**HAL Id: hal-01067821**

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**Mean curvature flow with obstacles: existence,** **uniqueness and regularity of solutions**

### Gwenael Mercier, Matteo Novaga

**To cite this version:**

### Gwenael Mercier, Matteo Novaga. Mean curvature flow with obstacles: existence, uniqueness and

### regularity of solutions. 2014. �hal-01067821�

## Mean curvature flow with obstacles:

## existence, uniqueness and regularity of solutions

### Gwenael Mercier

^{∗}

### , Matteo Novaga

^{†}

Abstract

We show short time existence and uniqueness ofC^{1}^{,}^{1} solutions to the mean curva-
ture flow with obstacles, when the obstacles are of class C^{1}^{,}^{1}. If the initial interface
is a periodic graph we show long time existence of the evolution and convergence to a
minimal constrained hypersurface.

### Contents

### 1 Introduction and main results 1

### 2 Mean curvature flow with a forcing term 3

### 2.1 Evolution of geometric quantities . . . . 3

### 2.2 The Monotonicity Formula . . . . 5

### 3 Proof of Theorem 1 7 4 Proof of Theorem 2 16 5 Proof of Theorem 3 20 A Viscosity solutions with obstacles 20 A.1 Definition of viscosity solution . . . 20

### A.2 Comparison principle . . . 21

### A.3 Existence . . . 22

### A.4 Regularity . . . 24

### 1 Introduction and main results

### Mean curvature flow is a prototypical geometric evolution, arising in many models from Physics, Biology and Material Science, as well as in a variety of mathematical problems.

### For such a reason, this flow has been widely studied in the past years, starting from the pioneristic work of K. Brakke [Bra78] (we refer to [GH86, Hui84, EH89, ES91, CGG91] for a far from complete list of references).

∗CMAP, École polytechnique, Palaiseau, France, email: gwenael.mercier@cmap.polytechnique.fr

†Dipartimento di Matematica, Università di Pisa, Largo Bruno Pontecorvo 5, 56127 Pisa, Italy; e-mail:

novaga@dm.unipi.it

### In some models, one needs to include the presence of hard obstacles, which the evolving surface cannot penetrate (see for instance [ESV12] and references therein). This leads to a double obstacle problem for the mean curvature flow, which reads

### v = H on M

_{t}

### ∩ U, (1)

### with constraint

### M

_{t}

### ⊂ U for all t, (2)

### where v, H denote respectively the normal velocity and d times the mean curvature of the interface M

_{t}

### , and the open set U ⊂

R^{d+1}

### represents the obstacle. Notice that, due to the presence of obstacles, the evolving interface is in general only of class C

^{1,1}

### in the space variable, differently from the unconstrained case where it is analytic (see [ISZ98]). While the regularity of parabolic obstacle problems is relatively well understood (see [Sha08]

### and references therein), a satisfactory existence and uniqueness theory for solutions is still missing.

### In [ACN12] (see also [Spa11]) the authors approximate such an obstacle problem with an implicit variational scheme introduced in [ATW93, LS95]. As a byproduct, they prove global existence of weak (variational) solutions, and short time existence and uniqueness of regular solutions in the two-dimensional case. In [Mer14] the first author adapts to this setting the theory of viscosity solutions introduced in [CIL92, CGG91], and constructs globally defined continuous (viscosity) solutions.

### Let us now state the main results of this paper.

### Theorem 1. Let M

0### ⊂ U be an initial hypersurface, and assume that both M

0### and ∂U are uniformly of class C

^{1,1}

### , with dist(M

_{0}

### , ∂U ) > 0. Then there exists T > 0 and a unique solution M

_{t}

### to (1), (2) on [0, T ), such that M

_{t}

### is of class C

^{1,1}

### for all t ∈ [0, T ).

### Notice that Theorem 1 extends a result in [ACN12] to dimensions greater than two.

### When the hypersurface M

_{t}

### can be written as the graph of a function u( · , t) :

R^{d}

### →

R### , equation (1) reads

### u

t### =

p### 1 + |∇ u |

^{2}

### div ∇ u

p### 1 + |∇ u |

^{2}

!

### . (3)

### If the obstacles are also graphs, the constraint (2) can be written as

### ψ

^{−}6

### u

6### ψ

^{+}

### , (4)

### where the functions ψ

^{±}

### :

R^{d}

### →

R### denote the obstacles.

### Theorem 2. Assume that ψ

^{±}

### ∈ C

^{1,1}

### (

R^{d}

### ), and let u

_{0}

### ∈ C

^{1,1}

### (

R^{d}

### ) satisfy (4). Then there exists a unique (viscosity) solution u of (3), (4) on

R^{d}

### × [0, + ∞ ), such that

### k∇ u( · , t) k

L^{∞}(Rd)

### ≤ max

### k∇ u

_{0}

### k

L^{∞}(Rd)

### , k∇ ψ

^{±}

### k

L^{∞}(Rd)

### k u

t### ( · , t) k

L^{∞}(R

^{d})

### ≤

p

### 1 + |∇ u

0### |

^{2}

### div ∇ u

_{0}p

### 1 + |∇ u

_{0}

### |

^{2}

!
L^{∞}(R^{d})

### for all t > 0. Moreover u is also of class C

^{1,1}

### uniformly on [0, + ∞ ).

### We observe that Theorem 2 extends previous results by Ecker and Huisken [EH89] in

### the unconstrained case (see also [CN13]).

### Theorem 3. Assume that u

0### and ψ

^{±}

### are Q-periodic, with periodicity cell Q = [0, L]

^{d}

### , for some L > 0. Then the solution u( · , t) of (3), (4) is also Q-periodic. Moreover there exists a sequence t

_{n}

### → + ∞ such that u( · , t

_{n}

### ) converges uniformly as n → + ∞ to a stationary solution to (3), (4).

### Our strategy will be to approximate the obstacles with “soft obstacles” modeled by a sequence of uniformly bounded forcing terms. Differently from [ACN12], where the existence of regular solution is derived from variational estimates on the approximating scheme, we obtain estimates on the evolving interface, in the spirit of [EH91a, EH91b, CNV11], which are uniform in the forcing terms.

