Existence and uniqueness for anisotropic and crystalline mean curvature flows

Abstract

An existence and uniqueness result, up to fattening, for crystalline mean curvature flows with forcing and arbitrary (convex) mobilities, is proven. This is achieved by introducing a new notion of solution to the corresponding level set formulation. Such solutions satisfy a comparison principle and stability properties with respect to the approximation by suitably regularized problems. The results are valid in any dimension and for arbitrary, possibly unbounded, initial closed sets. The approach accounts for the possible presence of a time-dependent bounded forcing term, with spatial Lipschitz continuity. As a result of our analysis, we deduce the convergence of a minimizing movement scheme proposed by Almgren, Taylor, and Wang (1993) to a unique (up to fattening) “flat flow” in the case of general, including crystalline, anisotropies, solving a long-standing open question.

1. Introduction

In this paper we deal with anisotropic and crystalline mean curvature flows; that is, flows of sets $t\mapsto E(t)$ (formally) governed by the law

$$\begin{equation} V(x,t) = -\psi (\nu ^{E(t)}) (\kappa ^{E(t)}_{\phi }(x) + g(x,t)), {\tag{1.1}} \end{equation}$$

where $V(x,t)$ stands for the (outer) normal velocity of the boundary $\partial E(t)$ at $x$, $\phi$ is a given norm on $\mathbb{R}^N$ representing the surface tension, $\kappa ^{E(t)}_{\phi }$ is the anisotropic mean curvature of $\partial E(t)$ associated with the anisotropy $\phi$, $\psi$ is a norm evaluated at the outer unit normal $\nu ^{E(t)}$ to $\partial E(t)$, and $g$ is a bounded spatially Lipschitz continuous forcing term. The factor $\psi$ plays the role of a mobility.⁠¹ We recall that when $\phi$ is differentiable in $\mathbb{R}^{N}\setminus \{0\}$, then $\kappa ^{E}_{\phi }$ is given by the surface divergence of a “Cahn-Hoffman” vector field 174748:

¹

Strictly speaking, the mobility is $\psi (\nu ^{E(t)})^{-1}$.

✖

$$\begin{equation} \kappa ^{E}_{\phi }=\operatorname {div}_{\tau }\left(\nabla \phi (\nu ^E)\right) . {\tag{1.2}} \end{equation}$$

However, in this work we will be interested mostly in the “crystalline case”, which is whenever the level sets of $\phi$ are polytopes and 1.2 should be replaced with

$$\begin{equation} \kappa ^{E}_{\phi }\in \operatorname {div}_{\tau }\left(\partial \phi (\nu ^E)\right), {\tag{1.3}} \end{equation}$$

as we will describe later on.

Equation 1.1 is relevant in materials science and the study of crystal growth; see for instance 434849 and the references therein. Its mathematical well-posedness is classical in the smooth setting, that is, when $\phi$, $\psi$, $g$, and the initial set are sufficiently smooth (and $\phi$ satisfies suitable ellipticity conditions). However, it is also well known that in dimensions $N\geq 3$ singularities may form in finite time even in the smooth case. When this occurs the strong formulation of 1.1 ceases to be applicable and one needs a weaker notion of solution leading to a (possibly unique) globally defined evolution.

Among the different approaches that have been proposed in the literature for the classical mean curvature flow (and for several other “regular” flows) in order to overcome this difficulty, we start by mentioning the so-called level set approach 2629303745, which consists in embedding the initial set in the one-parameter family of sets given by the sublevels of some initial function $u^0$, and then in letting all these sets evolve according to the same geometric law. The evolving sets are themselves the sublevels of a time-dependent function $u(x,t)$, which turns out to solve a (degenerate) parabolic equation for $u$ (with the prescribed initial datum $u^0$). The crucial point is that such a parabolic Cauchy problem is shown to admit a global-in-time unique viscosity solution for many relevant geometric motions: in fact, one only needs the continuity⁠² of the Hamiltonian of the level-set equation which corresponds to 1.1 2636. When this happens, the evolution of the sublevels of $u$ defines a generalized motion (with initial set given by the corresponding sublevels of $u^0$), which exists for all times and agrees with the classical one until the appearance of singularities (see 30). Moreover, such a generalized motion satisfies the comparison principle and is unique whenever the level sets of $u$ have an empty interior. Let us mention that the appearance of a nontrivial interior (the so-called fattening phenomenon) may in fact occur even starting from a smooth set (see for instance 8). On the other hand, such a phenomenon is rather rare: for instance, given any uniformly continuous initial function $u^0$, all its sublevels, with the exception of at most countably many, will not generate any fattening.

²

Which is of course weaker than the requirements for the existence of strong solutions, which at least include ellipticity properties.

✖

The second approach which is relevant for the present treatment is represented by the minimizing movements scheme devised by Almgren, Taylor, and Wang 3 and, independently, by Luckhaus and Sturzenhecker 44. It is variational in nature and hinges on the gradient flow structure of the geometric motion. More precisely, it consists in building a family of discrete-in-time evolutions by an iterative minimization procedure and in considering any limit of these evolutions (as the time step $\Delta t>0$ vanishes) as an admissible global-in-time solution to the geometric motion, usually referred to as a flat flow (or ATW flat flow). The problem which is solved at each step has the form 3, §2.6

$$\begin{equation} \min _E P_\phi (E) + \frac{1}{\Delta t}\int _{E\triangle E^{n-1}} \operatorname {dist}(x,\partial E^{n-1}) dx, {\tag{1.4}} \end{equation}$$

where $E\triangle E^{n-1}$ denotes the symmetric difference of the two sets $E$ and $E^{n-1}$, and the anisotropic perimeter $P_\phi (E)=\int _{\partial E}\phi (\nu )d\mathcal{H}^{N-1}$ is defined rigorously in 2.1 below. (We generalize slightly the scheme later on, in particular to deal with noncompact boundaries, of possibly infinite mass.) This scheme is studied in great detail in 3 and many convergence properties are proven, including to the previously mentioned viscosity solutions, under some technical assumption. If $\phi$ (and the initial set) is smooth enough, also convergence to strong solutions are proven. However, except in dimension 2 2, the convergence of this scheme in the crystalline case remains open.

In this paper, we show for the first time that, up to exceptional initial sets which might develop nonuniqueness, this discrete procedure converges to a unique motion in all cases, including crystalline. In practice, if we replace the distance in 1.4 with an anisotropic distance based on a norm “compatible” with $\phi$ (“$\phi$-regular”, Definition 4.1), it is relatively easy, though a bit technical, to extend our previous results in 22 and show convergence of the scheme. Our main result in this work is a stability result which shows additionally that even if this $\phi$-regularity is lost (as is always the case when the distance in 1.4 is Euclidean and $\phi$ crystalline), the discrete-in-time flows remain close to $\phi$-regular flows and their limit is still unique. While the stability for the limiting flow (only) is relatively simple 20, the stability at the discrete level, for $\Delta t>0$, which allows us to derive the uniqueness of the flat flow, is quite technical and requires precise estimates on the minimizers of 1.4, established in Section 3.4. The remainder of this introduction describes more closely the technical content of this paper.

Practically, it is somewhat convenient to combine the variational approach with the level set point of view, by implementing the Almgren-Taylor-Wang scheme (ATW) for all the sublevels of the initial function $u^0$ (level set ATW). As already mentioned, it turns out that the two approaches produce in general the same solutions. A very simple proof of convergence of the level set ATW to the viscosity solution of the level set equation in the case of anisotropic mean curvature flows (with smooth anisotropy) is given in 23 (see also 319); such a result implies in turn the convergence of the ATW to the aforementioned generalized motion whenever fattening does not occur.

When the anisotropy is crystalline, all the results mentioned before for regular anisotropies become much more difficult, starting from the very definition of crystalline curvature which cannot be given by 1.2 anymore, but rather by 1.3: one has to consider a suitable selection $z$ of the (multivalued) subdifferential map $x\mapsto \partial \phi (\nu ^E(x))$ (of “Cahn-Hoffmann fields”), such that the tangential divergence $\operatorname {div}_\tau z$ has minimal $L^2$-norm among all possible selections. The crystalline curvature is then given by the tangential divergence $\operatorname {div}_\tau z_{\mathrm{opt}}$ of the optimal Cahn-Hoffman field (see 1635) and thus, in particular, has a nonlocal character.

We briefly recall what is known about the mathematical well-posedness of 1.1 in the crystalline case. In two dimensions, the problem has been essentially settled in 34 (when $g$ is constant) by developing a crystalline version of the viscosity approach for the level-set equation; see also 273338484950 for important former work. The viscosity approach adopted in 34 applies in fact to more general equations of the form

$$\begin{equation} V=f(\nu ,-\kappa ^E_\phi )\,, {\tag{1.5}} \end{equation}$$

with $f$ continuous and nondecreasing with respect to the second variable, however, without spatial dependence. Former studies were rather treating the problem as a system of coupled ODEs describing the relative motion of each facet of an initial crystal 274849. We mention also the recent paper 25, where short time existence and uniqueness of strong solutions for initial “regular” sets (in a suitable sense) is shown.

In dimension $N\geq 3$ the situation was far less clear until very recently. Before commenting on the new developments, let us remark that before these, the only general available notion of global-in-time solution was that of a flat flow associated with the ATW scheme, defined as the limit of a converging subsequence of time discrete approximations. However, no general uniqueness and comparison results were available, except for special classes of initial data 131839 or for very specific anisotropies 35. As mentioned before, substantial progress in this direction has been made only very recently, in 41 and 22.

In 41, the authors succeed in extending the viscosity approach of 34 to $N=3$. They are able to deal with very general equations of the general form 1.5 establishing existence and uniqueness for the corresponding level set formulations. In an article just appeared 42, they show how to extend their approach to any dimension, which is a major breakthrough (moreover the new proof is considerably simpler than before). It seems that their method, as far as we know, still requires a purely crystalline anisotropy $\phi$ (so mixed situations are not allowed), bounded initial sets, and the only possible forcing term is a constant.

In 22, the first global-in-time existence and uniqueness (up to fattening) result for the crystalline mean curvature flow valid in all dimensions, for arbitrary (possibly unbounded) initial sets, and for general (including crystalline) anisotropies $\phi$ was established, but under the particular choice $\psi =\phi$ (and $g=0$) in 1.1. It is based on a new stronger distributional formulation of the problem in terms of distance functions, which is reminiscent of, but not quite the same as, the distance formulation proposed and studied in 46 (see also 46111828). Such a formulation enables the use of parabolic PDE arguments to prove comparison results, but of course makes it more difficult to prove existence. The latter is established by implementing the variant of the ATW scheme devised in 1819. The methods of 22 yield, as a byproduct, the uniqueness, up to fattening, of the ATW flat flow for the equation 1.1 with $\psi =\phi$ and $g=0$. But it leaves open the uniqueness issue for the general form of 1.1 and, in particular, for the constant mobility case

$$\begin{equation} V=-\kappa ^E_{\phi }\,, {\tag{1.6}} \end{equation}$$

originally appearing in 3, which is approximated by 1.4. The main reason is technical: the distributional formulation introduced in 22 becomes effective in yielding uniqueness results only if, roughly speaking, the level sets of the $\psi ^\circ$-distance function from any closed set ($\psi ^\circ$ being the norm polar to the mobility $\psi$) have (locally) bounded crystalline curvatures. This is certainly the case when $\phi =\psi$ (and explains such a restriction in 22).

As said, we remove in this paper the restriction $\phi =\psi$ and extend the existence and uniqueness results of 22 to the general equation 1.1. In order to deal with general mobilities, we cannot rely anymore on a distributional formulation in the spirit of 22, but instead we extend the notion of solution via an approximation procedure by suitable regularized versions of 1.1.

