Abstract
In this paper, we construct global distributional solutions to the volume-preserving mean-curvature flow using a variant of the time-discrete gradient flow approach proposed independently by Almgren et al. (SIAM J Control Optim 31(2):387–438, 1993) and Luckhaus and Sturzenhecker (Calc Var Partial Differ Equ 3(2):253–271, 1995).
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1 Introduction
A family of open sets with smooth boundary \(\{ E_t\}_{0\le t \le T}\) in \(\mathbb {R}^n\) is said to move according to volume-preserving mean-curvature flow if the motion law, expressed as an evolution equation for the boundaries \(\partial E_t\), takes the form
for all \(t\in [0,T]\). Here, at any point x on \(\partial E_t\), v(x) denotes the velocity component normal to the boundary, in the direction of the outer normal, H(x) is the scalar mean curvature (with the sign convention that H is positive for balls, see the next section), and the brackets \(\langle \cdot \rangle \) denote the average of a quantity over the boundary of \(E_t\).
It is immediately verified that the volume of the sets \(E_t\) (i.e., its n-dimensional Lebesgue measure, denoted by \(|E_t|\)) is indeed preserved under the smooth flow (1.1) because
And thus, upon rescaling variables, we may assume that \(|E_t|=|E_0|=1\) for any \(t\in [0,T]\). Moreover, the perimeter of the sets \(E_t\) is decreasing because
During a typical evolution, a volume-preserving mean-curvature flow exhibits singularities of different kinds, even in the case of smooth initial data. These singularities correspond to changes in the topology of the configuration and include shrinkage of islands to points and disappearance, collisions and merging of neighboring islands, pinch-offs etc... If the topology changes, the boundary of the evolving set looses regularity and, as a consequence, the formulation (1.1) of the evolution law is inadequate. The goal of the present work is the construction of a notion of a weak solution to the volume-preserving mean-curvature flow that is global in time and thus overcomes these singular moments.
Several solutions to volume-preserving mean-curvature flow have been proposed in the literature: existence and uniqueness of a global in time smooth solution and its convergence to a sphere is shown in [17, 22] for smooth convex initial data and in [15, 24] for initial data close to a sphere (for further related results see [4, 7, 8] and the references therein). In principle, these results also yield local in time existence and uniqueness of smooth solutions. In [30] and [31], the authors consider level-set and diffusion-generated solutions for the purpose of numerical studies. In Bellettini et al. [5], construct solutions for the volume-preserving anisotropic mean-curvature [or anisotropic variant of (1.1)] for convex sets using a method similar to ours (that will be outlined below). In this paper, it is also shown that the solution (the so-called flat flow) is unique and coincides with regular solutions when the latter are defined. We also mention a mean field approximation approach to the volume preserving mean-curvature flow as developed in [1, 6, 12].
It is well-known that volume-preserving mean-curvature can be (formally) interpreted as the \(L^2\)-gradient flow of the perimeter functional for configurations with a fixed volume, see, e.g., [29, Sect. 2]. This gradient flow structure, however, is for the purpose of well-posedness results impracticable, since the \(L^2\) (geodesic) distance is degenerate in the sense that two well separated configurations may have zero \(L^2\) distance [28]. In the present manuscript, we follow the method proposed independently by Almgren et al. [2] and Luckhaus and Sturzenhecker [26] in the study of (forced) mean-curvature flows to bypass this difficulty. The authors consider an implicit time-discretization of the flow, which comes as a gradient flow of the perimeter functional with respect to a new non-degenerate distance function that approximates the \(L^2\) distance. The limiting time-continuous flow constructed with this method is usually referred to as the flat flow. The main difference between the present work and [2, 26] relies on the non-locality of the volume-preserving mean-curvature flow. As an immediate consequence, there is no maximum-principle available for (1.1). Also related to this aspect, there is the problem of proving the consistency of the scheme, i.e. the coincidence of the flat volume-preserving mean-curvature flow with the smooth one when the latter exists. Under some assumption on the Lagrange multiplier of the flow, the consistency can be inferred from the arguments in [2], but we do not know if these conditions are generally satisfied and we do not discuss further the problem of the consistency in this paper. A more detailed discussion on the different features of the flows in [2, 26] and the one considered in the present manuscript will follow in Sect. 3 below.
We conclude this subsection with a short discussion on the background of this evolution. Volume-preserving mean-curvature flow can be considered as a simplified model for attachment-limited kinetics, and as such it plays an important role in the study of solidification processes, where solid islands grow in an undercooled liquid of the same substance. In such situations, solid particles melt at high-curvature regions and simultaneously precipitate at low-curvature regions, while the total mass of the solid remains essentially constant [11, 32, 33]. In this way, the total surface area of solid islands is decreasing, and thus, this process leads to the growth of larger islands at the expense of smaller ones: a phenomenon called coarsening [29]. In general, solidification processes are mathematically often modelled by Mullins–Sekerka equations (or a Stefan problem), where the Gibbs–Thompson relation is modified by a kinetic drift term [21, 25], and their phase-field counterparts respectively [9]. This model allows for both attachment kinetics (kinetic drift) and bulk diffusion (Mullins–Sekerka/Stefan). It turns out that attachment kinetics is the relevant mass transport mechanism in earlier stages of the evolution while bulk diffusion predominates the later stages [13]. In a certain sense, volume-preserving mean-curvature flow naturally arises as the singular limit of this more general solidification model in the regime of vanishing bulk diffusion. More recently, variants of volume-preserving mean curvature flow were also applied in the context of shape recovery in image processing [10].