### Acknowledgements

### We wish to thank to Antonin Chambolle for interesting discussions and useful comments on this work. The first author was partially supported by the ANR-12-BS01-0014-01 Project Geometrya.

### 2 Mean curvature flow with a forcing term

### 2.1 Evolution of geometric quantities

### Let M be a complete orientable d-dimensional Riemannian manifold without boundary, let F ( · , t) : M →

R^{d+1}

### be a smooth family of immersions, and denote by M

_{t}

### the image F(M, t). Since M

_{t}

### is orientable, we can write M

_{t}

### = ∂E(t) where E(t) is a family of open subsets of

R^{d+1}

### depending smoothly on t. We say that M

t### evolves by mean curvature with forcing term k if

### d

### dt F (p, t) = − H(p, t) + k(F(p, t))

### ν(p, t), (5)

### where k :

R^{d+1}

### →

R### is a smooth forcing term, ν is the unit normal to M

_{t}

### pointing outside E(t), and H is (d times) the mean curvature of M

_{t}

### , with the convention that H is positive whenever E(t) is convex.

### We shall compute the evolution of some relevant geometric quantities under the law (5). We denote by ∇

^{S}

### , ∆

^{S}

### respectively the covariant derivative and the Laplace-Beltrami operator on M . As in [Hui84], the metric on M

_{t}

### is denoted by g

_{ij}

### (t), it inverse is g

^{ij}

### (t), the scalar product (or any tensors contraction using the metric) on M

t### is denoted by h· , ·i whereas the ambiant scalar product is ( · , · ), the volume element is µ

_{t}

### , and the second fondamental form is A. In particular we have A (∂

_{i}

### , ∂

_{j}

### ) = h

_{ij}

### , where we set for simplicity

### ∂

i### =

_{∂x}

^{∂}

i

### , and H = h

ii### , using the Einstein notations (we implicitly sum every index which appears twice in an expression). We also denote by λ

_{1}

### , . . . , λ

_{d}

### the eigenvalues of A.

### Notice that, in terms of the parametrization F , we have g

_{ij}

### = (∂

_{i}

### F , ∂

_{j}

### F ) , h

_{ij}

### = − ∂

_{ij}

^{2}

### F , ν

### for all i, j ∈ { 1, . . . , d } . (6)

### Proposition 1. The following equalities hold:

### d

### dt g

_{ij}

### = − 2(H + k)h

_{ij}

### (7)

### d

### dt ν = ∇

^{S}

### (H + k) (8)

### d

### dt µ

t### = − H(H + k)µ

t### (9)

### d

### dt h

_{ij}

### = ∆

^{S}

### h

_{ij}

### + ∇

^{S}i

### ∇

^{S}j

### k − 2Hh

_{il}

### g

^{lm}

### h

_{mj}

### − kg

^{ml}

### h

_{im}

### h

_{jl}

### + | A |

^{2}

### h

_{ij}

### (10) d

### dt H = ∆

^{S}

### (H + k) + (H + k) | A |

^{2}

### (11)

### d

### dt | A |

^{2}

### = ∆

^{S}

### | A |

^{2}

### + 2kg

^{ij}

### g

^{sl}

### g

^{mn}

### h

_{is}

### h

_{lm}

### h

_{nj}

### + 2 | A |

^{4}

### − 2 |∇

^{S}

### A |

^{2}

### + 2

### A , ( ∇

^{S}

### )

^{2}

### k .(12) Proof. The proof follows by direct computations as in [Hui84, EH91b]. Recalling (6), we get

### d

### dt g

_{ij}

### = d

### dt (∂

_{i}

### F , ∂

_{j}

### F ) = − (H + k) ((∂

_{i}

### ν , ∂

_{j}

### F ) + (∂

_{i}

### F , ∂

_{j}

### ν)) = − 2(H + k)h

_{ij}

### d

### dt ν = d

### dt ν , ∂

i### F

### g

^{ij}

### ∂

j### F = −

### ν , d dt ∂

i### F

### g

^{ij}

### ∂

j### F

### = (ν , ∂

_{i}

### ((H + k)ν)) g

^{ij}

### ∂

_{j}

### F = ∂

_{i}

### (H + k)g

^{ij}

### ∂

_{j}

### F = ∇

^{S}

### (H + k).

### The evolution of the measure on M

_{t}

### µ

t### =

p### det[g]

### is given by d dt

p

### det[g] =

d dt

### det[g]

### 2

p### det[g] = det[g] · Tr g

^{ij d}

_{dt}

### g

_{ij}

### 2

p### det[g]

