The free-boundary Brakke flow

Lemma A.1.

Let u:U⊂Brn,1→RN-n be C1 in both variables, with

Suppose ϕ:B3⁢rN→RN satisfies

Then if we extend ϕ⁢(x,t):=(ϕ⁢(x),t) to act on RN,1 we have ϕ⁢(graph⁡(u))=graph⁡(u~), where

with the estimates

If further we have

then

Proof.

Write A⁢(x,t)=x+e1⁢(x,u⁢(x,t)) so that

By assumption, we have

and

Therefore, by the inverse function theorem an inverse At-1≡A⁢(⋅,t)-1 exists for each time slice, and we can set

where

From (A.4) and (A.5) we immediately obtain

This proves the C1 estimates on ζ. The required C1 estimate on η follows similarly, e.g.:

To prove the higher order estimates on ζ and η, we proceed as follows. First, by an easy induction one can show that Dℓ⁢∂tm⁡(A-1) is a linear combination of terms involving (D⁢A)-1, Da⁢e1|(Id,u)∘A-1, and Db⁢∂tc⁡(Id,u)|A-1, where |a|≥1 in each term. Using relation (A.1), and our assumed bounds for u, we have

for any a. Notice the left-hand side is the spacetime Hölder semi-norm, while the right-hand side is the regular Hölder semi-norm. This gives a Hölder bound of the form

which is the required estimate for ζ.

By similar reasoning, we have that Dℓ⁢∂tm⁡η is a linear combination of terms involving Da⁢e2|(Id,u)∘A-1 and Db⁢∂tc⁡((Id,u)∘A-1). Now use (A.6), (A.3) and our assumed regularity of u to obtain the Hölder estimate on η. ∎

Lemma A.2.

Let Mi,M be a sequence in S⁢(n,N), where Mi→M as free-boundary Brakke flows. Suppose 0 is a regular point of each Mi, and

Then 0 is a regular point of M, and the Mi converge to M in C2,α near 0.

If, additionally, the barriers converge in C∞, then convergence near 0 is smooth.

Proof.

If 0 is uniformly bounded away from the barriers Si, then this follows from the Arzela–Ascoli theorem and interior Schauder estimates. To handle points at the boundary, we will straighten the barrier.

Take 0∈S, and by replacing ℳi with ℳi-(ζi⁢(0),0) we can assume 0∈Si also. Pass to a subsequence, rotate by a fixed amount in space, and replace ℳi with ℳi-(0,ti) (with ti→0) as necessary, and we have

where u(i):Qi×I⊂Bρn,1→ℝ is uniformly bounded in C2,α, and I is either the interval [-ρ2,ρ2] or [-ρ2,0].

Let Φi, Φ be the map (2.2) straightening the barriers Si, S (respectively), centered at 0. Using Lemma A.1, we have

where u~(i) is uniformly bounded in C2,α also. There is a fixed half-space H⊂ℝn, so that u~(i):H×I→ℝ has Neumann boundary conditions in ∂⁡H.

The Arzela–Ascoli theorem implies u~(i) subsequentially converge in C2,α′ to some map u~:H×I→ℝ. By the definition of free-boundary convergence, Φi→Φ in C2,α. We deduce that

This shows ℳ is proper and C2,α′ near 0.

The limit u~ is a graphical mean curvature flow in the pullback metric γ=Φ*⁢δ. Therefore u~ satisfies

where gk⁢l is the inverse to the matrix γ⁢(ek+Dk⁢u,el+Dl⁢u). Provided ρ is sufficiently small, by (2.3) this is a parabolic equation, with coefficients as regular as D⁢u~. The usual bootstrap argument then gives C∞.

Suppose the barriers converge smoothly Si→S. Each u~(i) satisfies the graphical mean curvature equation (A.7), with the pullback metric γi=Φi*⁢δ:

for some analytic F. So the difference w(i)=u~(i)-u~ satisfies a linear PDE

Convergence of Φi and w(i) implies L is uniformly elliptic, with constant terms going to 0 in C∞. The usual Schauder estimates then imply w(i)→0 in C∞ also. ∎

Proposition A.3 (Allard [2, Lemma 3.1]).

Let V be an integral n-varifold with free-boundary in S⊂U. For any 0<τ<σ≤rS, and any h∈Cc1⁢(U,R), we have

Here d=d⁢(⋅,S).

In particular, by the dominated convergence theorem,

Proof.

Let X be the vector field

where ϕ is a cutoff function to be determined. We have

Therefore, if

then

Integrating the above relation between τ<σ, and then taking ϕ→1[0,1], we obtain

Apply the standard layer-cake formula to the measure

to obtain

which is the required equality. ∎