),$

where $c=c(E_{0},f,T,\psi)$ and $T$ is chosen such that
spt$\,\eta{\subset\joinrel\subset}\mathbb{R}^{N}\times[0,T]$. The conclusion
follows using the definition of $\alpha$ and taking the limit $h\to 0$.

∎

The proof of our main theorem of this section is now a consequence of the
previous results. In particular, hypothesis (34) and (35) imply that the
discrete flow converges to the flat flow in the sense of varifolds and this
allows to prove (5), while (6) is a consequence of Proposition 3.18. In order
to prove the convergence of the approximations in time of the forcing term, we
need to require additionally that (H2) holds.

###### Proof of Theorem 1.2.

Firstly, combining [32, Theorem 4.4.2] with the bounds contained in (37) and
in Proposition 3.14, we conclude the existence of functions
$v,H^{\phi},\tilde{f}:\mathbb{R}^{N}\times[0,\infty)\to\mathbb{R}$ satisfying

$\int_{0}^{T}\int_{\partial
E_{t}}|v|^{2}+|H^{\phi}|^{2}+|\tilde{f}|^{2}\,\textnormal{d}\mathcal{H}^{N-1}\,\textnormal{d}t\leq
C_{T}$

and the following properties

$\displaystyle\lim_{k}\int_{0}^{T}\int_{\partial
E_{t}^{(h_{k})}}v_{h_{k}}\eta\,\textnormal{d}\mathcal{H}^{N-1}\,\textnormal{d}t$
$\displaystyle=\int_{0}^{T}\int_{\partial E_{t}}\eta
v\,\textnormal{d}\mathcal{H}^{N-1}\,\textnormal{d}t$
$\displaystyle\lim_{k}\int_{0}^{T}\int_{\partial
E_{t}^{(h_{k})}}F_{h_{k}}(x,t)\eta\,\textnormal{d}\mathcal{H}^{N-1}\,\textnormal{d}t$
$\displaystyle=\int_{0}^{T}\int_{\partial
E_{t}}\eta\tilde{f}\,\textnormal{d}\mathcal{H}^{N-1}\,\textnormal{d}t$ (54)
$\displaystyle\lim_{k}\int_{0}^{T}\int_{\partial
E_{t}^{(h_{k})}}H^{\phi}_{E_{t}^{(h_{k})}}\eta\,\textnormal{d}\mathcal{H}^{N-1}\,\textnormal{d}t$
$\displaystyle=\int_{0}^{T}\int_{\partial E_{t}}\eta
H^{\phi}\,\textnormal{d}\mathcal{H}^{N-1}\,\textnormal{d}t,$

for any $\eta\in C^{0}_{c}(\mathbb{R}^{N}\times[0,T))$. We now employ an
approximation procedure to prove that $H^{\phi}(\cdot,t)$ is the $\phi-$mean
curvature of $E_{t}$ for a.e. $t\in[0,\infty)$, following the lines of [38,
41]. Fixed $t\in[0,+\infty)$ and $\varepsilon>0$, set $\nu_{\varepsilon}$ a
continuous function such that $\int_{\partial
E_{t}}(\nu_{E_{t}}-\nu_{\varepsilon})^{2}\,\textnormal{d}\mathcal{H}^{N-1}<\varepsilon.$
Then, by (34) one could prove that $\lim_{k\to\infty}\int_{\partial
E_{t}^{(h_{k})}}(\nu_{E_{t}^{(h_{k})}}-\nu_{\varepsilon})^{2}\,\textnormal{d}\mathcal{H}^{N-1}<\varepsilon.$
Considering test functions in (54) of the form $\eta(x,t)=a(t)g(x)$, one has
for a.e. $t\in[0,+\infty)$

$\lim_{k}\int_{\partial
E_{t}^{(h_{k})}}H^{\phi}_{E_{t}^{(h_{k})}}g\,\textnormal{d}\mathcal{H}^{N-1}=\int_{\partial
E_{t}}H^{\phi}g\,\textnormal{d}\mathcal{H}^{N-1}.$

Thus, for a.e. $t\in[0,+\infty)$ and for any $X\in
C^{0}_{c}(\mathbb{R}^{N};\mathbb{R}^{N})$ it holds

$\lim_{k}\int_{\partial
E_{t}^{(h_{k})}}H^{\phi}_{E_{t}^{(h_{k})}}\nu_{E_{t}^{(h_{k})}}\cdot
X\,\textnormal{d}\mathcal{H}^{N-1}=\int_{\partial
E_{t}}H^{\phi}\nu_{E_{t}}\cdot X\,\textnormal{d}\mathcal{H}^{N-1}$

by approximating the normal vectors of ${E_{t}^{(h_{k})}}$ with
$\nu_{\varepsilon}$. Furthermore, by the convergence (34) and the hypothesis
(35) we can use the Reshetnyak’s continuity theorem (see e.g. [4, Theorem
2.39]), ensuring

$\int_{\partial
E_{t}^{(h_{k})}}L(x,\nu_{E_{t}^{(h_{k})}})\,\textnormal{d}\mathcal{H}^{N-1}\to\int_{E_{t}}L(x,\nu_{E_{t}})\,\textnormal{d}\mathcal{H}^{N-1}$

as $k\to\infty$, for any $L\in C^{0}_{c}(\mathbb{R}^{N}\times\mathbb{R}^{N})$.
We choose $L(x,\nu)=\textnormal{div}_{\phi}X$ for some $X\in
C^{1}_{c}(\mathbb{R}^{N};\mathbb{R}^{N})$ to obtain

$\displaystyle\int_{\partial
E_{t}}\textnormal{div}_{\phi}X\,\textnormal{d}\mathcal{H}^{N-1}$
$\displaystyle=\lim_{k}\int_{\partial
E_{t}^{(h_{k})}}\textnormal{div}_{\phi}X\,\textnormal{d}\mathcal{H}^{N-1}$
$\displaystyle=\lim_{k}\int_{\partial
E_{t}^{(h_{k})}}X\cdot\nu_{E_{t}^{(h_{k})}}H^{\phi}_{E_{t}^{(h_{k})}}\,\textnormal{d}\mathcal{H}^{N-1}$
$\displaystyle=\int_{\partial
E_{t}}X\cdot\nu_{E_{t}}H^{\phi}\,\textnormal{d}\mathcal{H}^{N-1},$

which shows that $H^{\phi}(\cdot,t)$ is the $\phi-$mean curvature of the set
$E_{t}$ for a.e. $t\in[0,+\infty).$ Moreover, we remark that
$F_{h_{k}}(x,t)\to f(x,t)$ for every $(x,t)$, thus for any test function
$\eta\in C^{0}_{c}(\mathbb{R}^{N}\times[0,+\infty))$ and $t\in[0,+\infty)$ we
have

$\begin{split}\bigg{\lvert}\int_{\partial
E_{t}^{(h)}}F_{h_{k}}(x,t)\eta(x,t)\,\textnormal{d}\mathcal{H}^{N-1}_{x}&-\int_{\partial
E}f\eta\,\textnormal{d}\mathcal{H}^{N-1}_{x}\bigg{\rvert}\leq\left\lvert\int_{\partial
E_{t}^{(h)}}F_{h_{k}}\eta-\int_{\partial
E_{t}}F_{h_{k}}\eta\right\rvert+\int_{\partial E_{t}}|F_{h_{k}}-f|\eta\\\
&\leq\|f\|_{\infty}\|\eta\|_{\infty}\left(P(E_{t}^{(h)})-P(E_{t})\right)+\int_{\partial
E_{t}}|F_{h_{k}}-f|\eta\to 0\end{split}$

applying the dominated convergence theorem and recalling Lemma 3.9. Thus,
$\tilde{f}=f$. We then prove (5) by passing to the limit in the Euler-Lagrange
equation (22).

To prove (6) we employ Proposition 3.18: for every $\eta\in
C^{0}_{c}(\mathbb{R}^{N}\times[0,T))$, by a change of variables we have that

$\displaystyle\int_{h}^{T}$
$\displaystyle\left[\int_{E_{t}^{(h)}}\eta\,\textnormal{d}x-\int_{E_{t-h}^{(h)}}\eta\,\textnormal{d}x\right]\,\textnormal{d}t=\int_{h}^{T}\int_{E_{t}^{(h)}}\left(\eta(x,t)-\eta(x,t-h)\right)\,\textnormal{d}x\,\textnormal{d}t-h\int_{E_{0}}\eta\,\textnormal{d}x$

where we have used that $E^{(h)}_{t}=E_{0}$ for $t\in[0,h)$. Therefore, a
simple convergence argument yields

$\lim_{h\to
0}\frac{1}{h}\int_{h}^{T}\left[\int_{E_{t}^{(h)}}\eta\,\textnormal{d}x-\int_{E_{t-h}^{(h)}}\eta\,\textnormal{d}x\right]\,\textnormal{d}t=-\int_{h}^{T}\partial_{t}\eta(x,t)\,\textnormal{d}x\,\textnormal{d}t-\int_{E_{0}}\eta.$

Combining the previous estimate with Proposition 3.18 and passing to the
limit, we obtain (6). ∎

##  4\. Viscosity solutions

In this section we will prove the existence of another weak notion of solution
for the mean curvature flow starting from a compact set. We will follow the
so-called level set approach based on the theory of viscosity solution. We
recall that in the first part we work with the standing assumptions of the
paper (LABEL:standing_hp). Additionally, we require (H1).