We now describe in more detail the contributions and the methods of the paper. Before addressing the general mobilities, we consider the case where $\psi$ may be different from $\phi$ but satisfies a suitable regularity assumption, namely we assume that the Wulff shape 2.3 associated with $\psi$ (in short the $\psi$-Wulff shape) admits an inner tangent $\phi$-Wulff shape at all points of its boundary. We call such mobilities $\phi$-regular (see Definition 4.1). The $\phi$-regularity assumption implies in turn that the level sets of the $\psi ^\circ$-distance function from any closed set have locally bounded crystalline curvatures and makes it possible to extend the distributional formulation (and the methods) of 22 to 1.1 (Definition 2.2), to show that such a notion of solution bears a comparison principle (Theorem 2.7) and that the ATW scheme converges to it (Theorem 4.3). As is classical, we then use these results to build a unique level set flow (and a corresponding generalized motion), which satisfies comparison and geometricity properties (Theorem 4.8).

Having accomplished this, we deal with the general case of $\psi$ being any norm. As mentioned before, the idea here is to build a level set flow by means of approximation, after the easy observation that for any norm $\psi$ there exists a sequence $\{\psi _n\}$ of $\phi$-regular mobilities such that $\psi _n\to \psi$. More precisely, we say that $u$ is a solution to the level set flow associated with 1.1 if there exists an approximating sequence $\{\psi _n\}$ of $\phi$-regular mobilities such that the corresponding level set flows $u_n$ constructed in Section 4 locally uniformly converge to $u$ (Definition 5.6).

In Theorems 5.7 and 5.9 we establish the main results of the paper: we show that for any norm $\psi$ a solution-via-approximation $u$ always exists; moreover $u$ satisfies the following properties:

(i)

(Uniqueness and stability) The solution-via-approximation $u$ is unique in that it is independent of the choice of the approximating sequence of $\phi$-regular mobilities $\{\psi _n\}$. In fact, it is stable with respect to the convergence of any sequence of mobilities and anisotropies.

(ii)

(Comparison) If $u^0\leq v^0$, then the corresponding level set solutions $u$ and $v$ satisfy $u\leq v$.

(iii)

(Convergence of the level set ATW) $u$ is the unique limit of the level set ATW.

(iv)

(Generic nonfattening) As in the classical case, for any given uniformly continuous initial datum $u^0$ all but countably many sublevels do not produce any fattening.

(v)

(Comparison with other notions of solutions) Our solution-via-approximation $u$ coincides with the classical viscosity solution in the smooth case and with the Giga-Požár viscosity solution 4142 whenever such a solution is well-defined, that is, when $g$ is constant, $\phi$ is purely crystalline, and the initial set is bounded.

(vi)

(Phase-field approximation) When $g$ is constant, a phase-field Allen-Cahn type approximation of $u$ holds.

We finally mention that property (iii) implies the convergence of the ATW scheme, whenever no fattening occurs and thus settles the long-standing problem of the uniqueness (up to fattening) of the flat flow corresponding to 1.1 (and in particular for 1.6) when the anisotropy is crystalline. In our later paper 20 we show that it is also possible to build crystalline flows by approximating the anisotropies with smooth ones and relying for existence on the standard viscosity theory of generalized solutions. However, this variant, even if slightly simpler, does not show that flat flows are unique.

The plan of the paper is the following. In Section 2 we extend the distributional formulation of 22 to our setting and we study the main properties of the corresponding notions of sub- and supersolutions. The main result of the section is the comparison principle established in Section 2.3.

In Section 3 we set up the minimizing movements algorithm and we start paving the way for the main results of the paper by establishing some preliminary results. In particular, the density estimates and the barrier argument of Section 3.4, which do not require any regularity assumption on the mobility $\psi$, will be crucial for the stability analysis of the ATW scheme needed to deal with the general mobility case and developed in Section 5.1.

In Section 4 we develop the existence and uniqueness theory under the assumption of $\phi$-regularity for the mobility $\psi$. More precisely, we establish the convergence of the ATW scheme to a distributional solution of the flow, whenever fattening does not occur. Uniqueness then follows from the results of Section 2.

Finally, in Section 5 we establish the main results of the paper, namely the existence and uniqueness of a solution via approximation by $\phi$-regular mobilities. As already mentioned, the approximation procedure requires a delicate stability analysis of the ATW scheme with respect to varying mobilities. Such estimates are established in Section 5.1 and represent the main technical achievement of Section 5.

2. A distributional formulation of curvature flows

In this section we generalize the approach introduced in 22 by introducing a suitable distributional formulation of 1.1 and we show that such a formulation yields a comparison principle and is equivalent to the standard viscosity formulation when the anisotropy $\phi$ and the mobility $\psi$ are sufficiently regular.

The existence of the distributional solution defined in this section will be established in Section 4 under the additional assumption that the mobility $\psi$ satisfies a suitable regularity assumption (see Definition 4.1 below).

2.1. Preliminaries

We introduce the main objects and notation used throughout the paper.

Given a norm $\eta$ on $\mathbb{R}^N$ (a convex, even,⁠³ one-homogeneous real-valued function with $\eta (\nu )>0$ if $\nu \neq 0$), we define a polar norm $\eta ^\circ$ by $\eta ^\circ (\xi ):=\sup _{\eta (\nu )\le 1}\nu \cdot \xi$ and an associated anisotropic perimeter $P_\eta$ as

³

For simplicity we develop the theory in the symmetric case; see Remark 6.3.

✖

$$\begin{equation} P_\eta (E):=\sup \biggl \{\int _E\operatorname {div}\zeta \, dx: \zeta \in C^1_c(\mathbb{R}^N; \mathbb{R}^N),\, \eta ^\circ (\zeta )\leq 1\biggr \}\,. {\tag{2.1}} \end{equation}$$

As is well known, $(\eta ^\circ )^\circ =\eta$ so that when the set $E$ is smooth enough one has

$$\begin{equation*} P_\eta (E) = \int _{\partial E}\eta (\nu ^E)d\mathcal{H}^{N-1}\,, \end{equation*}$$

which is the perimeter of $E$ weighted by the surface tension $\eta (\nu ^E)$. The notation $\mathcal{H}^s$, $s>0$, stands for the $s$-dimensional Hausdorff measure. It is also useful to recall the notion of relative perimeter: given an open set $\Omega \subset \mathbb{R}^N$ we will denote by $P_\eta (E; \Omega )$ the $\eta$-perimeter of $E$ relative to $\Omega$; i.e.,

$$\begin{equation*} P_\eta (E; \Omega ):=\sup \biggl \{\int _E\operatorname {div}\zeta \, dx: \zeta \in C^1_c(\Omega ; \mathbb{R}^N),\, \eta ^\circ (\zeta )\leq 1\biggr \}\,. \end{equation*}$$

As before, note that if $E$ is sufficiently regular, then

$$\begin{equation*} P_\eta (E; \Omega )= \int _{\partial E\cap \Omega }\eta (\nu ^E)d\mathcal{H}^{N-1}\,. \end{equation*}$$

We will often use the following characterization of the subgradient:

$$\begin{equation} \partial \eta (\nu ) =\{\xi : \eta ^\circ (\xi )\le 1\text{ and }\xi \cdot \nu { \ge \eta (\nu )\} } {\tag{2.2}} \end{equation}$$

(and the symmetric statement for $\eta ^\circ$). In particular, if $\nu \neq 0$ and $\xi \in \partial \eta (\nu )$, then $\eta ^\circ (\xi )=1$ and $\partial \eta (0)=\{\xi : \eta ^\circ (\xi )\le 1\}$. For $R>0$ we denote

$$\begin{equation} W^{\eta }(x,R):=\{y : \eta ^\circ (y-x)\le R\}\,. {\tag{2.3}} \end{equation}$$

Such a set is called the Wulff shape (of radius $R$ and center $x$) associated with the norm $\eta$ and represents the unique (up to translations) solution of the anisotropic isoperimetric problem

$$\begin{equation*} \min \left\{P_\eta (E) : |E|=|W^{\eta }(0,R)|\right\}; \end{equation*}$$

see for instance 32. We let $W^\eta :=W^\eta (0,1)$, $B_1=W^{|\cdot |}$ be the unit ball, and more generally, for $r>0$, $B_r=\{x:|x|\le r\}$.

We denote by $\operatorname {dist}^\eta (\cdot , E)$ the distance from $E$ induced by the norm $\eta$, that is, for any $x\in \mathbb{R}^N$

$$\begin{equation*} \operatorname {dist}^\eta (x, E):=\inf _{y\in E}\eta (x-y) \end{equation*}$$

if $E\neq \emptyset$, and $\operatorname {dist}^\eta (x,\emptyset ):= + \infty$. Moreover, we denote by $d^\eta _E$ the signed distance from $E$ induced by $\eta$, i.e.,

$$\begin{equation} d^\eta _E(x):= \operatorname {dist}^\eta (x,E) - \operatorname {dist}^\eta (x,E^c)\,, {\tag{2.4}} \end{equation}$$

so that $\operatorname {dist}^\eta (x,E)=d^\eta _E(x)^+$ and $\operatorname {dist}^\eta (x,E^c)=d^\eta _E(x)^-$, where we adopted the standard notation $t^+:=t\lor 0$ and $t^-:=(-t)^+$. Note that by 2.2 we have $\eta (\nabla d^{\eta ^\circ }_{E})=\eta ^\circ (\nabla d^{\eta }_{E})=1$ a.e. in $\mathbb{R}^N\setminus \partial E$. We will write $\operatorname {dist}(\cdot , E)$ and $d_E$ without any superscript to denote the Euclidean distance and signed distance from $E$, respectively.

Finally we recall that a sequence of closed sets $(E_n)_{n\ge 1}$ in $\mathbb{R}^m$ converges to a closed set $E$ in the Kuratowki sense if the following conditions are satisfied:

(i)

if $x_n\in E_n$ for each $n$, any limit point of $(x_n)_{n\ge 1}$ belongs to $E$;

(ii)

any $x\in E$ is the limit of a sequence $(x_n)_{n\ge 1}$, with $x_n\in E_n$ for each $n$.

We write in this case:

$$\begin{equation*} E_n\stackrel{\mathcal{K}}{\longrightarrow } E\,. \end{equation*}$$

It is easily checked that $E_n\stackrel{\mathcal{K}}{\longrightarrow } E$ if and only if (for any norm $\eta$) $\operatorname {dist}^{\eta }(\cdot , E_n)\to \operatorname {dist}^{\eta }(\cdot , E)$ locally uniformly in $\mathbb{R}^m$. In particular, the Ascoli-Arzelà Theorem shows that any sequence of closed sets admits a converging subsequence in the Kuratowski sense.

2.2. The distributional formulation

In this subsection we give the precise formulation of the crystalline mean curvature flows we will deal with. Throughout the paper the norms $\phi$ and $\psi$ will stand for the anisotropy and the mobility, respectively, appearing in 1.1. Note that we do not assume any regularity on $\phi$ (nor on $\psi$) and in fact we are mainly interested in the case when $\phi$ is crystalline, that is, when the associated unit ball is a polytope.

Moreover, we will assume throughout the paper that the forcing term $g:\mathbb{R}^N\times [0, +\infty )\to \mathbb{R}$ satisfies the following two hypotheses:

H1)

$g\in L^{\infty }(\mathbb{R}^N\times (0, \infty ))$;

H2)

there exists $L>0$ such that $g(\cdot , t)$ is $L$-Lipschitz continuous (with respect to the metric $\psi ^\circ$) for a.e. $t>0$.

Remark 2.1.

Assumption H1) can be weakened and replaced by

H1)’

for every $T>0$, $g\in L^{\infty }(\mathbb{R}^N\times (0, T))$.

Indeed under the weaker assumption H1)’, all the arguments and the estimates presented throughout the paper continue to work in any time interval $(0,T)$, with some of the constants involved possibly depending on $T$. In the same way, if one restricts our study to the evolution of sets with compact boundary, then one could assume that $g$ is only locally bounded in space. We assume H1) instead of H1)’ to simplify the presentation.

In all that follows by the expression “admissible forcing term” we will mean a forcing term $g$ satisfying H1) and H2) above.

We are now ready to provide a suitable distributional formulation of the curvature flow 1.1.

Definition 2.2.