### = −

p### det[g] · (H + k)g

^{ij}

### h

_{ji}

### = − µ

_{t}

### H(H + k).

### In order to prove (10) we compute (as usual, we denote the Christoffel symbols by Γ

^{k}

_{ij}

### ) d

### dt h

_{ij}

### = − d

### dt ν , ∂

_{ij}

^{2}

### F

### = − ∇

^{S}

### (H + k) , ∂

_{ij}

^{2}

### F

### + ∂

_{ij}

^{2}

### (H + k)ν , ν

### = −

### g

^{kl}

### ∂

_{k}

### (H + k)∂

_{l}

### F , Γ

^{k}

_{ij}

### ∂

_{k}

### F − h

_{ij}

### ν +∂

_{ij}

^{2}

### (H + k) + (H + k)

### ∂

j

### h

im### g

^{ml}

### ∂

_{l}

### F , ν

### = ∂

_{ij}

^{2}

### (H + k) − Γ

^{k}

_{ij}

### ∂

_{k}

### (H + k) + (H + k)h

_{im}

### g

^{ml}

### Γ

^{k}

_{lj}

### ∂

_{k}

### F − h

_{lj}

### ν , ν

### = ∇

^{S}i

### ∇

^{S}j

### (H + k) − (H + k)h

_{il}

### g

^{lm}

### h

_{mj}

### . (13) Using Codazzi’s equations, one can show that

### ∆

^{S}

### h

_{ij}

### = ∇

^{S}i

### ∇

^{S}j

### H + Hh

_{il}

### g

^{lm}

### h

_{mj}

### − | A |

^{2}

### h

_{ij}

### , (14)

### so that (10) follows from (14) and (13). From (10) we deduce d

### dt H = d dt g

^{ij}

### h

_{ij}

### = 2(H + k)g

^{is}

### h

_{sl}

### g

^{lj}

### h

_{ij}

### + g

^{ij}

### ∇

^{S}i

### ∇

^{S}j

### (H + k) − (H + k)h

_{il}

### g

^{lm}

### h

_{mj}

### = ∆

^{S}

### (H + k) + (H + k) | A |

^{2}

### , which gives (11). In addition, we get

### d

### dt | A |

^{2}

### = d dt

### g

^{ik}

### g

^{jl}

### h

_{ij}

### h

_{kl}

### = 2 d

### dt g

^{jl}

### h

_{ij}

### h

_{kl}

### + 2g

^{ik}

### g

^{jl}

### d dt h

_{ij}

### h

_{kl}

### = 2

### 2(H + k)g

^{js}

### h

_{st}

### g

^{tl}

### g

^{jl}

### h

_{ij}

### h

_{kl}

### + 2g

^{ik}

### g

^{jl}

### ∆

^{S}

### h

ij### + ∇

^{S}i

### ∇

^{S}j

### k − 2Hh

_{il}

### g

^{lm}

### h

mj### − kg

^{ml}

### h

im### h

_{jl}

### + | A |

^{2}

### h

ij

### h

_{kl}

### = 2kg

^{js}

### h

_{st}

### g

^{tl}

### g

^{jl}

### h

_{ij}

### h

_{kl}

### + 2g

^{ik}

### g

^{jl}

### ∆

^{S}

### h

_{ij}

### h

_{kl}

### + 2 | A |

^{4}

### + 2

### A , ( ∇

^{S}

### )

^{2}

### k .

### (15)

### On the other hand, one has

### ∆

^{S}

### | A |

^{2}

### = 2

### ∆

^{S}

### A , A

### + 2 |∇

^{S}

### A |

^{2}

### = 2g

^{pq}

### g

^{mn}

### h

_{pm}

### ∆

^{S}

### h

_{qn}

### + 2 |∇

^{S}

### A |

^{2}

### . (16) so that (12) follows from (16) and (15).

### 2.2 The Monotonicity Formula

### We extend Huisken’s monotonicity formula [Hui90] to the forced mean curvature flow (5) (see also [CNV11, Section 2.2]).

### Given a vector field ω : M

_{t}

### →

R^{d+1}

### , we let

### ω

^{⊥}

### = (ω , ν) ν, ω

^{T}

### = ω − ω

^{⊥}

### .

### Letting X

_{0}

### ∈

R^{d+1}

### and t

_{0}

### ∈

R### , for (X, t) ∈

R^{d+1}

### × [t

_{0}

### , + ∞ ) we define the kernel ρ(X, t) = 1

### (4π(t

0### − t))

^{d/2}

### exp

### −| X

0### − X |

^{2}

### 4(t

_{0}

### − t)

### .

### A direct computation gives dρ

### dt = − ∆

^{S}

### ρ + ρ

### (X

_{0}

### − X , (H + k)ν)

### t

0### − t − | (X

_{0}

### − X)

^{⊥}

### |

^{2}

### 4(t

0### − t)

^{2}

### . (17)

### Proposition 2 (Monotonicty Formula).

### d dt

Z

Mt

### ρ = −

ZMt

### ρ H + k

### 2 + (X − X

0### , ν) 2(t

_{0}

### − t)

2

### − k

^{2}

### 4

!

### .

### Proof. Recalling (9), we compute

### d dt

Z

Mt

### ρ =

ZMt

### d

### dt ρ − H(H + k)ρ

### =

ZMt

### ρ

### − | X − X

0### |

^{2}

### 4(t

_{0}

### − t)

^{2}

### + d

### 2(t

_{0}

### − t) − (X − X

0### , ν)

### 2(t

_{0}

### − t) (H + k) − H(H + k)

### = −

ZMt

### ρ

### Hν + X − X

_{0}

### 2(t

_{0}

### − t) + kν

### 2

2

### − k

^{2}

### 4

!

### +

Z

Mt

### d

### 2(t

_{0}

### − t) ρ +

ZMt

### ρ (X − X

_{0}

### , ν) H 2(t

_{0}

### − t)

### We use the first variation formula: for all vector field

Y### on M

_{t}

### , we have

ZMt

### div

_{M}

_{t}Y

### =

ZMt

### h Hν ,

Yi### . As a result, with

Y### =

^{ρ(X−X}

_{2(t−t}

^{0}

^{)}

0)

### , we get d

### dt

ZMt

### ρ = −

ZMt

### ρ

### Hν + X − X

_{0}

### 2(t

_{0}

### − t) + kν

### 2

2

### − k

^{2}

### 4 − | (X − X

_{0}

### )

^{T}

### |

^{2}

### 4(t

_{0}

### − t)

^{2}

!

### = −

ZMt

### ρ

### H + (X − X

_{0}

### , ν ) 2(t

_{0}

### − t) + k

### 2

2

### − k

^{2}

### 4

!

### .

### In a similar way (see [EH89]) one can prove that for all functions f (X, t) defined on M

_{t}

### , one has

### ∂

_{t}Z

Mt

### ρf =

ZMt

### df

### dt − ∆

^{S}

### f

### ρ −

ZMt

### f ρ

### H + (X − X

_{0}

### , ν) 2(t

_{0}

### − t) + k

### 2

2

### − k

^{2}

### 4

!