###  4.1. The discrete scheme for unbounded sets

In this short subsection we will define the discrete evolution scheme for
unbounded sets having compact boundary. The idea would be to define this
evolution simply as the complement of the evolution of the complementary set,
but since the anisotropies we are considering are not symmetric, we need
additional care.

We recall that, given an anisotropy $\phi,$ we define
$\tilde{\phi}(x,\nu):=\phi(x,-\nu)$. This anisotropy has all the properties of
the original one, concerning regularity and bounds. We start remarking the
following simple fact. One can see that
$\textnormal{dist}^{\psi}(x,y)=\textnormal{dist}^{\tilde{\psi}}(y,x),$ since
for any curve $\gamma\in
W^{1,1}([0,1];\mathbb{R}^{N}),\gamma(0)=x,\gamma(1)=y,$ a simple change of
variable yields

$\int_{0}^{1}\psi^{\circ}(\gamma(t),\dot{\gamma}(t))\,\textnormal{d}t=\int_{0}^{1}\psi^{\circ}\left(\gamma(1-t),-\frac{\,\textnormal{d}}{\,\textnormal{d}t}\left(\gamma(1-t)\right)\right)\,\textnormal{d}t=\int_{0}^{1}\widetilde{(\psi^{\circ})}(\eta(t),\dot{\eta}(t))\,\textnormal{d}t,$

for $\eta(t)=\gamma(1-t)$, once one sees that

$\widetilde{(\psi^{\circ})}(\cdot,\nu)=\sup_{\psi(\cdot,\xi)\leq
1}\xi\cdot(-\nu)=\sup_{\tilde{\psi}(\cdot,-\xi)\leq
1}(-\xi)\cdot\nu=(\tilde{\psi})^{\circ}(\cdot,\nu).$

Therefore, by definition of signed distance we have

(55)
$\textnormal{sd}^{\psi}_{E}(x)=-\textnormal{sd}^{\tilde{\psi}}_{E^{c}}(x).$

For every compact set $F$ and $h>0,t\geq 0$, we will denote by
$\tilde{T}_{h,t}^{\pm}F$ the maximal and the minimal solution to problem (21),
according to Lemma 3.1 with $P_{\phi}$ and $\textnormal{sd}^{\psi}$,
respectively, replaced by $P_{\tilde{\phi}}$ and
$\textnormal{sd}^{\tilde{\psi}}$. Finally, for every set $E$ with compact
boundary we define

(56) $T_{h,t}^{\pm}E:=\left(\tilde{T}^{\mp}_{h,t}E^{c}\right)^{c}.$

As in the case for compact sets, we set $T_{h,t}E:=T_{h,t}^{-}E.$ Given an
open, unbounded set $E_{0}$ having compact boundary, we can then define the
discrete flow $\\{E_{t}^{(h)}\\}_{t\geq 0}$ as follows: $E^{(h)}_{t}:=E_{0}$
for $t\in[0,h)$ and

$E^{(h)}_{t}=T_{h,t}E^{(h)}_{t-h},\quad\forall t\in[h,+\infty).$

One easily checks that analogous results to Lemmas 3.2, 3.9 and 3.8 hold also
for this problem. We state the corresponding results.

######  Lemma 4.1.

Let $F_{1}\subseteq F_{2}$ be open, unbounded sets with compact boundary and
fix $h>0,t\geq 0$. Then, $T_{h,t}F_{1}\subseteq T_{h,t}F_{2}$.

######  Lemma 4.2.

For any $T>0$ there exists a constant $C_{T}(\phi,\psi,f,T)$ such that for
every $R>0$ the following holds. If the initial open set $E\supset B_{R}^{c}$,
then $E_{t}^{(h)}\supset B_{C_{T}R}^{c}$ for all $t\in[0,T].$

######  Lemma 4.3.

For every $R_{0}>0$ there exist $h_{0}(R_{0})>0$ and $C(R_{0},\phi,\psi,f)>0$
with the following property: For all $R\geq R_{0}$, $h\in(0,h_{0})$, $t>0$ and
$x\in\mathbb{R}^{N}$ one has

$T_{h,t}(\left(B_{R}(x)\right)^{c})\subseteq\left(B_{R-Ch}(x)\right)^{c}.$

We now state a comparison principle between bounded and unbounded sets,
following the line of [14, Lemma 6.10].

######  Lemma 4.4.

Let $E_{1}$ be a compact set and let $E_{2}$ be an open, unbounded set, with
compact boundary, and such that $E_{1}\subseteq E_{2}$. Then, for every
$h\in(0,1),t\geq 0$ it holds $T^{\pm}_{h,t}E_{1}\subseteq T^{\pm}_{h,t}E_{2}.$

###### Proof.

We fix $h\in(0,1),t\in[0,T]$ for $T>0$. Set $R>0$ such that
$E_{1},E_{2}^{c}\subseteq B_{R}$ and note that by Lemmas 3.2 and 3.9 (applied
to $P_{\tilde{\phi}}$ instead of $P_{\phi}$) we get

(57)
$\left(T^{+}_{h,t}E_{2}\right)^{c}\subseteq\tilde{T}^{-}_{h,t}E_{2}^{c}\subseteq
T^{-}_{h,t}B_{R}\subseteq B_{C_{T}R},$

for some $C_{T}(\phi,\psi,f,T)$. Since $\tilde{T}^{-}_{h,t}E_{2}^{c}$ is the
minimal solution of

$\min\left\\{P_{\tilde{\phi}}(E)+\dfrac{1}{h}\int_{E}\textnormal{sd}^{\tilde{\psi}}_{E^{c}_{2}}(x)\,\textnormal{d}x-\int_{E}F_{h}(x,t)\,\textnormal{d}x\right\\},$

considering the change of variables $\tilde{E}=E^{c}$ and using (55), we
easily conclude that
$T^{+}_{h,t}E_{2}=\left(\tilde{T}^{-}_{h,t}E_{2}^{c}\right)^{c}$ is the
maximal solution of

$\min\left\\{P_{\phi}(\tilde{E})+\dfrac{1}{h}\int_{B_{C_{T}R}}\textnormal{sd}^{\psi}_{E_{2}}-\dfrac{1}{h}\int_{\tilde{E}^{c}}\textnormal{sd}^{\psi}_{E_{2}}-\int_{\tilde{E}^{c}}F_{h}(x,t)\,\textnormal{d}x\right\\}-\dfrac{1}{h}\int_{B_{C_{T}R}}\textnormal{sd}^{\psi}_{E_{2}}.$

we then note that

$\int_{B_{C_{T}R}}\textnormal{sd}^{\psi}_{E_{2}}=\int_{\tilde{E}}\textnormal{sd}^{\psi}_{E_{2}}\chi_{B_{C_{T}R}}+\int_{\tilde{E}^{c}}\textnormal{sd}^{\psi}_{E_{2}},$

for every $\tilde{E}$ such that $\tilde{E}^{c}\subseteq B_{C_{T}R}.$ By (57),
we conclude that $T_{h,t}^{+}E_{2}$ is the maximal solution of

(58)
$\min\left\\{P_{\phi}(\tilde{E})+\dfrac{1}{h}\int_{\tilde{E}}\textnormal{sd}^{\psi}_{E_{2}}\chi_{B_{C_{T}R}}-\int_{\tilde{E}^{c}}F_{h}(x,t)\,\textnormal{d}x\
:\ \tilde{E}^{c}\subseteq B_{C_{T}R}\right\\}.$

Analogously, one proves that $T_{h,t}^{-}E_{2}$ is the minimal solution of
(58). Finally, we remark that
$\textnormal{sd}^{\psi}_{E_{s}}\chi_{B_{C_{T}R}}\leq\textnormal{sd}^{\psi}_{E_{1}}$
and that $T_{h,t}^{\pm}E_{1}\cup T_{h,t}^{\pm}E_{2},T_{h,t}^{\pm}E_{1}\cap
T_{h,t}^{\pm}E_{2}$ are both admissible competitors for (58), one argues
exactly as in the proof of Lemma 3.2 to conclude $T_{h,t}^{\pm}E_{1}\subseteq
T_{h,t}^{\pm}E_{2}.$ ∎

