Abstract
In this paper, we construct a family of integral varifolds, which is a global weak solution to the volume preserving mean curvature flow in the sense of \(L^2\)-flow. This flow is also a distributional BV-solution for a short time, when the perimeter of the initial data is sufficiently close to that of a ball with the same volume. To construct the flow, we use the Allen–Cahn equation with a non-local term motivated by studies of Mugnai, Seis, and Spadaro, and Kim and Kwon. We prove the convergence of the solution for the Allen–Cahn equation to the family of integral varifolds with only natural assumptions for the initial data.
Similar content being viewed by others
Avoid common mistakes on your manuscript.
1 Introduction
Let \(d\ge 2\) be an integer and \(\Omega := {\mathbb {T}}^d = ({\mathbb {R}}/{\mathbb {Z}}) ^d\). Assume that \(T>0\) and that \(U_t \subset \Omega \) is an open set with the smooth boundary \(M_t:=\partial U_t\) for any \(t \in [0,T)\). The family of the hypersurfaces \(\{ M_t \} _{t \in [0,T)}\) is called the volume preserving mean curvature flow if the normal velocity vector \(\vec {v}\) satisfies
Here, \({\mathscr {H}}^{d-1}\) is the \((d-1)\)-dimensional Hausdorff measure, and \(\vec {h}\) and \(\vec {\nu }\) are the mean curvature vector and the inner unit normal vector of \(M_t\), respectively. Note that the solution \(\{M_t\} _{t \in [0,T)}\) to (1) satisfies
where \({\mathscr {L}}^{d}\) is the d-dimensional Lebesgue measure. From (2), \(\{M_t\} _{t \in [0,T)}\) has the volume preserving property, that is, \({\mathscr {L}}^d (U_t)\) is constant with respect to t.
For when \(U_0\) is convex, Gage [15] and Huisken [19] proved that there exists a solution to (1) and it converges to a sphere as \(t\rightarrow \infty \). Escher and Simonett [10] showed the short time existence of the solution to (1) for smooth initial data \(M_0\) and they also proved that if \(M_0\) is sufficiently close to a sphere in the sense of the little Hölder norm \(h^{1+\alpha }\), then there exists a global solution and it converges to some sphere as \(t\rightarrow \infty \) (see also [3, 4, 29] for related results). Mugnai, Seis, and Spadaro [35] studied the minimizing movement for (1) and they proved the global existence of the flat flow, that is, there exist \(C=C(d,U_0) >0\) and a family of Caccioppoli sets \(\{ U_t \}_{t \in [0,\infty )}\) such that \({\mathscr {L}}^d (U_s \bigtriangleup U_t) \le C\sqrt{s-t}\) for any \(0\le t<s\), \({\mathscr {H}}^{d-1} (\partial ^*U_t)\) is monotone decreasing, and \({\mathscr {L}}^d (U_t)\) is constant. Here, \(\partial ^*U_t\) is the reduced boundary of \(U_t\). In addition, for \(d \le 7\), they proved the global existence of the weak solution to (1) in the sense of the distribution, under the reasonable assumption for the convergence, that is,
where \(\{ U_t ^k \}_{t \in [0,T)}\) is the time-discretized approximate solution to (1). This kind of condition was introduced in [31] (see also [1, 26]). Laux and Swartz [28] showed the convergence of the thresholding schemes to the distributional BV-solutions of (1) under an assumption of the convergence similar to (3). Laux and Simon [27] also proved similar results in the case of the phase field method. On the other hand, the author [43] proved the existence of the weak solution (family of integral varifolds) to (1) in the sense of \(L^2\)-flow for \(2\le d \le 3\) without any such convergence assumption, via the phase field method studied by Golovaty [17]. Recently, Kim and Kwon [24] proved the existence of the viscosity solution to (1) for the case where \(U_0\) satisfies a geometric condition called \(\rho \)-reflection. Moreover, they also proved that the viscosity solution converges to some sphere uniformly as \(t \rightarrow \infty \).
Let \(\{\delta _i \}_{i=1} ^\infty \) be a positive sequence with \(\delta _i \rightarrow 0\) as \(i\rightarrow \infty \) and we denote \(\delta _i\) as \(\delta \) for simplicity. Suppose that \(U_t ^\delta \) is an open set with smooth boundary \(M_t ^\delta \) for any \(t \in [0,T)\). The approximate solutions studied in [24, 35] correspond to the following mean curvature flow \(\{ M_t ^\delta \}_{t \in [0,T)}\) with non-local term:
where
One can check that (4) is a \(L^2\)-gradient flow of
that is,
Hence \(\{ M_t ^\delta \}_{t \in [0,T)}\) satisfies a relaxed volume preserving property, namely,
Therefore \(\{ M_t ^\delta \}_{t \in [0, T)}\) converges to the solution \(\{ M_t \}_{t \in [0,T)}\) to (1) as \(\delta \rightarrow 0\) formally. Note that we cannot directly obtain the monotonicity of \({\mathscr {H}}^{d-1} (M_t)\) by the energy estimates above. However, if we have a natural energy estimate \(\sup _i \int _0 ^T \vert \lambda ^{\delta _i} (t) \vert ^2 \, dt \le C_T\) for some constant \(C_T>0\), we can expect the property in some sense, because
by Fatou’s lemma (see Proposition 10). The reason why the \(L^2\)-estimate is natural is because the non-local term of the solution to (1) satisfies it (see Proposition 19). Mugnai, Seis, and Spadaro [35] used a minimizing movement scheme corresponding to (4), and Kim and Kwon [24] used (4) to prove the existence of the viscosity solution to (1). Based on these results, in this paper we show the global existence of the weak solution to (1), via the phase field method corresponding to (4).
We denote \(W(a):= \dfrac{(1-a^2)^2}{2}\) and \(k(s) = \int _{0} ^s \sqrt{2W (a)} \, \textrm{d}a= s -\dfrac{1}{3} s^3\). Let \(\varepsilon \in (0,1)\), \(T>0\), and \(\alpha \in (0,1)\). With reference to [35] and [24], in this paper we consider the following Allen–Cahn equation with non-local term:
where \(\lambda ^{\varepsilon }\) is given by
Note that if \(\varphi _0 ^\varepsilon \) satisfies suitable assumptions, the standard PDE theories imply the global existence and uniqueness of the solution to (5) (see Remark 8). Set
As above, one can check that the solution \(\varphi ^{\varepsilon } \) to (5) satisfies
and
for any \(t \in [0,\infty )\). Assume \( \sup _{\varepsilon \in (0,1)} E^\varepsilon _S (0) <\infty \) (this assumption corresponds to \({\mathscr {H}}^{d-1} (M_0) <\infty \) for (1)). Then, we can expect that \(\varphi ^\varepsilon (x,t) \approx 1\) or \(-1\) when x is outside the neighborhood of the zero level set \(M_t ^\varepsilon =\{x \in \Omega \mid \varphi ^\varepsilon (x,t)=0 \}\) for sufficiently small \(\varepsilon \). Then we have \(\int _{\Omega } k(\varphi ^\varepsilon )\, \textrm{d}x \approx \frac{2}{3} \int _{\Omega } \varphi ^\varepsilon \, \textrm{d}x\) and thus we can regard (10) as a relaxed volume preserving property. The function \(\sqrt{2W(\varphi ^\varepsilon )}\) expresses that the non-local term is almost zero when x is outside the neighborhood of \(M_t ^\varepsilon \). In addition, \(\sqrt{2W(\varphi ^\varepsilon )}\) plays important roles in \(L^\infty \)-estimates and energy estimates (see Proposition 6 and Theorem 12).
The first main result of this paper is that there exists a global-in-time weak solution to (1) for any \(d\ge 2\) in the sense of \(L^2\)-flow, under the assumptions on the regularity of \(M_0\) (see Theorem 3). We employ (5) to construct the solution. Note that we do not require assumptions such as (3). The second main result is that, when \(M_0\) is \(C^1\) and the value \({\mathscr {H}}^{d-1} (M_0)/ ({\mathscr {L}}^d (U_0))^{\frac{d-1}{d}}\) is sufficiently close to that of a ball, there exists \(T_1 >0\) such that the flow has a unit density for a.e. \(t \in [0,T_1)\) and is also a distributional BV-solution up to \(t=T_1\) (see Theorem 4). To obtain the main results, we need to prove that the varifold \(V_t ^\varepsilon \) defined by the Modica–Mortola functional [33] converges to a integral varifold for a.e. \(t\ge 0\) (roughly speaking, the condition (3) corresponds to this convergence). For the standard Allen–Cahn equation without non-local term, this convergence was shown by Ilmanen [23] and Tonegawa [47]. Therefore we can expect the convergence for (5) if \(\lambda ^\varepsilon \) has suitable properties. In fact, \(\lambda ^\varepsilon \) can be regarded as an error term when we consider the parabolic rescaled equation of (5). We explain this more precisely. Define \({\tilde{\varphi }} ^\varepsilon (\tilde{x},{\tilde{t}}) = \varphi ^\varepsilon (\varepsilon {\tilde{x}}, \varepsilon ^2 {\tilde{t}})\). Then \({\tilde{\varphi }} ^\varepsilon \) satisfies
where \(\Delta _{{\tilde{x}}} \) is a Laplacian with respect to \(\tilde{x}\). Assume \(\sup _{x} \vert \varphi ^\varepsilon _0 (x) \vert < 1\). Then Proposition 5 below yields \(\sup _{x,t} \vert \varphi ^\varepsilon (x,t) \vert < 1\). Thus we have
where we used \(\max _{s \in [-1,1]} \vert k (s) \vert =\frac{2}{3}\). Therefore, broadly speaking, the non-local term \(\varepsilon \lambda ^\varepsilon (\varepsilon ^2 {\tilde{t}})\sqrt{2W({\tilde{\varphi }} ^\varepsilon )}\) is a perturbation (to the best of our knowledge, for (1), no phase field model with such a property has been known). Hence we can show the rectifiability and the integrality of the varifold \(V_t\) with arguments similar to that in [23, 47] (see also [45]). However, the proofs are not exactly the same as those, because the monotonicity formula for (5) is different from the standard one (see Proposition 8). Therefore we give the proofs in Section 4. In addition, as another good property of \(\lambda ^\varepsilon \), the \(L^2\)-norm can be controlled (see Lemma 1). This property is useful when proving the monotonicity formula and the rectifiability of \(V_t\).
In [23], to construct the weak solution to the mean curvature flow (Brakke flow), the simplest Allen–Cahn equation was considered. As generalizations of the result, the equations with external forces (see [30, 34, 44, 45]) and with Laplace-Beltrami operators (see [36, 37]) have been studied. The most well-known phase field model for (1) studied by Rubinstein and Sternberg [39] is the equation
where \(\Lambda ^\varepsilon (t) = \frac{1}{{\mathscr {L}}^d (\Omega )}\int _\Omega \frac{W' (\varphi ^{\varepsilon } (x,t) )}{\varepsilon } \, dx\). As above, the solution to (13) has the volume preserving property \( \frac{d}{dt} \int _{\Omega } \varphi ^\varepsilon \, dx =0 \). Chen, Hilhorst, and Logak [8] proved that for the smooth solution \(\{M_t\} _{t \in [0,T)}\) to (1), there exists a family of functions \(\{ \varphi _0 ^{\varepsilon _i } \}_{i =1 } ^\infty \) with \(\varepsilon _i \rightarrow 0\) such that the level set \(M_t ^{\varepsilon _i} = \{ x \in \Omega \mid \varphi ^{\varepsilon _i} (x,t)=0 \}\) converges to \(M_t\), where \(\varphi ^\varepsilon \) is a solution to (13) with initial data \(\varphi ^\varepsilon _0\). In addition, as mentioned above, Laux and Simon [27] proved the convergence of the vector-valued version of (13) to the weak volume preserving multiphase mean curvature flow under an assumption that corresponds to (3). However, it is an open problem to show its convergence for (13) without such assumptions. One of the difficulties is that the boundedness of \(\sup _{\varepsilon >0} \int _0 ^T \vert \Lambda ^\varepsilon \vert ^2 \, dt\) proved by Bronsard and Stoth [7] does not immediately lead to
Note that (14) corresponds to \(\int _0 ^T \int _{M_t} \vert \vec {h} \vert ^2 \, \textrm{d}{\mathscr {H}}^{d-1} \textrm{d}t <\infty \) of the solution to (1) and is important to show the rectifiability of the varifold (see Theorem 12 and Theorem 14). As another phase field method for (1), the study of Brassel and Bretin [6] is known (see [43, Section 1] for a comparison of these equations).
Recently, the weak-strong uniqueness of the mean curvature flow and the relationship between weak solutions have been well studied. In [12], they proved the weak-strong uniqueness of the BV solution to the multiphase mean curvature flow. More precisely, if both BV solution and strong solution with the same initial data exist, then the BV solution agrees with the strong solution while both exist. As a result strongly related to our study, Laux [25] obtained the weak-strong uniqueness for the volume preserving mean curvature flow (see Remark 5). Hensel and Laux [18] proposed a new weak solution to the mean curvature flow via varifolds and the De Giorgi type inequality. Moreover, they proved the existence of the varifold solution and the weak-strong uniqueness. In [41], Stuvard and Tonegawa showed that the multiphase Brakke flow which they construct is a \(L^2\)-flow. In addition, they proved that the flow is also a BV solution for a short time, under the suitable assumptions for initial data (therefore, the results of [12] can be used in this case).
The organization of the rest of this paper is as follows: in Section 2, we set our notations and state the main results. In Section 3, to obtain the existence theorem we prove the energy estimates and \(L^\infty \)-estimates for the solution to (5). In addition, for \(d=2\) or 3, we give a short proof for the integrality of the limit measure \(\mu _t\) constructed as a weak solution to (1). In Section 4, we show the integrality of \(\mu _t\) for any \(d \ge 2\). In Section 5, we prove the main results. In Section 6, we give some supplements for this paper.
2 Preliminaries and Main Results
2.1 Notations and definitions
For \(r>0\), \(d\in {\mathbb {N}}\), and \(x \in {\mathbb {R}}^d\), we denote \(B_r ^d (x):= \{ y \in {\mathbb {R}}^d \mid \vert x-y \vert <r \}\) (we often write this as \(B_r (x)\) for simplicity). We define \(\omega _d:= {\mathscr {L}}^d (B_1 ^d (0))\). For \(d\times d\) matrix \(A =(a_{ij})\) and \(B=(b_{ij})\), we denote the usual matrix multiplication by \(A \circ B\) and \( A\cdot B:= \sum _{i,j} a_{ij} b_{ij}. \) For \(a=(a_1,a_2,\dots , a_d) \in {\mathbb {R}}^d\), we define a \(d\times d\) matrix \(a\otimes a\) by \(a\otimes a:= (a_i a_j)\). Next we recall notations and definitions from geometric measure theory and refer to [11, 16, 40, 47] for more details. For a Caccioppoli set \(E \subset {\mathbb {R}}^d\), we denote the reduced boundary of E by \(\partial ^*E\). For the characteristic function \(\chi _{E}\), we denote the total variation measure of the distributional derivative \(\nabla \chi _E\) by \(\Vert \nabla \chi _E \Vert \). Let \(U \subset {\mathbb {R}}^d\) be an open set. We write the space of bounded variation functions on U as BV(U). For any Radon measure \(\mu \) on U and \(\phi \in C_c (U)\), we often write \(\int \phi \, d\mu \) as \(\mu (\phi )\). For \(p\ge 1\), we write \(f \in L^p (\mu )\) if f is \(\mu \)-measurable and \(\int \vert f \vert ^p \, \textrm{d}\mu <\infty \). For \(d,k \in {\mathbb {N}}\) with \(k<d\), let \({\mathbb {G}}(d,k)\) be the space of k-dimensional subspace of \({\mathbb {R}}^d\). For an open set \(U \subset {\mathbb {R}}^d\), let \(G_k (U):= U \times {\mathbb {G}} (d,k)\). We say V is a general k-varifold on U if V is a Radon measure on \(G_k (U)\). We denote the set of all general k-varifolds on U by \({\mathbb {V}} _k (U)\). For a general varifold \(V \in {\mathbb {V}} _k (U)\), we define the weight measure \(\Vert V\Vert \) by
We call \(V \in {\mathbb {V}} _k (U)\) is rectifiable if there exist a \({\mathscr {H}}^k\)-measurable k-countably rectifiable set \(M \subset U\) and \(\theta \in L_{loc} ^1 ({\mathscr {H}}^{k}\lfloor _{M})\) such that \(\theta >0\) \({\mathscr {H}}^k\)-a.e. and
where \(T_x M\) is the approximate tangent space of M at x. Note that such x exists for \({\mathscr {H}}^k\)-a.e. on M. If \(\theta \in {\mathbb {N}}\) \({\mathscr {H}}^k\)-a.e. on M, we call V is integral. In addition, if \(\theta =1\) \({\mathscr {H}}^k\)-a.e. on M, we say V has unit density.