### . (18) Indeed, using (17)

### d dt

Z

Mt

### ρf =

ZMt

### f dρ dt + df

### dt ρ − H(H + k)f ρ

### =

ZMt

### f dρ

### dt − H(H + k)ρ

### + df dt ρ

### =

ZMt

### f

### − ∆

^{S}

### ρ + ρ

### (X

_{0}

### − X , (H + k)ν) t

_{0}

### − t − 1

### 4

### | (X

_{0}

### − X)

^{⊥}

### |

^{2}

### (t

_{0}

### − t)

^{2}

### − H(H + k)ρ

### + df dt ρ

### =

ZMt

### − ∆

^{S}

### f ρ +

### ρ

### (X

_{0}

### − X , (H + k)ν) t

0### − t − 1

### 4

### | (X

_{0}

### − X)

^{⊥}

### |

^{2}

### (t

0### − t)

^{2}

### − H(H + k)ρ

### + df dt ρ

### =

ZMt

### ρ d

### dt f − ∆

^{S}

### f

### −

Z### f ρ

### H + (X − X

_{0}

### , ν) 2(t

0### − t) + k

### 2

2

### − k

^{2}

### 4

!

### .

### Lemma 1. Let f be defined on M

t### and satisfy d

### dt f − ∆

^{S}

### f

6### a · ∇

^{S}

### f on M

t### (19) for some vector field a bounded on [0, t

1### ]. Then,

### sup

Mt, t∈[0,t1]

### f

6### sup

M0

### f.

### Proof. Denote by a

0### the bound on a, k := sup

_{M}

_{0}

### f and define f

_{l}

### = max(f − l, 0). Assump- tion (19) implies

### d dt − ∆

^{S}

### f

_{l}

^{2}6

### 2f

_{l}

### a · ∇

^{S}

### f

_{l}

### − 2 |∇

^{S}

### f

_{l}

### |

^{2}

### which, thanks to Young’s inequality, gives

### d dt − ∆

^{S}

### f

_{l}

^{2}6

### 1 2 a

^{2}

_{0}

### f

_{l}

^{2}

### . Applying (18) to f

_{l}

^{2}

### , we get

### d dt

Z

### f

_{l}

^{2}

### ρ

6### 1

### 2 (a

^{2}

_{0}

### + k k k

^{2}∞

### )

Z### f

_{l}

^{2}

### ρ. (20)

### Letting l = sup

_{M}

_{0}

### f , so that f

_{l}

### ≡ 0 on M

_{0}

### , from (20) and the Gronwall’s Lemma we obtain that f

_{l}

### ≡ 0 on M

_{t}

### for all t ∈ (0, t

_{1}

### ], which gives thesis.

### 3 Proof of Theorem 1

### We now prove short time existence for the mean curvature flow with obstacles (1), (2). Let M

0### = ∂E(0) ⊂ U , where we assume that U , E(0) are open sets with boundary uniformly of class C

^{1,1}

### , with dist(M

_{0}

### , ∂U ) > 0. In particular, M

_{0}

### satisfies a uniform exterior and interior ball condition, that is, there is R > 0 such that, for every x ∈ M

_{0}

### , one can find two open balls B

^{+}

### and B

^{−}

### of radius R which are tangent to M

0### at x and such that B

^{+}

### ⊂ E(0)

^{c}

### and B

^{−}

### ⊂ E(0). Let also Ω

^{−}

### := E(0) \ U , and Ω

^{+}

### := E(0)

^{c}

### \ (M

_{0}

### ∪ U ). Notice that Ω

^{±}

### are open sets with C

^{1,1}

### boundaries, with dist(Ω

^{−}

### , Ω

^{+}

### ) > 0. Let also

### k := 2N (χ

_{Ω}

^{−}

### − χ

_{Ω}

^{+}

### ) where N is bigger than (d times) the mean curvature of ∂U.

### We want to show that equation (5), with k as above, has a solution in an interval [0, T ). To this purpose, letting ρ

ε### be a standard mollifier supported in the ball of radius ε centered at 0, we introduce a smooth regularization k

_{ε}

### = k ∗ ρ

_{ε}

### of k. Notice that k k

_{ε}

### k

∞### = 2N, k

_{ε}

### (x) = 2N (resp. k

_{ε}

### (x) = − 2N ) at every x ∈ Ω

^{−}

### (resp. x ∈ Ω

^{−}

### ) such that dist(x, ∂U ) ≥ ε, and k

_{ε}

### (x) = 0 at every x ∈ U such that dist(x, ∂U ) ≥ ε.

### Using standard arguments (see for instance [EH91b, Theorem 4.1] and [EH91a, Prop.

### 4.1]) one can show existence of a smooth solution M

_{t}

^{ε}

### of (5), with k replaced by k

_{ε}

### , on a maximal time interval [0, T

_{ε}

### ).

### Let now

### Ω

^{±}

_{ε}

### := { x ∈ Ω

^{±}

### : dist(x, U) > ε } . The following result follows directly from the definition of k

_{ε}

### .

### Proposition 3. The hypersurfaces ∂Ω

^{±}

_{ε}

### are respectively a super and a subsolution of (5), with k replaced with k

_{ε}

### . In particular, by the parabolic comparison principle M

_{t}

^{ε}

### cannot intersect ∂Ω

^{±}

_{ε}

### .

### We will show that we can find a time T > 0 such that for every ε, there exists a smooth solution of (5) (with k replaced with k

ε### ) on [0, T ).