###  4.2. The level set approach

We recall that in this section we assume (LABEL:standing_hp), (H1). Consider a
function $u:\mathbb{R}^{N}\times[0,+\infty)\to\mathbb{R}$ whose spatial
superlevel sets $\\{u(\cdot,t)\geq s\\}$ evolve according to the mean
curvature equation

$V(x,t)=-\psi(x,\nu_{\\{u(\cdot,t)\geq s\\}})\left(H^{\phi}_{\\{u(\cdot,t)\geq
s\\}}(x)-f(x,t)\right)\quad{\text{for }x\in\partial\\{u(\cdot,t)\geq s\\}}.$

The function $u$ then satisfies (recalling that $-\nabla u/|\nabla u|$ is the
outer normal vector to the superlevel set $\\{u(\cdot,t)\geq u(x,t)\\}$) the
equation

$\displaystyle\partial_{t}u=|\nabla u|V(x)$ $\displaystyle=-\psi(x,-\nabla
u)\left(H^{\phi}_{\\{u(\cdot,t)\geq u(x,t)\\}}(x)-f(x,t)\right)$
$\displaystyle=-\psi(x,-\nabla
u)\left(\textnormal{div}\nabla_{p}\phi(x,-\nabla u)-f(x,t)\right)$
$\displaystyle=-\psi(x,-\nabla
u)\left(\sum_{i}\partial_{x_{i}}\partial_{p}\phi(x,-\nabla
u)-\nabla_{p}^{2}\phi(x,-\nabla u)\,:\,\nabla^{2}u-f(x,t)\right)$
$\displaystyle:$ $\displaystyle=-\psi(x,-\nabla u)\left(H(x,\nabla
u,\nabla^{2}u)-f(x,t)\right),$

where we defined the Hamiltonian $H\ :\
\mathbb{R}^{N}\times\mathbb{R}^{N}\setminus\\{0\\}\times{Sym_{N}}\to\mathbb{R}$
as

(59)
$H(x,p,X):=\sum_{i}\partial_{x_{i}}\partial_{p_{i}}\phi(x,-p)-\nabla_{p}^{2}\phi(x,-p)\,:\,X.$

We therefore focus on solving the parabolic Cauchy problem

(60) $\begin{cases}\partial_{t}u+\psi(x,-\nabla u)\left(H(x,\nabla
u,\nabla^{2}u)-f(x,t)\right)=0\\\ u(\cdot,t)=u_{0}.\end{cases}$

The appropriate setting for this type of geometric evolution equations is the
one of viscosity solutions, in the framework of [26, 35] (see also [14]). We
will focus on the evolution of sets with compact boundary on compact time
intervals of the form $[0,T]$. We now define the notion of admissible test
function. In the following, with a small abuse of language, we will say that a
function $u:\mathbb{R}^{n}\times[0,T]\to\mathbb{R}$ is constant outside a
compact set if there exists a compact set $K\subset\mathbb{R}^{N}$ such that
$u(\cdot,t)$ is constant in $\mathbb{R}^{N}\setminus K$ for every $t\in[0,T]$
(with the constant possibly depending on $t$).

######  Definition 4.5.

Let $\hat{z}=(\hat{x},\hat{t})\in\mathbb{R}^{N}\times(0,T)$ and let
$A\subseteq(0,T)$ be any open interval containing $\hat{t}$. We will say that
$\eta\in C^{0}(\mathbb{R}^{N}\times\bar{A})$ is admissible at the point
$\hat{z}$ if it is of class $C^{2}$ in a neighborhood of $\hat{z}$, if it is
constant out of a compact set, and, in case $\nabla\eta(\hat{z})=0,$ the
following holds: for all $(x,t)\in\mathbb{R}^{N}\times A$, and there exist
numbers $a,b>0$ such that

${|\eta(x,t)-\eta(\hat{z})-\eta_{t}(\hat{z})(t-\hat{t})|\leq
a|x-\hat{x}|^{3}+b|t-\hat{t}|^{2}.}$

We then recall one of the equivalent definitions of viscosity solutions.

######  Definition 4.6.

An upper semicontinuous function $u:\mathbb{R}^{N}\times[0,T]\to\mathbb{R}$
(in short, $u\in usc(\mathbb{R}^{N}\times[0,T])$), constant outside a compact
set, is a viscosity subsolution of the Cauchy problem (60) if $u(\cdot,0)\leq
u_{0}$ and for all $z:=(x,t)\in\mathbb{R}^{N}\times{(0,T)}$ and all
$C^{\infty}-$test functions $\eta$ such that $\eta$ is admissible at $z$ and
$u-\eta$ has a maximum at $z$ (in the domain of definition of $\eta$) the
following holds:

  * i)

If $\nabla\eta(z)=0$, then it holds

(61) $\eta_{t}(z)\leq 0$

  * ii)

If $\nabla\eta(z)\neq 0$, then

(62)
$\partial_{t}\eta(z)+\psi(z,-\nabla\eta(z))\left(H(z,\nabla\eta(z),\nabla^{2}\eta(z))-f(z,t)\right)\leq
0.$

A lower semicontinuous function $u:\mathbb{R}^{N}\times[0,T]\to\mathbb{R}$ (in
short, $u\in lsc(\mathbb{R}^{N}\times[0,T])$), constant outside a compact set,
is a viscosity supersolution of the Cauchy problem (60) if $u(\cdot,0)\geq
u_{0}$ and for all $z:=(x,t)\in\mathbb{R}^{N}\times[0,T]$ and all
$C^{\infty}-$test functions $\eta$ such that $\eta$ is admissible at $z$ and
$u-\eta$ has a minimum at $z$ (in the domain of definition of $\eta$) the
following holds:

  * i)

If $\nabla\eta(z)=0$, then $\eta_{t}(z)\geq 0$;

  * ii)

If $\nabla\eta\neq 0$ then

$\partial_{t}\eta(z)+\psi(z,-\nabla\eta(z))\left(H(z,\nabla\eta(z),\nabla^{2}\eta(z))-f(z,t)\right)\leq
0.$

Finally, a function $u$ is a viscosity solution for the Cauchy problem (60) if
it is both a subsolution and a supersolution of (60).

######  Remark.

By classical arguments, one could assume that the maximum of $u-\eta$ is
strict in the definition of subsolution above (an analogous remark holds for
supersolutions).

######  Remark.

We remark that, if $-u$ is a subsolution to (60) with initial datum $-u_{0}$,
then $u$ is a supersolution for (60) for the initial datum $u_{0}$ and where
$\phi,\psi$ are replaced by $\tilde{\phi},\tilde{\psi}$ respectively, as
defined in Section 4.1.

We will first prove existence for viscosity solutions of (60) via an
approximation-in-time technique, and then prove uniqueness of solutions to
(60) to link the approximate solution to the mean curvature flow equation. We
would like to proceed with the classical construction of e.g. [11, 14, 21],
but in our case the lack of continuity of the evolving functions forces us to
be particularly careful with the procedure.

We use the shorthand notation of lsc for lower semicontinuous and usc for
upper semicontinuous. Given a bounded, usc function $v$ which is constant
outside a compact set, we define the transformation

(63) $T_{h,t}^{+}v(x)=\sup\left\\{s\ :\ x\in T^{+}_{h,t}\\{v\geq
s\\}\right\\}.$

Firstly, we see that $T_{h,t}^{+}v(x)\in\mathbb{R},$ as $v$ is bounded.
Moreover, it turns out that the function $T^{+}_{h,t}v$ is usc, bounded and
constant outside a compact set. Indeed, definition (63) is equivalent to

$T^{+}_{h,t}v(x)=\inf\left\\{s\ :\ x\notin T^{+}_{h,t}\\{v\geq
s\\}\right\\}=\inf_{s\in\mathbb{R}}\left(s+\mathds{1}_{\left(T^{+}_{h,t}\\{v\geq
s\\}\right)^{c}}(x)\right),$

where $\mathds{1}_{A}(x)$ is the indicatrix function of a set $A$, being 0 on
the set and $+\infty$ outside. By definition, $\mathds{1}_{A}$ is an usc
function for any open set $A$. Thus, recalling Remark 3.4, in the equation
above we are taking the infimum of a family of usc functions, which is then a
usc function. The other two properties follows from the previous study of the
discrete evolution. Analogously, given a bounded lsc function $g$, we define

(64) $T_{h,t}^{-}g(x)=\sup\left\\{s\ :\ x\in
T^{-}_{h,t}\\{g>s\\}\right\\}=\sup_{s\in\mathbb{R}}\left(s-\mathds{1}_{T^{-}_{h,t}\\{g>s\\}}\right),$

which is now a bounded lsc function (as $\sup$ of lsc functions), constant
outside a compact set.