The article is organized as follows: in Sect. 2 we fix the notation and state the main results of the paper, which are then proved in Sects. 3 and 4 and are the existence of flat volume-preserving mean-curvature flows and the existence of distributional solutions, respectively.
2 Statements of the main results
2.1 Notation
For any Lebesgue measurable set \(E \subset \mathbb {R}^n\), we denote by |E| the n-dimensional Lebesgue measure of E and by \(\chi _E\) the characteristic function of E, i.e. \(\chi _E(x) = 0\) if \(x \notin E\) and \(\chi _E(x) =1\) if \(x \in E\). The perimeter of E in an open set \(\Omega \subset \mathbb {R}^n\) is defined as
and we write \(\mathrm {Per}(E):=\mathrm {Per}(E,\mathbb {R}^n)\). If the latter quantity is finite, we will call E a set of finite perimeter. In the case that E is an open set with \(\partial E\) of class \(C^1\), we simply have \(\mathrm {Per}(E,\Omega )=\mathcal {H}^{n-1}(\partial E\cap \Omega )\) and \(\mathrm {Per}(E)=\mathcal {H}^{n-1}(\partial E)\). The reduced boundary of a set of finite perimeter E is denoted by \(\partial ^* E\), cp. [16, Sect. 5.7], and for the unit outer normal to E we write \(\nu _{E}\). The tangential divergence of a vector field \(\Psi \in C^1(\mathbb {R}^n;\mathbb {R}^n)\) with respect to \(\partial E\) is defined by \(\mathrm{div}_{\partial E}\Psi := \mathrm{div} \Psi - \nu _E \cdot \nabla \Psi \nu _E\). We say that a set of finite perimeter E has a (generalized) mean-curvature \(H_E\in L^1(\partial ^* E,d\mathcal {H}^{n-1})\) provided that
Observe that with this sign convention it is \(H_{B_R}\equiv \frac{n-1}{R}\).
We write \(\mathrm {sd}_{ F}\) for the signed distance from a Lebesgue measurable set F, with the convention that \(\mathrm {sd}_{ F}\) is negative inside F, i.e.,
Here, \(F^c := \mathbb {R}^n{\setminus }F\) denotes the complement set of F, and the distance from a set F is by definition
We will sometimes use the notation \(\mathrm {d}_F=|\mathrm {sd}_F|\).
By [t] we denote the integer part of a real number t, that is the biggest integer m such that \(m\le t\).
Finally, we denote by \(c_n, C_n\) positive constants that depend on the space dimension only. Moreover, \(c_{n,0}\), \(C_{n,0}\), and \(C_{n,0,T}\) are constants that may additionally depend on the initial data or the time \(T>0\). During the computations, the value of these constants may change from line to line. However, for the sake of clarity we need to keep track of the dimensional constant in Proposition 3.2, therefore we make an exception to the above convention and denote it by \(\gamma _n\). The volume of the n-dimensional unit ball will be denoted by \(\omega _n\), and thus its surface area is \(n\omega _n\).
2.2 Approximate solutions
In this paper we introduce a notion of global flat solution to the volume-preserving mean-curvature flow which is based on the implicit time discretization of (1.1) in the spirit of Almgren et al. [2] and Luckhaus and Sturzenhecker [26]. That is, we consider a time-discrete gradient flow for the perimeter functional. For this purpose, we define
for any two sets of finite perimeter E and F in \(\mathbb {R}^n\). Here, h is a positive small number that plays the role of the time step of approximate solutions. The second term in the above functional approximates the degenerate \(L^2\) geodesic distance on the configuration space of hypersurfaces. The functional differs from the one considered in the original papers [2, 26] only in the last term: a weak penalization that favors unit-volume of minimizing sets.
Definition 2.1
Let \(E_0\) be a set of finite perimeter with \(|E_0|=1\), and \(h>0\). Let \(\{E^{(h)}_{kh}\}_{k\in \mathbb {N}}\) be a sequence of sets defined iteratively by
We furthermore define
and call \(\{E^{(h)}_t\}_{t\ge 0}\) an approximate flat solution to the volume-preserving mean-curvature flow with initial datum \(E_0\).
The existence of minimizers \(E^{(h)}_{kh}\) and thus the existence of an approximate flat solution is guaranteed by Lemma 3.1 below. Incorporating the volume constraint in a soft way into the energy functional rather than imposing a hard constraint on the admissible sets has the advantage that we are free to chose arbitrary competitors, most notably in the derivation of density estimates. Thanks to the penalizing factor \(1/\sqrt{h}\), the constraint becomes active in the limit \(h\downarrow 0\). Even more can be shown: the number of time steps in which approximate solutions violate the volume constraint \(|E^{(h)}_t|=1\) can be bounded uniformly in h, cf. Corollary 3.10 below. A similar functional including a soft volume constraint was recently considered by Goldman and Novaga [20] in the study of a prescribed curvature problem.
2.3 Main results
We can now state our main results. The first one is a convergence result for approximate solutions.
Theorem 2.2
(Existence of flat flows) Let \(E_0\) be a bounded set of finite perimeter with \(|E_0|=1\) and, for any \(h>0\), let \(\{E^{(h)}_t\}_{t \ge 0}\) be an approximate flat solution to the volume-preserving mean-curvature flow with initial datum \(E_0\). Then, there exists a family of sets of finite perimeter \(\{E_t\}_{t\ge 0}\) and a subsequence \(h_k\downarrow 0\) such that
and, for every \(0\le s\le t\),
Our next statement is the existence of a distributional solution in the sense of Luckhaus and Sturzenhecker [26] to the volume-preserving mean-curvature flow under the hypothesis that the perimeters of the approximate solutions converge to the perimeter of the limiting solutions identified in the previous theorem.