Let $E^0\subset \mathbb{R}^N$ be a closed set. Let $E$ be a closed set in $\mathbb{R}^N\times [0,+\infty )$ and for each $t\geq 0$ denote $E(t):=\{x\in \mathbb{R}^N : (x,t)\in E\}$. We say that $E$ is a superflow of 1.1 with initial datum $E^0$ if the following hold:

(a)

Initial condition: $E(0)\subseteq {E}^0$.

(b)

Left continuity: $E(s)\stackrel{\mathcal{K}}{\longrightarrow } E(t)$ as $s\nearrow t$ for all $t>0$.

(c)

If ${E}(t)=\emptyset$ for some $t\ge 0$, then $E(s)=\emptyset$ for all $s > t$.

(d)

Differential inequality: Set $T^*:=\inf \{t>0 : E(s)=\emptyset \text{ for $s\geq t$}\}$, and$$\begin{equation*} d(x,t):=\operatorname {dist}^{\psi ^\circ }(x, E(t)) \qquad \text{ for all } (x,t)\in \mathbb{R}^N\times (0,T^*)\setminus E. \end{equation*}$$

Then there exists $M>0$ such that the inequality$$\begin{equation} \partial _t d \ge \operatorname {div}z+g- Md {\tag{2.5}} \end{equation}$$

holds in the distributional sense in $\mathbb{R}^N\times (0,T^*)\setminus E$ for a suitable $z\in L^\infty (\mathbb{R}^N\times (0,T^*))$ such that $z\in \partial \phi (\nabla d)$ a.e., $\operatorname {div}z$ is a Radon measure in $\mathbb{R}^N\times (0,T^*)\setminus E$, and $(\operatorname {div}z)^+\in L^\infty (\{(x,t)\in \mathbb{R}^N\times (0,T^*):\, d(x,t)\geq \delta \})$ for every $\delta \in (0,1)$.

We say that $A$, an open set in $\mathbb{R}^N\times [0,+\infty )$, is a subflow of 1.1 with initial datum $E^0$ if $A^c$ is a superflow of 1.1 with $g$ replaced by $-g$ and with initial datum $(\mathring{E}^0)^c$.

Finally, we say that $E$, a closed set in $\mathbb{R}^N\times [0,+\infty )$, is a solution of 1.1 with initial datum $E^0$ if it is a superflow and if $\mathring{E}$ is a subflow, both with initial datum $E^0$, assuming in addition that both $E^0$ and $E$ coincide with the closure of their interior.

In Subsection 4.2 we will prove the existence of solutions satisfying 2.5 with $M=L$. In our definition, “super”flow refers to the fact that the distance function may grow faster than the distance to a solution of the mean curvature flow, which corresponds to a set shrinking also faster than expected with 1.1.

Remark 2.3.

Notice that the closedness of $E$ yields that $d$ is lower semicontinuous. Indeed, if $(x_k,t_k)\to (x,t)$, with $t_k,t\le T^*$, we can choose $y_k\in E(t_k)$ with $\psi ^\circ (x_k-y_k)=d(x_k,t_k)$. Then, since any limit point of $(y_k,t_k)$ is in $E$, one deduces $d(x,t)\le \liminf _k d(x_k,t_k)$. On the other hand, condition (b) implies that $d(\cdot ,t)$ is left-continuous. Moreover, by condition (d) of Definition 2.2, the distributional derivative $\partial _t d$ is a Radon measure in $\mathbb{R}^N\times (0,T^*)\setminus E$, so that $d$ is locally a function with bounded variation; using the fact that the distance functions are uniformly Lipschitz, we can deduce that for any $t\in [0,T^*)$, $d(\cdot ,s)$ converges locally uniformly in $\{x : d(x, t)>0\}$ as $s\searrow t$ to some function $d^r$ with $d^r \geq d(\cdot , t)$ in $\{x : d(x, t)>0\}$, while $d(\cdot ,s)$ converges locally uniformly to $d(\cdot , t)$ as $s\nearrow t$ (cf. 22, Lemma 2.4).

Remark 2.4.

Notice that the initial condition for subflows may be rewritten as $\mathring{E}^0 \subseteq A(0)$. In particular, if $E$ is a solution according to the previous definition, then $E(0)=E^0$.

We now introduce the corresponding notion of sub- and supersolution to the level set flow associated with 1.1.

Definition 2.5 (Level set subsolutions and supersolutions).

Let $u^0$ be a uniformly continuous function on $\mathbb{R}^N$. We will say that a lower semicontinuous function $u:\mathbb{R}^N\times [0, +\infty )\to \mathbb{R}$ is a supersolution to the level set flow corresponding to 1.1 (level set supersolution for short), with initial datum $u^0$, if $u(\cdot , 0)\geq u^0$ and if for a.e. $\lambda \in \mathbb{R}$ the closed sublevel set $\{(x,t) : u(x,t)\leq \lambda \}$ is a superflow of 1.1 in the sense of Definition 2.2, with initial datum $\{u_0\leq \lambda \}$.

We will say that an upper semicontinuous function $u:\mathbb{R}^N\times [0, +\infty )\to \mathbb{R}$ is a subsolution to the level set flow corresponding to 1.1 (level set subsolution for short), with initial datum $u^0$, if $-u$ is a level set supersolution in the previous sense, with initial datum $-u_0$ and with $g$ replaced by $-g$.

Finally, we will say that a continuous function $u:\mathbb{R}^N\times [0, +\infty )\to \mathbb{R}$ is a solution to the level set flow corresponding to 1.1 if it is both a level set subsolution and supersolution.

2.3. The comparison principle

In this subsection we establish a comparison principle between sub- and superflows as defined in the previous subsection. A first technical result is a (uniform) left-continuity estimate for the distance function to a superflow.

Lemma 2.6.

Let $E$ be a superflow in the sense of Definition 2.2, and $d(x,t)=\operatorname {dist}^{\psi ^\circ }(x,E(t))$ the associated distance function. Then, there exist $\tau _0,\chi$ depending on $N,\|g\|_\infty ,M$ such that for any $x,t\ge 0$ and any $s\in [0,\tau _0]$,

$$\begin{equation} d(x,t+s) \ge d(x,t)e^{-5Ms} - \chi \sqrt {s} {\tag{2.6}} \end{equation}$$

and (for any $s\in [0,\tau _0]$ with $s \le t$)

$$\begin{equation} d(x,t-s) \le d(x,t)e^{5Ms} + \chi \sqrt {s}. {\tag{2.7}} \end{equation}$$

Proof.

The proof follows the lines of the proof of 22, Lemma 3.2 up to minor changes that we will briefly describe in the following. By definition of a superflow we have

$$\begin{equation*} \partial _t d \ge \operatorname {div}z - Md - \|g\|_\infty , \end{equation*}$$

wherever $d>0$. Consider $(\bar{x},\bar{t})$ with $d(\bar{x},\bar{t})=R>0$. For $s>0$, let $\tau (s):=\log (1+Ms)/M$ and define

$$\begin{equation*} \delta (x,s) = d(x,\bar{t}+\tau (s))(1+Ms) + \|g\|_\infty s \ge 0. \end{equation*}$$

We have that $\delta (x,0)=d(x,\bar{t})$, while, in $\{d>0\}$, $\delta (x,\cdot )$ is $BV$ in time, the singular part $\partial _s^s\delta$ is nonnegative (as the singular part $\partial _t^s d$ is nonnegative thanks to 2.5 and the assumption on $\operatorname {div}z$), and the absolutely continuous part satisfies

$$\begin{equation*} \begin{split} {\partial _s^a} \delta (x,s)&= {\partial _t^a} d(x,\bar{t}+\tau (s))\tau '(s)(1+Ms) + Md(x,\bar{t}+\tau (s))+\|g\|_\infty \\ &\ge \operatorname {div}z (x,\bar{t}+\tau (s)). \end{split} \end{equation*}$$

As $z(x,\tau (s))\in \partial \phi (\nabla \delta (x,s))$, we obtain that $\delta$ is a supersolution of the $\phi$-total variation flow starting from $d(\cdot ,\bar{t})$, and we can reproduce the proof of 22, Lemma 3.2: we find that there exists a constant $\chi _N$ such that $\delta (\bar{x},s)\ge R-\chi _N\sqrt {s}$ for $s\ge 0$ as long as this bound ensures that $d(\bar{x},\bar{t}+\tau (s))>3R/4$, which is as long as

$$\begin{equation} \frac{R}{4} -\chi _N\sqrt {s} - \left(\|g\|_\infty + \frac{3MR}{4}\right)s>0. {\tag{2.8}} \end{equation}$$

Now we prove that for any $s\ge 0$

$$\begin{equation} d(\bar{x},\bar{t}+\tau (s))(1+Ms)\ge R-4\chi _N\sqrt {s}-(4\|g\|_\infty +3MR)s. {\tag{2.9}} \end{equation}$$

Indeed, as long as 2.8 holds true we have

$$\begin{multline*} R-4\chi _N\sqrt {s}-(4\|g\|_\infty +3MR)s \le R-\chi _N\sqrt {s}-\|g\|_\infty s \\ \le \delta (\bar{x},s) -\|g\|_\infty s = d(\bar{x},\bar{t}+\tau (s))(1+Ms). \end{multline*}$$

On the other hand, for later times the left-hand side of 2.9 is (always) nonnegative and the right-hand side becomes nonpositive. Notice that 2.9 can be rewritten as

$$\begin{equation*} d(\bar{x},\bar{t}+\tau (s))(1+Ms) \ge d(\bar{x},\bar{t})(1-3Ms) - 4\chi _N\sqrt {s} -4\|g\|_\infty s, \end{equation*}$$

and since this holds for any $s\ge 0$ and does not depend on the particular value of $R$, it holds in fact for any $\bar{x},\bar{t}$ and we denote this point simply by $x,t$ in the sequel.

Since $s=(e^{M\tau (s)}-1)/M$, we deduce that for any $x,t\ge 0,\tau \ge 0$,

$$\begin{equation*} d(x, t+\tau )\ge d( x, t)(4e^{-M\tau }-3) - 4\|g\|_\infty \frac{1-e^{-M\tau }}{M} - 4\chi _N \sqrt {\frac{e^{M\tau }-1}{M}}e^{-M\tau }. \end{equation*}$$

Inequality 2.6 follows by Taylor expansion, while 2.7 is in fact equivalent to 2.6, up to a change of constants.

■

We can now show the following important comparison result.

Theorem 2.7.

Let $E$ be a superflow with initial datum $E^0$, and let $F$ be a subflow with initial datum $F^0$ in the sense of Definition 2.2. Assume that $\operatorname {dist}^{\psi ^\circ }(E^0,{F^0}^c)=:\Delta >0$. Then,

$$\begin{equation*} \operatorname {dist}^{\psi ^\circ }(E(t),F^c(t))\ge \Delta e^{-Mt} \qquad \text{ for all } t\ge 0, \end{equation*}$$

where $M>0$ is as in 2.5 for both $E$ and $F$.

Proof.

Let $T^*_E$ and $T^*_F$ be the maximal existence time for $E$ and $F$. For all $t> \min \{T^*_E, T^*_F\}=:T^*$ we have that either $E$ or $F^c$ is empty. For all such $t$’s the conclusion clearly holds true.

Thus, we may assume without loss of generality that $T^*_E, T^*_F >0$ and we consider the case $t\leq T^*$. By iteration (thanks to the left-continuity of $d$) it is clearly enough to show the conclusion of the theorem for a time interval $(0, t^*)$ for some $0<t^*\leq T^*$.