### The following result will be useful in the sequel. We omit the proof which is a simple

### ODE argument.

### Figure 1: Main notations of Proposition 4.

### Lemma 2. Let M

_{0}

### = ∂B

_{R}

### (x

_{0}

### ) be a ball of radius R ≤ 1 centered at x

_{0}

### . Then, the evolution M

t### by (5), with constant forcing term k = 2N , is given by M

t### = B

_{R(t)}

### (x

0### ) with R(t)

>p### R

^{2}

### − (4N + 2d)t. In particular, the solution exists at least on

h### 0,

_{4N+2d}

^{R}

^{2}

### . Proposition 4. There exists r > 0, a collection of balls B

_{i}

### = B

_{r}

### (x

_{i}

### ) of radius r, and a positive time T

0### such that M

_{t}

^{ε}

### ⊂

Si

### B

i### for every t ∈ [0, min(T

0### , T

ε### )). In addition, we can choose the balls B

_{i}

### in such a way that, for every i, there exists ω

_{i}

### ∈

R^{d+1}

### such that

### ∂Ω

^{±}

### ∩ B

_{4r}

### (x

_{i}

### ) are graphs of some functions ψ

^{±}

_{i}

### :

R^{d}

### →

R### ∪ {±∞} over ω

_{i}

^{⊥}

### . In particular, one has

### ( ∇ k

_{ε}

### , ω

_{i}

### )

>### |∇ k

_{ε}

### | /2 on B

_{2r}

### (x

_{i}

### ).

### Most of these notations are summarized in Figure 1.

### Proof. By assumption, for every

x### ∈ M

_{0}

### there exist interior and exterior balls B

_{x}

^{±}

### of fixed radius R ≤ 1. Let B

_{x}

^{±}

### (t) be the evolution of B

_{x}

^{±}

### by (5) with forcing term k = 2N . By comparison, for every t ∈ [0, T

_{ε}

### ), B

_{x}

^{+}

### (t) ⊂ Ω(t)

^{c}

### and B

_{x}

^{−}

### (t) ⊂ Ω(t). Recalling Lemma 2, there exists δ > 0 and T

_{0}

### > 0, independent of ε, such that M

_{t}

### ⊂ { d

_{M}

_{0}6

### δ } =: C

_{δ}

### , for all t ∈ [0, min(T

ε### , T

0### )).

### We eventually reduce δ, T

_{0}

### such that C

_{δ}

### can be covered with a collection of balls B

_{i}

### = B

_{r}

### (x

_{i}

### ), centered at x

_{i}

### ∈ M

_{0}

### and with a radius r such that, for every i, there exists a unit vector ω

i### ∈

R^{d+1}

### satisfying

### ω

_{i}

### , ν

^{+}

### (x)

>

### 1

### 2 and ω

_{i}

### , ν

^{−}

### (y)

>

### 1 2

### for every x ∈ ∂Ω

^{+}

### ∩ B

4r### (x

i### ) and y ∈ ∂Ω

^{−}

### ∩ B

4r### (x

i### ), where ν

^{±}

### is the outer normal to Ω

^{±}

### .

### As a result, ∂Ω

^{±}

### ∩ B

_{4r}

### (x

_{i}

### ) are graphs of some functions ψ

_{i}

^{±}

### :

R^{d}

### →

R### ∪ {±∞} over ω

_{i}

^{⊥}

### (see Figure 1).

### Notice also that k is a BV function and Dk is a Radon measure concentrated on ∂U such that

### (Dk , ω

_{i}

### )

>### | Dk |

### 2 on B

_{4r}

### (x

_{i}

### ).

### Then, for every x ∈ B

2r### (x

i### ) and ε sufficiently small (such that ρ

ε### (x) = 0 as soon as

### | x |

>### 2r), we have

### ( ∇ k

ε### , ω

i### ) =

### ∇

ZR^{d+1}

### k(x − y)ρ

ε### (y)dy , ω

i

### =

ZRd+1

### (Dk(x − y) , ω

i### ) ρ

ε### (y)dy

>

Z

R^{d+1}

### | Dk | (x − y)

### 2 ρ

ε### (y)dy

>

### | Dk | ∗ ρ

_{ε}

### 2

>### |∇ k

_{ε}

### | 2 .

### In what follows, we will control the geometric quantities of M

_{t}

^{ε}

### inside each ball B

_{i}

### . As in [EH91a], we introduce a localization function φ

_{i}

### as follows: let η

_{i}

### (x, t) = | x − x

_{i}

### |

^{2}

### +(2d+Λ)t (Λ be a positive constant that will be fixed later) and, for R = 2r, φ

_{i}

### (x, t) = (R

^{2}

### − η

_{i}

### (x, t))

^{+}

### . We denote by φ

_{i}

### the quantity φ

_{i}

### (

x### , t), where

x### =

x### (p, t) will be a generic point in M

_{t}

### . Notice that there exists T

_{1}

### =

_{2d+Λ}

^{r}

^{2}

### such that for all t ∈ [0, min(T

_{1}

### , T

_{ε}

### )),

### M

_{t}

^{ε}

### ⊂

[i

### { φ

_{i}

### > r

^{2}

### } . (21)

### As a result, we have the following

### Lemma 3. Let f be a smooth function defined on M

_{t}

^{ε}

### . Assume that there is a C > 0 such that

### φ

_{i}

### f

6### C on M

_{t}

^{ε}

### ∀ t

6### min(T

_{ε}

### , T

_{1}

### ) and ∀ i ∈

N### . Then,

### f

6### αC on M

_{t}

^{ε}

### ∀ t

6### min(T

_{ε}

### , T

_{1}

### ), where α depends only on the C

^{1,1}

### norm of M

_{0}

### .

### Lemma 4. Let v := (ν , ω)

^{−1}

### . The quantity v

^{2}

### φ

^{2}

### satisfies d

### dt − ∆

^{S}

### v

^{2}

### φ

^{2}

### 2

6

### 1 2

### ∇

^{S}

### (v

^{2}

### φ

^{2}

### ) , ∇

^{S}

### φ

^{2}

### φ

^{2}

### +φ

^{2}

### v

^{3}

### ∇

^{S}

### k

_{ε}

### , ω

### + v

^{2}

### φ(2k

_{ε}

### (

x### , ν) − Λ). (22) Proof. In this proof and the proofs further, we use normal coordinates: we assume that g

ij### )δ

ij### (Kronecker symbol) and that the Christoffel symbols Γ

^{k}

_{ij}

### vanish at the computation point.

### We expand the derivatives d

### dt − ∆

^{S}

### v

^{2}

### φ

^{2}

### 2

### = v

^{2}

### d

### dt − ∆

^{S}

### φ

^{2}

### 2 + φ

^{2}

### d

### dt − ∆

^{S}

### v

^{2}

### 2 − 2

### ∇

^{S}

### φ

^{2}

### 2 , ∇

^{S}

### v

^{2}

### 2

### . First term. We start computing

### d dt − ∆

^{S}

### |x|

^{2}

### = − 2k

ε### (

x### , ν) − 2d.

