On obstacle problem for Brakke's mean curvature flow

We consider the obstacle problem of the weak solution for the mean curvature flow, in the sense of Brakke's mean curvature flow. We prove the global existence of the weak solution with obstacles which have $C^{1,1}$ boundaries, in two and three space dimensions. To obtain the weak solution, we use the Allen-Cahn equation with forcing term.


Introduction
Let T > 0 and d ≥ 2 be an integer. Assume that U t ⊂ R d is a bounded open set and M t is a smooth boundary of U t for any t ∈ [0, T ). We call the family of the hypersurfaces Here, v and h are the normal velocity vector and the mean curvature vector of M t , respectively. Brakke [5] proved the global existence of the multi-phase weak solution to (1.1) called Brakke's mean curvature flow. However, since the flow is defined by an integral inequality, its solution may become an empty set after a certain time. Subsequently, Kim-Tonegawa [21] proved the global existence of non-trivial Brakke's mean curvature flow, by showing that the each volume of the multi-phase is continuous with respect to t. The phase-field method and the elliptic regularization by Ilmanen [17,18] are known as another proofs of the global existence of the Brakke's mean curvature flow. Similar to the Brakke's mean curvature flow, the weak solution called L 2 -flow was studied by Mugnai-Röger [28,29]. In addition, the regularity of Brakke's mean curvature flow is studied by Brakke [5], White [44], Kasai-Tonegawa [20], and Tonegawa [42]. About results for other types of weak solutions, the existence theorem of the viscosity solutions via the level set method was presented independently by Chen-Giga-Goto [10] and Evans-Spruck [13] at the same period, and a weak solution using a variational method was studied by Almgren-Taylor-Wang [3] and Luckhaus-Sturzenhecker [24]. Since the mean curvature flow can be regarded as a simple model of the cell motility, it is natural to consider its obstacle problem (see [11,27]). In addition, the obstacle problems for elliptic equations including the minimal surface equation have been studied over a long period of time (see [8,30,32,35] and references therein).
About the obstacle problem for the mean curvature flow, Almeida-Chambolle-Novaga [4] showed the global existence of weak solutions for d ≥ 2 by a variational method. Moreover, they proved the short time existence and uniqueness of C 1,1 solutions for d = 2, when the obstacle has a compact C 1,1 boundary. Mercier-Novaga [26] extended the short time existence and uniqueness of C 1,1 solutions for d ≥ 2, and they also proved the global existence and uniqueness of the graphical viscosity solutions if the boundaries of obstacles are also graphs. In the case of the viscosity solution with the level set method, Mercier [25] showed the global existence and uniqueness of continuous viscosity solutions to where u − and u + are given uniformly continuous functions with u − ≤ u + , k is a given Lipschitz function, and the assumptions of F allow this equation to be the mean curvature flow with forcing term k, in the sense of the level set method. Ishii-Kamata-Koike [19] proved the global existence and uniqueness of Lipschitz viscosity solutions when k ≡ 0 and Giga-Tran-Zhang [15] studied the large time behavior of viscosity solutions with constant driving force k.
Let d = 2 or 3, Ω := T d , and the obstacles O + , O − ⊂ Ω have C 1,1 boundaries and satisfy dist (O + , O − ) > 0. In this paper, we prove the global existence of the weak solution to (1.1) with obstacles in the sense of Brakke (see Theorem 5.1 below). Note that the weak solution obtained in this paper has similar properties to the weak solution by the minimizing movement in [4,Theorem 4.6] (see Remark 5.3 below). However, since the uniqueness of the flow we obtain is not known, it is an open question whether Brakke's mean curvature flow coincides with the weak solution studied in [4].
To obtain the result, we use the phase-field method. Bretin-Perrier [6] studied the Allen-Cahn equation with a penalized double well potential depending on the obstacles. In contrast, roughly speaking, the Allen-Cahn equation considered in this paper (see (3.4) bellow) is formally an approximation to the following: where n is the outward unit normal vector of M t and g is given by where R 0 is given in (2.2). If the solution M t touches the obstacle at x, the absolute value of its mean curvature |h(x, t)| is less than d R 0 , hence the solution cannot move into the obstacle. Note that this argument was used in Mercier-Novaga [26]. In order to use this argument in the phase-field method, we give an appropriate forcing term for the Allen-Cahn equation and show simple sub and super solutions that correspond to obstacles (see Lemma 4.1 below).
To obtain the convergence of the Allen-Cahn equation to the Brakke flow, we need to prove that the Radon measure given by the energy of the Allen-Cahn equation has good properties, such as that it converges to the mass measure of an integral varifold (see [17]). In the case of d = 2 or 3, Röger-Schätzle [33] proved the properties under the suitable assumptions for the energies of the Allen-Cahn equation (this results have been used in [23,28,29,34,39]). The assumption for d in the main result of this paper comes from the use of [33] and [29] (see Remark 5.4 below).
The organization of the paper is as follows. In Section 2, we set out basic definitions and assumptions about the obstacles and the initial data. In Section 3 we introduce the Allen-Cahn equation we deal with in this paper. In addition we also show the standard estimates for the solution. In Section 4 we give supersolutions and subsolutions to the Allen-Cahn equation that are necessary to show that the solutions to (1.1) do not intrude into the obstacles. In Section 5 we prove the global existence of the weak solution to (1.1) with obstacles, in the sense of Brakke.