Let us fix $0<\eta _1<\eta _2<\eta _3<{\Delta }/2$. We denote by $z_E$ and $z_{F^c}$ the fields appearing in the definition of superflow (see Definition 2.2), corresponding to $E$ and $F^c$, respectively. Consider the set

$$\begin{equation*} S:=\{x\in \mathbb{R}^N : d^{\psi ^\circ }_E(x,0)>\eta _1\}\cap \{x\in \mathbb{R}^N : d^{\psi ^\circ }_{F^c}(x,0)>\eta _1\} \,. \end{equation*}$$

We now set

$$\begin{align*} & \widetilde{d}_E:=d^{\psi ^\circ }_E\lor (\eta _2+Ct)\,,\\ & \widetilde{d}_{F^c}:=d^{\psi ^\circ }_{F^c}\lor (\eta _2+Ct)\,, \end{align*}$$

with $C>0$ to be chosen later. By our assumptions $(\widetilde{d}_E+\widetilde{d}_{F^c})(\cdot , 0)\geq \Delta$. Moreover, since by construction

$$\begin{equation*} \widetilde{d}_E+\widetilde{d}_{F^c}\ge \Delta + (\eta _2-\eta _1)\text{ on }\partial S\times \{0\}\,, \end{equation*}$$

it follows from Lemma 2.6 that there exists $t^*\in (0,1\land T^*)$ such that

$$\begin{equation} \widetilde{d}_E+\widetilde{d}_{F^c}\geq \Delta \text{ on }\partial S\times (0, t^*)\,. {\tag{2.10}} \end{equation}$$

Relying again on Lemma 2.6 and arguing similarly we also have (for a possibly smaller $t^*$)

$$\begin{equation} S\subset \biggl \{x\in \mathbb{R}^N : d^{\psi ^\circ }_E(x,t)>\frac{\eta _1}{2}\biggr \}\cap \biggl \{x\in \mathbb{R}^N : d^{\psi ^\circ }_{F^c}(x,t)>\frac{\eta _1}{2}\biggr \}\quad \text{for all }t\in (0, t^*)\,, {\tag{2.11}} \end{equation}$$

$$\begin{equation*} E(t)\subset \subset F(t) \qquad \text{for }t\in (0, t^*), \end{equation*}$$ and

$$\begin{equation} \widetilde{d}_E=d^{\psi ^\circ }_E \quad \text{and}\quad \widetilde{d}_{F^c}=d^{\psi ^\circ }_{F^c}\qquad \text{in }S''\times (0, t^*)\,, {\tag{2.12}} \end{equation}$$

where

$$\begin{equation*} S'':=\{x\in \mathbb{R}^N : d^{\psi ^\circ }_E(x,0)>\eta _3\}\cap \{x\in \mathbb{R}^N : d^{\psi ^\circ }_{F^c}(x,0)>\eta _3\}\,. \end{equation*}$$

Since $d^{\psi ^\circ }_E$ is Lipschitz continuous in space and $\partial _t d^{\psi ^\circ }_E$ is a measure wherever $d^{\psi ^\circ }_E$ is positive, it follows that $d^{\psi ^\circ }_E$ (and in turn $\widetilde{d}_E$) is a function in $BV_{loc}(S\times (0,t^*))$ and its distributional time derivative has the form⁠⁴

⁴

With a slight abuse of notation, in the jump part at $t\in J$ we denote by $dx$ what should be the Hausdorff measure $\mathcal{H}^N$ on the hyperplane $\mathbb{R}^N\times \{t\}$.

✖

$$\begin{equation*} \partial _t d^{\psi ^\circ }_E = \sum _{t\in J}\left( d^{\psi ^\circ }_E(\cdot ,t+0)- d^{\psi ^\circ }_E(\cdot ,t-0)\right)dx + \partial _t^d d^{\psi ^\circ }_E, \end{equation*}$$

where $J$ is the (countable) set of times where $d^{\psi ^\circ }_E$ jumps and $\partial ^d_t d^{\psi ^\circ }_E$ is the diffuse part of the derivative. It turns out that (see Remark 2.3) $d^{\psi ^\circ }_E(\cdot ,t+0)-d^{\psi ^\circ }_E(\cdot ,t-0)\ge 0$ for each $t\in J$. Moreover, since the positive part of $\operatorname {div}z_E$ is absolutely continuous with respect to the Lebesgue measure (cf. Definition 2.2(d)), 2.5 entails

$$\begin{equation*} \partial ^d_t d^{\psi ^\circ }_E\ge \operatorname {div}z_E+g- M d^{\psi ^\circ }_E \end{equation*}$$

in $S\times (0,t^*)$. Using the chain rule (see for instance 5), in $S\times (0, t^*)$ we have

$$\begin{equation*} \partial ^d_t \widetilde{d}_E= \begin{cases} C & \text{a.e. in }\{(x,t) : \eta _2+Ct>d^{\psi ^\circ }_E(x,t)\}\,,\\ \partial ^d_t d^{\psi ^\circ }_E & \text{$|\partial ^d_t d^{\psi ^\circ }_E|$-a.e. in }\{(x,t) : \eta _2+Ct\leq d^{\psi ^\circ }_E(x,t)\}\,. \end{cases} \end{equation*}$$

An analogous formula holds for $\partial ^d_t \widetilde{d}_{F^c}$. Recalling that $(\operatorname {div}z_E)^+$ and $(\operatorname {div}z_{F^c})^+$ belong to $L^{\infty }(S\times (0,t^*))$ it follows that

$$\begin{equation} \partial ^d_t \widetilde{d}_E \geq \operatorname {div}z_E+g-M\widetilde{d}_E \quad \text{and}\quad \partial ^d_t \widetilde{d}_{F^c} \geq \operatorname {div}z_{F^c}-g-M\widetilde{d}_{F^c} {\tag{2.13}} \end{equation}$$

in the sense of measures in $S\times (0, t^*)$ provided that we have chosen (cf. 2.11 and the conditions in Definition 2.2(d))

$$\begin{equation*} C\geq \|(\operatorname {div}z_E)^+\|_{L^{\infty }(S\times (0,t^*))}+\|(\operatorname {div}z_{F^c})^+\|_{L^{\infty }(S\times (0,t^*))}+\|g\|_{L^{\infty }(S\times (0,t^*))}\,. \end{equation*}$$

Note also that a.e. in $S\times (0, t^*)$

$$\begin{equation} z_E\in \partial \phi (\nabla \widetilde{d}_E) \quad \text{and}\quad z_{F^c}\in \partial \phi (\nabla \widetilde{d}_{F^c})\,. {\tag{2.14}} \end{equation}$$

Fix $p>N$ and set $\Psi (s):=(s^+)^p$ and $w:=\Psi (\Delta - e^{Mt} (\widetilde{d}_E + \widetilde{d}_{F^c}))$. By 2.10 we have

$$\begin{equation} w=0 \qquad \text{on }\partial S\times (0, t^*)\,. {\tag{2.15}} \end{equation}$$

Using as before the chain rule for $BV$ functions, recalling 2.13 and the fact that the jump parts of $\partial _t\widetilde{d}_E$ and $\partial _t\widetilde{d}_{F^c}$ are nonnegative, in $S\times (0, t^*)$ we have

$$\begin{multline} \partial _t w\leq -\Psi '\big (\Delta - e^{Mt}(\widetilde{d}_E + \widetilde{d}_{F^c})\big )e^{Mt} \left(M(\widetilde{d}_E + \widetilde{d}_{F^c})+\partial ^d_t (\widetilde{d}_E + \widetilde{d}_{F^c})\right) \\ \le -\Psi '\big (\Delta - e^{Mt}(\widetilde{d}_E + \widetilde{d}_{F^c})\big )e^{Mt}\operatorname {div}(z_E+z_{F^c})\,, {\tag{2.16}} \end{multline}$$

where in the last inequality we have used 2.13. Choose a cut-off function $\eta \in C^{\infty }_c(\mathbb{R}^N)$ such that $0\leq \eta \leq 1$ and $\eta \equiv 1$ on $B_1$. For every $\varepsilon >0$ we set $\eta _\varepsilon (x):=\eta (\varepsilon x)$. Using 2.15 and 2.16, we have

$$\begin{align*} \partial _t \int _{S} w \eta _\varepsilon ^p dx &\le -e^{Mt} \int _{S}\eta _\varepsilon ^p\Psi '\big (\Delta - e^{Mt}(\widetilde{d}_E + \widetilde{d}_{F^c})\big )\operatorname {div}(z_E+z_{F^c})\\ &= -e^{Mt}\int _{S} \eta _\varepsilon ^p\Psi ''\big (\Delta - e^{Mt}(\widetilde{d}_E + \widetilde{d}_{F^c})\big )(\nabla \widetilde{d}_E+\nabla \widetilde{d}_{F^c})\cdot (z_E+z_{F^c})\, dx\\ &\hphantom {\leq }\,\,\, +pe^{Mt}\int _S\eta _\varepsilon ^{p-1}\,\Psi '\big (\Delta - e^{Mt}(\widetilde{d}_E + \widetilde{d}_{F^c})\big )\nabla \eta _\varepsilon \cdot (z_E+z_{F^c})\, dx\\ &\le pe^{Mt}\int _S\eta _\varepsilon ^{p-1}\,\Psi '\big (\Delta - e^{Mt}(\widetilde{d}_E + \widetilde{d}_{F^c})\big )\nabla \eta _\varepsilon \cdot (z_E+z_{F^c})\, dx\,, \end{align*}$$

where we have also used the inequality $(z_E+z_{F^c})\cdot (\nabla \widetilde{d}_E+\nabla \widetilde{d}_{F^c})\geq 0$, which follows from 2.14 and the convexity and symmetry of $\phi$. By Hölder’s inequality and using the explicit expression of $\Psi$ and $\Psi '$, we get

$$\begin{equation*} \partial _t \int _{S} w \, \eta _\varepsilon ^p dx\leq Cp^2 \|\nabla \eta _\varepsilon \|_{L^p(\mathbb{R}^N)}\left(\int _{S} w\, \eta _\varepsilon ^p dx\right)^{1-\frac{1}{p}} \end{equation*}$$

for some constant $C>0$ depending only on the $L^\infty$-norms of $z_E$ and $z_{F^c}$ and on $t^*$. Since $w=0$ at $t=0$, a simple ODE argument then yields

$$\begin{equation*} \int _{S} w \, \eta _\varepsilon ^p dx\leq \left(Cp\|\nabla \eta _\varepsilon \|_{L^p(\mathbb{R}^N)} t\right)^p \end{equation*}$$

for all $t\in (0, t^*)$. Observing that $\|\nabla \eta _\varepsilon \|_{L^p(\mathbb{R}^N)}^p=\varepsilon ^{p-N}\|\nabla \eta \|_{L^p(\mathbb{R}^N)}^p\to 0$ and $\eta _\varepsilon \nearrow 1$ as $\varepsilon \to 0^+$, we conclude that $w=0$, and in turn $\widetilde{d}_E+\widetilde{d}_{F^c}\geq \Delta e^{-Mt}$ in $S\times (0,t^*)$. In particular, by 2.12, we have shown that $d^{\psi ^\circ }_E+d^{\psi ^\circ }_{F^c}\geq \Delta e^{-Mt}$ in $S''\times (0,t^*)$. In turn, this easily implies that $\operatorname {dist}(E(t), F^c(t))\geq \Delta e^{-Mt}$ for $t\in (0, t^*)$ (see the end of the proof of 22, Theorem 3.3). This concludes the proof of the theorem.

■

The previous theorem easily yields a comparison principle also between level set subsolutions and supersolutions.

Theorem 2.8.

Let $u^0$, $v^0$ be uniformly continuous functions on $\mathbb{R}^N$ and let $u$, $v$ be respectively a level set subsolution with initial datum $u^0$ and a level set supersolution with initial datum $v^0$, in the sense of Definition 2.5. If $u^0\leq v^0$, then $u\leq v$.

Proof.