Theorem 2.3
(Existence of distributional solutions) Suppose that \(n\le 7\). Let \((h_k)_{k\in \mathbb {N}}\) and \(\{E_t\}_{t\ge 0}\) be as in Theorem 2.2. For any \(T>0\), if
then \(\{E_t\}_{0 \le t<T}\) is a distributional solution to the volume-preserving mean-curvature flow with initial datum \(E_0\) in the following sense:
-
(1)
for almost every \(t\in [0,T)\) the set \(E_t\) has (generalized) mean curvature in the sense of (2.1) satisfying
$$\begin{aligned} \int _0^T \int _{\partial ^* E_t}|H_{E_t}|^2 < +\infty ; \end{aligned}$$(2.3) -
(2)
there exists \(v:\mathbb {R}^n \times (0,T) \rightarrow \mathbb {R}\) with \(v(\cdot , t)\vert _{\partial ^* E_t} \in L^2_0( \partial ^*E_t,d\mathcal {H}^{n-1})\) for a.e. \(t \in (0,T)\) and \(\int _0^T \int _{\partial ^* E_t} v^2 d\mathcal {H}^{n-1}dt <+\infty \) such that
$$\begin{aligned} -\int _0^T\int _{\partial ^* E_t}v\,\phi \,d\mathcal {H}^{n-1}dt = \int _0^T\int _{\partial ^* E_t}\left( H_{E_t}\,\phi -\lambda \,\phi \right) d\mathcal {H}^{n-1}dt,\end{aligned}$$(2.4)$$\begin{aligned} \int _0^T\int _{E_t}\partial _t\phi \,dxdt+\int _{E_0}\phi (0,\, \cdot \,)\,dx= -\int _0^T\int _{\partial ^* E_t}v\,\phi \,d\mathcal {H}^{n-1}dt, \end{aligned}$$(2.5)for every \(\phi \in C_c^1([0,\infty )\times \mathbb {R}^n)\), where
$$\begin{aligned} \lambda (t):=\frac{1}{\mathcal {H}^{n-1}(\partial ^*E_t)} \int _{\partial ^* E_t} H_{E_t}\,d\mathcal {H}^{n-1} \quad for\,a.e.~t\in [0,T). \end{aligned}$$(2.6)
In the second part of the theorem, \(L^2_0\) is the set of all \(L^2\) functions with zero mean.
Note that (2.4) is a weak formulation of (1.1), while (2.5) establishes the link between v and the velocity of the boundaries of \(E_t\). It is straightforward to check that smooth solutions of (1.1) satisfy (2.4) and (2.5).
The restriction on the dimension \(n\le 7\) is technical and is needed in the proof of Corollary 4.2 where the Bernstein theorem for minimal surfaces is exploited.
3 Flat volume-preserving mean-curvature flows
In this section we prove the first main result in Theorem 2.2. We follow quite closely Luckhaus and Sturzenhecker [26], providing all the details for the readers’ convenience.
3.1 Existence of approximate solutions
We start remarking that
The existence of the approximate flat solutions is guaranteed by the following lemma.
Lemma 3.1
Let \(F \subset \mathbb {R}^n\) be a bounded set of finite perimeter. For every \(h>0\), there exists a minimizer E of \(\mathcal F_h(\, \cdot \,, F)\) and, moreover, E satisfies the discrete dissipation inequality
Proof
Since F is an admissible competitor, we obtain by (3.1) that
Let \(\{E_{\nu }\}_{\nu \in \mathbb {N}}\) denote a minimizing sequence of \(\mathcal F_h(\, \cdot \,, F)\). Without loss of generality we may assume that \(E_{\nu }\subset \subset B_R\) for a suitable \(R>0\). Since \(\{\chi _{E_{\nu }}\}_{\nu \in \mathbb {N}}\) is bounded in \(BV(B_R)\), there exists a subsequence (not relabeled) that converges weakly to a function \(\chi \) in \(BV(B_R)\), and thus strongly in \(L^1(B_R)\). In particular, \(\chi \) is the characteristic function of some set of finite perimeter E. Since \(\chi _{\tilde{E}}\mapsto \int _{\tilde{E}} \mathrm {sd}_F\, dx\) is continuous and the perimeter is lower semi-continuous with respect to \(L^1\) convergence, it follows that
Therefore, E minimizes \(\mathcal F_h(\, \cdot \,,F)\) and (3.2) follows from (3.3). \(\square \)
By standard results on minimal surfaces (see [27]), it holds that the minimizers E of \(\mathcal F_h(\, \cdot \,,F)\) can be chosen to be closed subsets with \(\partial E\) of class \(C^{1,\alpha }\) up to a (relatively closed) singular set of dimension at most \(n-7\). Using the Euler–Lagrange equation for \(\mathcal F_h(\, \cdot \,,F)\), one can also show that the regular part of the boundary \(\partial E\) is actually \(C^{2,\kappa }\) (cp. Lemma 3.7).
3.2 \(L^\infty \) and \(L^1\)-estimates
Our next statement gives a uniform bound on the distance between the boundary of the minimizing set and the boundary of the reference set.
