A general regularity theory for weak mean curvature flow

By Kota Kasai, Yoshihiro Tonegawa. 2011

64 citations recorded by the source corpus.

We give a new proof of Brakke’s partial regularity theorem up to $$C^{1,\varsigma }$$ 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.

Subfields: Applied Mathematics

View Paper: https://www.semanticscholar.org/paper/403aa8b2e2fd233dd538e06badb494a4612e8e9b

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In Applied Mathematics

Citations in the corpus

A sharp lower bound for the entropy of closed hypersurfaces up to dimension six (2014, methodology): “The reader may also consult the recent papers [18] and [26] which give a proo f using the monotonicity formula.”
On the mean curvature flow of grain boundaries (2015, methodology): “This can be proved by using a C1,α regularity theorem in [32] as demonstrated in [43, Theorem 2.”
A second derivative Hölder estimate for weak mean curvature flow (2012, methodology): “Just to avoid a possible confusion for the reader, we should point out that C 1;& regularity of [13] does not imply C 2; ̨ simply by the standard linear parabolic regularity theory.”
Mean Curvature Flow of Mean Convex Hypersurfaces (2013, background): “For general Brakke flows the proof of the local regularity theorem is very difficult [4,28].”