Recall that by Definition 2.5 there exists a null set $N_0\subset \mathbb{R}$ such that for all $\lambda \not \in N_0$ the sets $\{(x,t): u(x,t)<\lambda \}$ and $\{(x,t): v(x,t)\leq \lambda \}$ are respectively a subflow with initial datum $\{u^0\leq \lambda \}$ and a superflow with initial datum $\{v^0\leq \lambda \}$, in the sense of Definition 2.2. Fix now $\lambda \in \mathbb{R}$ and choose $\lambda <\lambda ''<\lambda '$, with $\lambda '$, $\lambda ''\not \in N_0$. Since $\{v^0\leq \lambda ''\}\subset \{v^0\leq \lambda '\}\subset \{u^0\leq \lambda '\}$, we have

$$\begin{equation*} \operatorname {dist}^{\psi ^\circ }(\{v^0\leq \lambda ''\}, \{u^0>\lambda '\})\geq \operatorname {dist}^{\psi ^\circ }(\{v^0\leq \lambda ''\}, \{v^0>\lambda '\}):=\Delta >0\,, \end{equation*}$$

where the last inequality follows from the uniform continuity of $v^0$. Thus, by Theorem 2.7, for all $t\ge 0$,

$$\begin{equation*} \{(x,t): v(x,t)\leq \lambda \}\subset \{(x,t): v(x,t)\leq \lambda ''\}\subset \{(x,t): u(x,t)<\lambda '\}\,. \end{equation*}$$

Letting $\lambda '\searrow \lambda$, with $\lambda '\not \in N_0$, we conclude that $\{(x,t): v(x,t)\leq \lambda \}\subseteq \{(x,t): u(x,t)\leq \lambda \}$ for all $\lambda \in \mathbb{R}$, which is clearly equivalent to $u\le v$.

■

2.4. Distributional versus viscosity solutions

We show here that in the smooth cases, the notion of solution in Definition 2.2 coincides with the definition of standard viscosity solutions for geometric motions, as for instance proposed in 12.

Lemma 2.9.

Assume $\phi ,\psi ,\psi ^\circ \in C^2(\mathbb{R}^N\setminus \{0\})$, and assume that $g$ is continuous also with respect to the time variable. Let $E$ be a superflow in the sense of Definition 2.2. Then, $-\chi _E$ is a viscosity supersolution of

$$\begin{equation} u_t = \psi (\nabla u)\big ( \operatorname {div}\nabla \phi (\nabla u)+g\big ) {\tag{2.17}} \end{equation}$$

in $\mathbb{R}^N\times (0,T^*)$, and in fact in $\mathbb{R}^N\times (0,T^*]$ whenever $T^*<+\infty$, where $T^*$ is the extinction time of $E$ introduced in Definition 2.2.

A converse statement is also true; see 2022.

Proof.

We follow the proof of a similar statement in 22, Appendix. Let $\varphi (x,t)$ be a smooth test function and assume $-\chi _E-\varphi$ has a (strict) local minimum at $(\bar{x},\bar{t})$, $0<\bar{t}\le T^*$. In other words, we can assume that near $(\bar{x},\bar{t})$, $-\chi _E(x,t)\ge \varphi (x,t)$, while $-\chi _E(\bar{x},\bar{t})=\varphi (\bar{x},\bar{t})$. We can also assume that the latter quantity is $-1$ (i.e., $(\bar{x},\bar{t})\in E$), since if it is zero, then we trivially deduce that $\nabla \varphi (\bar{x},\bar{t})=0$ while $\partial _t\varphi (\bar{x},\bar{t})\ge 0$.

If $\nabla \varphi (\bar{x},\bar{t})=0$, thanks to 10, Prop. 2.2 we can assume that also the spatial Hessian $D^2\varphi (\bar{x},\bar{t})=0$ (and then $D^3\varphi (\bar{x},\bar{t})=0$, $D^4\varphi (\bar{x},\bar{t})\le 0$). As usual, if we assume that $-a=\partial _t\varphi (\bar{x},\bar{t})<0$ and choose $a'<a$, we observe that near $\bar{x}$, $\partial _t\varphi (x,\bar{t})< -a'$ and, for $t\le \bar{t}$ close enough to $\bar{t}$ and $x$ close enough to $\bar{x}$,

$$\begin{equation*} \varphi (x,t)=\varphi (x,\bar{t})+\partial _t\varphi (x,\bar{t})(t-\bar{t})+o(|t-\bar{t}|) \ge \varphi (x,\bar{t}) + a'(\bar{t}-t). \end{equation*}$$

Hence, one has that near $(\bar{x},\bar{t})$ (for $t\le \bar{t}$), for some $\gamma >0$

$$\begin{equation*} \varphi (x,t)\ge -1+a'(\bar{t}-t)-\gamma |x-\bar{x}|^4. \end{equation*}$$

It follows that for such $t$, $\mathcal{N}\cap \{x:\gamma |x-\bar{x}|^4< a'(\bar{t}-t)\}\cap E(t)=\emptyset$, where $\mathcal{N}$ is a neighborhood of $\bar{x}$. For $\bar{t}-t>0$ small enough we deduce that $B(\bar{x}, (a'(\bar{t}-t)/\gamma )^{1/4})$ does not meet $E(t)$, in other words $d(\bar{x},t)\ge c((a'/\gamma )(\bar{t}-t))^{1/4}$ for constant $c$ depending only on $\psi$. It then follows from Lemma 2.6, and more precisely from 2.6, that (provided $\bar{t}-t\le \tau _0$, where $\tau _0$ is as in Lemma 2.6)

$$\begin{equation*} d(\bar{x},\bar{t}) \ge (\bar{t} - t)^{\frac{1}{4}} \left(c \left(\frac{a'}{\gamma }\right)^{1/4}e^{-5M(\bar{t}-t)} - \chi (\bar{t}-t)^{\frac{1}{4}}\right), \end{equation*}$$

which is positive if $t$ is close enough to $\bar{t}$, a contradiction. Hence $\partial _t \varphi (\bar{x},\bar{t})\ge 0$.

If, on the other hand, $\nabla \varphi (\bar{x},\bar{t})\neq 0$, then we can introduce the set $F=\{\varphi \le -1\}$, and we have that $F(t)$ is a smooth set near $\bar{x}$ for $t\le \bar{t}$ close to $\bar{t}$, which contains $E(t)$, with a contact at $(\bar{x},\bar{t})$. We then let $\delta (x,t) = \operatorname {dist}^{\psi ^\circ }(x,F(t))$, which is at least $C^2$ near $(\bar{x},\bar{t})$ (as $\psi ,\psi ^\circ$ are $C^2$) and is touching $d$ from below at all the points $(\bar{x}+s\nabla \phi (\nu _{F(\bar{t})}),\bar{t})$ for $s>0$ small.

Assume that

$$\begin{equation} \partial _t \delta < \psi (\nabla \delta )(\operatorname {div}\nabla \phi (\nabla \delta ) +g) = D^2\phi (\nabla \delta ):D^2\delta +g {\tag{2.18}} \end{equation}$$

at $(\bar{x},\bar{t})$. Then, by continuity, we can find $\bar{s}>0$ small and a neighborhood $B=\{ |x-\bar{x}|<\rho , \bar{t}-\rho <t\le \bar{t}\}$ of $(\bar{x},\bar{t})$ in $\mathbb{R}^N\times (0,\bar{t}]$ where

$$\begin{equation*} \partial _t \delta < D^2\phi (\nabla \delta ):D^2\delta +g-M\bar{s}. \end{equation*}$$

Possibly reducing $\rho$ and using (cf. Remark 2.3) the left-continuity of $d$, since $d(\bar{x},\bar{t})=0$, we can also assume that $d\le \bar{s}$ in $B$.

We choose then $s<\bar{s}$ small enough so that $\bar{x}^s=\bar{x}+s\nabla \phi (\nu _{F(\bar{t})})$ is such that $|\bar{x}^s-\bar{x}|<\rho$, and for $\eta >0$ small we define $\delta ^\eta (x,t) = \delta (x,t)-\eta (|x-\bar{x}^s|^2+|t-\bar{t}|^2)/2$. Then $d-\delta ^\eta$ has a unique strict minimum point at $(\bar{x}^s,\bar{t})$ in $B$. Moreover if $\eta$ is small enough, by continuity, we still have that

$$\begin{equation*} \partial _t \delta ^\eta < D^2\phi (\nabla \delta ^\eta ):D^2\delta ^\eta +g-M\bar{s} \end{equation*}$$

in $B$.

Then we continue as in 22, Appendix: given $\Psi \in C^\infty (\mathbb{R})$ nonincreasing, convex, vanishing on $\mathbb{R}_+$, and positive on $(-\infty ,0)$, we introduce $w=\Psi (d-\delta ^\eta -\varepsilon )\chi _B$ for $\varepsilon <d(\bar{x}^s,\bar{t}-\rho )-\delta ^\eta (\bar{x}^s,\bar{t}-\rho )$ small enough. We then show that, thanks to 2.5, for $\bar{t}-\rho <t< \bar{t}$,

$$\begin{multline*} \partial _t \int w dx \le \int _B \Psi '(d-\delta ^\eta -\varepsilon ) (\operatorname {div}z+g - Md - \operatorname {div}\nabla \phi (\nabla \delta ^\eta ) - g +M\bar{s}) dx \\ \le -\int _B \Psi ''(d-\delta ^\eta -\varepsilon )(\nabla d-\nabla \delta ^\eta )\cdot (z-\nabla \phi (\nabla \delta ^\eta )) dx + M\int _B \Psi '(d-\delta ^\eta -\varepsilon )(\bar{s}-d)dx \le 0 \end{multline*}$$

as we have assumed $d\le \bar{s}$ in $B$. This is in contradiction with

$$\begin{equation*} \Psi (d(\bar{x}^s,\bar{t})-\delta ^\eta (\bar{x}^s,\bar{t})-\varepsilon )=\Psi (-\varepsilon )>0, \end{equation*}$$

and it follows that 2.18 cannot hold: one must have

$$\begin{equation*} \partial _t \delta \ge \psi (\nabla \delta )(\operatorname {div}\nabla \phi (\nabla \delta ) +g) \end{equation*}$$

at $(\bar{x},\bar{t})$. Since this equation is geometric and the level set $\{\delta \le 0\}$ is $F$, which is the level $-1$ of $\varphi$ (near $(\bar{x},\bar{t})$), we also deduce that at the same point,

$$\begin{equation*} \partial _t \varphi \ge \psi (\nabla \varphi )(\operatorname {div}\nabla \phi (\nabla \varphi ) +g) \end{equation*}$$

so that $-\chi _E$ is a supersolution of 2.17.

■

3. Minimizing movements

As in 22, in order to build solutions to our geometric evolution problem, we implement a variant of the Almgren-Taylor-Wang 3 minimizing movements scheme 1.4 (in short the ATW scheme) introduced in 1819. In Section 3.2 we adapt this construction to take into account the forcing term, as in 24. We start by presenting some preliminary properties of the incremental problem.

3.1. The incremental problem

Given $z\in L^{\infty }(\mathbb{R}^N; \mathbb{R}^N)$ with $\operatorname {div}z\in L^2_{loc}(\mathbb{R}^N)$ and $w\in BV_{loc}(\mathbb{R}^N)\cap L^2_{loc}(\mathbb{R}^N)$, let $z\cdot Dw$ denote the Radon measure associated with the linear functional

$$\begin{equation*} L\varphi :=-\int _{\mathbb{R}^N}w\,\varphi \operatorname {div}z\, dx- \int _{\mathbb{R}^N}w\,z\cdot \nabla \varphi \, dx \qquad \text{ for all } \varphi \in C^{\infty }_c(\mathbb{R}^N); \end{equation*}$$

see 9. We recall the following result.

Proposition 3.1.

Let $p>\max \{N,2\}$, $f\in L^{p}_{loc}(\mathbb{R}^N)$, and $h>0$. There exist a field $z\in L^\infty (\mathbb{R}^N;\mathbb{R}^N)$ and a unique function $u \in BV_{loc}(\mathbb{R}^N)\cap L^{p}_{loc}(\mathbb{R}^N)$ such that the pair $(u,z)$ satisfies

$$\begin{equation} \left\{ \begin{array}{ll} -h \, \operatorname {div}z + u = f \qquad &\text{ in } \mathcal{D}'(\mathbb{R}^N), \\\phi ^\circ (z)\le 1 \quad & \text{ a.e. in } \mathbb{R}^N,\\z\cdot Du = \phi (Du) \qquad &\text{ in the sense of measures}. \end{array} \right. {\tag{3.1}} \end{equation}$$

Moreover, for any $R>0$ and $v\in BV(B_R)$ with $\operatorname {Supp}(u-v)\Subset B_R$,

$$\begin{equation*} \phi (Du)(B_R) +\frac{1}{2h} \int _{B_R}{(u-f)^2}\, dx \le \phi (Dv)(B_R) +\frac{1}{2h} \int _{B_R}{(v-f)^2}\, dx, \end{equation*}$$

and for every $s\in \mathbb{R}$ the set $E_s:=\{x\in \mathbb{R}^N : u(x)\le s\}$ solves the minimization problem

$$\begin{equation*} \min _{F\Delta E_s\Subset B_R} P_\phi (F;B_R) + \frac{1}{h} \int _{F\cap B_R} (f(x) -s) \, dx. \end{equation*}$$

If $f_1\leq f_2$ and if $u_1$, $u_2$ are the corresponding solutions to 3.1 (with $f$ replaced by $f_1$ and $f_2$, respectively), then $u_1\leq u_2$.