### Then,

### d dt − ∆

^{S}

### φ

^{2}

### = 2φ(2k

_{ε}

### (

x### − x

_{i}

### , ν) − Λ) − 2 |∇

^{S}

### |x|

^{2}

### |

^{2}

### . Second term. We are interested in

### 1 2

### d

### dt (ω , ν)

^{2}

### = (ω , ν) d

### dt ν , ω

### (23)

### = (ω , ν) ∇

^{S}

### (H + k

ε### ) , ω

### . (24)

### So,

### 1 2

### d

### dt (ω , ν)

^{−2}

### = − (ω , ν)

^{−3}

### ∇

^{S}

### (H + k

_{ε}

### ) , ω

### . (25)

### On the other hand, 1

### 2 ∆

^{S}

### ((ω , ν)

^{−2}

### ) = (ω , ν)

^{−1}

### ∆

^{S}

### (ω , ν)

^{−1}

### −

D### ∇

^{S}

### (ω , ν)

^{−1}

### , ∇

^{S}

### (ω , ν)

^{−1}E

### . (26) Let us note that

### ∂

_{ij}

### ν = ∂

_{i}

### h

_{jl}

### g

^{lm}

### ∂

_{m}

### F

### = ∂

_{i}

### (h

_{jl}

### )δ

_{lm}

### ∂

_{m}

### F − h

_{jl}

### δ

_{lm}

### ( − h

_{im}

### ν) = ∂

_{i}

### (h

_{jl}

### )∂

_{l}

### F − λ

^{2}

_{i}

### δ

_{ij}

### ν.

### We then get

### ∆

^{S}

### (ω , ν)

^{−1}

### = ∂

_{ii}

### (ω , ν)

^{−1}

### = ∂

_{i}

### − (ω , ∂

_{i}

### ν) (ω , ν)

^{−2}

### (27)

### = − (ω , ∂

_{ii}

### ν) (ω , ν)

^{−2}

### + 2 (ω , ∂

_{i}

### ν)

^{2}

### (ω , ν)

^{−3}

### (28)

### = − (ω , ν)

^{−2}

### ∂

_{i}

### h

_{il}

### ∂

_{l}

### F − λ

^{2}

_{i}

### ν , ω

### + 2 (ω , ν)

^{−3}

### (ω , λ

_{i}

### ∂

_{i}

### F )

^{2}

### . (29)

### = − (ω , ν)

^{−2}

### (∂

_{l}

### h

_{ii}

### ∂

_{l}

### F , ω) + | A |

^{2}

### (ν , ω)

^{−1}

### + 2 (ω , ν)

^{−3}

### (ω , λ

_{i}

### ∂

_{i}

### F )

^{2}

### . (30) We also have

D

### ∇

^{S}

### (ω , ν)

^{−1}

### , ∇

^{S}

### (ω , ν)

^{−1}E

### = (ω , ν)

^{−4}

### (ω , ∂

_{k}

### ν) (ω , ∂

_{k}

### ν) (31)

### = (ω , ν)

^{−4}

### (ω , h

_{ku}

### g

^{uv}

### ∂

_{v}

### F )

^{2}

### = (ω , ν)

^{−4}

### (ω , λ

_{k}

### ∂

_{k}

### F )

^{2}

### , (32) which leads to

### d dt − ∆

^{S}

### v

^{2}

### 2 = − v

^{3}

### ∇

^{S}

### (H + k

_{ε}

### ) , ω

### + v

^{3}

### ∂

_{m}

### (h

_{ii}

### ) (ω , ∂

_{m}

### F )

### −| A |

^{2}

### v

^{2}

### − 2v

^{4}

### λ

^{2}

_{k}

### (ω , ∂

_{k}

### F )

^{2}

### − v

^{4}

### (ω , λ

_{k}

### ∂

_{k}

### F )

^{2}

### Third term. We notice, as in [EH91a] that |∇

^{S}

### φ

^{2}

### |

^{2}

### = 4φ

^{2}

### |∇

^{S}

### ( |x|

^{2}

### ) |

^{2}

### and

### − ∇

^{S}

### (v

^{2}

### ) , ∇

^{S}

### φ

^{2}

### = − 3 v ∇

^{S}

### (v) , ∇

^{S}

### φ

^{2}

### + 1

### 2

### ∇

^{S}

### (v

^{2}

### φ

^{2}

### ) , ∇

^{S}

### φ

^{2}

### φ

^{2}

### − v

^{2}

### |∇

^{S}

### φ

^{2}

### |

^{2}

### φ

^{2}

### .

### Then, Young’s inequality gives 2

### v ∇

^{S}

### v , ∇

^{S}

### φ

^{2}

6

### 2φ

^{2}

### |∇

^{S}

### v

^{2}

### |

^{2}

### + 1

### 2φ

^{2}

### |∇

^{S}

### φ

^{2}

### |

^{2}6

### 2φ

^{2}

### |∇

^{S}

### v

^{2}

### |

^{2}

### + 2v

^{2}

### |∇

^{S}

### |x|

^{2}

### |

^{2}

### .

### Hence,

### − ∇

^{S}

### (v

^{2}

### ) , ∇

^{S}

### φ

^{2}

6

### − 3φ

^{2}

### |∇

^{S}

### v

^{2}

### |

^{2}

### − 3v

^{2}

### |∇

^{S}

### |x|

^{2}

### |

^{2}

### + 1 2

### ∇

^{S}

### (v

^{2}

### φ

^{2}

### ) , ∇

^{S}

### φ

^{2}

### φ

^{2}

### − v

^{2}

### |∇

^{S}

### φ

^{2}

### |

^{2}

### φ

^{2}

### .

### Summing the three terms, we get d

### dt − ∆

^{S}

### v

^{2}

### φ

^{2}

### 2

6

### 1

### 2

### ∇

^{S}

### (v

^{2}

### φ

^{2}

### ) , ∇

^{S}

### φ

^{2}

### φ

^{2}

### − φ

^{2}

### v

^{3}

### ∇

^{S}

### k

_{ε}

### , ω

### + v

^{2}

### φ(2k

_{ε}

### (

x### , ν) − Λ).