Proposition 3.2
(\(L^\infty \)-estimate) There exists a dimensional constant \(\gamma _n>0\) with the following property. Let \(F \subset \mathbb {R}^n\) be a bounded set of finite perimeter and let E be a minimizer of \(\mathcal F_h(\, \cdot \,,F)\). Then,
Proof
The proof of this proposition is based on the density estimates for one-side minimizing set which for readers’ convenience we prove in the Appendix 1. We claim indeed that the statement holds with
where \(c_n\) is the dimensional constant in Lemma 4.4. The argument is by contradiction. Let \(c>\max \Big \{3, \frac{4n\omega _n}{c_n}\Big \}\) and let \(x_0\in F\Delta E\) contradict (3.4) with \(\gamma _n\) replaced by c. Without loss of generality, we can assume that \(x_0 \in F{\setminus }E\): the other case is at all analogous. We then have that
Then any ball \(B_r(x_0)\) of radius \(r\le \frac{c\sqrt{h}}{2}\) is contained in F. By the minimality of E, we have \(\mathcal F_h(E,F)\le \mathcal F_h(E \cup B_r(x_0),F)\), and thus
We use (3.5) and \(r\le \frac{c\sqrt{h}}{2}\) to infer that
By assumption \(c>3\) and we can apply Lemma 4.4 with \(\mu =0\) and obtain
On the other hand, from (3.8) we deduce also that for a.e. \(0<r <\frac{c\sqrt{h}}{2}\)
Combining (3.9) and (3.10), we get that
for almost all \(0<r <\frac{c\sqrt{h}}{2}\), which gives the desired contradiction to the choice of c as soon as \(r \uparrow \frac{c\sqrt{h}}{2}\). \(\square \)
The following density estimates are now an immediate consequence.
Corollary 3.3
Let \(F \subset \mathbb {R}^n\) be a bounded set of finite perimeter and let E be a minimizer of \(\mathcal F_h(\, \cdot \,,F)\). Then, for every \(r \in (0, \gamma _n\,\sqrt{h})\) and for every \(x_0 \in \partial E\), it holds
Proof
Since E is a minimizer of \(\mathcal F_h(\, \cdot \,,F)\), for any \(x_0\in \partial E\), it holds that \(\mathcal F_h(E,F)\le \mathcal F_h(E{\setminus } B_{r}(x_0),F)\), which implies
Estimating the second term via Proposition 3.2, we obtain
A similar analysis shows that
Therefore, by Lemma 4.4 we deduce (by possibly redefining \(c_n\))
The first inequality in (3.12) is now an immediate consequence of the relative isoperimetric inequality (cf. [19, Corollary 1.29]). For the second inequality, we rewrite (3.13) as
Since \(r< \gamma _n \sqrt{h}\), the upper bound is obvious. \(\square \)
Next we prove an estimate on the volume of the symmetric difference of two consecutive sets of the approximate solutions.
Proposition 3.4
(\(L^1\)-estimate) Let \(F \subset \mathbb {R}^n\) be a bounded set of finite perimeter and let E be a minimizer of \(\mathcal F_h(\, \cdot \,,F)\). Then,
Proof
In order to estimate \(E\Delta F\), we split it into two parts:
The second term is easily estimated by
To estimate the first term, we use a simple covering argument to find a collection of disjoint balls \(\{B_{\ell }(x_i)\}_{i \in I}\) with \(x_i\in \partial ^*E\) and \(I \subset \mathbb {N}\) a finite set such that \(\partial ^* E \subset \cup _{i \in I}B_{2\ell }(x_i)\). Note that by (3.11) and the relative isoperimetric inequality (cf. [19, Corollary.1.29]) we have for every \(i\in I\)
Note finally that the set \(\{x\in E\Delta F:\, \mathrm {d}_{F}(x)\le \ell \}\) is covered by \(\{B_{3\ell }(x_i)\}_{i\in I}\). Summing over i and the choice of the balls \(\{B_{\ell }(x_i)\}_{i\in I}\) yields
\(\square \)
3.3 Hölder continuity in time
As an immediate consequence of the discrete dissipation inequality (3.2), we remark that
and, by iterating (3.15) and using \(|E_0|=1\),
for every \(T>h\). Similarly, using Proposition 3.4 with \(\ell = h < \gamma _n \sqrt{h}\), we also get
Proposition 3.5
(\(C^{1/2}\) regularity in time) Let \(h\le 1\) and let \(\{E_t\}_{t \ge 0}\) be an approximate flat flow. Then it holds
Proof
Clearly it is enough to consider the case \(s-t \ge {h}\). Let \(j\in \mathbb {N}\) and \(k\in \mathbb {N}{\setminus } \{0\}\) be such that \(t\in [jh, (j+1)h)\) and \(s\in [(j+k)h, (j+k+1)h)\). Then, we can use Proposition 3.4 with \(\ell =\gamma _n {h}/{|t-s|^1/2}\) (note that \( \ell \le \gamma _n\sqrt{h}\) by the assumption \(s-t \ge {h}\)) and (3.16), and estimate in the following way:
By using (3.15) we estimate the sum above by
Therefore, by (3.16) and (3.17), we get
where we used \(kh\le |t-s| + h\le 2|t-s|\), thus concluding the proof of the proposition. \(\square \)
3.4 First variations and first consequences
We now introduce the time-discrete normal velocity: for all \(t\ge 0\) and \(x \in \mathbb {R}^n\), we set
Lemma 3.6
(\(L^2\)-bound on the velocity) Let \(\{E^{(h)}_t\}_{t \ge 0}\) be an approximate flat flow. Then it holds
Proof
We fix \(t\in [h,+\infty )\) and consider for every \(\ell \in \mathbb {Z}\) with \(2^\ell \le {\gamma _n}/{\sqrt{h}}\) the sets
so that \(\mathbb {R}^n= \bigcup _{\ell } K(\ell )\). It follows from \(2^{\ell -1} h \le \gamma _n\sqrt{h}/2\) and from Corollary 3.3 that, for every \(x \in \partial E^{(h)}_t\),
Using \(2^{\ell -1}\le |v^{(h)}(t,y)| \le 4 \cdot 2^{\ell -1}\) for all \(y\in B_{2^{\ell -1} h}(x)\) with \(x\in \partial E^{(h)}_t\cap K(\ell )\), we obtain for every \(x\in \partial E^{(h)}_t\cap K(\ell )\)
Hence, combining these two estimates, we have
Now, by a simple application of Besicovitch’s covering theorem [16, Chapter 1.5.2] to \(\{ B_{2^{\ell -1} h}(x):~x\in \partial E^{(h)}_t\cap K(\ell )\}\), we obtain
Finally, summing up over \(\ell \in \mathbb {Z}\) with \(2^{\ell }\le \frac{\gamma _n}{\sqrt{h}}\) in (3.25) yields
We now show how the above estimate implies (3.22). In view of (3.15) we have
Integrating in time and using \(|E_0|=1\), we obtain
from which, by a simple limit for \(T \rightarrow +\infty \), (3.22) follows. \(\square \)
We now derive the Euler–Lagrange equations which constitute the weak motion law for the time-discrete evolution.