We are now ready to give the definition of the discrete-in-time approximations
of sub - and super solution to (60). Given an initial compact set $E_{0}$, set
$u_{0}$ as a (uniformly) continuous function, spatially constant outside a
compact set, such that $\\{u_{0}\geq 0\\}=E_{0}$. We remark that for every
$s\in\mathbb{R}$, the superlevel set $\\{u_{0}\geq s\\}$ is either compact or
it is unbounded with compact boundary. Then, for $h>0$ we introduce the
following family of maps as $u^{\pm}_{h}(\cdot,t)=u_{0}$ for $t\in[0,h)$ and

(65)
$u^{\pm}_{h}(\cdot,t):=T^{\pm}_{h,t-h}u^{\pm}_{h}(\cdot,t-h)\quad\text{for
}t\geq h.$

We easily see that the maps above are functions (as implied by the comparison
principle contained in Lemmas 3.2, 4.1 and 4.4) piecewise constant in time (as
$T_{h,t}^{\pm}=T_{h,[t/h]h}^{\pm}$). Moreover, by the previous remarks, we
have that $u^{+}_{h}(\cdot,t)$ is an usc function, while $u^{-}_{h}(\cdot,t)$
is a lsc function, for every $t\in[0,+\infty)$. Some further properties of the
approximating scheme are listed below.

######  Lemma 4.7.

For any $h>0$, $t\geq 0$ we have the following. It holds

(66) $u^{-}_{h}(\cdot,t)\leq u^{+}_{h}(\cdot,t).$

Furthermore, given any $\lambda\in\mathbb{R}$ and $t\geq h$ it holds

(67) $\displaystyle\\{u_{h}^{+}(\cdot,t)>\lambda\\}\subseteq
T^{+}_{h,t-h}\\{u_{h}^{+}(\cdot,t-h)$
$\displaystyle\geq\lambda\\}\subseteq\\{u_{h}^{+}(\cdot,t)\geq\lambda\\}$
$\displaystyle\\{u_{h}^{-}(\cdot,t)>\lambda\\}\subseteq
T^{-}_{h,t-h}\\{u_{h}^{-}(\cdot,t-h)$
$\displaystyle>\lambda\\}\subseteq\\{u_{h}^{-}(\cdot,t)\geq\lambda\\}.$

###### Proof.

Fix $x\in\mathbb{R}^{N}$, $t\in[0,h)$. For any given $\sigma<u_{h}^{-}(x,h)$
we have that there exists a sequence $(s_{n})\nearrow\sigma$ so that $x\in
T^{-}_{h,t-h}\\{u_{0}>s_{n}\\}\subseteq T^{+}_{h,t-h}\\{u_{0}\geq s_{n}\\}.$
Thus, $u_{h}^{+}(x,t)\geq\sigma.$ We then conclude by induction. Then, (67)
follows easily by the definition (65). ∎

We then prove that the half-relaxed limits (in the spirit of [6], see also the
references therein) of the families of functions $u_{h}^{\pm}$

(68) $\begin{split}u^{+}(x,t):=&\sup_{(x_{h},t_{h})\to(x,t)}\limsup_{h\to
0}u_{h}^{+}(x_{h},t_{h})\\\
u^{-}(x,t):=&\inf_{(x_{h},t_{h})\to(x,t)}\liminf_{h\to
0}u_{h}^{-}(x_{h},t_{h}),\end{split}$

are (respectively) sub - and supersolutions in the viscosity sense of (60),
see Theorem 1.3 (note that, by definition, $u^{+}$ is usc, while $u^{-}$ is
lsc). The proof of this result is the subject of the following section and we
recall that the hypothesis required are (LABEL:standing_hp), (H1) and $f\in
C^{0}(\mathbb{R}^{N}\times[0,\infty))$ only. Once the existence of sub - and
super-solutions to the equation is settled, we need to properly define the
notion of level-set solution to the mean curvature flow. To do so, we first
prove uniqueness for (60) via a comparison principle and under additional
hypothesis. Then, we show that the evolution of the zero superlevel set of the
solution does not depend on the choice of the initial function $u_{0}$.

We start with a comparison result between $u^{+},u^{-}$ and $u_{0}$ at the
initial time: it will ensure that the classical hypothesis for the comparison
principle are satisfied. We first prove an estimate for the speed of decay of
the level sets of the evolving functions. While it will only be needed in the
following section, in the proof of the forthcoming Lemma 4.9 we will use
similar techniques, so we preferred to state it here.

######  Lemma 4.8.

Let $u^{+}(x,t)$ be the function defined in (68), let $\sigma\in\mathbb{R}$.
Assume that, for a suitable $x_{0}$ and $R>0$, it holds
$B(x_{0},R)\subseteq\\{u^{+}(\cdot,t_{0})\geq\sigma\\}$ . Then, there exists
$C=C(R,\phi,\psi,f)$ such that
$B(x_{0},R-C(t-t_{0}))\subseteq\\{u^{+}(\cdot,t)\geq\sigma\\}$ for every
$t\leq t_{0}+R/(2C)$. An analogous statement holds for $u^{-}$ by considering
its open sublevel sets.

###### Proof.

We focus on the case $\\{u^{+}(\cdot,t_{0})\geq\sigma\\}$ bounded, the other
case being analogous. By assumption, for any $R_{0}<R$, if $h$ is small
enough, we have
$B(x_{0},R_{0})\subseteq\\{u_{h}^{+}(\cdot,t_{0})\geq\sigma\\}$. Set
$C=C(R_{0}/2,\phi,\psi,f)$ as the constant of Lemma 3.8. Let $R_{n}$ be
defined recursively following law (27), that is $R_{n+1}=R_{n}-Ch$, as long as
$R_{n}\geq R_{0}/2$. By simple iteration we find that $R_{n}=R_{0}-nCh,$ as
long as $R_{n}\geq R_{0}/2$, which can be ensured enforcing
$h\,n\leq{R_{0}}/(2C).$ Therefore, for any $t\geq t_{0}$ such that
$t-t_{0}\leq R_{0}/(2C)$, we set $n=[(t-t_{0})/h]$ and send $h\to 0$ to deduce
(recalling also Lemma 3.2)

$\\{u^{+}(\cdot,t)\geq\sigma\\}\supset B(x_{0},R_{0}-C(t-t_{0})).$

Since the choice of $R_{0}$ is arbitrary, we conclude. ∎

We are now ready to prove a comparison result for the functions $u^{\pm}$ and
a continuity estimate at the initial time $t=0$.

######  Lemma 4.9.

For any $(x,t)\in\mathbb{R}^{N}\times[0,+\infty)$ it holds

$u^{-}(x,t)\leq u^{+}(x,t).$

Moreover $u^{-}(\cdot,0)=u^{+}(\cdot,0)=u_{0},$ so that there exists a modulus
of continuity $\omega$ such that $\forall x,y\in\mathbb{R}^{N}$

$u^{+}(x,0)-u^{-}(y,0)\leq\omega(|x-y|).$

###### Proof.

The proof of the first inequality essentially follows from (66) and the
definition of $u^{\pm}$. To prove the equality at the initial time $t=0$, we
start by remarking that $u^{+}(\cdot,0)\geq u_{0}$ as can be seen taking
sequences of the form $(x_{h},0)$ in (68). Then, consider $\omega$ as a
continuous, strictly increasing modulus of continuity for $u_{0}$. We can also
see that $\forall\varepsilon>0\ \\{u_{0}\leq u_{0}(x)+\varepsilon\\}\supseteq
B(x,\omega^{-1}(\varepsilon))$ by uniform continuity. Thus, reasoning
iteratively as in Lemma 4.8 and using (67), we obtain that there exists
$h_{0}(\varepsilon)$ such that $\forall h\leq h_{0}$ it holds

$\\{u_{h}^{+}(\cdot,t)\leq
u_{h}^{+}(x,0)+\varepsilon\\}\supseteq\left(T_{h,t-h}^{+}\\{u_{0}>u_{0}(x)+\varepsilon\\}\right)^{c}=T_{h,t-h}^{-}\\{u_{0}\leq
u_{0}(x)+\varepsilon\\}\supseteq B(x,\omega^{-1}(\varepsilon/2)),$

as long as
$t\leq(\omega^{-1}(\varepsilon)-\omega^{-1}(\varepsilon/2))/(2C)=:t_{\varepsilon}$,
and where we recalled that $u_{h}^{\pm}(\cdot,0)=u_{0}$. Now, fix
$\sigma>0,x\in\mathbb{R}^{N}$ such that $u(x,0)>\sigma$ and a sequence
$(x_{h_{k}},t_{h_{k}})\to(x,0)$ such that
$\lim_{k}u_{h_{k}}^{+}(x_{h_{k}},t_{h_{k}})>\sigma$. Then, for $k$ large
enough $(x_{h_{k}},t_{h_{k}})\in
B(x,\omega^{-1}(\varepsilon/2))\times[0,t_{\varepsilon})$ and so we conclude