### For γ > 0, we let

### ψ(v

^{2}

### ) := γv

^{2}

### 1 − γv

^{2}

### . Lemma 5. For ε

6### r, we have

### d dt − ∆

^{S}

### φ

^{2}

### | A |

^{2}

### ψ(v

^{2}

### )

### 2

6### φ

^{2}

### ψ(v

^{2}

### )( − γ | A |

^{4}

### − 2k

_{ε}X

i

### λ

^{3}

_{i}

### − 2

### A , ( ∇

^{S}

### )

^{2}

### k

_{ε}

### )

### − φ

^{2}

### | A |

^{2}

### v

^{3}

### ψ

^{′}

### (v

^{2}

### ) ∇

^{S}

### k

_{ε}

### , ω

### − φ

^{2}

### | A |

^{2}X

i

### (λ

_{i}

### ω

^{i}

### )

^{2}

### 2v

^{4}

### + γv

^{6}

### (1 − γv

^{2}

### )

^{3}

### .

### Proof. We denote V =

^{φ}

^{2}

^{|A|}

^{2}

_{2}

^{ψ(v}

^{2}

^{)}

### and compute d

### dt − ∆

^{S}

### φ

^{2}

### | A |

^{2}

### ψ(v

^{2}

### )

### 2 = | A |

^{2}

### ψ(v

^{2}

### ) d

### dt − ∆

^{S}

### 1

### 2 φ

^{2}

### + φ

^{2}

### ψ(v

^{2}

### ) d

### dt − ∆

^{S}

### 1

### 2 | A |

^{2}

### +φ

^{2}

### | A |

^{2}

### d dt − ∆

^{S}

### 1

### 2 ψ(v

^{2}

### ) − 2

### 1/2 ∇

^{S}

### | A |

^{2}

### , 1/2 ∇

^{S}

### φ

^{2}

### − 2

### 1/2 ∇

^{S}

### | A |

^{2}

### , 1/2 ∇

^{S}

### ψ(v

^{2}

### )

### − 2

### 1/2 ∇

^{S}

### φ

^{2}

### , 1/2 ∇

^{S}

### ψ(v

^{2}

### ) . The two first terms have already been computed. Let us consider in the third one.

### 1 2

### d

### dt ψ(v

^{2}

### ) = v dv

### dt ψ

^{′}

### (v

^{2}

### ) = − v

^{3}

### ψ

^{′}

### (v

^{2}

### ) ∇

^{S}

### (H + k

_{ε}

### ) , ω ,

### 1

### 2 ∆

^{S}

### ψ(v

^{2}

### ) = 1

### 2 ∂

_{ii}

### ψ(v

^{2}

### ) = ∂

_{i}

### (v∂

_{i}

### vψ

^{′}

### (v

^{2}

### )) = v∆

^{S}

### vψ

^{′}

### (v

^{2}

### ) + 2v

^{2}

### |∇

^{S}

### v |

^{2}

### ψ

^{′′}

### (v

^{2}

### ) + |∇

^{S}

### v |

^{2}

### ψ

^{′}

### (v

^{2}

### )

### = (3 |∇

^{S}

### v |

^{2}

### − v

^{3}

### (∂

_{l}

### (h

_{kk}

### )w

^{l}

### ) + v

^{2}

### | A |

^{2}

### )ψ

^{′}

### (v

^{2}

### ) + 2 |∇

^{S}

### v |

^{2}

### ψ

^{′′}

### (v

^{2}

### ).

### Hence d

### dt − ∆

^{S}

### 1

### 2 ψ(v

^{2}

### ) = − v

^{3}

### ψ

^{′}

### (v

^{2}

### ) ∇

^{S}

### k

_{ε}

### , ω

### − (3 |∇

^{S}

### v |

^{2}

### + v

^{2}

### | A |

^{2}

### )ψ

^{′}

### (v

^{2}

### ) − 2v

^{2}

### |∇

^{S}

### v |

^{2}

### ψ

^{′′}

### (v

^{2}

### ).

### As above, we want to conclude the proof using the weak maximum principle. So, we want to rewrite the last terms (which are gradient terms) using the gradient of V . Let us expand ∇