Lemma 3.7
(Euler–Lagrange equations) For every \(t\in [h,+\infty )\) and \(\Psi \in C^1_c(\mathbb {R}^n;\mathbb {R}^n)\), it holds
where
Moreover, if \(\vert E^{(h)}_t\vert \ne 1\), then it also holds \(\lambda ^{(h)}(t)=\frac{1}{\sqrt{h}}\,\mathrm {sgn}(1-\vert E^{(h)}_t\vert )\).
As we shall see in the proof below, the constants \(\lambda ^{(h)}(t)\) defined in (3.27) are Lagrange multipliers corresponding to the volume constraint, whenever it is active. Since this constraint is satisfied up to a finite number of times (uniformly in h) by Corollary 3.10 below, by a slight abuse of terminology, we call \(\lambda ^{(h)}(t)\) a Lagrange multiplier, even if the volume constraint is not active.
Proof
If \(\vert E^{(h)}_t\vert \ne 1\), it is very simple to compute the variations of \(\mathcal F_h(\cdot ,E^{(h)}_{t-h})\) along the vector field \(\Psi \in C^1_c(\mathbb {R}^n;\mathbb {R}^n)\) and see that they are given by (3.26) with \(\lambda ^{(h)}(t)=\frac{1}{\sqrt{h}}\,\mathrm {sgn}(1-\vert E^{(h)}_t\vert )\). In the case \(\vert E^{(h)}_t\vert = 1\), we have
Hence, performing variations of
within the class of sets of unit volume, for every \(\Psi \in C^1_c(\mathbb {R}^n;\mathbb {R}^n)\), we again find (3.26), where \(\lambda ^{(h)}(t)\) is the Lagrange multiplier related to the constraint \(\vert F\vert =1\). Observe that in both cases, we can choose a sequence of \(\Psi \in C^1_c(\mathbb {R}^n;\mathbb {R}^n)\) approximating \(\nu _{E^{(h)}_t}\) on \(\partial E^{(h)}_t\), and conclude that \(\lambda ^{(h)}(t)\) is given by (3.27). \(\square \)
It is now clear that the regular part of \(\partial ^* E^{(h)}_t\) is of class \(C^{2,\kappa }\). Indeed, by choosing a suitable system of coordinates, \(\partial ^* E^{(h)}_t\) can be written in a neighbourhood of any regular point as the graph of a \(C^{1,\kappa }\) function u solving the following equation in the sense of distributions:
Since \(v^{(h)}\) is Lipschitz continuous, by standard elliptic regularity theory (cp., e.g., [18]), one then deduce that \(u \in C^{2,\kappa }\) for every \(\kappa \in (0, 1)\).
Next we prove that the whole family of sets defining the discrete flow up to time \(T>0\) is contained in a large ball, whose radius does not depend on the discrete time-step h but may depend on the T.
Lemma 3.8
(Boundedness of minimizing sets) Let \(\{E^{(h)}_t\}_{t \ge 0}\) be an approximate solution and let \(T>0\). Then there exist \(h_0, R_T>0\) (depending on T, n, and \(E_0\) only) such that, if \(h\le h_0\), then \(E^{(h)}_t\subset B_{R_T}\) for all \(t\in [0,T]\).
