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Epsilon-regularity for the Brakke flow with boundary

Carlo Gasparetto

TL;DR

The paper advances boundary regularity for mean curvature flow by establishing an epsilon-regularity result for integral Brakke flows with fixed boundary: if the flow is $\varepsilon$-close to a unit-density half-plane in a unit ball, then it enjoys $C^{1,\alpha}$ regularity up to the boundary in a smaller region after a waiting time. The authors adapt Savin's viscosity approach to the parabolic Brakke setting, proving an improvement-of-flatness through a compactness argument that yields a heat-equation limit, and then applying boundary Schauder estimates to obtain higher regularity. A boundary-appropriate maximum principle and barrier arguments underpin the analysis, ensuring the flow cannot develop singularities near the fixed boundary under the flatness hypotheses. The results provide backward regularity (smoothness in a backward time neighborhood) for flows with boundary tangent to a unit-density half-plane and offer a novel, viscosity-based route to boundary regularity in mean curvature flow, complementing variational approaches like Allard-type theory.

Abstract

We prove that, if a Brakke flow with boundary is close enough to a stationary half-plane with density one, then it is $C^{1,α}$. Our approach is based on viscosity techniques introduced by Savin in the context of elliptic equations. The same techniques can be used to give a proof of Brakke's (interior) regularity theorem which is alternative to the original one.

Epsilon-regularity for the Brakke flow with boundary

TL;DR

The paper advances boundary regularity for mean curvature flow by establishing an epsilon-regularity result for integral Brakke flows with fixed boundary: if the flow is -close to a unit-density half-plane in a unit ball, then it enjoys regularity up to the boundary in a smaller region after a waiting time. The authors adapt Savin's viscosity approach to the parabolic Brakke setting, proving an improvement-of-flatness through a compactness argument that yields a heat-equation limit, and then applying boundary Schauder estimates to obtain higher regularity. A boundary-appropriate maximum principle and barrier arguments underpin the analysis, ensuring the flow cannot develop singularities near the fixed boundary under the flatness hypotheses. The results provide backward regularity (smoothness in a backward time neighborhood) for flows with boundary tangent to a unit-density half-plane and offer a novel, viscosity-based route to boundary regularity in mean curvature flow, complementing variational approaches like Allard-type theory.

Abstract

We prove that, if a Brakke flow with boundary is close enough to a stationary half-plane with density one, then it is . Our approach is based on viscosity techniques introduced by Savin in the context of elliptic equations. The same techniques can be used to give a proof of Brakke's (interior) regularity theorem which is alternative to the original one.
Paper Structure (19 sections, 25 theorems, 226 equations)

This paper contains 19 sections, 25 theorems, 226 equations.

Table of Contents

  1. Introduction
  2. Structure of the paper
  3. Preliminaries, notation and definitions
  4. Space-time
  5. Linear functions and subspaces of the euclidean space
  6. Hölder regularity
  7. Integral Varifolds
  8. Integral Brakke flows with boundary
  9. Properties of Integral Brakke flows with boundary
  10. Monotonicity properties
  11. Maximum principle
  12. Improvement of flatness
  13. Preliminaries to the proof of the improvement of flatness
  14. Proof of the improvement of flatness
  15. Boundary behavior
  16. ...and 4 more sections

Key Result

Theorem 1.1

Let $\Gamma$ be a $C^{1,\alpha}$, $(m-1)$-dimensional submanifold of $B_1$ and let $\{V_t\}_{t\in[-\Lambda,0]}$ be an integral Brakke flow with boundary $\Gamma$ in $B_1\times[-\Lambda,0]$. Assume the following: If $\varepsilon$ and $\Lambda$ are small enough, then there exist small constants $\eta,\beta$ and a family $\{N_t\}_{t\in(-\eta^2,0]}$ of $C^{1,\beta}$ surfaces with boundary $\Gamma$ su

Theorems & Definitions (35)

  • Theorem 1.1: $\varepsilon$-regularity
  • Proposition 1.2: Improvement of flatness
  • Remark 2.1
  • Definition 2.2
  • Definition 2.3: Integral Brakke flow
  • Remark 2.4: Scaling properties
  • Proposition 3.1: Huisken monotonicity formula
  • Lemma 3.2
  • Definition 3.3: Maximal density ratio
  • Proposition 3.4: Clearing-out lemma
  • ...and 25 more