For \(V \in {\mathbb {V}} _k (U)\) and a smooth diffeomorphism \(f:U\rightarrow U\), we define the push-forward of V by
for any \(\phi \in C_c (G_k(U))\), where \(\vert \Lambda _k \nabla f (x) \circ S \vert \) is the Jacobian of the map.
For \(V \in {\mathbb {V}} _k (U)\), we define the first variation \(\delta V\) by
Here, we identify \(S \in {\mathbb {G}} (d,k)\) with the corresponding orthogonal projection of \({\mathbb {R}}^d\) onto S. When the total variation \(\Vert \delta V \Vert \) of \(\delta V\) is locally bounded and absolutely continuous with respect to \(\Vert V \Vert \), there exists a measurable vector field \(\vec h\) such that
The vector valued function \(\vec h\) is called the generalized mean curvature vector of V. In addition, a Radon measure \(\mu \) is called k-rectifiable if there exists a k-rectifiable varifold such that \(\mu \) is represented by \(\mu = \Vert V \Vert \). Note that this V is uniquely determined, so the first variation and the generalized mean curvature vector of \(\mu \) is naturally determined by V. The definition of an integral Radon measure is determined in the same way.
The formulation of the following is similar to that of the Brakke flow [5, 48]:
Definition 1
(\(L^2\)-flow [34]) Let \(T>0\), \(U\subset {\mathbb {R}}^d\) be an open set, and \(\{\mu _t\} _{t \in [0,T)}\) be a family of Radon measures on U. Set \(d\mu := d\mu _t dt\). We call \(\{\mu _t\} _{t \in [0,T)}\) an \(L^2\)-flow with a generalized velocity vector \(\vec {v}\) if the following hold:
-
1.
For a.e. \(t \in (0,T)\), \(\mu _t\) is \((d-1)\)-integral, and also has a generalized mean curvature vector \(\vec {h} \in L^2 (\mu _t; {\mathbb {R}}^d)\).
-
2.
The vector field \(\vec {v}\) belongs to \(L^2 (0,T; (L^2 (\mu _t))^d)\) and
$$\begin{aligned} \vec {v}(x,t) \perp T_x \mu _t \quad \text {for} \ \mu \text {-a.e.} \ (x,t) \in U \times (0,T), \end{aligned}$$where \(T_x \mu _t \in {\mathbb {G}} (d,d-1)\) is the approximate tangent space of \(\mu _t\) at x.
-
3.
There exists \(C_T>0\) such that
$$\begin{aligned} \left| \int _0 ^T \int _U (\partial _t \eta + \nabla \eta \cdot \vec {v}) \, d\mu _t dt \right| \le C_T \Vert \eta \Vert _{C^0 (U\times (0,T))} \end{aligned}$$(16)for any \(\eta \in C_c ^1 (U\times (0,T))\).
Remark 1
If there exists a family of smooth hypersurfaces \(\{M_t\}_{t \in [0,T)}\) with the normal velocity vector \(\vec {w}\), then (16) holds with \(\vec {v} =\vec {w}\) and \(\mu _t = {\mathscr {H}} ^{d-1} \lfloor _{M_t}\). In addition, if \(\vec {v}\) satisfies (16) with \(\mu _t = {\mathscr {H}} ^{d-1} \lfloor _{M_t}\), then \(\vec {v} =\vec {w}\). This proof is almost identical to the proof in [48, Proposition 2.1].
The \(L^2\)-flow has the following property:
Proposition 1
(See Proposition 3.3 of [34]) Assume that \(\{ \mu _t \} _{t \in (0,T)}\) is an \(L^2\)-flow with the generalized velocity vector \(\vec {v}\) and set \(d\mu :=d\mu _t dt\). Then
at \(\mu \)-a.e. \( (x_0,t_0) \in \Sigma (\mu )\), where \(T_{(x_0,t_0)} \mu \in {\mathbb {G}} (d+1,d)\) is the approximate tangent space of \(\mu \) at \((x_0, t_0)\) and \(\Sigma (\mu )= \{ (x,t) \mid T_{(x,t)} \mu \ \text {exists at} \ (x,t) \}\).
2.2 Assumptions for initial data
Let \(U _0 \subset \subset (0,1)^d\) be a bounded open set with the following properties:
-
1.
There exists \(D_0 >0\) such that
$$\begin{aligned} \sup _{x \in (0,1) ^d, 0<R <1} \frac{{\mathscr {H}}^{d-1} (M_0 \cap B_r (x))}{\omega _{d-1} r^{d-1}} \le D_0, \end{aligned}$$(17)where \(M_0 = \partial U_0\).
-
2.
There exists a family of open sets \(\{ U_0 ^i\}_{i=1} ^\infty \) such that \(U_0 ^i\) has a \(C^3\) boundary \(M_0 ^i = \partial U_0 ^i\) for any i and it holds that
$$\begin{aligned} \lim _{i\rightarrow \infty } {\mathscr {L}}^d (U_0 \triangle U_0 ^i)=0 \ \text{ and } \ \lim _{i\rightarrow \infty } \Vert \nabla \chi _{U_0 ^i} \Vert = \Vert \nabla \chi _{U_0} \Vert \ \text{ as } \text{ Radon } \text{ measures }.\nonumber \\ \end{aligned}$$(18)
Note that the second assumption is satisfied when \(U_0\) is a Caccioppoli set, and both conditions are fulfilled when \(M_0\) is \(C^1\) (see [16]).
We denote \(q^\varepsilon (r):= \tanh (r/\varepsilon )\) for \(r \in {\mathbb {R}}\). Then denoting derivatives by subscript r, \(q^\varepsilon \) satisfies
and
In addition, (19) yields
This means that the Radon measure \(\mu _t ^\varepsilon \) defined below needs to be normalized by \(\sigma \).
Next we extend \(U_0 ^i \) and \(M_0 ^i \) periodically to \({\mathbb {R}}^d\) with period \(\Omega ={\mathbb {T}}^d\) and define
Then \(\vert \nabla r _i (x) \vert \le 1\) for a.e. \(x \in {\mathbb {R}}^d\) and there exists \(b_i >0\) such that \(r_i \) is \(C^3\) on \(N_{b_i}:= \{ x \mid \textrm{dist}\,(x, M_0 ^i)<b_i \}\) (see [9]). Let \(d_i\) be a smooth monotone non-decreasing function such that
and \(\vert \frac{d}{dr}d_i \vert \le 1\). Set \(\overline{r_i}:= d_i (r_i) \). Then \(\overline{r_i} \in C^3 (\Omega )\), \(\overline{r_i} =r_i\) on \(N_{b_i/4}\), and \(\vert \nabla \overline{r_i} (x) \vert \le 1\) for any \(x \in {\mathbb {R}}^d\). Let \(\{ \varepsilon _i \}_{i=1} ^\infty \) be a positive sequence with \(\varepsilon _i \rightarrow 0\) and \(\displaystyle \frac{ \varepsilon _i }{b_i ^2} \rightarrow 0\) as \(i \rightarrow \infty \), and
Note that (21) corresponds to the condition (29) below. We define a periodic function \(\varphi _0 ^{\varepsilon _i} \in C^3 (\Omega ) \) by
We define a Radon measure \(\mu _t ^{\varepsilon _i}\) by
where \(\varphi ^{\varepsilon _i}\) is the solution to (5) with initial data \(\varphi _0 ^{\varepsilon _i}\) defined by (22) and \(\sigma = \int _{-1} ^1 \sqrt{2W(s)} \, ds\).
Remark 2
In this paper we choose a typical function \(W(a)= \frac{(1-a^2)^2}{2}\) as a double-well potential for simplicity. Since more general potentials have been considered for the convergence of the standard Allen–Cahn equation (see [23, 45, 47]), generalizations of our results regarding W can be made as well.
For \(\varphi ^{\varepsilon _i} _0\) and \(\mu _0 ^{\varepsilon _i}\), we have the following properties (see [23, p. 423] and [30, Section 5]):
Proposition 2
There exists a subsequence \(\{ \varepsilon _i \}_{i=1} ^\infty \) (denoted by the same index and the subsequence is taken only for \(\{ \varepsilon _i\}_{i=1} ^\infty \), not for \(\{ M_0 ^i\} _{i=1} ^\infty \)) such that the following hold:
-
1.
For any \(i \in {\mathbb {N}}\) and \(x \in \Omega \), we have \(\displaystyle \frac{\varepsilon _i \vert \nabla \varphi ^{\varepsilon _i} _0 (x) \vert ^2 }{2} \le \frac{W(\varphi ^{\varepsilon _i} _0 (x))}{\varepsilon _i} \).
-
2.
There exists \(D_1= D_1 (D_0) >0\) such that
$$\begin{aligned} \max \left\{ \sup _{i \in {\mathbb {N}}} \mu _0 ^{\varepsilon _i} (\Omega ), \sup _{i \in {\mathbb {N}}, \ x \in \Omega , \ r \in (0,1)} \frac{\mu _0 ^{\varepsilon _i} (B_r (x))}{\omega ^{d-1} r^{d-1}} \right\} \le D_1. \end{aligned}$$(24) -
3.
\(\mu _0 ^{\varepsilon _i} \rightharpoonup {\mathscr {H}}^{d-1} \lfloor _{M_0}\) as Radon measures, that is,
$$\begin{aligned} \int _{\Omega } \phi \, \textrm{d} \mu _0 ^{\varepsilon _i} \rightarrow \int _{M_0} \phi \, d{\mathscr {H}}^{d-1} \qquad \text {for any} \ \phi \in C_c(\Omega ). \end{aligned}$$ -
4.
For \(\psi ^{\varepsilon _i} =\frac{1}{2} ( \varphi ^{\varepsilon _i} +1)\), \(\lim _{i\rightarrow \infty } \psi ^{\varepsilon _i} = \chi _{U_0} \) in \(L^1\) and \(\lim _{i\rightarrow \infty } \Vert \nabla \psi ^{\varepsilon _i } \Vert = \Vert \nabla \chi _{U_0} \Vert \) as Radon measures.
Remark 3
The first property (1) is obtained from \( \vert \nabla \overline{r_i} \vert \le 1\) (see the proof of Proposition 6). The assumption \(\frac{ \varepsilon _i }{b_i ^2} \rightarrow 0\) is used to show \(\int _{\Omega {\setminus } N_{b_i /4} } \left( \frac{\varepsilon _i \vert \nabla \varphi ^{\varepsilon _i} _0 \vert ^2}{2} + \frac{W(\varphi ^{\varepsilon _i} _0 )}{\varepsilon _i} \right) \, \textrm{d}x\rightarrow 0 \).
2.3 Main results
We denote the approximate velocity vector \(\vec {v} ^{\, \varepsilon _i}\) by
The first main result of this paper is
Theorem 3
Suppose that \(d\ge 2\) and \(U_0\) satisfies (17) and (18). For any \(i \in {\mathbb {N}}\), let \(\varphi _0 ^{\varepsilon _i}\) be defined so that all the claims of Proposition 2 are satisfied and \(\varphi ^{\varepsilon _i} \) be a solution to (5) with initial data \(\varphi _0 ^{\varepsilon _i}\). Then there exists a subsequence \(\{ \varepsilon _i \}_{i=1} ^\infty \) (denoted by the same index) such that the following hold:
-
(a)
There exist a countable subset \(B \subset [0,\infty )\) and a family of \((d-1)\)-integral Radon measures \(\{\mu _t\}_{t \in [0,\infty )}\) on \(\Omega \) such that
$$\begin{aligned} \mu _0 = {\mathscr {H}}^{d-1} \lfloor _{M_0}, \qquad \mu _t ^{\varepsilon _i} \rightharpoonup \mu _t \ \ \ \text {as Radon measures for any} \ t\ge 0, \end{aligned}$$and
$$\begin{aligned} \mu _s (\Omega ) \le \mu _t (\Omega ) \qquad \text {for any} \ s,t \in [0,\infty ) \setminus B \ \text {with} \ 0 \le t<s <\infty . \end{aligned}$$ -
(b)
There exists \(\psi \in BV_{loc} (\Omega \times [0,\infty )) \cap C_{loc} ^{\frac{1}{2}} ([0,\infty ); L^1 (\Omega ))\) such that the following hold:
-
(b1)
\(\psi ^{\varepsilon _i} \rightarrow \psi \) in \(L^1 _{loc} (\Omega \times [0,\infty ))\) and a.e. pointwise, where \(\psi ^{\varepsilon _i}=\frac{1}{2} (\varphi ^{\varepsilon _i} +1)\).
-
(b2)
\(\psi \vert _{t=0}=\chi _{U_0}\) a.e. on \(\Omega \).
-
(b3)
For any \(t \in [0,\infty )\), \(\psi (x,t) =1\) or 0 for \({\mathscr {L}}^d\)-a.e. \(x \in \Omega \) and \(\psi \) satisfies the volume preserving property, that is,
$$\begin{aligned} \int _{\Omega } \psi (x,t) \, \textrm{d}x = {\mathscr {L}}^{d} (U_0) \qquad \text {for all} \ t \in [0,\infty ). \end{aligned}$$ -
(b4)
For any \(t \in [0,\infty )\) and for any \(\phi \in C_c (\Omega ; [0,\infty ))\), we have \(\Vert \nabla \psi (\cdot ,t) \Vert (\phi ) \le \mu _t (\phi )\).
-
(b1)
-
(c)
For \(\lambda ^{\varepsilon _i}\) given by (6), we have
$$\begin{aligned} \sup _{i \in {\mathbb {N}}} \int _0 ^T \vert \lambda ^{\varepsilon _i} \vert ^2 \, dt <\infty \qquad \text {for any} \ T>0 \end{aligned}$$and there exists \(\lambda \in L_{loc} ^2 (0,\infty )\) such that \(\lambda ^{\varepsilon _i} \rightharpoonup \lambda \) weakly in \(L^2 (0,T)\) for any \(T>0\).
-
(d)
There exists \(\vec {f} \in L_{loc} ^2 ([0,\infty ); (L^2 (\mu _t))^d)\) such that
$$\begin{aligned} \begin{aligned}&\lim _{i \rightarrow \infty } \frac{1}{\sigma } \int _0 ^\infty \int _{\Omega } -\lambda ^{\varepsilon _i} \sqrt{2W (\varphi ^{\varepsilon _i})} \nabla \varphi ^{\varepsilon _i} \cdot \vec {\phi } \, \textrm{d}x\textrm{d}t \\ =&\int _{0} ^\infty \int _{\Omega } \vec {f} \cdot \vec {\phi } \, d \mu _t \textrm{d}t = \int _{0} ^\infty \int _{\Omega } -\lambda \vec {\nu }\cdot \vec {\phi } \, d \Vert \nabla \psi (\cdot ,t) \Vert \textrm{d}t \end{aligned} \end{aligned}$$(25)for any \(\vec {\phi } \in C_c (\Omega \times [0,\infty ); {\mathbb {R}}^d)\), where \(\vec \nu \) is the inner unit normal vector of \(\{ \psi (\cdot ,t) =1 \}\) on \(\textrm{spt}\,\Vert \nabla \psi (\cdot ,t) \Vert \).
