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A short proof of Allard's and Brakke's regularity theorems

Guido De Philippis, Carlo Gasparetto, Felix Schulze

Abstract

We give new short proofs of Allard's regularity theorem for varifolds with bounded first variation and Brakke's regularity theorem for integral Brakke flows with bounded forcing. They are based on a decay of flatness, following from weighted versions of the respective monotonicity formulas, together with a characterization of non-homogeneous blow-ups using the viscosity approach introduced by Savin.

A short proof of Allard's and Brakke's regularity theorems

Abstract

We give new short proofs of Allard's regularity theorem for varifolds with bounded first variation and Brakke's regularity theorem for integral Brakke flows with bounded forcing. They are based on a decay of flatness, following from weighted versions of the respective monotonicity formulas, together with a characterization of non-homogeneous blow-ups using the viscosity approach introduced by Savin.
Paper Structure (8 sections, 14 theorems, 125 equations)

This paper contains 8 sections, 14 theorems, 125 equations.

Table of Contents

  1. Introduction
  2. Allard's regularity theorem
  3. Decay of oscillations
  4. Allard's theorem
  5. Brakke's regularity theorem
  6. Decay of oscillations
  7. Brakke's theorem
  8. Maximum principles

Key Result

Theorem 2.1

For every $\alpha\in(0,1)$, there are $\delta_0>0$ and $C>0$ with the following property. Let $M\in\mathcal{M}_m^\infty(B_1)$ and assume that $0\in\mathop{\mathrm{supp}}\nolimits M$, $\Theta^m(M,x)\geq1$ for $M$-almost every $x$, Then $\mathop{\mathrm{supp}}\nolimits M\cap B_{1/2}$ is the graph of some function $u\in C^{1,\alpha}(B_1^m;{\mathbb R}^{d-m})$ with

Theorems & Definitions (20)

  • Theorem 2.1: Allard's regularity theorem
  • Proposition 2.3: Weighted monotonicity formula
  • Remark 2.4
  • Remark 2.5
  • Proposition 2.6: Harnack inequality
  • Proposition 2.7
  • Theorem 2.8: Improvement of flatness
  • Lemma 2.9
  • Definition 3.1: Brakke flow with transport term
  • Remark 3.2
  • ...and 10 more