$\sigma<\lim_{k}u_{h}^{+}(x_{h_{k}},t_{h_{k}})\leq u_{0}(x,0)+\varepsilon.$

Letting $\varepsilon\to 0$ we conclude $u(\cdot,0)^{+}\leq u_{0}$. The proof
for $u^{-}$ is essentially the same. The last claim follows from the previous
one, recalling that $\omega$ is a modulus of uniform continuity for $u_{0}$. ∎

In order to prove a comparison principle for (60), we will need to assume
(H3). Under these additional hypotheses, we are able to prove uniqueness for
the parabolic Cauchy problem (60). The proof of this result follows from [26,
Theorem 4.2]: we will just show in detail that the assumption of the
aforementioned theorem hold in our case, following [8, Proposition 6.1] and
[26, pag. 463].

###### Proof of Theorem 1.4.

The proof of this result essentially follows from [26, Theorem 4.2], combined
with the existence result of Theorem 1.3. Referring to the notation of [26],
we firstly remark that in our case $\Omega=\mathbb{R}^{N}$, thus the parabolic
boundary of $U=\Omega\times[0,T]$ is simply
$\partial_{p}U=\mathbb{R}^{N}\times\\{0\\}$. Therefore, the initial conditions
$(A1)-(A3)$ are all verified by Lemma 4.9. We then define the continuous
Hamiltonian
$F:[0,T]\times\mathbb{R}^{N}\times(\mathbb{R}^{N}\setminus\\{0\\})\times
M^{N\times N}\to\mathbb{R}$ as follows

(69)
$F(t,x,p,X):=\psi(x,-p)\left(-\sum_{i}\partial_{x_{i}}\partial_{{p_{i}}}\phi(x,-p)+\nabla_{p}^{2}\phi(x,-p):X+f(x,t)\right),$

and focus on the conditions $(F1),(F3)-(F5),(F6^{\prime}),(F7),(F9),(F10)$
that $F$ must satisfy. The assumptions $(F1),$ $(F3)-(F5),(F9)$ are easily
checked. $(F6^{\prime})$ follows from the Lipschitz regularity of $\phi$ and
$\psi$, as $\forall t\in[0,T],x\in\mathbb{R}^{N},|p|\geq\rho,|q|+|X|\leq R$
one has

$\displaystyle|F(t,x,p,X)-F(t,x,q,X)|\leq
c_{\psi}|p-q|\left\lvert-\sum_{i}\partial_{x_{i}}\partial_{{p_{i}}}\phi(x,-p)+\nabla^{2}_{p}\phi(x,-p):X\right\rvert$
$\displaystyle+\psi(x,-q)\left\lvert-\sum_{i}\left(\partial_{x_{i}}\partial_{{p_{i}}}\phi(x,-p)-\partial_{x_{i}}\partial_{{p_{i}}}\phi(x,-q)\right)+\left(\nabla^{2}_{p}\phi(x,-p)-\nabla^{2}_{p}\phi(x,-q)\right):X\right\rvert$
$\displaystyle\leq c_{R}|p-q|\left(1+\dfrac{1}{|p|}\right)+c_{R}|p-q|\leq
c_{R,\rho}|p-q|.$

For $(F7)$, we remark that the first term in the parenthesis in (69) is
$0-$homogeneous in $p$, while the second one is $(-1)-$homogeneous in $p$ but
$1-$homogeneous in $X$. Lastly, we sketch how to prove $(F10)$. Since it
concerns the $X$-terms, we focus simply on

$\nabla^{2}_{p}\phi(x,-p):X=\text{tr}\left(\nabla^{2}_{p}\phi(x-,p)\,X^{T}\right).$

Multiplying by $\phi(x,-p)$, we rewrite
$\phi(x,-p)\text{tr}\left(\nabla^{2}_{p}\phi(x-,p)\,X^{T}\right)=\text{tr}(A(x,-p)X^{T}),$
where $A=B-\left(\nabla_{p}\phi\otimes\nabla_{p}\phi\right),$ with $B$ being
the uniformly elliptic operator $\frac{1}{2}\nabla^{2}_{p}\phi^{2}$. We can
then factorize $B=\tilde{L}\tilde{L}^{T}$, with $\tilde{L}$ being a
nondegenerate, lower triangular matrix. Then, following the proof of [8,
Proposition 6.1] and [26, pg. 463], we obtain $(F10)$. ∎

Once uniqueness is settled, one can finally define the notion of level set
solution to the mean curvature flow as follows.

######  Definition 4.10.

Let $E_{0}$ be a compact initial set. Define a uniformly continuous, bounded
function $u_{0}:\mathbb{R}^{N}\to\mathbb{R}$ such that $\\{u_{0}\geq
0\\}=E_{0}$. Then, let $u:\mathbb{R}^{N}\times[0,+\infty)\to\mathbb{R}$ be the
unique continuous viscosity solution to (60) given by Theorem 1.4. Then, the
family $E_{t}:=\\{u^{+}(\cdot,t)\geq 0\\}_{t\geq 0}$ will be called the level
set solution to the mean curvature flow.

This definition is well posed since the Hamiltonian defined in (59) satisfies
the so-called geometricity condition. Namely, one can easily check that for
any $\lambda\neq 0,p\in\mathbb{R}^{N}\setminus 0,q\in\mathbb{R}^{N}$ and any
symmetric $N\times N$ matrix $X$ one has

$H(x,\lambda p,\lambda X+p\otimes q+q\otimes
p)=\dfrac{\lambda}{|\lambda|}H(x,p,X).$

Thus, one can prove by classical arguments (see e.g. [14, Remark 3.9]) the
following result.

######  Lemma 4.11.

Let $u_{0},\tilde{u}_{0}$ two initial data for (60) such that $\\{u_{0}\geq
0\\}=\\{\tilde{u}_{0}\geq 0\\}$. Then, denoting by $u,\tilde{u}$ the
corresponding solutions to (60), one has

$\\{u(\cdot,t)\geq 0\\}=\\{\tilde{u}(\cdot,t)\geq 0\\}\quad\text{ for all
}t\in[0,T],$

and the same identity holds for the open superlevel sets.

###  4.3. Proof of Theorem 1.3

In this section we will prove that the limiting functions $u^{\pm}$ are
respectively a viscosity sub - and supersolutions to (60). We remark that we
work assuming (LABEL:standing_hp), (H1) and that $f\in
C^{0}(\mathbb{R}^{N}\times[0,+\infty))$. We will be following the structure of
the proof of [14, Theorem 6.16], but taking into account the weaker definition
of $u^{+}$ holding in our case. We will be using the $O,o$ notations with
respect to $h\to 0$ and focus on proving that $u^{+}$ is a subsolution. The
proof for $u^{-}$ is analogous.

###### Proof of Theorem 1.3.

Consider $u^{+}$ as defined in (8): we need to prove that it is a subsolution.
In the following, we will denote $u:=u^{+}$ and $u_{h}:=u_{h}^{+}$. Let
$\eta(x,t)$ be an admissible test function in
$\bar{z}:=(\bar{x},\bar{t}){\in\mathbb{R}^{N}\times(0,T)}$ and assume that
$(\bar{x},\bar{t})$ is a strict maximum point for $u-\eta$. Assume furthermore
that $u-\eta=0$ in such point. We need to show that either (61) or (62) holds
at $\bar{z}$.
Case 1. Let us first assume that $\nabla\eta(\bar{z})\neq 0$. By classical
arguments, we can assume that $\bar{z}$ is a strict maximum point and that
$\eta$ is smooth. By the definition of $u$, there exists a sequence
$\tilde{z}_{k}:=(\tilde{x}_{h_{k}},\tilde{t}_{h_{k}})\to\bar{z}$ such that
$\lim_{k}u_{h_{k}}(\tilde{z}_{k})=u(\bar{z}).$ We remark that we can
substitute the functions $u_{h_{k}}$ for $t>0$ with their usc envelope in
time, without changing the value of $u$. Indeed, the usc envelope of
$u_{h_{k}}$ is the function at all discrete times $lh_{k}$ is given by