Finally if in addition $f$ is Lipschitz with $\psi (\nabla f)\le 1$ for some norm $\psi$, then the unique solution $u$ of 3.1 is also Lipschitz and satisfies $\psi (\nabla u)\le 1$ a.e. in $\mathbb{R}^N$. As a consequence, 3.1 is equivalent to

$$\begin{equation} \left\{ \begin{array}{lll} -h \, \operatorname {div}z + u = f & \text{in }\mathcal{D}'(\mathbb{R}^N),\\z \in \partial \phi (\nabla u) & \text{a.e. in $\mathbb{R}^N$}.\\\end{array} \right. {\tag{3.2}} \end{equation}$$

Proof.

See 18, Theorem 2, 1, Theorem 3.3.

■

The comparison property in the previous proposition has the following “local” version, which results from the geometric character of 3.1.

Lemma 3.2.

Let $f_1$, $f_2$ be Lipschitz functions and let $(u_i, z_i)$, $i=1,2$, be solutions to 3.1 with $f$ replaced by $f_i$. Assume also that for some $\lambda \in \mathbb{R}$,

$$\begin{equation} \left|(\{ u_1 \leq \lambda \}\cup \{u_2\leq \lambda \})\setminus \{ f_1 \leq f_2 \} \right|=0\qquad \text{for }i=1,2\,. {\tag{3.3}} \end{equation}$$

Then, $\min \{u_1,\lambda \}\leq \min \{u_2 ,\lambda \}$ a.e.

Proof.

Let us set $v_i:=\min \{u_i, \lambda \}$ for $i=1,2$ and observe that

$$\begin{equation} z_i\in \partial \phi (\nabla v_i)\quad {\text{\textit{a.e.}} } {\tag{3.4}} \end{equation}$$

Writing 3.1 for $u_i$ and subtracting the equations we get

$$\begin{equation} -h \operatorname {div}(z_1-z_2) + (u_1-u_2) = f_1-f_2\,. {\tag{3.5}} \end{equation}$$

Let $\psi$ be a smooth, increasing, nonnegative function with support in $(0, +\infty )$, let $\eta \in C_c^\infty (\mathbb{R}^N;\mathbb{R}_+)$, and let $p>N$. First notice that

$$\begin{align*} \int _{\mathbb{R}^N} (v_1-v_2)\psi (v_1-v_2) \eta ^p\, dx&= \int _{\{v_1>v_2\}} (v_1-v_2)\psi (v_1-v_2) \eta ^p\, dx \\ &\leq \int _{\mathbb{R}^N} (u_1-u_2 )\psi (v_1-v_2) \eta ^p\, dx \end{align*}$$

since it can be easily checked that $v_1-v_2\leq u_1-u_2$ in $\{v_1>v_2\}$. Thus, from 3.5 we deduce that

$$\begin{multline*} p h\int _{\mathbb{R}^N} (z_1-z_2)\cdot \nabla \eta \, \psi (v_1-v_2)\eta ^{p-1} dx + h\int _{\mathbb{R}^N} (z_1-z_2)\cdot (\nabla v_1-\nabla v_2)\psi '(v_1-v_2) \eta ^p\, dx \\ +\int _{\mathbb{R}^N} (v_1-v_2) \psi (v_1-v_2) \eta ^p\, dx \leq \int _{\mathbb{R}^N} (f_1-f_2)\psi (v_1-v_2) \eta ^p\, dx. \end{multline*}$$

Notice that the last integral in this equation is nonpositive, since by 3.3, the set $\{f_1>f_2\}$ is contained (up to a negligible set) in $\{u_1>\lambda \}\cap \{u_2>\lambda \}\subseteq \{v_1=v_2\}$. Hence, using also that $(z_2-z_1)\cdot (\nabla v_2-\nabla v_1)\ge 0$ thanks to 3.4, we deduce

$$\begin{equation*} p h\int _{\mathbb{R}^N} (z_1-z_2)\cdot \nabla \eta \, \psi (v_1-v_2)\eta ^{p-1} dx +\int _{\mathbb{R}^N} (v_1-v_2)\psi (v_1-v_2)\eta ^p\, dx \le 0. \end{equation*}$$

Letting $\psi (s)\to (s^+)^{p-1}$ we obtain

$$\begin{multline*} \|(v_1-v_2)^+\eta \|_{L^p(\mathbb{R}^N)}^p \le - p h \int _{\mathbb{R}^N} (z_1-z_2)\cdot \nabla \eta \, \left((v_1-v_2)^+\eta \right)^{p-1}dx \\ \le p h \|(z_1-z_2)\nabla \eta \|_{L^p(\mathbb{R}^N)} \|(v_1-v_2)^+\eta \|_{L^p(\mathbb{R}^N)}^{p-1} \end{multline*}$$

so that

$$\begin{equation*} \|(v_1-v_2)^+\eta \|_{L^p(\mathbb{R}^N)} \le 2 p h C \|\nabla \eta \|_{L^p(\mathbb{R}^N)}, \end{equation*}$$

where $C=\max _{\partial \phi (0)} |z|$. Replacing now $\eta (\cdot )$ with $\eta (\cdot /R)$, $R>0$, assuming $\eta (0)=1$, we obtain

$$\begin{equation*} \|(v_1-v_2)^+\eta (\cdot /R)\|_{L^p(\mathbb{R}^N)} \le 2 p h C R^{d/p-1}\|\nabla \eta \|_{L^p(\mathbb{R}^N)} \stackrel{R\to \infty }{\longrightarrow } 0 \end{equation*}$$

as we have assumed $p>N$. It follows that $(v_1-v_2)^+=0$ a.e., which is the thesis of the lemma.

■

Remark 3.3.

The result remains true if $f_i\in L^p_{loc}(\mathbb{R}^N)$, $p>N$, are not assumed to be Lipschitz. This is because again, in this case, the pairing $(z_1-z_2)\cdot D(\psi (v_1-v_2))$ is nonnegative, which may be shown for instance by first approximating the functions $f_i$ with Lipschitz functions.

We conclude this subsection with the following useful computation, proved in 18, p. 1576.

Lemma 3.4.

Let $R> 0$ and let $u$ be the solution to 3.2, with $f:=c_1(\phi ^\circ -R) \lor c_2 (\phi ^\circ -R)$, where $0<c_1\leq c_2$. Then $u$ is given by

$$\begin{equation*} u(x)= \begin{cases} \sqrt {c_1h}\frac{2N}{\sqrt {N+1}}-c_1R & \text{ if } \phi ^\circ (x)\le \sqrt {\frac{h}{c_1}(N+1)},\\ f(x)+h\frac{N-1}{\phi ^\circ (x)} & \text{ otherwise,} \end{cases} \end{equation*}$$

as long as $h/c_1\leq R^2/(N+1)$.

3.2. The ATW scheme

Let $\phi$, $\psi$, and $g$ satisfy all the assumptions stated in Subsection 2.2. Set

$$\begin{equation*} G(\cdot ,t):= \int _0^t g(\cdot ,s) \, ds\,. \end{equation*}$$

Let $E^0\subset \mathbb{R}^N$ be closed. Fix a time-step $h>0$ and set $E^0_h:=E^0$. We then inductively define $E_h^{k+1}$ (for all $k\in \mathbb{N}$) according to the following procedure: if $E_h^{k}\not \in \{ \emptyset$, $\mathbb{R}^N\}$, then let $(u_h^{k+1},z_h^{k+1}) :\mathbb{R}^N\to \mathbb{R}\times \mathbb{R}^N$ satisfy

$$\begin{equation} \left\{ \begin{array}{lll} -h \, \operatorname {div}z_h^{k+1} + u_h^{k+1} = d^{\psi ^\circ }_{E_h^k} + G(\cdot ,(k+1)h)-G(\cdot , kh) &\text{in }\mathcal{D}'(\mathbb{R}^N), \\z_h^{k+1} \in \partial \phi (\nabla u_h^{k+1}) & \text{a.e. in $\mathbb{R}^N$},\\\end{array} \right. {\tag{3.6}} \end{equation}$$

and set $E_h^{k+1}:=\{x:u_h^{k+1}\le 0\}$.⁠⁵ If either $E_h^{k}=\emptyset$ or $E_h^{k}=\mathbb{R}^N$, then set $E_h^{k+1}:=E_h^{k}$. We denote by $T^*_h$ the first discrete time $hk$ such that $E_h^k=\emptyset$, if such a time exists; otherwise we set $T^*_h=+\infty$. Analogously, we denote by ${T'_h}^*$ the first discrete time $hk$ such that $E_h^k=\mathbb{R}^N$, if such a time exists; otherwise we set ${T'_h}^*=+\infty$.

⁵

Choosing $E_h^{k+1}=\overline{\{u_h^{k+1}<0\}}$ might provide a different (smaller) solution, which would enjoy exactly the same properties as the one we (arbitrarily) chose.

✖

Remark 3.5.

In the following, when changing mobilities, forcing terms, and initial data, we will sometimes write $(E^0)_{g,h}^{\psi ,k}$ in place of $E_h^k$ in order to highlight the dependence of the scheme on $\psi$, $g$, and $E^0$. More generally, given any closed set $H$, $H_{g,h}^{\psi ,k}$ will denote the $k$th minimizing movements starting from $H$ with mobility $\psi$, forcing term $g$ and time-step $h$, as described by the algorithm above.

Remark 3.6 (Monotonicity of the scheme).

From the comparison property stated in Proposition 3.1 it easily follows that if $E^0\subseteq F^0$ are closed sets, then (with the notation introduced in the previous remark) $(E^0)_{g,h}^{\psi , k}\subseteq (F^0)_{g,h}^{\psi , k}$ for all $k\in \mathbb{N}$. In addition, note that $\overline{\bigl ((E^0)_{g,h}^{\psi , k}\bigr )^c}=\bigl (\overline{(E^0)^c}\bigr )_{-g,h}^{\psi , k}$ for all $k$. Thus, if $\operatorname {dist}(E^0, F^0)>0$, then we may apply Lemma 5.2 below with $g_1=g_2=g$, $c=0$, and with $\eta$ the Euclidean norm to deduce that $\operatorname {dist}\left((E^0)_{g,h}^{\psi , k}, (F^0)_{-g,h}^{\psi , k}\right)>0$ for all $k\in \mathbb{N}$.

The assumption that the sets are at positive distance is necessary, otherwise one could only conclude, for instance, that the smallest solution of the ATW scheme from $E^0$ is in the complement of any solution from $F^0$, etc.