Proof
We fix \(h>0\), and for every \(t\in [0,T)\) we let
We notice \({\bar{B}_{r_t}}\cap \partial E^{(h)}_t\ne \emptyset \) is made of regular points (because there are no singular minimizing cones contained in a half space, cp. [19, Theorem 15.5]), and moreover
By this observation and the Euler–Lagrange equation \(v^{(h)}(t,y)= \lambda ^{(h)}(t) - H_{E^{(h)}_t}(y)\), it follows that
Iterating the above estimate, we then deduce that
To get some control on \(\lambda ^{(h)}(t)\) we consider \(\Psi \in C_c^1(\mathbb {R}^n;\mathbb {R}^n)\) such that \(\Psi (x)=x\) in \(B_{r_t}\), and, since for h small enough \(|E^{(h)}_t|\ge \frac{1}{2}\) by (3.17), using \(\Psi \) as test in (3.26), we obtain
where we used \(|\Psi |\le r_t\) on \(\partial E^{(h)}_t\). Integrating in time and using the Cauchy–Schwarz inequality together with (3.16) and (3.22), we obtain
Combining (3.28) and (3.30), it follows that
The remainder of the proof is a standard ODE argument. Indeed, squaring both sides of the equation and redefining \(C_{n,0}\) yields
where \(F(\tau ) = \int _0^{\tau } r_t^2\, dt\). Integration in \(\tau \) over the interval [0, T] yields
and thus the statement in Lemma 3.8 follows via (3.31). \(\square \)
Corollary 3.9
For every \(h>0\) small enough, it holds
Proof
The integrability of \(\lambda ^{(h)}\) follows from (3.29) and from Lemma 3.8; the one of \(H_{E^{(h)}_t}\) follows taking into account the first variation (3.26) and the integrability of the discrete velocity, Lemma 3.6. \(\square \)
For every \(h>0\) we set
Corollary 3.10
For every \(h>0\) small enough, we have
Proof
In view of Lemma 3.7, it is
and thus we have by Corollary 3.9
\(\square \)
3.5 Proof of Theorem 2.2
The proof of the existence of a flat flow is now a simple consequence of the results above. Indeed, by (3.16), (3.17) and Lemma 3.8, one can find sets \(\{E_t\}_{t\in \mathbb {Q}^+}\), where \(\mathbb {Q}^+\) denotes the set of positive rational numbers, and a subsequence \(h_k \downarrow 0\) such that
Using the triangular inequality and Proposition 3.5, we deduce that
Now a simple continuity argument implies that the sequence \(E^{(h_k)}_t\) converges to sets \(E_t\) for all times \(t\ge 0\) and satisfies (3.32) for all \(s,\, t\in [0,+\infty )\). Finally, note that passing to the limit in (3.15) yields that \(|E_t|=1\) and \(\mathrm {Per}(E_t)\le \mathrm {Per}(E_0)\) (cf. [16, Sect 5.2.1]) for a.e. \(t\in (0,T)\).
Remark 3.11
It is also possible to show that the sequence of characteristic functions
are precompact in \(L^1((0,T)\times \mathbb {R}^n)\) for every \(T>0\), thus giving an alternative proof of the theorem.
4 Distributional solutions
In this section we prove Theorem 2.3 on the existence of distributional solutions. The two main ingredients of the proof besides the estimates of the previous section are the hypothesis (2.2) on the continuity of the perimeters of the approximate solutions and the following proposition which links the discrete velocity to the distributional time derivative of the flat flow.
Proposition 4.1
Let \(n\le 7\) and \(\{E^{(h)}_t\}_{t \ge 0}\) be an approximate solution to the volume-preserving mean-curvature flow. Then, for every \(\phi \in C_c^\infty ([0,+\infty ) \times \mathbb {R}^n)\) it holds
Assuming the proposition we give a proof of the theorem.
4.1 Proof of Theorem 2.3
It follows straightforwardly from (2.2) (cp., for instance, [3, Proposition 1.80]) that \(\mathcal {H}^{n-1}\partial ^* E_t^{(h_k)}\) weakly converges to \(\mathcal {H}^{n-1} \partial ^* E_t\) for almost every \(t \in [0, +\infty )\). In particular this implies that, for almost every \(t \in [0, +\infty )\), the boundaries \(\partial ^* E_t^{(h_k)}\) converge to \(\partial ^* E_t\) in the sense of varifolds: namely, for a.e. \(t\in [0, +\infty )\) it holds
for every \(F\in C_c(\mathbb {R}^n \times \mathbb {R}^n)\). Indeed, by a simple approximation argument it is easy to verify that it is enough to consider \(F\in C_c^1(\mathbb {R}^n \times \mathbb {R}^n)\). Then, for every \(\varepsilon >0\) we pick a continuous function \(\nu _\varepsilon :\mathbb {R}^n \rightarrow \mathbb {R}^n\) such that
and estimate as follows
where we used (2.2) and the weak convergence of the the vector valued measures \(\nu _{E^{(h_k)}_t}\mathcal {H}^{n-1} \partial ^*E^{(h_k)}_t {\mathop {\rightharpoonup }\limits ^{*}}\,\nu _{E_t}\mathcal {H}^{n-1} \partial ^*E_t\) in the following way:
Next we use Lemma 3.6 and Corollary 3.9 in conjunction with the results in Hutchinson [23, Theorem 4.4.2] to deduce the existence of functions \(v:[0,+\infty ) \times \mathbb {R}^n \rightarrow \mathbb {R}\), \(\widehat{\lambda }:[0,+\infty ) \rightarrow \mathbb {R}\) and \(H:[0,+\infty ) \times \mathbb {R}^n \rightarrow \mathbb {R}\) such that
and
for every \(\phi \in C^0_c([0,T)\times \mathbb {R}^n)\) and every \(\Phi \in C^0_c([0,T)\times \mathbb {R}^n;\mathbb {R}^n)\).
In particular, testing (4.2) with \(F(x, \nu ) := \mathrm{div} \Psi - \nu \cdot \nabla \Psi \nu \) for some \(\Psi \in C_c^1(\mathbb {R}^n;\mathbb {R}^n)\) and using (4.5), by a simple approximation argument we conclude that, for a.e. \(t\in [0, +\infty )\),
thus showing that \(H(t, \cdot )\) is the generalized mean-curvature of \(E_t\) for a.e. \(t \in [0,+\infty )\) (cp. (2.1)) and proving (2.3) of Theorem 2.3.
Similarly, (2.4) and (2.6) follows from (3.26) and (3.27) by using (4.3) and (4.4).