^{S}

### V .

### ∇

^{S}

### φ

^{2}

### | A |

^{2}

### ψ(v

^{2}

### )

### 2 = φ

^{2}

### | A |

^{2}

### 1

### 2 ∇

^{S}

### ψ(v

^{2}

### ) + | A |

^{2}

### ψ(v

^{2}

### ) 1

### 2 ∇

^{S}

### φ

^{2}

### + φ

^{2}

### ψ(v

^{2}

### ) 1

### 2 ∇

^{S}

### | A |

^{2}

### .

### So,

### ∇

^{S}

### φ

^{2}

### | A |

^{2}

### ψ(v

^{2}

### ) 2

2

### = φ

^{4}

### | A |

^{4}

### |∇

^{S}

### ψ(v

^{2}

### ) |

^{2}

### 4 + | A |

^{4}

### ψ

^{2}

### (v

^{2}

### ) |∇

^{S}

### φ

^{2}

### |

^{2}

### 4 + φ

^{4}

### ψ

^{2}

### (v

^{2}

### ) |∇

^{S}

### | A |

^{2}

### |

^{2}

### 4 +φ

^{2}

### | A |

^{4}

### ψ(v

^{2}

### )

### ∇

^{S}

### ψ(v

^{2}

### ) , ∇

^{S}

### φ

^{2}

### + φ

^{4}

### | A |

^{2}

### ψ(v

^{2}

### )

### ∇

^{S}

### ψ(v

^{2}

### ) , ∇

^{S}

### | A |

^{2}

### + | A |

^{2}

### ψ

^{2}

### (v

^{2}

### )φ

^{2}

### ∇

^{S}

### φ

^{2}

### , ∇

^{S}

### | A |

^{2}

### . As a matter of fact,

### 1 φ

^{2}

### | A |

^{2}

### ψ(v

^{2}

### )

### ∇

^{S}

### φ

^{2}

### | A |

^{2}

### ψ(v

^{2}

### ) 2

2

### = φ

^{2}

### | A |

^{2}

### |∇

^{S}

### ψ(v

^{2}

### ) |

^{2}

### 4ψ(v

^{2}

### ) + | A |

^{2}

### ψ(v

^{2}

### ) |∇

^{S}

### φ

^{2}

### |

^{2}

### 4φ

^{2}

### +φ

^{2}

### ψ(v

^{2}

### ) |∇

^{S}

### | A |

^{2}

### |

^{2}

### 4 | A |

^{2}

### + 2 | A |

^{2}

### ∇

^{S}

### ψ(v

^{2}

### )/2 , ∇

^{S}

### φ

^{2}

### /2 +2φ

^{2}

### ∇

^{S}

### ψ(v

^{2}

### )/2 , ∇

^{S}

### | A |

^{2}

### /2

### + 2ψ(v

^{2}

### )

### ∇

^{S}

### φ

^{2}

### /2 , ∇

^{S}

### | A |

^{2}

### /2 . We use the last equality to rewrite

### d dt − ∆

^{S}

### φ

^{2}

### | A |

^{2}

### ψ(v

^{2}

### ) 2

### = | A |

^{2}

### ψ(v

^{2}

### ) φ(2k

_{ε}

### (

x### , ν ) − Λ) − |∇

^{S}

### | x |

^{2}

### |

^{2}

### + φ

^{2}

### ψ(v

^{2}

### )

### −

### ∇

^{S}

### A , ∇

^{S}

### A

### + | A |

^{4}

### − 2k

_{ε}

### g

^{js}

### h

_{st}

### g

^{tl}

### g

^{jl}

### h

_{ij}

### h

_{kl}

### − 2

### A , ∇

^{2}

### k

_{ε}

### + φ

^{2}

### | A |

^{2}

### − v

^{3}

### ψ

^{′}

### (v

^{2}

### ) ∇

^{S}

### k

ε### , ω

### − (3 |∇

^{S}

### v |

^{2}

### + v

^{2}

### | A |

^{2}

### )ψ

^{′}

### (v

^{2}

### ) − 2v

^{2}

### |∇

^{S}

### v |

^{2}

### ψ

^{′′}

### (v

^{2}

### )

### − 1

### φ

^{2}

### | A |

^{2}

### ψ(v

^{2}

### )

### ∇

^{S}

### φ

^{2}

### | A |

^{2}

### ψ(v

^{2}

### ) 2

2

### + φ

^{2}

### | A |

^{2}

### |∇

^{S}

### ψ(v

^{2}

### ) |

^{2}

### 4ψ(v

^{2}

### ) + | A |

^{2}

### ψ(v

^{2}

### ) |∇

^{S}

### φ

^{2}

### |

^{2}

### 4φ

^{2}

### + φ

^{2}

### ψ(v

^{2}

### ) |∇

^{S}

### | A |

^{2}

### |

^{2}

### 4 | A |

^{2}

### .

### (33) Let us precise some terms:

### |∇

^{S}

### φ

^{2}

### |

^{2}

### = 4φ

^{2}

### · | − 2

x^{T}

### |

^{2}

### = 4φ

^{2}

### (4 |x|

^{2}

### − 4 (

x### , ν)),

### |∇

^{S}

### ψ(v

^{2}

### ) |

^{2}

### = ψ

^{′}

### (v

^{2}

### )

^{2}

### |∇

^{S}

### v

^{2}

### |

^{2}

### = 4ψ

^{′}

### (v

^{2}

### )

^{2}

### v

^{6}X

k

### (λ

_{k}

### ω

^{k}

### )

^{2}

### ,

### |∇

^{S}

### | A |

^{2}

### |

^{2}

### = 4

Xi

### (∂

_{i}

### (h

_{ll}

### )λ

_{l}

### )

^{2}

### ,

### |∇

^{S}

### A |

^{2}

### =

Xi,k,l

### (∂

i### (h

_{km}

### ))

^{2}

### .

### In addition, we have the obvious estimate

### |∇

^{S}

### | A |

^{2}

### |

^{2}6

### 4 | A |

^{2}

### |∇

^{S}

### A |

^{2}

### . So,

### φ

^{2}

### | A |

^{2}

### |∇

^{S}

### ψ(v

^{2}

### ) |

^{2}

### 4ψ(v

^{2}

### ) + | A |

^{2}

### ψ(v

^{2}

### ) |∇

^{S}

### φ

^{2}

### |

^{2}

### 4φ

^{2}

### + φ

^{2}

### ψ(v

^{2}

### ) |∇

^{S}

### | A |

^{2}

### |

^{2}

### 4 | A |

^{2}6

### φ

^{2}

### | A |

^{2}

### ψ

^{′}

### (v

^{2}

### )

^{2}

### v

^{6}P

k

### (λ

_{k}

### ω

^{k}

### )

^{2}

### ψ(v

^{2}

### ) + 4 | A |

^{2}

### ψ(v

^{2}

### )( |x|

^{2}

### − (

x### , ν)

^{2}

### ) + φ

^{2}

### ψ(v

^{2}

### ) |∇

^{S}

### A |

^{2}