-
(e)
The family of Radon measures \(\{ \mu _t \}_{t \in [0,\infty )}\) is an \(L^2\)-flow with a generalized velocity vector \(\vec {v} = \vec {h} +\vec {f}\), where \(\vec {h} \in L_{loc} ^2 ([0,\infty ); (L ^2 (\mu _t) )^d) \) is the generalized mean curvature vector of \(\mu _t\). Moreover, for any \(\vec {\phi } \in C_c (\Omega \times [0,\infty ); {\mathbb {R}}^d)\),
$$\begin{aligned} \lim _{i \rightarrow \infty } \int _0 ^\infty \int _{\Omega } \vec {v}^{ \, \varepsilon _i} \cdot \vec {\phi } \, \textrm{d}\mu _t ^{\varepsilon _i}\textrm{d}t = \int _{0} ^\infty \int _{\Omega } \vec {v} \cdot \vec {\phi } \, \textrm{d} \mu _t \textrm{d}t. \end{aligned}$$(26)
Remark 4
From (b4), (d), and (e), we have
for any \(\vec {\phi } \in C_c (\Omega \times (0,\infty ); {\mathbb {R}}^d)\), where \(\frac{d \Vert \nabla \psi (\cdot ,t) \Vert }{d\mu _t}\) is the Radon–Nikodym derivative. Hence we have \(\vec {v} = \vec {h} -\lambda \vec {\nu }\) in the sense of \(L^2\)-flow if \(\mu _t = \Vert \nabla \psi (\cdot ,t) \Vert \) for a.e. t (from Theorem 4 below, this is correct for a short time if the initial data is sufficiently close to a ball). Since \(\mu _t\) is integral for a.e. t, for such t, we have \(\left( \frac{d \Vert \nabla \psi (\cdot ,t) \Vert }{d\mu _t} \right) ^{-1} \in {\mathbb {N}}\) for \(\mu _t\)-a.e. \(x \in \Omega \), if \(\frac{d \Vert \nabla \psi (\cdot ,t) \Vert }{d\mu _t}\not =0\).
Set \(U_t:= \{ x \in \Omega \mid \psi (x,t)=1 \}\) for \(t>0\). Let \(B \subset \subset (0,1) ^d\) be an open ball. We also show that if \(U_0 \approx B\), then there exists \(T_1 >0\) such that \(\{\partial ^*U _t\} _{t \in [0,T_1)}\) is a distributional solution to (1) in the framework of BV functions.
Theorem 4
For any \(r \in (0,\frac{1}{4})\), there exists \(\delta _1 >0\) depending only on d and r with the following properties. Assume that \( U_0 \subset (\frac{1}{4},\frac{3}{4})^d \) satisfies \({\mathscr {L}}^d (U_0)={\mathscr {L}}^d (B_{r} (0))\) and has a \(C^1\) boundary \(M_0\) with \({\mathscr {H}}^{d-1} (M_0) \le 2 {\mathscr {H}}^{d-1} (\partial B_{r} (0))\) and
Then there exists \(T_1 =T_1(d,r,M_0) >0\) such that the following hold:
-
(a)
For a.e. \(t \in [0,T_1)\), \( \mu _t = \Vert \nabla \psi (\cdot ,t) \Vert = {\mathscr {H}}^{d-1} \lfloor _{\partial ^{*} U_t} \), where \(\{ \mu _t\} _{t \in [0,\infty )}\) is the \(L^2\)-flow with initial data \(\mu _0 ={\mathscr {H}}^{d-1}\lfloor _{M_0}\), given by Theorem 3.
-
(b)
Let \(\vec {v}\), \(\vec {h}\), \(\vec {\nu }\), and \(\lambda \) be functions given by Theorem 3. Then \(\{ \partial ^*U_t \}_{t \in [0,T_1)}\) is a distributional solution to (1) with initial data \(\partial U_0 = M_0\) in the following sense.
-
(b1)
For any \(t \in [0,T_1)\), \({\mathscr {L}}^d (U_t) = {\mathscr {L}}^d (U_0)\).
-
(b2)
For a.e. \(t\in [0,T_1)\), \(\vec {h}\) is also a generalized mean curvature vector of \({\mathscr {H}}^{d-1} \lfloor _{\partial ^*U_t}\).
-
(b3)
For any \(\vec {\phi } \in C_c (\Omega \times [0,T_1);{\mathbb {R}}^d)\), we have
$$\begin{aligned} \int _0 ^{T_1} \int _{\partial ^*U_t} \{ \vec {v} -\vec {h} + \lambda \vec {\nu } \} \cdot \vec {\phi } \, \textrm{d}{\mathscr {H}} ^{d-1} \textrm{d}t =0. \end{aligned}$$ -
(b4)
For any \(\phi \in C_c ^1 (\Omega \times (0,T_1))\), we have
$$\begin{aligned} \int _0 ^{T_1} \int _{U_t} \partial _t \phi \, \textrm{d}x\textrm{d}t = \int _0 ^{T_1} \int _{\partial ^*U_t} \vec {v} \cdot \vec {\nu } \phi \, \textrm{d}{\mathscr {H}}^{d-1} \textrm{d}t. \end{aligned}$$ -
(b5)
(Additional volume preserving property). For a.e. \(t \in [0,T_1)\), we have
$$\begin{aligned} \int _{\partial ^*U_t} \vec {v} \cdot \vec {\nu } \, d {\mathscr {H}}^{d-1} = \int _{\Omega } \vec {v} \cdot \vec {\nu } \, d \Vert \nabla \psi (\cdot ,t)\Vert =0. \end{aligned}$$ -
(b6)
For a.e. \(t \in [0,T_1)\), we have
$$\begin{aligned} \lambda (t) = \frac{1}{{\mathscr {H}}^{d-1} (\partial ^*U_t)} \int _{\partial ^*U_t} \vec {h} \cdot \vec {\nu } \, d {\mathscr {H}}^{d-1}. \end{aligned}$$
-
(b1)
Remark 5
Recently, Laux [25] showed that if both strong solution and BV solution to the volume preserving mean curvature flow with same initial data exist, then the strong solution matches the BV solution while both exist (see also [12, 18]).
Remark 6
The isoperimetric inequality tells us that \(d \omega _d ^{\frac{1}{d}} ({\mathscr {L}}^d (U))^{\frac{d-1}{d}} \le {\mathscr {H}}^{d-1} (\partial ^*U) \) for any Caccioppoli set \(U \subset {\mathbb {R}}^d\) with \({\mathscr {L}}^d (U) <\infty \) and the equality holds if and only if there exists a ball \(B \subset {\mathbb {R}}^d\) such that \({\mathscr {L}}^d (U \triangle B )=0\) (see [13, 46] and references therein). Moreover, by the quantitative isoperimetric inequality (see [14, Theorem 1.1]) and the assumption (27), we have
where \(r>0\) is a constant given by Theorem 4 and \(C>0\) depends only on d (thus, \(U_0\) needs to be close to a sphere in the above sense). On the other hand, \(M_0\) does not have to be close to a sphere in \(C^0\) (for example, \(U_0\) does not have to be connected).
Remark 7
The property (b4) claims that \(\vec v\) is a normal velocity vector in a weak sense, since
holds for any \(\phi \in C_c ^1 (\Omega \times (0,T_1))\) and \(t \in (0,T_1)\), where \(\{ U_t \}_{t\in [0,T_1)}\) is a family of open sets and the smooth boundary \(\partial U_t\) moves by the normal velocity vector \(\vec {v}\). By (2), we can regard (b5) as a volume preserving property in a weak sense.
3 Energy and Pointwise Estimates
In this section we show standard estimates for (5) such as the uniform \(L^2\)-estimate for \(\lambda ^\varepsilon \) and the monotonicity formula.
3.1 Assumptions
Let \(\{ \varepsilon _i \}_{i=1} ^\infty \) be a positive sequence with \(\varepsilon _i \rightarrow 0\) as \(i \rightarrow \infty \). In this section, we assume that there exist \(D_1>0\) and \(\omega >0\) such that (24) and
hold for any \(i \in {\mathbb {N}}\). The set \(\{ x \in \Omega \mid \varphi ^{\varepsilon _i} _0 (x) = 0 \}\) corresponds to the initial data \(M_0\) of (1), and (28) yields that \({\mathscr {L}}^d (\{ x \in \Omega \mid \varphi ^{\varepsilon _i} _0 \approx 1 \}) >0\) formally, since \(\int _{\Omega } k(\pm 1)\, dx = \pm \frac{2}{3} {\mathscr {L}}^d(\Omega ) =\pm \frac{2}{3} \). For some \({{C}_1} >0\), we also assume that the initial data \(\varphi _0 ^{\varepsilon _i}\) of the solution to (5) satisfies
for any \(i \in {\mathbb {N}}\) and \(j=1,2,3\). In addition, to control the discrepancy measure \(\xi _t ^\varepsilon \) defined below, we assume
Note that the function \(\varphi _0 ^{\varepsilon _i}\) defined by (22) satisfies all the assumptions above, for sufficiently large i. Throughout this paper, we often write \(\varepsilon \) as \(\varepsilon _i\) for simplicity.
3.2 Pointwise estimates
The comparison principle implies
Proposition 5
The solution \(\varphi ^\varepsilon \) to (5) with (29) satisfies
Remark 8
The estimate (31) implies \(\sqrt{2W (\varphi ^\varepsilon )} =1-(\varphi ^\varepsilon )^2 \). By a priori estimates including Proposition 6 below, standard PDE theories imply the global existence and uniqueness of the classical solution to (5) with initial data \(\varphi ^\varepsilon _0\) satisfying (29).
Proof
Suppose that \(t_0:= \inf \{ t \in [0,\infty ) \mid \sup _{x \in \Omega } \varphi ^\varepsilon (x,t) \ge 1 \} <\infty \). Then \(t_0 > 0 \) since \(\sup _{x \in \Omega } \varphi ^\varepsilon _0 (x) <1 \). We may assume that there exists \(t_1 \in (t_0, \infty )\) such that \(\sup _{x \in \Omega } \varphi ^\varepsilon (x,t) \le 2\) for any \(t <t_1\). Let \(\varphi ^\varepsilon _+ \) be a solution to
with initial data \(\varphi ^\varepsilon _+ (x,0) = \sup _{x \in \Omega } \varphi ^\varepsilon _0 (x) \), where \(L^\varepsilon := 2 \varepsilon ^{-\alpha } \max _{ \vert s \vert \le 2} \vert k(s) \vert \). Note that \(\sup _{ t \in (0, t_1)} \vert \lambda ^\varepsilon (t) \vert \le L^\varepsilon \), where \(\lambda ^\varepsilon \) is given by the solution \(\varphi ^\varepsilon \) to (5), and this implies that \(\varphi ^{\varepsilon } _+\) is a supersolution to (5) if we regard \(\lambda ^\varepsilon \) as a given function. Since the initial data is constant and \(W'(s), \sqrt{2W(s)} \rightarrow 0\) as \(s\rightarrow 1\), one can easily check that the solution \(\varphi _+ ^\varepsilon \) to (32) depends only on t and satisfies \(\varphi _+ ^\varepsilon (t) <1\) for any \(t \in (0,t_1)\). Therefore the comparison principle implies that \(\varphi ^\varepsilon (x,t) \le \varphi _+ ^\varepsilon (t) <1\) for any \((x,t) \in \Omega \times (0,t_1)\). This yields a contradiction. Hence \(\varphi ^\varepsilon (x,t) <1\) for any \((x,t) \in \Omega \times [0,\infty )\) and the remaining inequality \(\varphi ^\varepsilon (x,t)>-1\) can be proved similarly. \(\square \)
In addition, by Proposition 5 and the maximum principle we have the following proposition (see [23]):
Proposition 6
If the solution \(\varphi ^\varepsilon \) to (5) satisfies (29) and (30), then we have
Proof
By (31), we can define a function \(r^\varepsilon \) by
since \(q^\varepsilon : {\mathbb {R}}\rightarrow (-1, 1)\) is one to one and surjective. Denoting the derivatives of \(q ^\varepsilon \) by subscript r, we compute that
where we used (19) and (20). Then we obtain
where we used \(W' (q^\varepsilon ) / \sqrt{2W (q^\varepsilon )} =-2 q^\varepsilon \). We compute
where we used \(\nabla \lambda ^\varepsilon =0\). Set \(w^\varepsilon = \vert \nabla r ^\varepsilon \vert ^2 -1\). Then \(w^\varepsilon \) satisfies
In addition, we have \(w ^\varepsilon (x,0) \le 0\), because
by (30). Hence the maximum principle implies \(w^\varepsilon (x,t) \le 0\) for any \(x \in \Omega \) and \(t \in [0,\infty )\), and we obtain (33) by \(\frac{\varepsilon \vert \nabla \varphi ^\varepsilon \vert ^2 }{2} - \frac{W(\varphi ^\varepsilon )}{\varepsilon } = \frac{W(q ^\varepsilon )}{\varepsilon } w^\varepsilon \le 0. \) \(\square \)
3.3 Energy estimates
By \(E_S ^\varepsilon (t)=\sigma \mu _t ^\varepsilon (\Omega )\), (8), (9), and (24), we can easily obtain the following estimates:
Proposition 7
For any \(\varepsilon >0\) and \(T>0\), we have
and
Remark 9
Generally, “\(\mu _{s} ^\varepsilon (\Omega ) \le \mu _{t} ^\varepsilon (\Omega )\) for any \(0\le t<s <\infty \)” can not be shown from the energy estimates above. However, there exists a countable set B such that \(\mu _{s} (\Omega ) \le \mu _{t} (\Omega )\) holds for any \(t,s \in [0,\infty ) {\setminus } B\) with \(t<s\), where \(\mu _{t} (\Omega ) =\lim _{\varepsilon \rightarrow 0} \mu _{t} ^\varepsilon (\Omega ) \) (see Proposition 10).
Set \(D' _1:= \sup _{\varepsilon \in (0,1)} \mu _0 ^\varepsilon (\Omega )\). Note that \(D' _1 \le D_1\). By an argument similar to that in [7], we have the following lemma:
Lemma 1
There exist constants \(C_2= C_2(\omega ,d,D' _1)>0\), \(C_3= C_3(\omega ,d,D' _1)>0\), and \(\varepsilon _1 =\varepsilon _1 (\omega , d, D' _1, \alpha ) >0\) such that
and
Proof
Let \(\vec {\zeta }=(\zeta ^1,\zeta ^2,\dots , \zeta ^d): \Omega \times [0,\infty ) \rightarrow {\mathbb {R}}^d\) be a smooth periodic test function. By integration by parts, we have
and
Multiply (5) by \(\nabla \varphi ^\varepsilon \cdot \vec {\zeta }\) and integrate over \(\Omega \). Then, using (39) and (40), we have
The Cauchy–Schwarz inequality, (35), and (36) imply
where \(C_4>0\) depends only on d. Let \(\eta \in C ^\infty _c (B_1 (0))\) be a smooth non-negative function with \(\int _{B_1 (0)} \eta \, dx =1\) and define the standard mollifier \(\eta _\delta \) by \(\eta _\delta (x) = \delta ^{-d} \eta (x/\delta )\) for \(\delta >0\). Let \(u=u(x,t)\) be the periodic solution to
Note that
and there exists \(C>0\) depending only on \({\mathscr {L}}^d (\Omega )\) such that
where we used \(\Vert \varphi ^\varepsilon \Vert _{L^\infty } \le 1\). Therefore the standard PDE arguments imply the existence and uniqueness of the solution u and
where \(\beta \in (0,1)\) and \(C_5>0\) depends only on \(\beta \), d, and \(\delta \). Set \(\vec \zeta (x,t) = \nabla u(x,t)\). Then, by (41) and (42), we have
We compute
By \((k(s))^2 -\frac{4}{9} \ge -W(s)\) for any \(s \in [-1,1]\), we have
By using
\(\Vert \varphi ^\varepsilon \Vert _{L^\infty } \le 1\), and Proposition 18, we have
and
where \(C_6 >0\) depends only on \(D' _1\) and \({\mathscr {L}}^d (\Omega )\). Set \(\delta = \frac{\omega ^2}{4C_6 {\mathscr {L}}^d (\Omega )}\). By (10), (28), (44), (45), (46), and (47), there exists \(\varepsilon _1 >0\) depending only on \(\alpha \), \(D' _1\), \({\mathscr {L}}^d (\Omega )\), and \(\omega \) such that
holds for any \(\varepsilon \in (0,\varepsilon _1)\), where we used \((\int _{\Omega } k (\varphi _0 ^\varepsilon ) \, \textrm{d}x)^2 -(\int _{\Omega } k (\varphi ^\varepsilon ) \, dx)^2 \le \frac{4\sqrt{2}}{3} \varepsilon ^{\frac{\alpha }{2}} (D' _1)^{\frac{1}{2}}\) by (10). From (35), (36), (43), and (48), we obtain (37) and (38). \(\square \)
Remark 10
For the classical solution to the volume preserving mean curvature flow, we can obtain a similar estimate for the non-local term (see Proposition 19).