We now study the space regularity of the functions $u_h^k$ constructed above. In the following computations, given any function $f:\mathbb{R}^N\to \mathbb{R}^m$ and $\tau \in \mathbb{R}^N$, we denote $f_\tau (\cdot ):= f(\cdot +\tau )$. Then, the function $(u_h^{k+1})_\tau$ satisfies

$$\begin{multline} -h \, \operatorname {div}(z_h^{k+1})_\tau + (u_h^{k+1})_\tau = (d^{\psi ^\circ }_{E_h^k})_\tau + (G(\cdot ,(k+1)h))_\tau - (G(\cdot , kh))_\tau \\ \le d^{\psi ^\circ }_{E_h^k} + G(\cdot ,(k+1)h)-G(\cdot , kh) + \psi ^\circ (\tau )(1+Lh)\,, \tag{3.7} \end{multline}$$

where in the last inequality we also used the Lipschitz-continuity of $g$. By the comparison property stated in Proposition 3.1 we deduce that $(u_h^{k+1})_\tau - (1+Lh) \psi ^\circ (\tau ) \le u_h^{k+1}$. By the arbitrariness of $\tau \in \mathbb{R}^N$, we get $\psi (\nabla u_h^{k+1}) \le 1+Lh$, and in turn

$$\begin{equation} \begin{array}{ll} u_h^{k+1} \le (1+Lh) d^{\psi ^\circ }_{E_h^{k+1}} & \text{ in } \bigl \{x : d^{\psi ^\circ }_{E_h^{k+1}}(x)>0\bigr \}\,,\\[5.69054pt] u_h^{k+1} \ge (1+Lh) d^{\psi ^\circ }_{E_h^{k+1}} & \text{ in } \bigl \{x : d^{\psi ^\circ }_{E_h^{k+1}}(x)<0\bigr \}\,. \end{array} {\tag{3.8}} \end{equation}$$

We are now in a position to define the discrete-in-time evolutions constructed via minimizing movements. Precisely, we set

$$\begin{equation} \left\{ \begin{array}{l} E_h(t):=E_h^{[t/h]},\\[2.0pt] E_h:=\{(x,t): x\in E_h(t)\},\\[2.0pt] d_h(x,t):=d^{\psi ^\circ }_{E_h(t)}(x),\\[2.0pt] u_h(x,t):=u_h^{[t/h]}(x),\\[2.0pt] z_h(x,t):=z_h^{[t/h]}(x), \end{array} \right. {\tag{3.9}} \end{equation}$$

where $[\cdot ]$ stands for the integer part.

We conclude this subsection with the following remark.

Remark 3.7 (Discrete comparison principle).

Remark 3.6 now reads as follows: if $E^0\subseteq F^0$ are closed sets and if we denote by $E_h$ and $F_h$ the discrete evolutions with initial datum $E^0$ and $F^0$, respectively, then $E_h(t)\subseteq F_h(t)$ for all $t\geq 0$. Analogously, if $\operatorname {dist}(E^0,F^0)>0$, $E_h$ is defined with a forcing $g$, and $F_h$ is defined with the forcing $-g$, then $\operatorname {dist}(E_h(t), F_h(t))>0$ for all $t\geq 0$.

3.3. Evolution of $\phi$-Wulff shapes

We start paving the way for the convergence analysis of the scheme by deriving some estimates on the minimizing movements starting from a Wulff shape. We consider as initial set the $\phi$-Wulff shape $W^\phi (0, R)$ for $R>0$. First, thanks to Lemma 3.4 (with $c_1=c_2=1$) (cf. also 18, Appendix B, Eq. (39)), the solution of 3.1 with $f=d^{\phi ^\circ }_{W^\phi (0,R)}=\phi ^\circ -R$ is given by $\phi ^\circ _h-R$, where

$$\begin{equation} \phi ^\circ _h(x): =\begin{cases} \sqrt {h}\frac{2N}{\sqrt {N+1}} & \text{ if } \phi ^\circ (x)\le \sqrt {h(N+1)},\\ \phi ^\circ (x)+h\frac{N-1}{\phi ^\circ (x)} & \text{ else.} \end{cases} {\tag{3.10}} \end{equation}$$

Observe then that there exist two positive constants $c_1\leq c_2$ such that

$$\begin{equation} c_1\phi ^\circ \leq \psi ^\circ \leq c_2\phi ^\circ \,, {\tag{3.11}} \end{equation}$$

and in particular

$$\begin{equation} d^{\psi ^\circ }_{W^\phi (0,R)} \le c_1(\phi ^\circ -R) \lor c_2(\phi ^\circ -R). {\tag{3.12}} \end{equation}$$

Thus, for any $k\in \mathbb{N}$ we have

$$\begin{equation} d^{\psi ^\circ }_{W^\phi (0,R)} + G(\cdot ,(k+1)h)-G(\cdot , kh)\leq c_1(\phi ^\circ -R)\lor c_2 (\phi ^\circ -R)+\|g\|_\infty h=:f\,. {\tag{3.13}} \end{equation}$$

Denoting by $u$ the solution to 3.2, with $f$ defined above, then Lemma 3.4 yields

$$\begin{equation*} u(x)=\|g\|_\infty h + \begin{cases} \sqrt {c_1h}\frac{2N}{\sqrt {N+1}}-c_1R & \text{ if } \phi ^\circ (x)\le \sqrt {\frac{h}{c_1}(N+1)},\\ f(x)+h\frac{N-1}{\phi ^\circ (x)} & \text{ otherwise,} \end{cases} \end{equation*}$$

provided that $h/c_1\leq C(N)R^2$ (here and in the following $C(N)$ denotes a positive constant that depends only on the dimension $N$ and may change from line to line). Notice that $\{u\leq 0\} = W^{\phi }(0, \bar{r})$ for

$$\begin{equation*} \bar{r}:=\frac{R-\frac{h}{c_1}\|g\|_\infty +\sqrt {\big (R-\frac{h}{c_1}\|g\|_\infty \big )^2-4 \frac{h}{c_1}(N-1)}}{2}. \end{equation*}$$

Taking into account 3.13, we may apply the comparison principle stated in Proposition 3.1 to infer that if $k=[t/h]$ and $E^k_h=E_h(t)=W^\phi (0,R)$, then

$$\begin{equation*} W^{\phi }(0, \bar{r})=\{u\leq 0\}\subseteq \{u_h^{k+1}\leq 0\}=E_h(t+h)\,, \end{equation*}$$

provided that $h/c_1\leq C(N) R^2$. Since

$$\begin{equation*} \bar{r}\geq \sqrt {\big (R-\tfrac{h}{c_1}\|g\|_\infty \big )^2-4 \tfrac{h}{c_1}(N-1)}\stackrel{(R\leq 1)}{\geq } \sqrt {R^2-2\tfrac{h}{c_1}\big (2(N-1)+\|g\|_\infty \big )} \end{equation*}$$

and setting for $0\leq s-t\leq \frac{c_1R^2}{4\big (2(N-1)+\|g\|_\infty \big )}$

$$\begin{equation} r^R(s):=\sqrt {R^2-2\frac{s-t}{c_1}\big (2(N-1)+\|g\|_\infty \big )}\geq \frac{R}{\sqrt 2}\,, {\tag{3.14}} \end{equation}$$

by iteration we deduce that

$$\begin{equation} W^{\phi }(0, r^R(s))\subseteq E_h(s) {\tag{3.15}} \end{equation}$$

for all $0\leq s-t\leq \frac{c_1R^2}{4\big (2(N-1)+\|g\|_\infty \big )}$ and $h\leq c_1C(N)R^2$.

In particular, there exist a constant $C$ depending only on $\|g\|_\infty$, $c_1$, and the dimension $N$, and $h_0>0$ depending also on $R$, such that for any $y\in \mathbb{R}^N$ and for all $h\leq h_0$

$$\begin{equation} W^\phi (y, R-\tfrac{C}{R}h)\subseteq \bigl ( W^\phi (y, R) \bigr )_{{g,h}}^{{\psi ,1}}\subseteq W^\phi (y, R). {\tag{3.16}} \end{equation}$$

3.4. Density estimates and barriers

In this section we collect some preliminary estimates on the incremental problem that will be crucial for the stability properties established in Section 5.1. The following density lemma and the subsequent corollary show that the solution to the incremental problem starting from a closed set $E$ cannot be too “thin” in $\mathbb{R}^N\setminus E$. The main point is that the estimate turns out to be independent of $h$ and $\psi$; see also Lemma 1.3 and Remark 1.4 in 44. We observe that there exist positive constants $a_1$, $a_2$ such that

$$\begin{equation} a_1|\xi |\leq \phi (\xi )\leq a_2 |\xi | \quad \text{for all $\xi \in \mathbb{R}^N$.} {\tag{3.17}} \end{equation}$$

Lemma 3.8.

Let $E\subset \mathbb{R}^N$ be a closed set, let $h>0$, and let $g_h\in L^\infty (\mathbb{R}^N)$ with $\|g_h\|_\infty \le G h$ for some $G>0$. Let $E'$ be a solution to

$$\begin{equation} \min _{ F\Delta E'\Subset B_R} P_\phi (F;B_R) + \frac{1}{h} \int _{F\cap B_R} (d_E^{\psi ^\circ }(x) + g_h(x)) \, dx {\tag{3.18}} \end{equation}$$

for all positive $R$. Then, there exist $\sigma >0$ depending only on $N$ and $a_1$, and $r_0>0$ depending on $N$ and $G$, with the following property: if $\bar{x}$ is such that $|E'\cap W^\phi (\bar{x}, s)|>0$ for all $s>0$ and $W^\phi (\bar{x}, r)\cap E=\emptyset$ with $r\le r_0$, then

$$\begin{equation*} |E'\cap W^\phi (\bar{x}, r)| \ge \sigma r^N. \end{equation*}$$

Proof.

We adapt to our context a classical argument from the regularity theory of the (quasi)minimizers of the perimeter 44, Lemma 1.3, Remark 1.4. As mentioned before, the main point is to use the fact that the Wulff shapes $W^\phi (\bar{x},r)$ lie outside $E$ to deduce that the constants $\sigma ,r_0$ are independent of $h$ and $\psi ^\circ$. Let $\bar{x}$ and $W^\phi (\bar{x}, r)$ be as in the statement. Fix $R>0$ such that $W^\phi (\bar{x},r) \subset B_R$. For all $s\in (0,r)$, set $E'(s):= E'\setminus W^\phi (\bar{x},s)$. For a.e. $s$, we have:⁠⁶

⁶

We use here that $\phi$ is even, otherwise the constant $2$ in the formula should be replaced by $1+c_\phi$, where $c_\phi$ is such that $\phi (\xi )\le c_\phi \phi (-\xi )$; see also Remark 6.3.

✖

$$\begin{equation*} P_\phi (E'(s);B_R) = P_\phi (E' ;B_R) - P_\phi (E' \cap W^\phi (\bar{x},s)) + { 2\int _{E'\cap \partial W^\phi (\bar{x},s)} \phi (\nu )d\mathcal{H}^{N-1}} \end{equation*}$$

with $\nu$ the outer normal vector to $\partial W^\phi (\bar{x},s)$. Using also the fact that

$$\begin{equation*} \int _{E'(s)\cap B_R}d^{\psi ^\circ }_E\, dx\le \int _{E'\cap B_R}d^{\psi ^\circ }_E\, dx \end{equation*}$$

(since $d^{\psi }_E>0$ in $E^c$), by minimality of $E'$ in 3.18 we find

$$\begin{equation*} P_\phi (E'\cap W^\phi (\bar{x},s))+\frac{1}{h}\int _{E'\cap B_R} g_h(x)dx \le \begin{aligned}[t] & 2\int _{E'\cap \partial W^\phi (\bar{x},s)} \phi (\nu )d\mathcal{H}^{N-1} \\ & +\frac{1}{h}\int _{E'(s)\cap B_R}g_h(x)dx. \end{aligned} \end{equation*}$$

The above inequality and the (anisotropic; see for instance 32) isoperimetric inequality yield

$$\begin{align} 2\int _{E'\cap \partial W^\phi (\bar{x},s)} \phi (\nu )d\mathcal{H}^{N-1} &\ge N |W_\phi |^{\frac{1}{N}}|E'\cap W^\phi (\bar{x},s)|^{\frac{N-1}{N}} - G |E'\cap W^\phi (\bar{x},s)| \\ &\ge (N -Gs)|W_\phi |^{\frac{1}{N}}|E'\cap W^\phi (\bar{x},s)|^{\frac{N-1}{N}}\,, {\tag{3.19}} \end{align}$$

where we have used that $|E'\cap W^\phi (\bar{x},s)|^{\frac{1}{N}}\le |W_\phi |^{\frac{1}{N}}s$. We observe also that (using $\phi (\nabla \phi ^\circ )=1$ and the coarea formula)

$$\begin{align*} |E'\cap W^\phi (\bar{x},s)|&= \int _{E'\cap \{\phi ^\circ (\cdot -\bar{x})\le s\}} \phi \left(\frac{\nabla \phi ^\circ }{|\nabla \phi ^\circ |}\right)|\nabla \phi ^\circ | dx\\ &= \int _0^s\int _{E'\cap \partial W^\phi (\bar{x},t)} \phi (\nu )d\mathcal{H}^{N-1} dt \end{align*}$$

so that for a.e. $s$, $\int _{E'\cap \partial W^\phi (\bar{x},s)} \phi (\nu )d\mathcal{H}^{N-1} =\frac{d}{ds} |E'\cap W^\phi (\bar{x},s)|$. Using that $|E'\cap W^\phi (\bar{x},s)|>0$ for all $s>0$, the inequality 3.19 implies in turn that

$$\begin{equation*} \frac{d}{ds} |E'\cap W^\phi (\bar{x},s)|^{\frac{1}{N}} \ge \frac{1}{2}|W_\phi |^{\frac{1}{N}}\left(1-\tfrac{G}{N}s\right) \text{ for a.e. } s\ge 0. \end{equation*}$$

If $r\le r_0:=N/G$ we obtain, integrating the above inequality on $(0,r)$,

$$\begin{equation*} |E'\cap W^\phi (\bar{x},r)|^{\frac{1}{N}} \ge \frac{r}{4}|W_\phi |^{\frac{1}{N}} \ge \frac{a_1r}{4}|B_1|^{\frac{1}{N}}, \end{equation*}$$

where we have used 3.17 for the last inequality. The thesis follows.