$\max\\{u_{h_{k}}(\cdot,(l-1)h_{k}),u_{h_{k}}(\cdot,lh_{k})\\}$

and coincides with $u_{h_{k}}$ elsewhere. Since now $u_{h_{k}}$ is usc in time
and space, by standard arguments (compare e.g. [5, Lemma 6.1]), there exists a
radius $\rho>0$ such that all functions $u_{h_{k}}-\eta$ achieve a local
maximum in $B_{\rho}(\bar{z})$ at points $z_{k}=(x_{k},t_{k})$. Then, passing
to a further subsequence we can ensure that $z_{k}\to w\in B_{\rho}(\bar{z})$,
and we use the definition of $u$ to obtain

$(u-\eta)(w)\geq\limsup_{k}(u_{h_{k}}-\eta)(z_{k})\geq\limsup_{k}(u_{h_{k}}-\eta)(\tilde{z}_{k})=(u-\eta)(\bar{z}).$

Therefore, $w=\bar{z}$ by maximality. Thus we can assume that each function
$u_{h_{k}}-\eta$ achieves a local maximum in $B_{\rho}(\bar{z})$ at a point
$z_{h_{k}}=:(x_{k},t_{k})$ and that $u_{h_{k}}(z_{h_{k}})\to u(\bar{z})$ as
$k\to\infty.$ Finally, we can assume also that $\nabla\eta(x_{k},t_{k})\neq 0$
for $k$ large enough.
Step 1. We start defining an appropriate set which is then used as a
competitor for the minimality of the level sets of the functions $u_{h}$. From
the previous computations, one has in particular that

(70) $u_{h}(x,t)\leq\eta(x,t)+c_{k}$

where $c_{k}:=u_{h_{k}}(x_{k},t_{k})-\eta(x_{k},t_{k})$, with equality if
$(x,t)=(x_{k},t_{k})$. Let $\sigma>0$ and set

$\eta_{h_{k}}^{\sigma}(x):=\eta(x,t_{k})+c_{k}+\dfrac{\sigma}{2}|x-x_{k}|^{2}.$

Then, for all $x\in\mathbb{R}^{N}$,

$u_{h_{k}}(x,t_{k})\leq\eta^{\sigma}_{h_{k}}(x)$

with equality if and only if $x=x_{k}$. We set
$l_{k}=u_{h_{k}}(x_{k},t_{k})=\eta^{\sigma}_{h_{k}}(x_{k})$. We fix
$\varepsilon>0$, to be chosen later, and write
$E_{\varepsilon,k}:=\left\\{u_{h_{k}}(\cdot,t_{k}-h_{k})\geq
l_{k}-\varepsilon\right\\}$. We define111 We need to define the sets
$W_{\varepsilon}$ in this way (compare the different definition in [14]) since
firstly, we can not rule out that the inclusions in (72) are strict, and
secondly it is not clear if otherwise $|W_{\varepsilon}|>0$.

(71)
$W_{\varepsilon}:=\left(T_{h,t_{k}-h_{k}}^{+}E_{\varepsilon,k}\right)\setminus\left\\{\eta^{\sigma}_{h_{k}}(\cdot)>l_{k}+\varepsilon\right\\}.$

We immediately see that $W_{\varepsilon}\to\\{x_{k}\\}$ in the Kuratowski
sense as $\varepsilon\to 0$ since by (67)

(72)
$\left\\{u_{h_{k}}(\cdot,t_{k})>l_{k}-\varepsilon\right\\}\setminus\left\\{\eta^{\sigma}_{h_{k}}(\cdot)>l_{k}+\varepsilon\right\\}\subseteq
W_{\varepsilon}\subseteq\left\\{u_{h_{k}}(\cdot,t_{k})\geq
l_{k}-\varepsilon\right\\}\setminus\left\\{\eta^{\sigma}_{h_{k}}(\cdot)>l_{k}+\varepsilon\right\\},$

see also (78) below. Then, we check that $|W_{\varepsilon}|>0$ for all
$\varepsilon$ small enough. By the continuity of $\eta^{\sigma}$ and
$|\nabla\eta(\bar{z})|\neq 0,$ for any $\varepsilon$ there exist a radius
$r_{\varepsilon}$ such that $W_{\varepsilon}\supseteq
B(x_{k},r_{\varepsilon})\cap T_{h,t_{k}-h_{k}}^{+}E_{\varepsilon,k}$.
Furthermore, for any $\varepsilon>0$, using (67) again yields $x_{k}\in
T^{+}_{h_{k},t_{k}-h_{k}}\\{u_{h_{k}}(\cdot,t_{k}-h_{k})\geq
l_{k}-\varepsilon\\},$ and the latter set coincides with the closure of its
points of density 1 by Lemma 3.3. Thus, $x_{k}$ satisfies lower density
estimates and so we conclude that $|W_{\varepsilon}|>0$. Now, assume
$E_{\varepsilon,k}$ is bounded. By minimality we have

$\displaystyle
P_{\phi}(T_{h,t_{k}-h_{k}}^{+}E_{\varepsilon,k})+\dfrac{1}{h_{k}}\int_{T_{h,t_{k}-h_{k}}^{+}E_{\varepsilon,k}}\textnormal{sd}^{\psi}_{E_{\varepsilon,k}}(x)\,\textnormal{d}x+\int_{W_{\varepsilon}}F_{h_{k}}(x,t_{k}-h_{k})\,\textnormal{d}x$
(73) $\displaystyle\leq
P_{\phi}\left(\left(T_{h,t_{k}-h_{k}}^{+}E_{\varepsilon,k}\right)\cap\\{\eta^{\sigma}_{h_{k}}>l_{k}+\varepsilon\\}\right)+\dfrac{1}{h_{k}}\int_{\left(T_{h,t_{k}-h_{k}}^{+}E_{\varepsilon,k}\right)\cap\\{\eta^{\sigma}_{h_{k}}>l_{k}\\}}\textnormal{sd}^{\psi}_{E_{\varepsilon,k}}.$

Adding to both sides the term
$P_{\phi}\left(\\{\eta^{\sigma}_{h_{k}}>l_{k}+\varepsilon\\}\cup
T_{h,t_{k}-h_{k}}^{+}E_{\varepsilon,k}\right)$ and using the submodularity
(12), we obtain

$\displaystyle P_{\phi}(\\{\eta^{\sigma}_{h_{k}}>l_{k}+\varepsilon\\}\cup
W_{\varepsilon})$ $\displaystyle-
P_{\phi}(\\{\eta^{\sigma}_{h_{k}}>l_{k}+\varepsilon\\})+\dfrac{1}{h_{k}}\int_{W_{\varepsilon}}\textnormal{sd}^{\psi}_{E_{\varepsilon,k}}(x)\,\textnormal{d}x$
$\displaystyle+\int_{W_{\varepsilon}}F_{h_{k}}(x,t_{k}-h_{k})\,\textnormal{d}x\leq
0.$

By (70), $\\{u_{h_{k}}(\cdot,t_{k}-h_{k})\geq
l_{k}-\varepsilon\\}\subseteq\\{\eta(\cdot,t_{k}-h_{k})\geq
l_{k}-c_{k}-\varepsilon\\}$, therefore it holds

$\displaystyle P_{\phi}(\\{\eta^{\sigma}_{h_{k}}>l_{k}+\varepsilon\\}\cup
W_{\varepsilon})$ $\displaystyle-
P_{\phi}(\\{\eta^{\sigma}_{h_{k}}>l_{k}+\varepsilon\\})+\dfrac{1}{h_{k}}\int_{W_{\varepsilon}}\textnormal{sd}^{\psi}_{\\{\eta(\cdot,t_{k}-h_{k})\geq
l_{k}-c_{k}-\varepsilon\\}}(x)\,\textnormal{d}x$ (74)
$\displaystyle+\int_{W_{\varepsilon}}F_{h_{k}}(x,t_{k}-h_{k})\,\textnormal{d}x\leq
0.$

If instead $E_{\varepsilon,k}$ is an unbounded set with compact boundary, we
replace inequality (73) by