We define the discrepancy measure \(\xi _t ^\varepsilon \) on \(\Omega \) by
where
Proposition 6 implies the following lemma.
Lemma 2
Assume (30). Then \(\xi _\varepsilon (x,t) \le 0\) for any \( (x,t) \in \Omega \times [0,\infty )\). In addition, \(\xi _t ^\varepsilon \) is a non-positive measure for any \(t \ge 0\).
We denote the backward heat kernel \(\rho =\rho _{(y,s)} (x,t)\) by
With exactly the same proof as in [43, p. 2028], we obtain the following estimates similar to the monotonicity formula obtained by Huisken [20] and Ilmanen [23] (for convenience, we call thus the monotonicity formula)
Proposition 8
(See [43]) Let \(\xi _\varepsilon (x,0) \le 0\) for any \( x \in \Omega \). Assume (30). Then
holds for any \(0\le t<s<\infty \) and for any \(y \in {\mathbb {R}}^d\). Here, \(\mu _t ^\varepsilon \) and \(\xi ^\varepsilon _t\) are extended periodically to \({\mathbb {R}}^d\). In addition, we have
for any \(y \in {\mathbb {R}}^d\), \(0\le t_1< t_2 <\infty \), and \(\varepsilon \in (0,\varepsilon _1)\).
Remark 11
Ilmanen [23] proved the monotone decreasing of \(\int _{{\mathbb {R}}^d} \rho _{(y,s)} (x,t) \, \textrm{d}\mu _t ^\varepsilon (x)\) with respect to t, for the solution to the Allen–Cahn equation without the non-local term under suitable assumptions. In general, one can show that the Brakke flow with smooth initial data has unit density for a short time by using the monotonicity formula see [45]). However, in order to show a similar conclusion for our problem, it is necessary that \(\mu _0 ^\varepsilon (\Omega ) -\mu _ {t} ^\varepsilon (\Omega )\) is small enough, due to (37) (see Lemma 12 below).
As a corollary of the monotonicity formula, we can obtain the following upper bounds of the densities of \(\mu _t ^\varepsilon \):
Corollary 1
(See [23, 42]) There exists \(0<D_2 <\infty \) depending only on d, \(C_3\), \(D_1\), and T such that
for all \(y \in {\mathbb {R}}^d\), \(R \in (0,1)\), \(\varepsilon \in (0,\varepsilon _1)\), and \(t \in [0, T]\).
Proof
Using \( \int _0 ^1 \left( \log \frac{1}{k} \right) ^{\frac{d-1}{2}} \, dk =\Gamma (\frac{d-1}{2} +1) =\pi ^{\frac{d-1}{2}} /\omega _{d-1} \) and the same calculation as (155) below, we have
for any \(s>0\) and \(y \in {\mathbb {R}}^d\). By (51) and (53),
for any \(t \in [0,T)\) with \(0<t<s\) and \(y \in {\mathbb {R}}^d\). Set \(R=2\sqrt{s-t}\). We compute
Therefore we have (52) by (54) and (55). \(\square \)
By integration by parts, we have the following estimate:
Lemma 3
For any non-negative test function \(\phi \in C_c ^2 (\Omega )\), there exists \(C_7 >0\) depending only on \(D_1\), \(\Vert \phi \Vert _{C^2 (\Omega )}\), \(\omega \), and d such that
Proof
By integration by parts, we have
By Cauchy’s mean-value theorem, there exists \(c=c(d) >0\) such that \(\sup _{\phi >0} \frac{\vert \nabla \phi \vert ^2}{\phi } \le c(d) \Vert \nabla ^2 \phi \Vert _{C^0 (\Omega )}\). Hence,
where \(C>0\) depends only on \(\Vert \phi \Vert _{C^2 (\Omega )}\) and d. Thus (35), (38), and (58) imply (56). \(\square \)
With an argument similar to that of [23], we can show the following proposition by using (56):
Proposition 9
There exist a subsequence \(\{ \varepsilon _{i_j} \}_{j=1} ^\infty \) and a family of Radon measures \(\{ \mu _t \}_{t\ge 0}\) such that
for any \(t \in [0,\infty )\) and for any \(d\ge 2\). In addition, there exists a countable set \(B \subset [0,\infty )\) such that \(\mu _t (\Omega )\) is continuous on \([0,\infty ) {\setminus } B\).
Proof
Let \(\{ \phi _k \} _{k=1} ^\infty \subset C_c (\Omega ) \) be a dense subset with \(\phi _k \in C_c ^2 (\Omega )\) for any k, and for any \(x \in \Omega \cap {\mathbb {Q}}^d\) and \(r \in (0,1) \cap {\mathbb {Q}}\), there exists \(k \in {\mathbb {N}}\) such that \(\phi _k \in C_c ^2 (B_r (x))\). Let \(f _k ^i (t)=\mu _0 ^{\varepsilon _i} (\phi _k) + \int _0 ^t \left( \frac{d}{ds} \mu _s ^{\varepsilon _i} (\phi _k) \right) _+ \, ds\) and \( g _k ^i (t) = \int _0 ^t \left( \frac{d}{ds} \mu _s ^{\varepsilon _i} (\phi _k) \right) _- \, ds \). Then
and \(f _k ^i (t)\) and \(g _k ^i (t)\) are non-decreasing functions with
for any i and \(t \in [0,T)\), where \(C_k >0\) depends only on \(D_1\), \(\Vert \phi _k \Vert _{C^2 (\Omega )}\), \(\omega \), and d, by (56). Then Helly’s selection theorem implies that there exist a subsequence \(\varepsilon _i \rightarrow 0\) (denoted by the same index), \(f_k,g_k:[0,T) \rightarrow [0,\infty )\) such that \(\lim _{i\rightarrow \infty } f_k ^i (t) = f_k (t)\) and \(\lim _{i\rightarrow \infty } g_k ^i (t) = g_k (t)\) for any \(t \in [0,T)\). Therefore we have
for any \(t \in [0,T)\). By this and the diagonal argument, we can choose a subsequence such that (60) holds for any \(t \in [0,T)\) and \(k \in {\mathbb {N}}\). On the other hand, for any \(t \in [0,T)\), the compactness of Radon measures yields that there exist \(\mu _t \) and a subsequence \(\varepsilon _i \rightarrow 0\) (depending on t) such that \(\mu _t ^{\varepsilon _i} \rightharpoonup \mu _t\) as Radon measures. However, \(\mu _t\) is uniquely determined by (60). Hence we obtain (59) for any \(t \in [0,T)\). By the diagonal argument with \(T\rightarrow \infty \), we have (59) for any \(t \in [0,\infty )\).
From a similar argument as to that above, there exists monotone increasing functions f and g such that \(\mu _t (\Omega ) = f(t) - g(t)\). By the monotonicity, there exists a countable set B such that f and g are continuous on \([0,\infty ) \setminus B\). This concludes the proof. \(\square \)
Proposition 10
Let B be the countable set given by Proposition 9. For any \(t,s \in [0,\infty ) {\setminus } B\) with \(t<s\), we have \(\mu _s (\Omega ) \le \mu _t (\Omega )\).
Proof
From Proposition 9, we may assume that \(\mu _t ^{\varepsilon _i} (\Omega ) \rightarrow \mu _t (\Omega )\) for any \(t \in [0,\infty )\). We recall that \(E_S ^{\varepsilon _i}\) and \(E_P ^{\varepsilon _i}\) are energies defined by (7). By (8) and \(E ^{\varepsilon _i} (t) \le E_S ^{\varepsilon _i} (0) \le D_1\), Helly’s selection theorem yields that there exist a subsequence \(\varepsilon _i \rightarrow 0\) (denoted by the same index) and a monotone decreasing function E(t) such that \(E^{\varepsilon _i} (t) \rightarrow E (t)\) for any \(t \in [0,\infty )\). For any \(T>0\), the estimate (38) and Fatou’s lemma imply
Therefore \(\liminf _{i\rightarrow \infty } E_P ^{\varepsilon _i} (t)=0\) a.e. \(t \ge 0\) and hence \( E(t) = \sigma \mu _t (\Omega ) \) for a.e. \(t\ge 0\). By this, the monotonicity of E(t), and the continuity of \(\mu _t (\Omega ) \) on \([0,\infty ) {\setminus } B\), we obtain the claim. \(\square \)
We define a Radon measure \(\mu \) on \(\Omega \times [0,\infty )\) by \(d\mu :=d\mu _t dt\). By the boundedness of \(\sup _i \mu _t ^{\varepsilon _i} (\Omega )\), the dominated convergence theorem implies
For our measures \(\mu \) and \(\mu _t\), we have the following property:
Proposition 11
There exists a countable set \({\tilde{B}} \subset [0,\infty )\) such that
for any \(t \in (0,\infty ) \setminus {\tilde{B}}\).
Proof
Let \(f_k\) and \(g_k\) be monotone increase functions given by Proposition 9. Then there exists a countable set \({\tilde{B}}\) such that \(f_k\) and \(g_k\) are continuous on \([0,\infty ) {\setminus } {\tilde{B}}\) for any k. Suppose that there exists \(t_0 \in [0,\infty ) {\setminus } {\tilde{B}}\) such that \(x \in \textrm{spt}\,\mu _{t_0}\) and \((x,t_0) \not \in \textrm{spt}\,\mu \). Then we may assume that there exists k such that \(x \in \textrm{spt}\,\phi _k \) and \(\mu (\phi _k \times (t_0 -\delta , t_0 +\delta ))=0\) for sufficiently small \(\delta >0\), where \(\phi _k\) is a function given by Proposition 9. From \(x \in \textrm{spt}\,\mu _{t_0}\), \(\mu _{t_0 } (\phi _k)>0\) and there exists \(\delta ' >0\) such that \( \mu _{t } (\phi _k)>0\) for any \(t \in (t_0 -\delta ', t_0 +\delta ')\) by the continuity of \(f_k\) and \(g_k\). However, this contradicts \(\mu (\phi _k \times (t_0 -\delta , t_0 +\delta ))=0\). Therefore we obtain (61) for \(t \in [0,\infty ) {\setminus } {\tilde{B}}\). \(\square \)
3.4 Integrality of \(\mu _t\) for \(d \le 3\)
In the case of \(d\le 3\), we can use the results of [38]. For \(d\ge 4\), we employ the arguments of [23, 32, 45] in Section 4 below.
Theorem 12
Assume that \(d=2\) or 3 and (59). Then \(\mu _t\) is integral for a.e. \(t \ge 0\).
Proof
The estimates (35), (36), and (38) imply
for any \(T>0\). Then Fatou’s lemma yields
Therefore
By this, \(2\le d \le 3\), and (36), \(\mu _t\) is integral for a.e. \(t \ge 0\) (see [38, Theorem 5.1]). \(\square \)
4 Rectifiability and Integrality of \(\mu _t\)
We already proved the rectifiability and integrality of \(\mu _t\) with \(d\le 3\) in Theorem 12. Next we consider the case of \(d\ge 2\) and basically follow [23, 32, 45].
4.1 Assumptions
We assume (24) and (28–30) again in this section. Let \(\{ \varepsilon _i \}_{i=1} ^\infty \) be a positive sequence such that \(\varepsilon _i \rightarrow 0 \) as \(i \rightarrow \infty \). By the weak compactness of the Radon measures and Proposition 9, we may assume that there exist Radon measures \(\mu \), \(\vert \xi \vert \) and a family of Radon measures \(\{ \mu _t\} _{t \in [0,T)}\) such that
for \(\phi \in C_c (\Omega \times (0,T))\), and
Remark 12
In the discussion above, we proved that there exists \(\mu _t = \lim _{\varepsilon \rightarrow 0} \mu _t ^\varepsilon \) for any \(t\ge 0\), however such a property does not necessarily hold for \(\xi _t ^\varepsilon \), because we do not know anything about the monotonicity of \(\xi _t ^\varepsilon \) (which was the key to the argument for \(\mu _t ^\varepsilon \)).
By the standard PDE theories and the rescaling arguments, we obtain the following lemma. The proof is almost the same as [45, Lemma 4.1]. So, we skip this.
Lemma 4
There exists \(C_8>0\) depending only on d and \(C_1\) such that
for any \(\varepsilon \in (0,1)\).
4.2 Vanishing of \(\xi \)
First we show \(\vert \xi \vert =0\) for any \(d \ge 2\).
Lemma 5
Assume \((x',t') \in \textrm{spt}\,\mu \) and \(\alpha _1 \in (0,1)\). Then there exist a sequence \(\{ (x_j,t_j) \}_{j=1} ^\infty \) and a subsequence \(\{ \varepsilon _{i_j} \}_{j=1} ^\infty \) such that \(\vert (x_j,t_j)-(x', t') \vert < \frac{1}{j}\) and \(\vert \varphi ^{\varepsilon _{i_j}} (x_j,t_j) \vert <\alpha _1 \) for all j.
Proof
Define \(Q_r = \overline{ B_r (x') \times (t' -r, t' +r)}\) for \(r>0\). If the claim is not true, then there are \(r>0\) and \(N>1\) such that \(\inf _{Q_r} \vert \varphi ^{\varepsilon _i } \vert \ge \alpha _1 \) for any \(i >N\). Without loss of generality, we may assume that \(\inf _{Q_r} \varphi ^{\varepsilon _i } \ge \alpha _1 \) for any \(i >N\). For \(s \in [\alpha _1,1)\), we have \(W(s) = \frac{1}{4\,s}W'(s) (s^2 -1) \le \frac{1}{4\alpha _1 }W'(s) (s^2 -1)\). Note that \( W' (s) (s^2 -1) \ge 0\) for \(s \in [\alpha _1,1)\). Assume that \(\phi \in C_c ^\infty (B_r (x'))\) satisfies \(0 \le \phi \le 1\) and \(\phi =1 \) on \(B_{r/2} (x')\). We compute
Now we estimate the three terms on the right hand side above. We compute
where \(C>0\) depends only on r. Here we used \(\Vert \varphi ^\varepsilon \Vert _{L^\infty } \le 1\) and \(0\le \phi \le 1\). By \(\inf _{Q_r} \varphi ^{\varepsilon _i } \ge \alpha _1 \), integration by parts, and Young’s inequality,
where \(C>0\) depends only on \(\alpha _1\), r, and \(\Vert \nabla \phi \Vert _{L^\infty }\). Here we used \(0\le \phi \le 1\). By (36), (38), and \(\sqrt{2W (\varphi ^\varepsilon )} ( (\varphi ^\varepsilon )^2 - 1 ) =-2 W (\varphi ^\varepsilon ) \),
Therefore there exists \(C>0\) depending only on \(\alpha _1\), r, \(C_3\), \(\Vert \nabla \phi \Vert _{L^\infty }\), and \(D_1\) such that
By (33), \(\mu _t ^\varepsilon (B_{r/2} (x')) \le 2\sigma ^{-1} \int _{B_{r/2} (x')} \frac{W}{\varepsilon } \, \textrm{d}x\). Thus
where \(C>0\) depends only on \(\alpha _1\), r, \(C_3\), \(\Vert \nabla \phi \Vert _{L^\infty }\), and \(D_1\). However, this implies \((x',t') \not \in \textrm{spt}\,\mu \). This is a contradiction. \(\square \)
Set
Note that \(\rho _{(y,s)}(x,t)=\rho _y ^r (x)=\rho _x ^r (y)\) for \(r=\sqrt{2(s-t)}\).
Lemma 6
There exist \(\gamma _1, \eta _1, \eta _2 \in (0,1)\) depending only on d, W, T, \(D_2\), and \(C_3\) such that the following hold. For \(t,s \in [0,T/2)\) with \(0<s-t\le \eta _1\), we denote \(r = \sqrt{2(s-t)}\) and \(t' = s+ r^2/2\). If \(x \in \Omega \) satisfies
then \((B_{\gamma _1 r} (x) \times \{ t' \}) \cap \textrm{spt}\,\mu = \emptyset \).