■

Remark 3.9.

The same argument shows that a similar but $h$-dependent density estimate holds inside $E$.

We introduce the following notation: for any set $A\subset \mathbb{R}^N$, for any norm $\eta$, and for $\rho \in \mathbb{R}$ we denote

$$\begin{equation} (A)^{\eta }_\rho :=\{x\in \mathbb{R}^N : d_{A}^{\eta }(x)\leq \rho \}\,, {\tag{3.20}} \end{equation}$$

and we will omit $\eta$ in the notation if $\eta$ is the Euclidean norm. We also recall the notation $E_{g,h}^{\psi ,k}$ introduced in Remark 3.5 to denote the $k$th minimizing movement starting from $E$, with mobility $\psi$, forcing term $g$ and time-step $h$.

Corollary 3.10.

Let $g$ and $\psi$ be an admissible forcing term and a mobility, respectively, and let $h>0$. Denote by $E_{g,h}^{\psi ,1}$ the corresponding (single) minimizing movement starting from $E$ (see Section 3.2 and Remark 3.5). Let $\sigma$ and $r_0$ be the constants provided by Lemma 3.8 for $G:= \|g\|_\infty +1$. If $\bar{x} \in E_{g,h}^{\psi ,1}$ and $W^\phi (\bar{x}, r)\cap E=\emptyset$ with $r\le r_0$, then

$$\begin{equation*} |E_{g,h}^{\psi ,1} \cap W^\phi (\bar{x}, r)| \ge \sigma r^N. \end{equation*}$$

Proof.

Recall that $E_{g,h}^{\psi ,1}= \{u(\cdot ) \le 0\}$, where $u$ solves

$$\begin{equation*} \left\{ \begin{array}{ll} -h \, \operatorname {div}z + u = d_E^{\psi ^\circ } + \int _0^h g(\cdot ,s) \, ds \qquad &\text{ in } \mathcal{D}'(\mathbb{R}^N), \\\phi ^\circ (z)\le 1 \quad & \text{ a.e. in } \mathbb{R}^N,\\z\cdot Du = \phi (Du) \qquad &\text{ in the sense of measures}. \end{array} \right. \end{equation*}$$

Thus, by virtue of Proposition 3.1, setting $E'_\eta := \{u(\cdot ) \le \eta \}$ for $\eta \in (0,h)$, we have that $E'_\eta$ solves 3.18 with $g_h:= \int _0^h g(\cdot ,s) - \eta$. Since $\bar{x}$ belongs to the interior of $E'_\eta$ and $\|g_h\|_\infty \leq G h$, from Lemma 3.8 we deduce that

$$\begin{equation} |E_\eta '\cap W^\phi (\bar{x}, r)| \ge \sigma r^N. {\tag{3.21}} \end{equation}$$

The thesis follows by monotone convergence by letting $\eta \searrow 0^+$.

■

Lemma 3.11.

Let $F\subset \mathbb{R}^N$ be a convex set and let $r>0$. Then,

$$\begin{equation*} | ((F)_\varepsilon \setminus F ) \cap B_r|\le C(N) \varepsilon r^{N-1} \qquad \text{ for all } \varepsilon \ge 0, \end{equation*}$$

where $C(N)$ depends only on the dimension $N$.

Proof.

Notice that $(F)_s$ is convex for all positive $\varepsilon$, so that $(F)_s\cap B_r$ is a convex set contained in $B_r$, and

$$\begin{equation*} \mathcal{H}^{N-1}(\partial ((F)_s \cap B_r)) \le \mathcal{H}^{N-1}(\partial B_r) = C(N) r^{N-1} \qquad \text{ for all } s>0. \end{equation*}$$

Therefore, thanks to the coarea formula,

$$\begin{align*} | ((F)_\varepsilon \setminus F ) \cap B_r(0)| &= \int _0^\varepsilon \mathcal{H}^{N-1}(\partial (F)_s \cap B_r) \, ds\\ &\le \int _0^\varepsilon \mathcal{H}^{N-1}(\partial ((F)_s \cap B_r)) \, ds \le C(N) \varepsilon r^{N-1}. \end{align*}$$ ■

The next lemma provides a crucial estimate on the “expansion” of any closed set $E$ under a single minimizing movement, provided that $E$ satisfies a uniform exterior Wulff shape condition. The result is achieved by combining a barrier argument with the density estimate established in Corollary 3.10.

Lemma 3.12.

For any $\beta$, $G$, $\Delta >0$, there exists $h_0 > 0$ depending on the previous constants, on the anisotropy $\phi$, and the dimension $N$, and there exists $M_0>0$ depending on the same quantities but $\Delta$, with the following property: let $\psi$ be a mobility satisfying

$$\begin{equation} \psi \leq \beta \phi {\tag{3.22}} \end{equation}$$

and let $g$ be an admissible forcing term with $\|g\|_\infty \leq G$. Then for any closed set $E\subseteq \mathbb{R}^N$ such that $\mathbb{R}^N\setminus E = \bigcup _{W\in \mathcal{G}} W$, where $\mathcal{G}$ is a family of (closed) $\phi$-Wulff shapes of radius $\Delta$, and for all $h\le h_0$, we have $E_{g,h}^{\psi ,1}\subset (E)^{\phi ^\circ }_{\frac{M_0 h}{\Delta }}$.

Proof.

First notice that 3.22 is equivalent to $\frac{1}{\beta }\phi ^\circ \leq \psi ^\circ \,.$ Hence, recalling 3.16, there exists a constant $C$ depending only on $\|g\|_\infty$, $\beta$ ($=1/c_1$ in 3.16), and $N$, such that

$$\begin{equation*} W^\phi (y, \Delta -\tfrac{C}{\Delta }h)\subseteq \bigl ( W^\phi (y, \Delta ) \bigr )_{{-g},h}^{\psi ,1}\subseteq W^\phi (y, \Delta ) \end{equation*}$$

for all $h\leq h_0$. By Lemma 3.11 it follows that for a possibly different constant $C$ depending only on $\|g\|_\infty$, $\beta$, and $N$, we have

$$\begin{equation} \Bigl |\bigl (W^\phi (y, \Delta )\setminus \bigl ( W^\phi (y, \Delta ) \bigr )_{{-g},h}^{\psi ,1} \bigr )\cap B_r(x)\Bigr | \le \frac{C}{\Delta } h r^{N-1} {\tag{3.23}} \end{equation}$$

for all $x, y\in \mathbb{R}^N,\, r>0,\, h\le h_0$.

Observe now that there exists $\theta >0$ depending only on $W^{\phi }(0,1)$, such that

$$\begin{equation} \frac{|W^\phi (y, R)\cap Q_r(x)|}{|Q_r(x)|}\ge \theta \text{ for all $y\in \mathbb{R}^N$, $x\in W^\phi (y, R)$, $R\geq 1$, and $r\in (0,1)$, } {\tag{3.24}} \end{equation}$$

where $Q_r(x)$ stands for the cube of side $r$ centered at $x$. Let $\sigma >0$ be the constant provided by Corollary 3.10, and let $\mathcal{N}$ be the constant provided by the covering Lemma 3.13 below, corresponding to the constant $\theta$ in 3.24, $\delta =\sigma /4$, and $\mathcal{C}=W^\phi$. Set $M_0 := (2C \mathcal{N})/\sigma$, where $C$ is the constant in 3.23, and note that for $r= \frac{M_0 h}{\Delta }$ we have

$$\begin{equation} \mathcal{N} \frac{C}{\Delta } h r^{N-1} = \mathcal{N} \frac{C}{\Delta ^N} h^N M_0^{N-1} =\frac{\sigma }{2} \left(\frac{M_0 h}{\Delta }\right)^N = \frac{\sigma }{2}r^N. {\tag{3.25}} \end{equation}$$

Let $x\in \mathbb{R}^N$ be such that $W^\phi \left(x, \frac{M_0h}{\Delta }\right)\cap E=\emptyset$ for some $h\leq h_0$, and assume by contradiction that $x\in E_{g,h}^{\psi ,1}$. Without loss of generality we may assume $x=0$.

By taking $h_0$ smaller if needed, we can also assume that $\frac{\Delta ^2}{M_0 h} \ge 1$ and $\frac{M_0h}{\Delta }\leq r_0$ for all $h\leq h_0$, where $r_0$ is the radius provided by Corollary 3.10. Thus, recalling also 3.24, for $h\leq h_0$, applying Lemma 3.13 below (with $\delta =\sigma /4$) to the family

$$\begin{equation*} \mathcal{F}=\frac{\Delta }{M_0h}\mathcal{G}= \left\{\frac{\Delta }{M_0h}W: W\in \mathcal{G}\right\} \end{equation*}$$

which by assumption on $\mathcal{G}$ and $E$ covers $\mathcal{C}=W^\phi (0,1)$, we find a finite subfamily

$$\begin{equation*} \mathcal{F}'= \left\{\frac{\Delta }{M_0h}W^{\phi }(x_1,\Delta ) , \ldots , \frac{\Delta }{M_0h}W^{\phi }(x_{\mathcal{N}}, \Delta )\right\} \end{equation*}$$

of $\mathcal{N}$ elements such that

$$\begin{equation*} \biggl |W^\phi (0,1)\setminus \bigcup _{i=1}^{\mathcal{N}}\frac{\Delta }{M_0h}W^{\phi }(x_i,\Delta )\biggr |\leq \frac{\sigma }{4}\,. \end{equation*}$$

By scaling back we obtain

$$\begin{equation} \Bigr |W^\phi \Bigl (0, \frac{M_0h}{\Delta }\Bigr )\setminus \bigcup _{i=1}^{\mathcal{N}} W^{\phi }(x_i,\Delta )\Bigl |\le \frac{\sigma }{4} \left(\frac{M_0 h}{\Delta }\right)^N. {\tag{3.26}} \end{equation}$$

Note now that by the comparison principle (see Remark 3.6)

$$\begin{equation*} E_{g,h}^{\psi ,1}\cap W^\phi \Bigl (0, \frac{M_0h}{\Delta }\Bigr )\subset W^\phi \Bigl (0, \frac{M_0h}{\Delta }\Bigr )\setminus \bigcup _{i=1}^{\mathcal{N}} \bigl (W^{\phi }(x_i,\Delta )\bigr )_{{-g},h}^{\psi ,1}\,. \end{equation*}$$

Thus, using also 3.23, 3.25, and 3.26 we deduce that