We need only to show (2.5). To this aim we use Proposition 4.1. For every \(\phi \in C_c^1([0,+\infty ) \times \mathbb {R}^n)\), by a change of variables we have that
where we used that \(E^{(h)}_t = E_0\) for \(t\in [0, h)\). Therefore it follows by a simple convergence argument that
In view of (4.3) and (4.1), we conclude (2.5) straightforwardly.
4.2 Tilting of the tangent planes
In this subsection and in the next one we give the proof of Proposition 4.1. We follow closely the arguments in [26] and for the sake of completeness we provide a detailed proof in different steps.
This subsection is devoted to the estimate of the tilting of the normals around points of small curvature. We recall that we assume in this section \(n\le 7\) (in particular, the approximate solutions of the volume-preserving mean-curvature flow are everywhere of class \(C^{2,\kappa }\)).
Lemma 4.2
For given constants \(\frac{1}{2}<\beta <\alpha <1\), there exists a continuous increasing function \(\omega :[0,1] \rightarrow \mathbb {R}\) with \(\omega (0) =0\) with the following property. Let \(t\in [2h,+\infty )\), \(\{E^{(h)}_t\}_{t \ge 0}\) be an approximate solution to the volume-preserving mean-curvature flow, and let \(x_0\in \partial E^{(h)}_t\) be such that
Then there exists \(\nu \in \mathbb {R}^n\) such that \(\vert \nu \vert =1\) and
Proof
Let \(0<R \le h^{\frac{1}{2}-\beta }\) and let \(F\subset \mathbb {R}^n\) be any set such that \(E^{(h)}_t\Delta F\subset \subset B_{Rh^\beta }(x_0)\). By the minimizing property of \(E^{(h)}_t\) we have that
A straightforward computation yields
where we used that \(|v^{(h)}(t,y)|\le Rh^{\beta -1} + \gamma _n \, h^{-1/2}\le (\gamma _n+1)\,h^{-\frac{1}{2}}\) for all \(y\in B_{Rh^\beta }(x_0) \cap (E^{(h)}_t\Delta F)\) thanks to the fact that \(x_0 \in \partial E^{(h)}_t\), Proposition 3.2 and the 1-Lipschitz continuity of the signed distance \(\mathrm {sd}_{E^{(h)}_{t-h}}\). Combining the above estimates with (4.9), we obtain
Next we introduce the sets
By a simple rescaling argument, from (4.10) and from the analogous estimates at time \(t-h\) (recall that \(t\ge 2h\)) we deduce that for \(s=t, t-h\)
This implies that \(E^{(h),\beta }_t\) and \(E^{(h),\beta }_{t-h}\) are both \((\Lambda _h,r_h)\)-minimizers of the perimeter on \(\Lambda _h:=(\gamma _n+2) h^{\beta -\frac{1}{2}}\) and \(r_h:=h^{\frac{1}{2}-\beta }\). By the precompactness for sequences of \(\Lambda _h\)-minimizers (cf. [27, Proposition 21.13]), we conclude that we can find a subsequence (not relabeled) verifying
Moreover, using the lower semicontinuity of the perimeter with respect to \(L^1\) convergence and \(\beta >\frac{1}{2}\), we deduce that \(E^\beta _1\) and \(E^\beta _2\) are locally minimizing the perimeter. By the assumption \(n\le 7\) and a Bernstein theorem (see [19, Theorem 17.3]), \(E^\beta _1,\, E^\beta _2\) are half-spaces. Moreover, by hypothesis it holds
thus implying that \(E^\beta _1= E^\beta _2\), and by the fact that both are hyperplanes there exists \(\nu \in \mathbb {R}^n\) with \(|\nu |=1\) such that
To reach the conclusion of the lemma we need only to invoke the regularity theory of \(\Lambda \)-minimizing set (cp. [27, Theorem 26.3]) and conclude that \(\partial E^{(h),\beta }_s\) is uniformly \(C^{1,\kappa }\) in \(B_{1}\) for \(s= t, t-h\), thus leading straightforwardly to (4.7) and (4.8). \(\square \)
Corollary 4.3
Under the hypotheses of Lemma 4.2, let \({\mathbf {C}}_{{h^\beta }/{2}}(x_0,\nu )\subset \mathbb {R}^n\) be the open cylinder defined as
Then, there exists a dimensional constant \(C>0\) such that
Proof
From Lemma 4.2 we know that, for h sufficiently small, \(\partial E^{(h)}_t\) and \(\partial E^{(h)}_{t-h}\) in \({\mathbf {C}}_{{h^\beta }/{2}}(x_0,\nu )\) can both be written as graphs of functions of class \(C^{1,\kappa }\). Namely, by an affine change of coordinates we can assume without loss of generality that \(x_0 =0\) and \(\nu = e_n\), and for simplicity we set \({\mathbf {C}}:= {\mathbf {C}}_{{h^\beta }/{2}}(0, e_n)\). With this assumption we then have that for \(s=t, t-h\)
where \(f_s:B_{{h^\beta }/{2}}\subset \mathbb {R}^{n-1} \rightarrow \mathbb {R}\) are \(C^{1,\kappa }\) functions with
In view of Fubini’s theorem it is then clear that
Moreover, from (4.12) it follows that there exists a geometric constant \(C>0\) such that, for every \(y \in B_{{h^\beta }/{2}}\),
Therefore, one infers (4.11) as follows
where we used that (4.12). \(\square \)
We are finally ready for the proof of Proposition 4.1.