Proof
First we remark that \(0\le t< s<t' <T\), \(s=\frac{t+t'}{2}\), and \(r=\sqrt{2(s-t)}=\sqrt{2(t'-s)}\). Assume that \(x \in \Omega \) satisfies (65), \((x',t') \in \textrm{spt}\,\mu \), and \(x' \in B_{\gamma _1 r} (x)\). We choose \(\gamma _1\), \(\eta _1\), and \(\eta _2\) later. Let \(\alpha _1 \in (0,1)\) be a constant. By Lemma 5, there exist a sequence \(\{ (x_j,t_j) \}_{j=1} ^\infty \) and a subsequence \(\varepsilon _j \rightarrow 0\) such that \(\lim _{j\rightarrow \infty } (x_j,t_j) = (x',t' )\) and \( \vert \varphi ^{\varepsilon _j} (x_j,t_j) \vert <\alpha _1\) for all j. Then we may assume that for \(\alpha ' = (\alpha _1 + 1)/2 >\alpha _1\), there exists \(\gamma _2 =\gamma _2 (W,\alpha _1) >0\) such that \( \frac{W(\varphi ^{\varepsilon _j} (y,t_j) )}{\varepsilon _j} \ge \frac{W(\alpha ')}{\varepsilon _j} \) for any j and for any \(y \in B_{\gamma _2 \varepsilon _j } (x_j)\), because \(W(\alpha _1) >W(\alpha ')\) and
for any \(y \in B_{\gamma _2 \varepsilon _j } (x_j)\), where we used (33). Thus, there exists \(\eta _3 = \eta _3 (d, \gamma _2)>0\) such that
Here we used
where \(C_9>0\) depends only on d and \(\gamma _2\). By the monotonicity formula, we have
Choose \(\eta _2 = \eta _2 (d,\gamma _2, T, C_3) >0\) such that
and letting \(j\rightarrow \infty \), we have
Changing the center of the backward heat kernel by using (153), we have
when \(\vert x-x' \vert \le \gamma _1 r\). Here \(\gamma _1\) depends only on \(\eta _2\) and \(D_2\). This is a contradiction to (65). Therefore \((x', t') \not \in \textrm{spt}\,\mu \). \(\square \)
We can also show the following using the estimate (66):
Lemma 7
There exists \(C_{10} \le 0\) depending only on d, T, \(C_3\), and \(D_2\) such that
for any \(t \in (0,T) \setminus {\tilde{B}}\) and for any open set \(U \subset \Omega \), where \({\tilde{B}}\) is the countable set given by Proposition 11.
Proof
We only need to prove (67) for any compact set \(K \subset U\). Let \(X_t:= \{ x \in K \mid (x,t) \in \textrm{spt}\,\mu \}\) with \(t \in (0,T) {\setminus } {\tilde{B}}\). For any \(x \in X_t\), by (66), we have
for sufficiently small \(r>0\). By (154), we deduce that
for any \(L>0\). Therefore for sufficiently large \(L>0\) depending only on d, \(\gamma _2\), T, \(C_3\), and \(D_2\), we have
where we used \(\rho _{(x,t)} (y,t-r^2) \le (4\pi )^{-\frac{d-1}{2}} r^{1-d} \). Hence there exists \(C_{11} >0\) depending only on d, \(\gamma _2\), T, \(C_3\), and \(D_2\) such that
holds for any sufficiently small \(r >0\). Set \({\mathcal {B}}:=\{ {\overline{B}} _{Lr} (x) \subset U \mid x \in X_t \}\). By the Besicovitch covering theorem, there exists a finite sub-collection \({\mathcal {B}}_1\), \({\mathcal {B}}_2\), ..., \({\mathcal {B}}_{N(d)}\) such that each \({\mathcal {B}}_i\) is a family of the disjoint closed balls and
Let \({\mathscr {H}}^{d-1} _\delta \) be defined in [11, Chapter 2]. Note that \({\mathscr {H}}^{d-1} = \lim _{\delta \downarrow 0} {\mathscr {H}}^{d-1} _\delta \). By (68) and (69), we compute
Letting \(r\downarrow 0\), we have \({\mathscr {H}}^{d-1} (X_t) \le N(d) L^{d-1} C_{11} \liminf _{r\downarrow 0} \mu _{t-r^2} (U)\). By this and (61), we obtain (67). \(\square \)
By Lemma 6, we obtain
Lemma 8
(see [23, 32]) For \(T \in [1,\infty )\), let \(\eta _2\) be a constant as in Lemma 6. Set
Then \(\mu (Z_T) =0\) holds.
Proof
Let \(\eta _1\), \(\eta _2\), and \(\gamma _1\) be constants as in Lemma 6. For \(\tau \in (0, \eta _1)\), we denote
Let \(\{ \tau _m \}_{m=1} ^\infty \) be a positive sequence with \(\tau _m \rightarrow 0\) as \(m \rightarrow \infty \). Then \(Z_T \subset \cup _{m=1} ^{\infty } Z ^{\tau _m}\). Therefore we need only show \(\mu (Z ^\tau ) =0\) for any \(\tau \in (0,\eta _1)\). Set
We now show that if \((x,t) \in Z^\tau \), then
Assume that \((x', t') \in P_\tau (x,t) \cap Z^\tau \) for a contradiction. First we consider the case of \(t' >t\). Set \(s=\frac{t'+t}{2}\) and \(r=\sqrt{t' -t} = \sqrt{2(s-t)}\). Since \((x,t) \in Z^\tau \),
Therefore Lemma 6 yields \((x', t') \not \in \textrm{spt}\,\mu \), because \(x' \in B_{\gamma _1 r} (x)\) by the definition of \(P_\tau (x,t)\). This yields a contradiction. In the case of \(t' <t\), we can show \((x,t) \not \in \textrm{spt}\,\mu \) similarly. This is a contradiction. Therefore (70) holds.
For \((x_0, t_0) \in \Omega \times [\tau /2, T/2]\), we denote
We can choose a countable set \(\{ (x_j, t_j ) \}_{j=1} ^\infty \) such that \( Z^\tau \subset \cup _{j=1} ^\infty Z^{\tau ,x_j,t_j} \). Thus we only need to prove \(\mu (Z^{\tau ,x_0,t_0}) =0\). Let \(P: {\mathbb {R}}^{d+1} \rightarrow {\mathbb {R}}^d\) be a projection such that \(P(x,t)=x\). For \(\rho \in (0,1)\) and \(r \le \rho \), let \(\{{\overline{B}}_{r/5} (x_\lambda ) \}_{\lambda \in \Lambda } \) be a covering of \(P(Z^{\tau ,x_0,t_0}) \subset B_{\frac{\gamma _1}{2} \sqrt{\tau }} (x_0)\). Then, we may choose a countable covering \({\mathcal {F}}=\{ {\overline{B}}_{r} (x_i) \}_{i=1} ^\infty \) of \(P(Z^{\tau ,x_0,t_0})\) with \((x_i, t_i) \in Z^{\tau ,x_0,t_0}\) for some \(t_i\), by Vitali’s covering theorem. Let A be a set of centers of all balls in \(\{ {\overline{B}}_{r} (x_i) \}_{i=1} ^\infty \). Then, by Besicovitch’s covering theorem, there exist N(d) and subcollections \({\mathcal {F}} _1\), \({\mathcal {F}} _2\), ..., \({\mathcal {F}} _{N(d)} \subset {\mathcal {F}}\) of disjoint balls such that
Note that \({\mathcal {F}}_k\) is finite (\({\mathcal {F}} _k = \{ B_{k,1}, \dots ,B_{k,n_k} \}\)) and
since each balls in \({\mathcal {F}} _k\) are disjoint and \(B_{k,i} \subset {\overline{B}}_{\frac{\gamma _1}{2} \sqrt{\tau } +\rho } (x_0)\). Therefore
If \((x,t) \in Z^{\tau ,x_0,t_0}\), then there exists \(B_{k,i} = {\overline{B}}_r (x_{k,i}) \in {\mathcal {F}}_k\) for some k and i such that \(x \in {\overline{B}}_{2r} (x_{k,i})\) and \( \vert t_{k,i} -t \vert \le \gamma _1 ^{-1} \vert x_{k,i} -x \vert ^2 \le 4 \gamma _1 ^{-1} r ^2\) by (70) and (71) (note that we should change the radius because A is not a covering of \(Z^{\tau ,x_0,t_0}\)). Hence, we have
By this, (52), and (72) we obtain
Letting \(\rho \rightarrow 0\), we have \(\mu (Z^{\tau ,x_0,t_0})=0\). Thus \(\mu (Z_T) =0\) holds. \(\square \)
Theorem 13
(see [23]) We see that \(\vert \xi \vert =0\) and \(\lim _{i \rightarrow \infty } \vert \xi _t ^{\varepsilon _i} \vert (\Omega )=0\) for a.e. \(t\in [0,T)\).
Proof
First we show that
for some \(C>0\). By (33) and (51), integrating (50) on \((0, s-\delta )\) with \(\delta >0\), we obtain
Letting \(\delta \rightarrow 0\) and \(\varepsilon \rightarrow 0\), we obtain (73). Next, integrating (73) on \(\Omega \times (0,T)\) by \(d\mu _s ds\) we have
where we used Fubini’s theorem. Then this boundedness implies
Next we claim
Define \(\beta := \log (s-t)\) and
Assume that (x, t) satisfies (74). Then
Let \(\theta \in (0,1]\) and \(\{ \beta _i \} _{i=1} ^\infty \) be a negative monotone decreasing sequence such that
For any \(\beta \in (-\infty , \beta _1)\), choose i such that \(\beta \in [\beta _{i},\beta _{i-1})\) holds. One can check that
for \(M>0\), where \(r=\sqrt{2(2 e ^{\beta } -e ^{\beta _i})}\). We compute
where we used (51), (154), and
Thus, for any \(\delta >0\), we can choose \(\theta \in (0,1]\) and \(M>0\) such that \(h (t + e^{\beta }) \le \delta \) for any \(\beta < \beta _1\). This proves (75). Set
Then \(\Omega \times (0,T)= A\cup B\) and \(\vert \xi \vert (B)=0\) by (75). Moreover, Lemma 8 and (154) imply \(\mu (A) =0\) and thus \(\vert \xi \vert (A) =0\), because \(\vert \xi \vert \) is absolute continuous with respect to \(\mu \). Therefore \(\vert \xi \vert (\Omega \times (0,T)) =0\). The rest of the claim can be shown from the dominated convergence theorem. \(\square \)
4.3 Rectifiability
Next we show the rectifiability of \(\mu _t\).
Definition 2
For \(\phi \in C_c (G_{d-1} (\Omega ))\), we define \(V_t ^\varepsilon \in {\mathbb {V}} _{d-1} (\Omega )\) by
Here, \(\varphi ^\varepsilon \) is a solution to (5).
Note that the first variation of \(V_t ^\varepsilon \) is given by
for \(\vec {\phi } \in C_c ^1 (\Omega ; {\mathbb {R}}^d)\). By integration by parts, we have
Note that the second and third terms of the right hand side converges to 0 for a.e. \(t \in [ 0,T)\) by Theorem 13. By (35) and (38), we have
for some \(C>0\) (see the proof of Theorem 12). Thus Fatou’s lemma implies
for a.e. \(t \in [ 0, T)\). Hence, by (80), (81), and the Cauchy–Schwarz inequality, we have
for a.e. \(t \in [ 0, T)\) and for any \(\vec {\phi } \in C_c ^1 (\Omega ;{\mathbb {R}}^d)\) with \(\sup \vert \vec {\phi } \vert \le 1\). Let \(t \in [ 0, T) {\setminus } {\tilde{B}}\) satisfy (82), where \({\tilde{B}}\) is given by Proposition 11. Taking a subsequence \(i_j \rightarrow \infty \) (note that the subsequence depends on t), there exists a varifold \(V_t\) such that \(V_t ^{\varepsilon _{i_j}} \rightharpoonup V_t\) as Radon measures and \(\delta V_t\) is a Radon measure by (82). In addition, Proposition 11, Lemma 7, and the standard measure theoretic argument imply
Therefore Allard’s rectifiability theorem yields the following theorem.
Theorem 14
For a.e. \(t \ge 0\), \(\mu _t\) is rectifiable. In addition, for a.e. \(t \ge 0\), \(\mu _t\) has a generalized mean curvature vector \(\vec {h} (\cdot ,t)\) with
and
for any \(\phi \in C_c (\Omega ;[0,\infty ))\) and \(\vec \phi \in C_c (\Omega ;{\mathbb {R}}^d)\).
Detailed proof of this is in [23, 45], so we omit it (however, the essential part has already been discussed above). Note that (59) and \(\mu _t =\Vert V_t\Vert \) imply that \(V_t\) does not depend on the choice of subsequence \(\{V_t ^{\varepsilon _{i_j}} \}_{j=1} ^\infty \) above.
4.4 Integrality
To prove the integrality, we mainly follow [22, 45, 47]. The propositions that are directly applicable to our problem are in Appendix for readers’ convenience. Let \(\{ r _i\} _{i=1} ^\infty \) be a positive sequence with \(r _i \rightarrow 0\) and \(\frac{\varepsilon _i}{r_i} \rightarrow 0\) as \(i \rightarrow \infty \). Set \(u ^{{\tilde{\varepsilon }}} ({\tilde{x}}, {\tilde{t}}) = \varphi ^\varepsilon (x,t)\) and \(g ^{{\tilde{\varepsilon }}} ({\tilde{t}}) = r \lambda ^\varepsilon (t)\) for \({\tilde{x}}= \frac{x}{r}\), \({\tilde{t}}= \frac{t}{r^2}\), and \({\tilde{\varepsilon }} = \frac{\varepsilon }{r}\). Then, \(u ^{{\tilde{\varepsilon }}}\) is a solution to
We remark that the monotonicity formula (51) and the upper bound of the density (52) hold for \(d\tilde{\mu }_{{\tilde{t}}} ^{{\tilde{\varepsilon }}} ({\tilde{x}})= \sigma ^{-1} (\frac{{\tilde{\varepsilon }} \vert \nabla _{{\tilde{x}}} u ^{\tilde{\varepsilon }} \vert ^2}{2} + \frac{W(u ^{{\tilde{\varepsilon }}})}{\tilde{\varepsilon }} )\, d{\tilde{x}}\), because the value
is invariant under this rescaling, and for any \(s>0\) we have
by (52). We subsequently drop \({\tilde{\cdot }}\) for simplicity. First we consider the energy estimate on \(\{x \in B_1 (0) \mid \vert u ^\varepsilon (x,t) \vert \ge 1 -b \}\).
Proposition 15
(See [47]) For any \(s >0\) and \(a \in (0,T)\), there exist positive constants b and \(\varepsilon _2\) depending only on \(D_1\), \(D_2\), \(C_3\), a, \(\alpha \), and s such that
for all \(t \in (a, T)\) whenever \(\varepsilon \in (0,\varepsilon _2)\).
To prove Proposition 15, we prepare following two lemmas:
Lemma 9
(See [47]) For any \(\delta \in (0,T)\), there exist positive constants \(C_{12}\) and \(\varepsilon _3\) depending only on d, \(\delta \), \(\alpha \), and \(C_1\) with the following property. assume that there exist \((x_0,t_0) \in B_1 (0) \times (\delta ,T)\) and \(\gamma \in (0,\frac{2}{3}]\) such that
and
Then
for any \(\varepsilon \in (0,\varepsilon _3)\).
