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$L^2$ normal velocity implies strong solution for graphical Brakke flows

Kotaro Motegi

Abstract

We prove that if a one-parameter family of varifolds has an $L^2$ normal velocity $v$ in the sense of Brakke, and if the family is represented as the graph of a continuous function $f$ with continuous spatial derivative $\nabla f$, then $f$ has weak derivatives $\partial_t f, \nabla^2 f \in L^2$, and $v$ coincides with the usual normal velocity of the graph. Moreover, by combining this result with parabolic regularity theory, we show that graphical Brakke flows with forcing term in $L^{p,q}$ and $C^{0,α}$ are strong and classical solutions to the forced mean curvature flow equation, respectively.

$L^2$ normal velocity implies strong solution for graphical Brakke flows

Abstract

We prove that if a one-parameter family of varifolds has an normal velocity in the sense of Brakke, and if the family is represented as the graph of a continuous function with continuous spatial derivative , then has weak derivatives , and coincides with the usual normal velocity of the graph. Moreover, by combining this result with parabolic regularity theory, we show that graphical Brakke flows with forcing term in and are strong and classical solutions to the forced mean curvature flow equation, respectively.

Paper Structure

This paper contains 8 sections, 6 theorems, 48 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and main results
  3. Basic notation
  4. Varifolds
  5. Generalized notion of normal velocity
  6. Main results
  7. Proof of Theorem \ref{['thm:main']}
  8. Regularity for Brakke flows with forcing term

Key Result

Theorem 1

Let $\Omega \subset \mathbb{R}^k$ be an open set and $I \subset \mathbb{R}$ an open interval. Suppose that a family $\{V_t\}_{t \in I}$ of varifolds in $\Omega \times \mathbb{R}^{n-k}$ has a normal velocity $v \in L^2_\mathrm{loc}(\mu;\mathbb{R}^n)$; that is, $\{V_t\}_{t \in I}$ and $v$ satisfy eq:i

Theorems & Definitions (15)

  • Theorem 1
  • Theorem 2
  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3
  • Remark 2.4
  • Remark 2.5
  • Theorem 2.6
  • Remark 2.7
  • Theorem 2.8
  • ...and 5 more