A general regularity theory for weak mean curvature flow

Abstract

We give a new proof of Brakke's partial regularity theorem up to for weak varifold solutions of mean curvature flow by utilizing parabolic monotonicity formula, parabolic Lipschitz approximation and blow-up technique. The new proof extends to a general flow whose velocity is the sum of the mean curvature and any given background flow field in a dimensionally sharp integrability class. It is a natural parabolic generalization of Allard's regularity theorem in the sense that the special time-independent case reduces to Allard's theorem

Mean Curvature Flow of Mean Convex Hypersurfaces

“…Remark C.3. For general Brakke flows the proof of the local regularity theorem is very difficult [4,28]. However, as pointed out by White [39] (see also [1,11]), there is a fairly simple argument in the smooth setting.…”

On the mean curvature flow of grain boundaries

“…We note that the claim of Theorem 3.6 is slightly different from [32,45] in that it is stated for (x, t) ∈ R n+1 \ S t here instead of spt V t \ S t , allowing a possibility of O (x,t) ∩ spt µ being empty. But exactly the same proof of [32] gives this slightly stronger claim of partial regularity and we write the result in this form.…”

“…t ∈ R + . Under this unit density assumption, the results of partial regularity theory of [8,32,45] (see also [34]) apply to this flow. Theorem 3.6.…”

“…The following is a variant of well-known Huisken's monotonicity formula [26]. We include the outline of proof and the reader is advised to see [32,Lemma 6.1] for more details.…”

“…, N }, i = i , such that x ∈ ∂E 0,i ∩ ∂E 0,i ), then there exists a space-time neighborhood of (x, 0) in which the constructed flow is C 1 in the parabolic sense up to t = 0 and C ∞ for t > 0. This can be proved by using a C 1,α regularity theorem in [32] as demonstrated in [43, Theorem 2.3(4)] for a phase field setting.…”