$\displaystyle
P_{\phi}(T_{h,t_{k}-h_{k}}E_{\varepsilon,k})+\dfrac{1}{h_{k}}\int_{\left(T_{h,t_{k}-h_{k}}^{+}E_{\varepsilon,k}\right)\cap
B_{R}}\textnormal{sd}^{\psi}_{E_{\varepsilon,k}}(x)\,\textnormal{d}x+\int_{W_{\varepsilon}}F_{h_{k}}(x,t_{k}-h_{k})\,\textnormal{d}x$
$\displaystyle\leq
P_{\phi}(\left(T_{h,t_{k}-h_{k}}^{+}E_{\varepsilon,k}\right)\cap\\{\eta^{\sigma}_{h_{k}}>l_{k}+\varepsilon\\})+\dfrac{1}{h_{k}}\int_{\left(T_{h,t_{k}-h_{k}}^{+}E_{\varepsilon,k}\right)\cap\\{\eta^{\sigma}_{h_{k}}>l_{k}+\varepsilon\\}\cap
B_{R}}\textnormal{sd}^{\psi}_{E_{\varepsilon,k}},$

for $R>0$ sufficiently large, see (58). Then, one can argue as before to
obtain (74).
Step 2. We estimate the first two terms in (74). The quantity
$P_{\phi}(\\{\eta^{\sigma}_{h_{k}}>l_{k}+\varepsilon\\}\cup
W_{\varepsilon})-P_{\phi}(\\{\eta^{\sigma}_{h_{k}}>l_{k}+\varepsilon\\})$ can
be estimated as done in Lemma 3.8. Indeed, we consider the vector field
$v=\nabla_{p}\phi(x,\nabla\eta^{\sigma}_{h_{k}})$ in (13) and we use the
divergence theorem to get

(75) $\begin{split}P_{\phi}(\\{\eta^{\sigma}_{h_{k}}>l_{k}+\varepsilon\\}\cup
W_{\varepsilon})-P_{\phi}(\\{\eta^{\sigma}_{h_{k}}\geq
l_{k}+\varepsilon\\})&\geq\int_{\partial(\\{\eta^{\sigma}_{h_{k}}>l_{k}+\varepsilon\\}\cup
W_{\varepsilon})}v\cdot\nu-\int_{\partial\\{\eta^{\sigma}_{h_{k}}>l_{k}+\varepsilon\\}}v\cdot\nu\\\
&=|W_{\varepsilon}|\fint_{W_{\varepsilon}}\textnormal{div}\,v,\end{split}$

where $\nu$ denotes the unit outer vector to the set we are integrating on. We
then remark that $\fint_{W_{\varepsilon}}\textnormal{div}\,v\to
H^{\phi}_{\\{\eta^{\sigma}_{h_{k}}>l_{k}\\}}(x_{k})$ and
$\fint_{W_{\varepsilon}}F_{h_{k}}(x,t_{k}-h_{k})\,\textnormal{d}x\to
F_{h_{k}}(x_{k},t_{k}-h_{k})$ as $\varepsilon\to 0$ by continuity.
Step 3. We bound the distance term in (74) by showing that

(76)
$\dfrac{1}{h_{k}}\textnormal{sd}^{\psi}_{\\{\eta(\cdot,t_{k}-h_{k})=l_{k}-c_{k}-\varepsilon\\}}(z)\geq\dfrac{\partial_{t}\eta(z,t_{k})-O(h_{k})}{\psi(y,-\nabla\eta(y,t_{k}-h_{k}))+O(h_{k})}.$

For any $z\in W_{\varepsilon}$, we have

(77) $\eta(z,t_{k})+c_{k}+\frac{\sigma}{2}|z-x_{k}|^{2}\leq
l_{k}+\varepsilon.$

Since, in turn, $\eta(z,t_{k})+c_{k}>l_{k}-\varepsilon$ it follows that
$\sigma|z-x_{k}|^{2}<4\varepsilon$ and thus, for $\varepsilon$ small enough,

(78) $W_{\varepsilon}\subseteq B_{c\sqrt{\varepsilon}}(x_{k}).$

By a Taylor expansion, for every $z\in W_{\varepsilon}$ we have

(79)
$\eta(z,t_{k}-h_{k})=\eta(z,t_{k})-h_{k}\partial_{t}\eta(z,t_{k})+h^{2}_{k}\int_{0}^{1}(1-s)\partial^{2}_{tt}\eta(z,t_{k}-sh_{k})\,\textnormal{d}s.$

Then, we consider
$y,y_{e}\in\\{\eta(\cdot,t_{k}-h_{k})(y)=l_{k}-c_{k}-\varepsilon\\}$ being
respectively, a point of minimal $\psi-$distance and Euclidean distance from
$z$.
Claim: We claim that it holds

(80) $|z-y|=O(h_{k}).$

In order to prove this result, we start remarking that for $k\to\infty$ and
choosing $\varepsilon\ll h_{k}$, one has
$\textnormal{sd}^{\psi}_{\\{\eta(\cdot,t_{k}-h_{k})\geq
l_{k}-c_{k}-\varepsilon\\}}(z)\to 0$ (as $z\to x_{k}$ for $\varepsilon\to 0$
and $x_{k}\in\\{\eta(\cdot,t_{k})\geq l_{k}-c_{k}\\}$). In particular,
recalling the bounds (11) one has

$|z-y_{e}|\leq c_{\psi}^{2}|z-y|\leq
c_{\psi}^{3}|\textnormal{sd}^{\psi}_{\\{\eta(\cdot,t_{k}-h_{k})\geq
l_{k}-c_{k}-\varepsilon\\}}(z)|\to 0$

as $k\to\infty.$ By (77) we deduce in particular
$\eta(z,t_{k})+c_{k}<l_{k}+\varepsilon$, that is,

(81) $0\leq\eta(z,t_{k})-\eta(y,t_{k}-h_{k})\leq 2\varepsilon,$

and the same inequality substituting $y_{e}$ to $y$. Thus, one has

$\displaystyle\eta(z,t_{k})-\eta(y_{e},t_{k}-h_{k})$
$\displaystyle=\nabla\eta(y,t_{k}-h_{k})\cdot(z-y_{e})-h_{k}\partial_{t}\eta(y,t_{k}-h_{k})+O(|z-y_{e}|^{2}+h_{k}^{2})$

which we combine with
$\nabla\eta(y,t_{k}-h_{k})\cdot(z-y_{e})=\pm|\nabla\eta(y,t_{k}-h_{k})|\,|z-y_{e}|$
(see [14] for details) and (81) to get

$|z-y_{e}|\,|\nabla\eta(y,t_{k}-h_{k})|\leq
2\varepsilon+O(h_{k})+O(|z-y_{e}|^{2}).$

Recalling that $|\nabla\eta(y,t_{k}-h_{k})|\geq c>0$ for $h_{k}$ small enough,
we divide by $|\nabla\eta(y,t_{k}-h_{k})|$ to conclude $|z-y_{e}|=O(h_{k})$ as
$\varepsilon\ll h_{k}.$ Finally, employing again (11), we prove the claimed
(80).

Then, we consider a geodesic curve for the definition of
$\textnormal{sd}^{\psi}_{\\{\eta(\cdot,t_{k}-h_{k})\geq
l_{k}-c_{k}-\varepsilon\\}}(z)$: if this distance is positive, we choose
$\gamma:[0,1]\to\mathbb{R}^{N}$ with $\gamma(0)=z,\gamma(1)=y$, with $y$ as
before, otherwise we take $\gamma$ such that $\gamma(0)=y,\gamma(1)=z$. In the
following, we will assume
$\textnormal{sd}^{\psi}_{\\{\eta(\cdot,t_{k}-h_{k})\geq
l_{k}-c_{k}-\varepsilon\\}}(z)>0$, the other case being analogous. Recalling
(9), we have

$\displaystyle\eta(z,t_{k}-h_{k})$
$\displaystyle=\eta(y,t_{k}-h_{k})+\int_{0}^{1}\nabla\eta(\gamma,t_{k}-h_{k})\cdot\dot{\gamma}\,\textnormal{d}t$
$\displaystyle\geq\eta(y,t_{k}-h_{k})-\int_{0}^{1}\psi(\gamma,-\nabla\eta(\gamma,t_{k}-h_{k}))\psi^{\circ}(\gamma,\dot{\gamma})\,\textnormal{d}t$
$\displaystyle\geq\eta(y,t_{k}-h_{k})-\psi(y,-\nabla\eta(y,t_{k}-h_{k}))\,\textnormal{sd}^{\psi}_{\\{\eta(\cdot,t_{k}-h_{k})=l_{k}-c_{k}-\varepsilon\\}}(z)$
$\displaystyle\quad-\int_{0}^{1}\left(\psi(\gamma,-\nabla\eta(\gamma,t_{k}-h_{k}))-\psi(y,-\nabla\eta(y,t_{k}-h_{k}))\right)\psi^{\circ}(\gamma,\dot{\gamma})\,\textnormal{d}t$
$\displaystyle\geq\eta(y,t_{k}-h_{k})-\big{(}\psi(y,-\nabla\eta(y,t_{k}-h_{k}))+c|z-y|\big{)}\,\textnormal{sd}^{\psi}_{\\{\eta(\cdot,t_{k}-h_{k})=l_{k}-c_{k}-\varepsilon\\}}(z),$

where in the last line we reasoned as in (47) to obtain the bound
$\sup_{t}|\gamma(t)-y|\leq c|z-y|$. Recalling (80) one has