Notation and assumptions
First we recall some notions and definitions from the geometric measure theory and refer to [2,5,14,36,43] for more details. Let d be a positive integer. For y ∈ R d and r > 0, we define B r (y) := {x ∈ R d | |x − y| < r}. We denote the space of bounded variation functions on U ⊂ R d as BV (U). For a function ψ ∈ BV (U), we write the total variation measure of the distributional derivative ∇ψ by ∇ψ . Let µ be a Radon measure on R d .
e. on M, then we say µ is k-integral. Note that if M is a countably k-rectifiable set with locally finite and H k -measurable, then there exists the approximate tangent space The left-hand side is called the first variation of µ. The weak solution to the mean curvature flow considered in this paper is as follows.
Here h is the generalized mean curvature vector of µ t . Note that (2.1) is called Brakke's inequality.
Next we state assumptions for the initial data and the obstacles.
2) is satisfied for some R 0 > 0 (see [1]). Let U 0 ⊂ Ω be a bounded open set and we denote M 0 := ∂U 0 . Throughout this paper, we assume the following: Here ω d−1 is a (d − 1)-dimensional volume of the unit ball in R d−1 . (

Allen-Cahn equation with forcing term
In this section, we consider the Allen-Cahn equation forcing term and give basic energy estimates for the solution.
Next we define the measures that correspond to the surface M t in Section 1. 2W (s) ds. Assume that ϕ ε i is a solution to (3.4). We denote Radon measures µ ε i t ,μ ε i t , andμ ε i t by Remark 3.5. If there exist t ≥ 0 and a Radon measure µ t on Ω such that and µ ε t → µ t as Radon measures, namely, for any φ ∈ C c (Ω), thenμ ε t andμ ε t also converge to µ t as Radon measures.
Set D 1 = sup i∈N µ ε i 0 (Ω). Proposition 3.6 implies D 1 < ∞. The integration by parts implies the following standard estimates: Proposition 3.7. Let ϕ ε be a solution to (3.4). Then we have and Proof. By the integration by parts and Young's inequality, we have By this and sup x∈Ω |g| ≤ d R 0 we obtain (3.9) and (3.11). Similarly we can obtain (3.10).
Next we show the monotonicity formula. Set Similar to the proof in [38, p. 2028], we obtain the following monotonicity formula.
Proposition 3.8. Let ϕ ε i be a solution to (3.4) with initial data ϕ ε i which satisfies (3.2) and µ ε i t be a Radon measure defined in (3.7). Then we have (3.12) Here, µ ε t is extended periodically to R d .
In order to use the comparison principle, we need the following estimates for the initial data.
for sufficiently large i ≥ 1.
Proof. To show the first inequality of (4.5), we only need to prove that for sufficiently large i ≥ 1. We assume that for any N ≥ 1 there exist i ≥ N and x ′ ∈ Ω such that r y (x ′ ) >r ε i 0 (x ′ ) with r y (x ′ ) ≥ 0. In addition, we may assume that sup x |r ε i 0 (x) −r ε i 0 (x)| < δ 1 4 . Then x ′ ∈ B R 0 (y) and 0 ≤ r y (x ′ ) ≤ dist (x ′ , ∂B R 0 (y)) (4.7) by max x∈B R 0 (y) |∇r y (x)| ≤ 1 and r y = 0 on ∂B R 0 (y). The assumptions Then (4.7) and (4.8) imply . This is a contradiction to r y (x ′ ) >r ε i 0 (x ′ ). In the case of r y (x ′ ) < 0, we may obtain a contradiction similarly, by using min x∈(B R 0 (y)) c |∇r y (x)| = 1. Therefore we obtain (4.6). We can show the second inequality of (4.5) by the similar argument.
For the proof, we only need to use the standard comparison principle. Therefore we omit it.