4.3 Proof of Proposition 4.1
We fix any time \(t \in [2h, +\infty )\). For every \(x_0 \in \partial E^{(h)}_t\), we fix \(\alpha \in \big (\frac{1}{2}, \frac{n+2}{2(n+1)}\big )\) and consider the following open set \(A_{x_0}\) defined as follows:
-
(i)
if (4.6) holds, then we set \(A_{x_0} := {\mathbf {C}}_{{h^\beta }/{2}}(x_0, \nu )\) where \(\nu \in \mathbb {R}^n\) is the unit vector in Lemma 4.2;
-
(ii)
otherwise we set \(A_{x_0} := B_{\gamma _n \sqrt{h}}(x_0)\).
Note that by Proposition 3.2 we have that \(\{A_{x_0}\}_{x_0 \in \partial E^{(h)}_t}\) is a covering of \(E^{(h)}_t \Delta E^{(h)}_{t-h}\). Moreover, by a simple application of Besicovitch’s covering theorem, cp. [16, Chapter 1.5.2] (applied, for example, to the balls to \(B_{{h^\beta }/{2}}(x_0) \subset A_{x_0}\)), there exists a finite collections of points \(I \subset \partial E^{(h)}_t\) such that \(\{A_{x_0}\}_{x_0 \in I}\) is a covering of \(E^{(h)}_t \Delta E^{(h)}_{t-h}\).
We estimate the contribution of the integrals in (4.1) in every \(A_{x_0}\) with \(x_0 \in I\) in two steps, depending on whether (i) above applies or (ii).
Estimate in case (i). We use Corollary 4.3 and deduce that
where we used the fact that \(A_{x_0} = {\mathbf {C}}_{{h^\beta }/{2}}(x_0, \nu )\).
Estimate in case (ii). By assumption there exists a point \(y_0 \in B_{\gamma _n\sqrt{h}}(x_0)\cap (E^{(h)}_t\Delta E^{(h)}_{t-h})\) such that \(|v^{(h)}(t,y_0)|>h^{\alpha -1}\). Without loss of generality we can assume that \(y_0 \in E^{(h)}_t\) (the other case can be treated analogously and we leave the details to the reader). It is then clear that \(B_{{h^{\alpha }}/{2}}(y_0) \subset \mathbb {R}^n{\setminus }E^{(h)}_{t-h}\) and \(v^{(h)}(t,y)>{h^{\alpha -1}}/{2}\) for every \(y \in B_{{h^{\alpha }}/{2}}(y_0)\). Since \(h^{\alpha } < 2\gamma _n h^{1/2}\), we can apply the density estimate in (3.11) and deduce that
Similarly, by the density estimate in (3.12) and Proposition 3.2 we deduce that
From (4.14), (4.15) and \(B_{{h^{\alpha }}/{2}}(y_0) \subset B_{2\gamma _n \sqrt{h}}(x_0)\) we then deduce that
We can then sum (4.13) and (4.16) over \(x_0 \in I\) and, recalling (3.15), (3.16) and (3.17), we get
where we used the finite finiteness of the covering.
Finally, integrating in time and using (3.18) and (3.19) we get
where \(T>0\) is such that \(supp (\phi ) \subset [0,T] \times \mathbb {R}^n\). Recalling the definition of \(\alpha \) and taking the limit as h goes to 0, we conclude (4.1).
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Open access funding provided by Max Planck Society (or associated institution if applicable). C.S. acknowledges the kind hospitality of the Max-Planck-Institut für Mathematik in den Naturwissenschaften in Leipzig, Germany, where part of this work was performed.
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Appendix: A density lemma
Appendix: A density lemma
We premise the following density estimate for one-sided minimizers of the perimeter. The estimate can be easily deduced from the original arguments by De Giorgi exploited for minimizers [14].
Lemma 4.4
There exists a dimensional constant \(c_n>0\) with this property. Let \(E\subset \mathbb {R}^n\) be a set of finite perimeter, \(R, \mu >0\) and \(x_0\in \partial E\) be such that
Then,
Recall that \(x_0 \in \partial E\) if \(\min \{|B_r(x_0){\setminus } E|,|E\cap B_r(x_0)|\}>0\) for every \(r>0\).
Proof
Without loss of generality, we can assume \(x_0 = 0\). We use the following indentity which are true a.e. \(r>0\):
Indeed, if E were smooth, these formulas follow for all the r such that \(B_r\) and E have transversal intersections. Otherwise one can argue by approximation. Using now (4.18), we deduce that, for a.e. \(r>0\),
By the isoperimetric inequality [19, Corollary 1.29], there exists a dimensional constant \(C>0\), such that
Setting \(f(r):=|B_r{\setminus } E|\), by the coarea formula [16, 3.4.4], it holds
Hence, (4.24) reads as
Finally, note that \(f(r) \le \omega _n\,r^n\), from which \(f(r) \le \omega _n^{{1}/{n}} r\,f(r)^{{n-1}/{n}}\). Therefore, there exists a dimensional constant \(C_n>0\) such that if \(0< r < \min \{R, C_n\,\mu ^{-1}\}\), then the last term in (4.25) can be absorbed in the left hand side and deduce that
Integrating (4.24) (recall that \(f(r)>0\) for every \(r>0\)) we get the desired (4.19) for every \(0< r < \min \{R, C_n\,\mu ^{-1}\}\) and, by changing the dimensional constant \(c_n>0\), for every \(0< r < \min \{R, \mu ^{-1}\}\). \(\square \)
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Mugnai, L., Seis, C. & Spadaro, E. Global solutions to the volume-preserving mean-curvature flow. Calc. Var. 55, 18 (2016). https://doi.org/10.1007/s00526-015-0943-x
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DOI: https://doi.org/10.1007/s00526-015-0943-x