Proof
We may assume that \(B_{\varepsilon {\tilde{r}}} (x_0) \times (t_0 -\varepsilon ^2 {\tilde{r}}^2, t_0) \subset B_2 (0) \times (0,T)\) by (85). We consider the rescaling of (83) by \({\tilde{x}}= \frac{x-x_0}{\varepsilon }\) and \({\tilde{t}}= \frac{t-t_0}{\varepsilon ^2}\). Then we obtain
where \({\tilde{u}} ^{\varepsilon } ({\tilde{x}},{\tilde{t}}) = u ^\varepsilon (x,t)\) and \({\tilde{g}} ^{\varepsilon } ({\tilde{t}}) = g ^\varepsilon (t)\). Note that (12) and \({\mathscr {L}}^d (\Omega )=1\) yield
for \(\alpha \in (0,1)\). Let \(\psi \) be a function with
for some constant \(C_{13}>0\). For example, \(\psi = e^{-\frac{{\tilde{t}}}{100} -1} e^{\frac{1}{100d}\sqrt{1+\vert {\tilde{x}} \vert ^2}}\) satisfies (88). Set \({\tilde{r}}:= C_{13} \gamma \vert \log \varepsilon \vert \). We may assume that \({\tilde{r}}\ge 1\) for sufficiently small \(\varepsilon \). Note that
The assumption (84) is equivalent to
For a contradiction, we assume that
Set \(\phi ^\varepsilon := 1 - \varepsilon ^\gamma \psi \). Then (88) and (90) imply
and
Moreover, by \({\tilde{r}} \ge 1\),
Therefore
by (89) and (91). We consider a function \(w = \phi ^\varepsilon - {\tilde{u}}^\varepsilon \) on \(B_{{\tilde{r}}} (0) \times (-{\tilde{r}} ^2,0)\). By (92) and (93), w attains its positive maximum at an interior point \((x',t') \in B_{{\tilde{r}}} (0) \times (-{\tilde{r}} ^2,0)\), and hence \(\partial _{{\tilde{t}}} w - \Delta _{{\tilde{x}}} w \ge 0\) and \(w >0\) at \((x',t')\). At \((x',t')\), we compute that
for sufficiently small \(\varepsilon \), where we used (87) and \(1>\phi ^\varepsilon > {\tilde{u}} ^\varepsilon \ge \frac{1}{2}\) at \((x',t')\). This is a contradiction. The other case can be proved similarly. \(\square \)
Lemma 10
(See [47]) For any \(\delta \in (0,T)\), there exist positive constants \(C_{14}\) and \(\varepsilon _4\) depending only on \(\delta \), \(\alpha \), d, \(C_3\), and \(D_2\) such that the following holds. For \(t \in (\delta , T)\) and \(r \in (0,\frac{1}{2})\), set
Then for any \(\varepsilon \in (0,\varepsilon _4)\), we have
Proof
First we claim that there exist some constants \(\varepsilon _4\), \(C_{15}\), and \(C_{16}\) such that if \(x_0 \in Z_{r, t_0}\) and \(\varepsilon \in (0,\varepsilon _4)\) then
holds for any \(r \in [\varepsilon ,\frac{1}{2})\). We may assume that \((x_1, t_1) \in B_r (x_0) \times (t_0 -r^2, t_0)\) with \(\vert u ^\varepsilon (x_1,t_1) \vert <\frac{1}{2}\). By the monotonicity formula (51), for any \(\varepsilon \in (0,\varepsilon _1)\) we have
By \(\vert u ^\varepsilon (x_1,t_1)\vert <\frac{1}{2}\), repeating the proof of Lemma 6, there exists \( \eta =\eta (\alpha ,d) >0\) such that
Then (96), (97), and (154) imply
where \(\eta ' =\eta '(\alpha ,d,C_3) >0\). By \(\vert t_1 -t_0 \vert <r^2\) and \(\varepsilon \le r\), we have \(e^{-\frac{3R^2}{16(t_1 +\varepsilon ^2 -t_0 +2 r^2)}} \le e^{-\frac{3R^2}{64 r^2}}\). Thus there exists \(\gamma >0\) depending only on \(\alpha ,d,C_3\), and \(D_2\) such that
Note that since \(t_1 +\varepsilon ^2 - (t_0 -2r^2) \ge 2r^2\) there exists \(C_{17}>0\) depending only on d such that
Hence we obtain (95) for some \(C_{15}\) and \(C_{16}\).
Finally we prove (94). The inequality (95) yields that there exists \(C_{17}>0\) depending only on \(\alpha ,d,C_3\), and \(D_2\) such that
for any \(x _0 \in Z_{r,t_0}\) and \(r \in [\varepsilon ,\frac{1}{2})\). Set \({\tilde{r}}:= C_{15} r\). By an argument similar to that in the proof of Lemma 8, there exist \({\mathcal {F}}_1,\dots ,{\mathcal {F}} _{N(d)}\) such that N(d) depends only on d, \({\mathcal {F}} _k= \{ {\overline{B}}_{{\tilde{r}}} (x_{k,1}), \dots , {\overline{B}}_{{\tilde{r}}} (x_{k,n_k}) \}\) is a family of disjoint closed balls for any k, and
Therefore
where we used (52), (98), and the property that \({\mathcal {F}} _k\) is a family of disjoint balls. Hence, we obtain (94). \(\square \)
Proof of Proposition 15
First, we restrict \(b\in (0,1)\) to be small enough so that
and restrict \(\varepsilon \) to be small enough to use Lemmas 9 and 10. We choose a positive integer J such that
Then, (85), (99), and (100) imply
Set \(t_0 \in (\delta ,T)\) and
For \(x_0 \in A_j\), we use Lemma 9 with \(\gamma =\frac{1}{2^j}\). Note that (85) holds with \({\tilde{r}} = \frac{1}{2^j} C_{12} \vert \log \varepsilon \vert \) by (101). Then we obtain
and hence
By (85), we have \(\varepsilon \le \varepsilon {\tilde{r}} <\frac{1}{2}\) for sufficiently small \(\varepsilon \). Therefore (94) and (102) yield
for any \(j=1,\dots ,J\). On the other hand, since \(\vert u ^\varepsilon (x,t_0) \vert \ge 1- \varepsilon ^{\frac{1}{2^{j+1}}}\) for any \(x \in A_j\), we obtain
for some constant \(C_{18}\) depending only on W. We define \( Y:= \{ x \in B_1(0) \mid 1-b \le \vert u ^\varepsilon (x,t_0) \vert \le 1-\sqrt{\varepsilon } \} \). Note that
by (100). Set \(p(t) = 2^{-t} \varepsilon ^{2^{-t}}\). Then p satisfies
because \(2^{-J-1} \log \varepsilon \le \log \sqrt{b} \le -1\) by (99) and (100). Set \(C_{19} = C_{12} C_{14} C_{18}\). Then from (100), (103), (104), (105), and (106) we have
Using the same argument above, we can show that
where we used Lemma 9 with \(\gamma = \frac{2}{3}\). Since \(\vert u ^\varepsilon \vert \le 1\), we have
By (107), (108), and (109), Proposition 15 holds for sufficiently small b and \(\varepsilon \). \(\square \)
Now we prove the integrality of \(\mu _t\).
Theorem 16
For a.e. \(t>0\), there exist a countably \((d-1)\)-rectifiable set \(M_t\) and \({\mathscr {H}}^{d-1}\)-measurable function \(\theta _t: M_t \rightarrow {\mathbb {N}}\) with \(\theta _t \in L_{loc} ^{1} ({\mathscr {H}}^{d-1} \lfloor _{M_t})\) such that \(\mu _t = \theta _t {\mathscr {H}}^{d-1} \lfloor _{M_t}\) holds.
Proof
Set \(H ^\varepsilon := \Delta \varphi ^\varepsilon -\frac{W' (\varphi ^\varepsilon )}{\varepsilon ^2}\). Then for a.e. \(t _0 > 0\), we can choose a subsequence \(\{ V_{t_0} ^{\varepsilon _{i_j}} \} _{j=1} ^\infty \) such that \(V_{t_0} ^{\varepsilon _{i_j}} \rightharpoonup V_{t_0}\),
and
hold by Theorem 13 and (81). Note that \(V_{t_0}\) is a countably \((d-1)\)-rectifiable varifold and determined by \(\mu _{t_0}\) uniquely from Theorem 14. We fix such \(t_0 >0\) and show the claim for \(\mu _{t_0}\). In this proof, even if we take a subsequence \(\varepsilon _{i_j}\), we always abbreviate \(\varepsilon _{i_j}\) by \(\varepsilon _i\) for simplicity. Set
and
for any \(m \in {\mathbb {N}}\). Then the Besicovitch covering theorem implies
where \(c(d) >0\) is a constant depending only on d. Set \( A:= \cup _{m=1} ^\infty A_m. \) Next we prove
If (113) is not true, there exists a compact set \(K \subset \Omega {\setminus } A\) with \(\mu _{t_0} (K)>\frac{1}{2} \mu _{t_0} (\Omega {\setminus } A) >0\). Since \(A_1 \subset A_2 \subset A_3 \subset \cdots \), we have \(K \subset \Omega {\setminus } A_m\) for any \(m \in {\mathbb {N}}\). For any \(x \in K\), there is a neighborhood \(B_r (x)\) such that \(B_r (x) \cap A_{i,m} =\emptyset \) for sufficiently large i, by the definition of \(A_m\). This and the compactness of K imply that there exist an open set \(O_m\) and \(i _0 \in {\mathbb {N}}\) such that \(K \subset O_m\) and \(O_m \cap A_{i,m} =\emptyset \) for any \(i \ge i_0\). Let \(\phi _m \in C_c (O_m)\) be a non-negative test function such that \(0\le \phi _m \le 1\) and \(\phi _m =1\) on K. We compute
for any \(k \ge i_0\). Combining (112) and (114), we obtain \(\mu _{t_0} (K) =0\). Therefore we have proved (113).
By the rectifiability of \(\mu _{t_0}\) and (113), for \(\mu _{t_0}\) a.e. \(x \in \textrm{spt}\,\mu _{t_0}\), it has an approximate tangent space P and \(x \in A_m\) for some m. Fix such x. We may assume that \(x=0\) and \(P= \{ x \in {\mathbb {R}}^d \mid x_d=0 \}\) by a parallel translation and a rotation. Set \(\theta := \lim _{r \downarrow 0} \frac{\mu _{t_0} (B_r (0))}{\omega _{d-1} r^{d-1}}\). We only need to prove \(\theta \in {\mathbb {N}}\). Let \(\Phi _{r} (x) = \frac{x}{r}\) for \(r>0\) and \((\Phi _r)_\# V_{t_0}\) be the push-forward of the varifold (see (15)). Then for any positive sequence \(r_i \rightarrow 0\), we have \(\lim _{i\rightarrow \infty } (\Phi _{r_i})_\# V_{t_0} = \theta \vert P \vert \), where \(\vert P \vert \) is the unit density varifold generated by P. By the assumption \(0 \in A_m\), there exists \(\{ x_i \}_{i=1} ^\infty \) such that \(x_i \in A_{i,m}\) and \(x_i \rightarrow 0\) as \(i\rightarrow \infty \). Passing to a subsequence if necessary, we may assume that
and
Set \(u ^{{\tilde{\varepsilon }}_i } ({\tilde{x}}, {\tilde{t}}) = \varphi ^{\varepsilon _i} (x,t)\) and \(g ^{{\tilde{\varepsilon }}_i } ({\tilde{t}}) = r_i \lambda ^{\varepsilon _i} (t)\) for \({\tilde{x}}= \frac{x}{r_i}\), \({\tilde{t}}= \frac{t-t_0}{r^2 _i}\), and \({\tilde{\varepsilon }}_i = \frac{\varepsilon _i}{r_i}\) (another functions \({\tilde{\xi }} _{\tilde{\varepsilon }_i}\) and \({\tilde{H}} ^{{\tilde{\varepsilon }} _i}\) are defined in the same way). Note that \({\tilde{x}}_i:= \frac{x_i}{r_i} \rightarrow 0\) and \({\tilde{\varepsilon }} _i \rightarrow 0\) by (115) and \(u ^{\tilde{\varepsilon }_i }\) is a solution to (83) with \(\tilde{\varepsilon }_i\) instead of \({\tilde{\varepsilon }}\). We compute
Thus, by (110) we may assume that
passing to a subsequence if necessary. We compute
where we used (52), (115), and \(x_i \in A_{i,m}\). Let \({\tilde{V}}_{{\tilde{t}}} ^{{\tilde{\varepsilon }} _i} \) be a varifold defined by (79) with \(u ^{{\tilde{\varepsilon }}_i}\) instead of \(\varphi ^\varepsilon \). Then \({\tilde{V}}_0 ^{\tilde{\varepsilon }_i} = (\Phi _{r_i})_\# V_{t_0} ^{\varepsilon _i}\). Next we show
where \(\nu ^i = (\nu _1 ^i,\nu _2 ^i,\dots , \nu _d ^i) = \frac{\nabla _{{\tilde{x}}} u ^{{\tilde{\varepsilon }} _i}}{\vert \nabla _{{\tilde{x}}} u ^{{\tilde{\varepsilon }} _i} \vert }\). For \(S \in {\mathbb {G}}(d,d-1)\), set \(\psi (S):=1- \nu _d ^2\), where \(\nu \in {\mathbb {S}}^{d-1}\) be one of the unit normal vectors to S. Then \(\psi :{\mathbb {G}}(d,d-1) \rightarrow {\mathbb {R}}\) is well-defined, continuous, and \(\psi (P)=0\). Hence, for any \(\phi \in C_c ({\mathbb {R}}^d)\), \(\phi \psi \in C_c (G_{d-1}({\mathbb {R}}^d) )\) and
where we used (116) and \(\psi (P)=0\). Thus (120) proves (119). We subsequently fix the subsequence and drop \({\tilde{\cdot }}\) and time variable (for example, we write \(u^{\tilde{\varepsilon }_i} ({\tilde{x}}, 0)\) as \(u^{\varepsilon _i} \)) for simplicity. We assume that \(N \in {\mathbb {N}}\) is the smallest positive integer grater than \(\theta \), namely,
Let \(s >0\) be an arbitrary number. Then Proposition 15 and (33) imply that there exists \(b >0\) such that
for sufficiently large i. Note that we may use Proposition 15 with \(t=0\) since \({\tilde{t}} = \frac{t -t_0}{r_i ^2}\) in this proof. For these \(s >0\), \(b>0\), and \(c>0\) given by Lemma 4, we choose \(\varrho \) and L given by Propositions 20 and 21 in the Appendix with \(R=2\) (we may restrict \(\varrho \) to be small if necessary). We choose \(a=L\varepsilon _i\) as a constant in Proposition 20. Set
for sufficiently large i. The Besicovitch covering theorem, (117), (118), and (119) yield
Next we show that for sufficiently large i
Note that all the assumptions in Proposition 21 are satisfied by Lemma 4, (33), and (123). Thus we have (125) with \(r=L \varepsilon _i\). By integration by parts, we have
Thus we can compute
where we used (52), (123), and \(\xi _{\varepsilon _i} \le 0\). Therefore we obtain (125) for sufficiently large i by restricting \(\varrho \) to be small. Let \(\delta >0\) and \(\phi \in C_c (B_3 (0))\) be a non-negative test function such that \(\phi =1\) on \(B_2(0) \cap \{ \vert x_d \vert > \delta \}\). Then there exists \(i_0 \ge 1\) such that
since \(\mu _0 ^{\varepsilon _i} =\Vert V_0 ^{\varepsilon _i} \Vert \rightharpoonup \theta {\mathscr {H}}^{d-1} \lfloor _{P}\). Assume that \(x \in G_i \cap \{ \vert x_d \vert > 2 \delta \}\) for \(i \ge i_0\). Then (125) and (126) imply
This is a contradiction. Thus
Set \(Y:=P^{-1} (x) \cap G_i \cap \{ x \mid u ^{\varepsilon _i} (x) =l \}\) for \(x \in P \cap B_1 (0)\). Next we show that for sufficiently large i
For a contradiction, assume that \(\# Y \ge N\) and choose \(y_j \in Y\) for \(j=1,2,\dots , N\). We use Proposition 20 with \(R=1\), \(a=L\varepsilon _i\) and \(Y' =\{ y_j \}_{j=1} ^N\) instead of Y. Note that the smallness of \(\textrm{diam}\,Y'\) is true from (127) and \(\vert y_j -y_k \vert >3 L\varepsilon _i\) for any \(1\le j < k \le N\) holds by (159). Then (158) yields
for sufficiently large i. By (127) and \(\mu _0 ^{\varepsilon _i} =\Vert V_0 ^{\varepsilon _i} \Vert \rightharpoonup \theta {\mathscr {H}}^{d-1} \lfloor _{P}\),
By this, \(\# Y' = N\), (125), and (129) we have
However, this contradicts (121) by restricting s to be small. Thus (128) holds for sufficiently large i.