(82)
$\eta(z,t_{k}-h_{k})\geq\eta(y,t_{k}-h_{k})-\psi(y,-\nabla\eta(y,t_{k}-h_{k}))\,\textnormal{sd}^{\psi}_{\\{\eta(\cdot,t_{k}-h_{k})=l_{k}-c_{k}-\varepsilon\\}}(z)+o(h_{k}).$

Combining (79) with (82) and using (81), we deduce

$\displaystyle\textnormal{sd}^{\psi}$
${}_{\\{\eta(\cdot,t_{k}-h_{k})=l_{k}-c_{k}-\varepsilon\\}}(z)\,\psi(y,-\nabla\eta(y,t_{k}-h_{k}))+o(h_{k})$
$\displaystyle\geq-2\varepsilon+h_{k}\partial_{t}\eta(z,t_{k})-h_{k}^{2}\int_{0}^{1}(1-s)\partial^{2}_{tt}\eta(z,t_{k}-sh_{k})\,\textnormal{d}s.$

Note that, in view of (77) and (11), $|\eta(z,t_{k})-\eta(y,t_{k})|\leq
c\varepsilon+ch_{k}=O(h_{k})$, provided $\varepsilon\ll h_{k}$ and small
enough. We then conlude (76) by combining the previous inequality with
(78),(80) as

$\displaystyle\dfrac{1}{h_{k}}\textnormal{sd}^{\psi}_{\\{\eta(\cdot,t_{k}-h_{k})=l_{k}-c_{k}-\varepsilon\\}}(z)$
$\displaystyle\geq\dfrac{\partial_{t}\eta(z,t_{k})-\frac{2\varepsilon}{h_{k}}-O(h_{k})-O_{h_{k}}(1)}{\psi(y,-\nabla\eta(y,t_{k}-h_{k}))}$
$\displaystyle=\dfrac{\partial_{t}\eta(x_{k},t_{k})+O(\sqrt{\varepsilon})-\frac{2\varepsilon}{h_{k}}-O(h_{k})-O_{h_{k}}(1)}{\psi(x_{k},-\nabla\eta(x_{k},t_{k}-h_{k}))+O(\sqrt{\varepsilon})+O(h_{k})}.$

Step 4. We conclude the proof by employing (74), (75) and (76), dividing by
$|W_{\varepsilon}|$ and sending $\varepsilon\to 0$ to obtain

$\dfrac{\partial_{t}\eta(x_{k},t_{k})-O_{h_{k}}(1)}{\psi(x_{k},-\nabla\eta(x_{k},t_{k}))+O(h_{k})}+H^{\phi}_{\\{\eta^{\sigma}_{h_{k}}\geq\eta^{\sigma}_{h_{k}}(x_{k})\\}}(x_{k})-F_{h_{k}}(x_{k},t_{k}-h_{k})\leq
0.$

Letting simultaneously $\sigma\to 0$ and $k\to\infty$, recalling the
continuity properties of $H^{\phi}$, we deduce (62). Indeed the sets
$\\{\eta^{\sigma}_{h_{k}}>\eta^{\sigma}_{h_{k}}(x_{k})\\}$ are converging in
$C^{2}$ to the set $\\{\eta>\eta(x)\\}$, $x_{k}\to x$ and thus

$H^{\phi}_{\\{\eta^{\sigma}_{h_{k}}>\eta^{\sigma}_{h_{k}}(x_{k})\\}}(x_{k})\to
H^{\phi}_{\\{\eta>\eta(x)\\}}(x),$

and we conclude the proof of this step.
Case 2. Now we consider the case $\nabla\eta(\bar{x},\bar{t})=0$ and we show
that $\partial_{t}\eta(\bar{x},\bar{t})\leq 0$. The proof follows the line of
the one in [14], we just highlight the differences.

Since $\nabla\eta(\bar{z})=0,$ there exist $a,b>0$ such that

$|\eta(x,t)-\eta(\bar{z})-\partial_{t}\eta(\bar{z})(t-\bar{t})|\leq{a|x-\bar{x}|^{3}+b|t-\bar{t}|^{2},}$

thus, we can define

$\displaystyle\tilde{\eta}(x,t)=\partial_{t}\eta(\bar{z})(t-\bar{t})+2{a|x-\bar{x}|^{3}}+2{b|t-\bar{t}|^{2}}$
$\displaystyle\tilde{\eta}_{k}(x,t)=\tilde{\eta}(x,t)+\dfrac{1}{k(\bar{t}-t)}.$

We remark that $u-\tilde{\eta}$ achieves a strict maximum in $\bar{z}$ and the
local maxima of $u-\tilde{\eta}_{k}$ in $\mathbb{R}^{N}\times[0,\bar{t}]$ are
in points $(x_{k},t_{k})\to\bar{z}$ as $k\to\infty$, with $t_{n}\leq\bar{t}$.
From now on, the only difference from [14] is in the case $x_{k}=\bar{x}$ for
an (unrelabeled) subsequence. We assume $x_{k}=\bar{x}$ $\forall k>0$ and
define $b_{k}=\bar{t}-t_{k}>0$ and the radii

$r_{k}:={2\sqrt{Cb_{k}}},$

where $C$ is the constant of Lemma 4.8. Taking $k$ large enough, by Lemma 4.8
the balls $B(\cdot,r_{k})$ have an extinction time greater than
$2(\bar{t}-t_{k})$. We then have

$\displaystyle B(\bar{x},r_{k})$
$\displaystyle\subseteq\\{\tilde{\eta}_{k}(\cdot,t_{k})\leq\tilde{\eta}_{k}(\bar{x},t_{k})+2{ar_{k}^{3}}\\}$
$\displaystyle\subseteq\\{u(\cdot,t_{k})\leq
u(\bar{x},t_{k})+2{ar_{k}^{3}}\\},$

by maximality of $u-\tilde{\eta}_{k}$ at $z_{k}$. Since the balls
$B(\cdot,r_{k})$ are not vanishing, we conclude

$\bar{x}\in\\{u(\cdot,\bar{t})\leq u(\bar{x},t_{k})+2{ar_{k}^{3}}\\}.$

Finally, we use again the maximality of $u-\eta$ at $\bar{z}$ and the choice
of $r_{k}$ to obtain

$\dfrac{\eta(\bar{x},t_{k})-\eta(\bar{z})}{t_{k}-\bar{t}}=\dfrac{\eta(\bar{x},t_{k})-\eta(\bar{z})}{-b_{k}}\leq\dfrac{u(\bar{x},t_{k})-u(\bar{x},\bar{t})}{-b_{k}}\leq\dfrac{-2{ar_{k}^{3}}}{-b_{k}}=c\,\sqrt{b_{k}}.$

Passing to the limit $k\to\infty$, we conclude that
$\partial_{t}\eta(\bar{z})\leq 0$. ∎

We conclude with two remarks concerning some possible generalizations of the
results presented.

######  Remark 4.12.

The results presented in this work can be immediately extended to unbounded
initial open sets $E_{0}$, whose boundary is compact. Indeed, defining the
discrete flow as $E_{t}^{(h)}=E_{0}$ if $t\in[0,h)$, otherwise by induction
$E_{t}^{(h)}=T^{-}_{h,t}E_{t-h}^{(h)},$ where the operator $T_{h,n}^{-}$ is
the one defined in (56), this evolution is uniquely characterized by the one
of the complement. Thus, all the results presented in this paper can be
extended to this particular unbounded case.

######  Remark 4.13.

Following the lines of [8] (in the spirit of [3]) one can see that the results
of this paper may be extended to prove existence of flat flows and level set
solutions to the mean curvature flow on $\mathbb{R}^{N}$ endowed with the
geometric structure induced by a Finsler metric $\phi^{\circ}.$ For example,
the perimeter functional in this setting is defined as follows. Given a set
$E$ of finite perimeter, its (intrinsic) perimeter is

$\mathcal{P}_{\phi^{\circ}}(E)=\int_{\partial^{*}E}\phi(x,\nu_{E}(x))\,\textnormal{d}\mathcal{H}^{N-1}_{\phi^{\circ}}(x),$

where the Hausdorff measure $\mathcal{H}^{N-1}_{\phi^{\circ}}$ is the one
induced by the metric ${\phi^{\circ}}.$ In particular, one can compute
$\,\textnormal{d}\mathcal{H}^{N-1}_{\phi^{\circ}}(x)=\omega_{N}|B^{\phi^{\circ}}(x)|^{-1}\,\textnormal{d}\mathcal{H}^{N-1}(x)$
(see [8]), thus this approach is equivalent to consider in our framework a
slightly different (but still regular) anisotropy, namely