Finally, we complete the proof. Set \({\hat{V}}_0 ^{\varepsilon _i}:= V_0 ^{\varepsilon _i} \lfloor _{\{ \vert x_d \vert \le 1 \} \times {\mathbb {G}}(d,d-1)}\). We regard P as a diagonal matrix \((p_{jk})\) with \(p_{kk}=1\) for \(1\le k\le d-1\) and \(p_{dd}=0\). Then the push-forward of \({\hat{V}}_0 ^{\varepsilon _i}\) by P is given by
for any \(\phi \in C_c (P \cap B_2 (0) \times {\mathbb {G}}(d,d-1))\). Here \(\vert \Lambda _{d-1} \nabla Px \circ (I -\nu ^i \otimes \nu ^i) \vert \) is the Jacobian and \(\nu ^ i _d = \frac{\partial _{x_d} u ^{\varepsilon _i}}{\vert \nabla u ^{\varepsilon _i} \vert }\). Due to (116), \(P_\# {\hat{V}}_0 ^{\varepsilon _i} \rightharpoonup P_\# (\theta {\mathscr {H}}^{d-1} \lfloor _{P}) =\theta {\mathscr {H}}^{d-1} \lfloor _{P}\) as \(i\rightarrow \infty \). By (117),
holds (see (133) below). We compute
where we used (122), (124), and (130). By the co-area formula and the area formula, we have
where we used (128) and \(\sigma = \int _{-1} ^{1} \sqrt{2W (\tau )} \, d\tau \). Hence \( \theta \le N-1 \) due to (131) and (132) and the arbitrariness of s. By this and (121), \(\theta =N-1\). \(\square \)
5 Proofs of Main Theorems
In this section we prove Theorem 3 and Theorem 4 on the existence of the weak solution in the sense of \(L^2\)-flow and distributional BV-solution.
Proof of Theorem 3
Let \(\{ \varphi ^{\varepsilon _i} _0 \} _{i=1} ^\infty \) be a family of functions such that all the claims of Proposition 2 are satisfied. Then one can check that all the assumptions in Sections 3 and 4 are fulfilled. Therefore (a) holds by Propositions 2, 9, and 10. By taking a subsequence \(\varepsilon _i \rightarrow 0\), we obtain (b) (the proof is standard and is exactly the same as that in [43], so we omit it). By Lemma 1 and the weak compactness of \(L^2 (0,T)\), we may take a subsequence \(\varepsilon _i \rightarrow 0\) such that (c) holds (for the weak convergence for all \(T>0\), we only need to use the diagonal argument).
Next we show (d). We compute
Set \(\textrm{d} {\hat{\mu }} ^\varepsilon := \frac{1}{\sigma }\sqrt{2W(\varphi ^\varepsilon )} \vert \nabla \varphi ^\varepsilon \vert \, \textrm{d}x\textrm{d}t\). By (133), Proposition 9, and Theorem 13, we have
where \(\textrm{d}\mu := \textrm{d}\mu _t \textrm{d}t\). By (35), (38), (133), and (134), we obtain
Then there exist \(\vec {f} \in (L_{loc} ^2 (\mu ))^d\) and the subsequence \(\varepsilon _i \rightarrow 0\) such that
for any \(\vec {\phi } \in C_c (\Omega \times [0,T); {\mathbb {R}}^d)\) (see [21, Theorem 4.4.2]). Moreover, if \(\vec {\phi } \) is smooth, we have
where we used (c), \(k'(s) =\sqrt{2W(s)}\), \(\lim _{i\rightarrow \infty } k(\varphi ^{\varepsilon _i}) = \sigma (\psi -\frac{1}{2}) \) for a.e. (x, t), and the dominated convergence theorem. Hence we have (25).
Now we prove (e). By replacing \(d{{\hat{\mu }}} ^{\varepsilon }\) with \(d {\tilde{\mu }}^\varepsilon := \frac{\varepsilon }{\sigma } \vert \nabla \varphi ^{\varepsilon }\vert ^2 \, dxdt\), the convergence (26) is obtained in the same way as (d). In addition, for any \(\vec {\phi } \in C_c (\Omega \times [0,T); {\mathbb {R}}^d)\), we compute
where we used Theorem 14 and (d). Thus \(\vec {v}=\vec {h} +\vec {f}\). One can check that \(\{ \mu _t \}_{t \in [0,\infty )}\) is an \(L^2\)-flow with the generalized velocity vector \(\vec {v}\) (see [43, Proposition 4.3] for the inequality (16) and [34, Lemma 6.3] for the perpendicularity). \(\square \)
To prove Theorem 4, we use the next proposition and lemmas. In the original proof of the proposition, \(2\le d\le 3\) is assumed to use the results of [38, Proposition 4.9, Theorem 5.1]. however, we already know that \(\vert \xi \vert =0\) and \(\mu _t\) is integral for a.e. t, so we can show the claim in the same way.
Proposition 17
(See Proposition 4.5 of [34]) Let \(\psi \), \(\vec {v}\), and \(\vec {\nu }\) are given by Theorem 3. Then \(\int \vert \vec {v} \cdot \vec {\nu } \vert \textrm{d} \Vert \nabla \psi (\cdot , t)\Vert \textrm{d}t <\infty \) and
for any \(\phi \in C_c ^1 (\Omega \times (0,T))\) and for any \(T>0\).
Proof
Set \(\nabla _{x,t} = (\nabla , \partial _t)\) in the sense of BV. One can check that \(\Vert \nabla _{x,t} \psi \Vert \ll \mu \), \( \mu \lfloor _{\partial ^*\{ \psi =1 \}}\) is rectifiable, \(\int \vert \vec {v} \cdot \vec {\nu } \vert \, d \Vert \nabla \psi (\cdot , t)\Vert dt <\infty \), and the approximate tangent space coincides with that of \(\Vert \nabla _{x,t} \psi \Vert \) for \(\mu \)-a.e. and \(\Vert \nabla _{x,t} \psi \Vert \)-a.e. (see [34, Proposition 8.1–8.3] and [2, Proposition 2.85]). By this and Proposition 1, we have
for any \(\phi \in C_c ^1 (\Omega \times (0,T))\), where \(\vec \nu _{x,t}\) is the inner unit normal vector of \(\{ (x,t) \mid \psi (x,t)=1 \}\). Therefore we obtain (135). \(\square \)
Lemma 11
Let \(\gamma \) and \(\delta \) be positive constants with \(\delta < \gamma \). Under the same assumptions of Theorem 4, there exist \(T_2 \in ( 0,1)\) and \(\varepsilon _5 \in (0,1)\) depending only on \(\gamma \), \(\delta \), and \(C_3(\omega ,d,D' _1)\) with the following property. Let \(g: {\mathbb {R}}\rightarrow [0,\infty )\) be a smooth even function such that \(g(0)=0\), \(0 \le g'' (s) \le 2\) for any \(s \in {\mathbb {R}}\), and \(g(s)=\vert s \vert -\frac{1}{2}\) if \(\vert s \vert \ge 1\). Set \(g^\delta (s):= \delta g(s/\delta )\) and define \(\tilde{r}^{\varepsilon ,\delta } \in C^\infty ({\mathbb {R}}^d \times [0,\infty ))\) by
where \(\lambda ^\varepsilon \) is given by (6). Set \({\tilde{\phi }}^{\varepsilon ,\delta }:= q^\varepsilon (\tilde{r}^{\varepsilon ,\delta } )\) and assume that \({\tilde{\phi }} ^\varepsilon (x,0) \ge \varphi ^\varepsilon _0 (x) \) for any \(x \in {\mathbb {R}}^d\). Then
for any \(\varepsilon \in (0,\varepsilon _5)\).
Proof
We denote \({\tilde{r}} ={\tilde{r}}^{\varepsilon ,\delta }\) for simplicity. By (34) and the comparison principle, we only need to prove
for sufficiently small \(T_2>0\) and \(\varepsilon >0\), since \(\tilde{\phi }^{\varepsilon ,\delta } \ge \varphi ^\varepsilon \) if and only if \({\tilde{r}} \ge r ^\varepsilon \). In the case of \(\vert x_1 \vert \ge \delta \), (137) holds by \(\partial _t {\tilde{r}} = 2\delta ^{-1} + \lambda ^\varepsilon \), \(\vert \nabla {\tilde{r}} \vert =1\), and \(\Delta {\tilde{r}} =0\). Next we consider the case of \(\vert x_1 \vert \le \delta \). Set \(O_\delta :=\{ x \in {\mathbb {R}}^d \mid \vert x_1 \vert \le \delta \}\). By \({\tilde{r}} (x,0) \le -\gamma +\frac{\delta }{2} \) on \(O_\delta \) and
there exists \(T_2 >0\) such that
By (138) and \(\vert \nabla {\tilde{r}} \vert \le 1\), \(\dfrac{2 q ^\varepsilon ({\tilde{r}}) }{\varepsilon } ( \vert \nabla {\tilde{r}} \vert ^2 -1) \ge 0\). By using this, for any \((x,t) \in O_\delta \times [0,T_2)\),
where we used \(\Delta {\tilde{r}} \le \delta ^{-1} g'' (x_1 /\delta ) \le 2 \delta ^{-1}\). Therefore we obtain (137). \(\square \)
Lemma 12
Let \(r \in (0,\frac{1}{4})\). Then there exists \(T_3 >0\) depending only on d and r with the following property. Let \(U_0 \subset \subset (\frac{1}{4},\frac{3}{4})^d\) satisfies \({\mathscr {L}}^d (U_0) ={\mathscr {L}}^d (B_{r} (0))\) and has a \(C^1\) boundary \(M_0\) with (27) for \(\delta _1 >0\). In addition, we assume \({\mathscr {H}}^{d-1} (M_0) \le 2 {\mathscr {H}}^{d-1} (\partial B_{r} (0))\). Then we have
where \(\mu _t\) is a weak solution to (1) with initial data \(M_0\).
Proof
First we claim that there exists \(T_3 >0\) depending only on d and r such that
for any \( t \in [0,T_3)\), where \(\psi = \lim _{i\rightarrow \infty } \psi ^{\varepsilon _i} = \lim _{i\rightarrow \infty } \frac{1}{2} ( \varphi ^{\varepsilon _i} +1)\). Let \({\tilde{\phi }} ^{\varepsilon ,\delta }\) be a function given by Lemma 11 with \(\gamma = \frac{1}{10}\) and \(\delta = \frac{1}{20}\). By (136) and (138), one can check that there exists \(T_3=T_3 (C_3(\omega ,d,D' _1)) >0\) such that \(\lim _{ i\rightarrow \infty }\varphi ^{\varepsilon _i} (x,t) = -1 \) on \(\{ x \in {\mathbb {R}}^d \mid \vert x_1 \vert \le \frac{1}{10} \}\) for any \(t \in [0,T_3)\). Note that \(\omega \) and \(D' _1\) depend only on r by \({\mathscr {L}}^d (U_0) ={\mathscr {L}}^d (B_{r} (0))\) and \({\mathscr {H}}^{d-1} (M_0) \le 2 {\mathscr {H}}^{d-1} (\partial B_{r} (0))\). Hence \(T_3\) depends only on d and r. Therefore \(U_t \cap \{ x \in {\mathbb {R}}^d \mid \vert x_1 \vert \le \frac{1}{10} \} =\emptyset \) for any \(t \in [0,T_3)\). Similarly we have (141). Thus \(\partial ^*(U_t \cap (0,1)^d) = \partial ^*U_t\) for any \(t \in [0,T_3)\). Hence, by using the isoperimetric inequality for Caccioppoli sets (see [13, 46]), and (b3) and (b4) of Theorem 3, we have
By (27) and (142), we obtain (140). \(\square \)
Finally we prove Theorem 4.
Proof of Theorem 4
First we show (a). From (37),
for any \(T>0\) and for any \(\varepsilon \in (0,\varepsilon _1)\). By this and (140), we can choose \(\delta _1 = \delta _1 (C_2(\omega ,d,D' _1)) >0\) so that
where \(T_4 = T_4 (d,r) = \min \{ T_3, \frac{2}{ C_2 } \log \frac{100}{99} \} >0\) and \(\delta _1\) also depends only on d and r since \({\mathscr {L}}^d (U_0) ={\mathscr {L}}^d (B_{r} (0))\) and \({\mathscr {H}}^{d-1} (M_0) \le 2 {\mathscr {H}}^{d-1} (\partial B_{r} (0))\). Then (51) and (143) imply
for any \(y \in {\mathbb {R}}^d\), \(t\in [0,T_4)\), and \(s>0\) with \(0\le t <s\). Recall that \(\rho _{(y,s)} (x,0)\) converges to \((d-1)\)-dimensional delta function at y as \(s\downarrow 0\). Therefore, since \(M_0\) is \(C^1\), we may assume that there exists \(s_0>0\) depending only on \(M_0\) such that
for any \(y\in {\mathbb {R}}^d\) and \(s \in (0,s_0)\). Set \(T_1= T_1 (d,r,M_0):= \min \{ T_4, s_0 \}\). Let \(t_0 \in (0,T_1)\) be a number such that \(\mu _{t_0}\) is integral. Then there exist a countably \((d-1)\)-rectifiable set \(M_{t_0}\) and \(\theta _{t_0} \in L ^1 _{loc} ({\mathscr {H}}^{d-1} \lfloor _{M_{t_0}})\) such that \(\mu _{t_0} = \theta _{t_0} {\mathscr {H}}^{d-1} \lfloor _{M_{t_0}}\). Assume that there exist \(x_0 \in M_{t_0}\) and \(N\ge 2\) such that \(M_{t_0}\) has an approximate tangent space at \(x_0\) and \(\displaystyle \lim _{r\rightarrow 0} \frac{ \mu _{t_0} (B_r (x_0)) }{\omega _{d-1} r^{d-1}} =\theta _{t_0} (x_0) =N\). Set \(r=\sqrt{2(s-t_0)}\) for \(t_0<s <T_1\). Using the same calculation as (155), for any \(\delta \in (0,1)\), we obtain
By this and \( \int _0 ^1 \left( \log \frac{1}{k} \right) ^{\frac{d-1}{2}} \, \textrm{d}k =\Gamma (\frac{d-1}{2} +1) =\pi ^{\frac{d-1}{2}} /\omega _{d-1} \), we have
Then we have a contradiction by (144) and (145). Therefore \(\theta _{t_0} (x)=1\) \({\mathscr {H}}^{d-1}\)-a.e. on \(M_{t_0}\). By an argument similar to that in [45, Theorem 2.2 (2d)], we have \({\mathscr {H}}^{d-1} (\partial ^*U_{t_0} {\setminus } M_{t_0})=0\) and \({\mathscr {H}}^{d-1} (M_{t_0} {\setminus } \partial ^*U_{t_0})=0\) because \(\theta _{t_0} (x)\) is an even integer for \({\mathscr {H}}^{d-1}\)-a.e. \(x \in M_{t_0} {\setminus } \partial ^*U_{t_0}\). Hence we obtain (a).
The claim (b1) and (b2) are clear and (b3) is also obvious by Remark 4 and \(\mu _t= \Vert \nabla \psi (\cdot ,t)\Vert \) for a.e. \(t \in (0,T_1)\). By (133), we have (b4).
Next we prove (b5). By (135), for any \(\phi \in C_c ^1 ((0,T))\), we compute
where we used (b3) of Theorem 3. Thus \(\int _{\Omega } \vec v \cdot \vec \nu \, d \Vert \nabla \psi (\cdot ,t) \Vert =0\) for a.e. \(t \in (0,T_1)\).
Now we prove (b6). Set \(d\nu :=d{\mathscr {H}}^{d-1} \lfloor _{\partial ^*U_t} dt\). Since the space \(C_c (\Omega )\) is dense in \(L^2 (\nu )\), for any \(\eta \in C_c ((0,T_1))\) we have
where we used (b3) and (b5). Hence we obtain (b6). \(\square \)
Data Availability
Data sharing not applicable to this article as no datasets were generated or analysed during the current study.
References
Almgren, F., Taylor, J.E., Wang, L.: Curvature-driven flows: a variational approach. SIAM J. Control. Optim. 31, 387–438, 1993
Ambrosio, L., Fusco, N., Pallara, D.: Functions of Bounded Variation and Free Discontinuity Problems. The Clarendon Press, Oxford University Press, New York (2000)
Antonopoulou, D., Karali, G., Sigal, I.M.: Stability of spheres under volume-preserving mean curvature flow. Dyn. Partial Differ. Equ. 7, 327–344, 2010
Athanassenas, M.: Volume-preserving mean curvature flow of rotationally symmetric surfaces. Comment. Math. Helv. 72, 52–66, 1997
Brakke, K.A.: The Motion of a Surface by Its Mean Curvature. Princeton University Press, Princeton (1978)
Brassel, M., Bretin, E.: A modified phase field approximation for mean curvature flow with conservation of the volume. Math. Methods Appl. Sci. 34, 1157–1180, 2011
Bronsard, L., Stoth, B.: Volume-preserving mean curvature flow as a limit of a nonlocal Ginzburg-Landau equation. SIAM J. Math. Anal. 28, 769–807, 1997
Chen, X., Hilhorst, D., Logak, E.: Mass conserving Allen-Cahn equation and volume preserving mean curvature flow. Interfaces Free Bound 12, 527–549, 2010
Delfour, M.C., Zolésio, J.P.: Shape analysis via oriented distance functions. J. Funct. Anal. 123, 129–201, 1994
Escher, J., Simonett, G.: The volume preserving mean curvature flow near spheres. Proc. Am. Math. Soc. 126, 2789–2796, 1998
Evans, L.C., Gariepy, R.F.: Measure Theory and Fine Properties of Functions. CRC Press, Boca Raton (2015)
Fischer, J., Hensel, S., Laux, T., Simon, T.: The local structure of the energy landscape in multiphase mean curvature flow: Weak-strong uniqueness and stability of evolutions, arXiv preprint arXiv:2003.05478, 2020
Fusco, N.: The classical isoperimetric theorem. Rend. Accad. Sci. Fis. Mat. Napoli. 71, 63–107, 2004
Fusco, N., Maggi, F., Pratelli, A.: The sharp quantitative isoperimetric inequality. Ann. Math. 168, 941–980, 2008
Gage, M., On an area-preserving evolution equation for plane curves, Nonlinear problems in geometry (Mobile, Ala.,: Contemp. Math., 51 Amer. Math. Soc. Providence, R I1986, 51–62, 1985
Giusti, E.: Minimal Surfaces and Functions of Bounded Variation. Birkhäuser, Boston (1984)
Golovaty, D.: The volume-preserving motion by mean curvature as an asymptotic limit of reaction-diffusion equations. Quart. Appl. Math. 55, 243–298, 1997
Hensel, S., Laux, T.: A new varifold solution concept for mean curvature flow: convergence of the Allen-Cahn equation and weak-strong uniqueness, arXiv preprint arXiv:2109.04233, 2021
Huisken, G.: The volume preserving mean curvature flow. J. Reine Angew. Math. 382, 35–48, 1987
Huisken, G.: Asymptotic behavior for singularities of the mean curvature flow. J. Differ. Geom. 31, 285–299, 1990
Hutchinson, J.E.: Second fundamental form for varifolds and the existence of surfaces minimising curvature. Indiana Univ. Math. J. 35, 45–71, 1986
Hutchinson, J.E., Tonegawa, Y.: Convergence of phase interfaces in the van der Waals-Cahn-Hilliard theory. Calc. Var. Partial Differ. Equ. 10, 49–84, 2000
Ilmanen, T.: Convergence of the Allen-Cahn equation to Brakke’s motion by mean curvature. J. Differ. Geom. 38, 417–461, 1993
Kim, I., Kwon, D.: Volume preserving mean curvature flow for star-shaped sets. Calc. Var. Partial Differ. Equ., 2020. https://doi.org/10.1007/s00526-020-01738-0
Laux, T.: Weak-strong uniqueness for volume-preserving mean curvature flow, arXiv preprint arXiv:2205.13040, 2022
Laux, T., Otto, F.: Convergence of the thresholding scheme for multi-phase mean-curvature flow. Calc. Var. Partial Differ. Equ., 2016. https://doi.org/10.1007/s00526-016-1053-0
Laux, T., Simon, T.M.: Convergence of the Allen-Cahn equation to multiphase mean curvature flow. Comm. Pure Appl. Math. 71, 1597–1647, 2018
Laux, T., Swartz, D.: Convergence of thresholding schemes incorporating bulk effects. Interfaces Free Bound. 19, 273–304, 2017
Li, H.: The volume-preserving mean curvature flow in Euclidean space. Pac. J. Math. 243, 331–355, 2009
Liu, C., Sato, N., Tonegawa, Y.: On the existence of mean curvature flow with transport term. Interfaces Free Bound 12, 251–277, 2010
Luckhaus, S., Sturzenhecker, T.: Implicit time discretization for the mean curvature flow equation. Calc. Var. Partial Differ. Equ. 3, 253–271, 1995
Mizuno, M., Tonegawa, Y.: Convergence of the Allen-Cahn equation with Neumann boundary conditions. SIAM J. Math. Anal. 47, 1906–1932, 2015
Modica, L., Mortola, S.: Il limite nella \(\Gamma \)-convergenza di una famiglia di funzionali ellittici. Boll. Un. Mat. Ital. A 14, 526–529, 1977
Mugnai, L., Röger, M.: The Allen-Cahn action functional in higher dimensions. Interfaces Free Bound 10, 45–78, 2008
Mugnai, L., Seis, C., Spadaro, E.: Global solutions to the volume-preserving mean-curvature flow. Calc. Var. Partial Differ. Equ., 2016. https://doi.org/10.1007/s00526-015-0943-x
Pisante, A., Punzo, F.: Allen-Cahn approximation of mean curvature flow in Riemannian manifolds, II: Brakke’s flows. Commun. Contemp. Math. 17, 1450041, 2015
Pisante, A., Punzo, F.: Allen-Cahn approximation of mean curvature flow in Riemannian manifolds I, uniform estimates. Ann. Sc. Norm. Super. Pisa Cl. Sci. 15, 309–341, 2016
Röger, M., Schätzle, R.: On a modified conjecture of De Giorgi. Math. Z. 254, 675–714, 2006
Rubinstein, J., Sternberg, P.: Nonlocal reaction-diffusion equations and nucleation. IMA J. Appl. Math. 48, 249–264, 1992
Simon, L.: Lectures on geometric measure theory, Proc. Centre Math. Anal. Austral. Nat. Univ. 3, 1983
Stuvard, S., Tonegawa, Y.: On the existence of canonical multi-phase Brakke flows, arXiv preprint arXiv:2109.14415, 2021
Takasao, K.: Convergence of the Allen-Cahn equation with constraint to Brakke’s mean curvature flow. Adv. Differ. Equ. 22, 765–792, 2017
Takasao, K.: Existence of weak solution for volume preserving mean curvature flow via phase field method. Indiana Univ. Math. J. 66, 2015–2035, 2017
Takasao, K.: On obstacle problem for Brakke’s mean curvature flow. SIAM J. Math. Anal. 53, 6355–6369, 2021
Takasao, K., Tonegawa, Y.: Existence and regularity of mean curvature flow with transport term in higher dimensions. Math. Ann. 364, 857–935, 2016
Talenti, G.: The standard isoperimetric theorem, Handbook of convex geometry, vol. A, pp. 73–123. North-Holland, Amsterdam (1993)
Tonegawa, Y.: Integrality of varifolds in the singular limit of reaction-diffusion equations. Hiroshima Math. J. 33, 323–341, 2003
Tonegawa, Y.: Brakke’s Mean Curvature Flow. SpringerBriefs in Mathematics. Springer, Singapore (2019)
Acknowledgements
The author expresses his gratitude to the referee for his or her useful comments and suggestions that helped him to improve the original manuscript. The author would like to thank Professor Takashi Kagaya for his helpful comments. This work was supported by JSPS KAKENHI Grant Numbers JP20K14343, JP18H03670, and JSPS Leading Initiative for Excellent Young Researchers (LEADER) operated by Funds for the Development of Human Resources in Science and Technology.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by F. Lin.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Appendix
Appendix
Proposition 18
(See Lemma 3.24 in [2]) Let \(U \subset {\mathbb {R}}^d\) be an open set. Assume that \(u \in BV(U)\) and \(K \subset U\) is a compact set. Then
for all \(\delta \in (0, \textrm{dist}\,(K,\partial U))\), where \(\eta _\delta \) is the standard mollifier defined in Section 3.
As in Lemma 1, we can obtain the following estimate for the classical solution to the volume preserving mean curvature flow (see also [7, 24, 27, 28, 35] for the weak solutions).
Proposition 19
Let \(\Omega = {\mathbb {T}}^d\) and \(U_t \subset \Omega \) be a bounded open set with smooth boundary \(M_t\) for any \(t\ \in [0,T)\) and \(0< {\mathscr {L}}^d (U_0) < {\mathscr {L}}^d(\Omega )\). Assume that \(\{M_t \} _{t\in [0,T)}\) is the volume preserving mean curvature flow. Then there exists \(C_\lambda >0\) depending only on d, \({\mathscr {L}}^d(U_0)\), and \({\mathscr {H}}^{d-1}(M_0)\) such that
where \(\lambda (t) =\frac{1}{{\mathscr {H}}^{d-1} (M_t)} \int _{M_t} \vec {h} \cdot \vec {\nu } \, \textrm{d} {\mathscr {H}}^{d-1}\).
Proof
Let \(\vec {\zeta }:\Omega \times [0,\infty ) \rightarrow {\mathbb {R}}^d\) be a smooth periodic function. By (1), (2), the divergence theorem, and the property of the mean curvature, we have
and
Let \(\alpha , \delta \in (0,1)\) and \(u=u(x,t)\) be a periodic solution to
Then the standard PDE arguments imply the existence and uniqueness of the solution u and
where \(C_\delta >0\) depends only on d and \(\delta \). Set \(\vec {\zeta } (x,t) = \nabla u(x,t)\). We compute that
where we used Proposition 18, \(\Vert \nabla \chi _{U_t}\Vert (\Omega )= {\mathscr {H}}^{d-1} (M_t)\), (147), and the volume preserving property. We choose \(\delta >0\) such that
By this and (149),
where \(\omega ' = \frac{1}{2} {\mathscr {L}}^d (U_0) \left( 1- \frac{{\mathscr {L}}^d (U_0)}{{\mathscr {L}}^d (\Omega )} \right) \). The equality (147) implies
Therefore we obtain (146) by (150) and (151). \(\square \)
Next we show some properties of the backward heat kernel.
Lemma 13
(See [23]) Let \(D>0\) and \(\nu \) be a Radon measure on \({\mathbb {R}}^d\) satisfying
Then the following hold:
-
1.
For any \(a>0\) there exists \(\gamma _1 =\gamma _1 (a)>0\) such that for any \(r>0\) and for any \(x,x_1 \in {\mathbb {R}}^d\) with \(\vert x-x_1 \vert \le \gamma _1 r\), we have the estimate
$$\begin{aligned} \int _{{\mathbb {R}}^d} \rho _{x_1 } ^r (y) \,\textrm{d}\nu (y) \le \int _{{\mathbb {R}}^d} \rho _{x} ^r \,\textrm{d}\nu (y) +aD, \end{aligned}$$(153)where \(\rho _x ^r\) is given by (64).
-
2.
For any \(r,R>0\) and for any \(x\in {\mathbb {R}}^d\), we have
$$\begin{aligned} \int _{{\mathbb {R}}^d \setminus B_R(x)} \rho ^r _x (y) \,d\nu (y) \le 2^{d-1} e^{-3R^2 /8r^2}D. \end{aligned}$$(154)
Proof
We only show (153) here (the estimate (154) can be shown more easily). For \(\beta \in (0,1)\), we have
where we used (152). By \( \int _0 ^1 \left( \log \frac{1}{k} \right) ^{\frac{d-1}{2}} \, dk =\Gamma (\frac{d-1}{2} +1) =\pi ^{\frac{d-1}{2}} /\omega _{d-1} \), we have
We choose \(\gamma _1 >0\) depending only on \(\beta \) such that
For any \(x \in {\mathbb {R}}^d\) with \(\vert x-x_1 \vert \le \gamma _1 r\), we have \( B_{r \sqrt{2 \log \frac{1}{k}}} (x_1) \subset B_{(\sqrt{2 \log \frac{1}{k}} + \gamma _1) r} (x) \subset B_{r \sqrt{2 \log \frac{1}{k-\beta }}} (x) \) for \(k \in (\beta ,1]\). Therefore
Hence (155), (156), and (157) imply (153). \(\square \)
Let \(u ^\varepsilon =u ^\varepsilon (x)\) be a smooth function and define
The following propositions are used in the proof of the integrality of \(\mu _t\).
Proposition 20
(See [22, 45, 47]) For any \(R \in (0,\infty )\), \(E_0 \in (0,\infty )\), \(s\in (0,1)\), and \(N \in {\mathbb {N}}\), there exists \(\varrho \in (0,1)\) with the following property: Assume that a set \(Y \in {\mathbb {R}}^d\) has no more than \(N+1\) elements and \(Y \subset \{ (0,\dots ,0, x_d) \in {\mathbb {R}}^d \mid x_d \in {\mathbb {R}}\}\), \(\textrm{diam}\,Y \le \varrho R\), and there exists \(a \in (0,R) \) such that \(\vert y-z \vert >3a\) holds for any \(y,z \in Y\) with \(y\not =z\). Moreover, we assume the following:
-
1.
\(u ^\varepsilon \in C^2 (\{ y \in {\mathbb {R}}^d \mid \textrm{dist}\,(y, Y) <R \})\).
-
2.
For any \(x \in Y\) and \(r \in [a,R]\),
$$\begin{aligned} \int _{B_r (x)} \vert \xi _\varepsilon \vert + (1-(\nu _d) ^2) \varepsilon \vert \nabla u ^\varepsilon \vert ^2 + \varepsilon \vert \nabla u ^\varepsilon \vert \left| \Delta u ^\varepsilon - \frac{W'(u ^\varepsilon )}{\varepsilon ^2} \right| \, \textrm{d}y \le \varrho r^{d-1}. \end{aligned}$$Here \(\nu = (\nu _1,\dots , \nu _d) = \frac{\nabla u ^\varepsilon }{\vert \nabla u ^\varepsilon \vert }\).
-
3.
For any \(x \in Y\),
$$\begin{aligned} \int _ a ^R \frac{d \tau }{\tau ^d} \int _{B_\tau (x)} (\xi _\varepsilon )_{+} \, \textrm{d}y \le \varrho . \end{aligned}$$ -
4.
For any \(x \in Y\) and \( r \in [a,R]\),
$$\begin{aligned} \int _{B_r (x)} \varepsilon \vert \nabla u ^\varepsilon \vert ^2 \, \textrm{d}y \le E_0 r^{d-1}. \end{aligned}$$
Then, we have
Proposition 21
(See [22, 45, 47]) For any \(s,b,\beta \in (0,1)\), and \(c \in (1,\infty )\), there exist \(\varrho , \varepsilon \in (0,1)\) and \(L \in (1,\infty ) \) with the following property: Assume that \(\varepsilon \in (0,\varepsilon )\), \(u ^\varepsilon \in C^2 (B_{4\varepsilon L} (0))\) and
and
Then we have
where \(J= B_{3\varepsilon L} (0) \cap \{ (0,\dots , 0, x_d) \in {\mathbb {R}}^d \mid x_d \in {\mathbb {R}}\}\). In addition, we have
Remark 13
Note that the assumptions for \((\xi _\varepsilon )_+\) in the propositions above hold for the solution to (5) with suitable initial data (see 33).
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Takasao, K. The Existence of a Weak Solution to Volume Preserving Mean Curvature Flow in Higher Dimensions. Arch Rational Mech Anal 247, 52 (2023). https://doi.org/10.1007/s00205-023-01881-w
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00205-023-01881-w