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On a Problem Posed by Brezis and Mironescu

Fanghua Lin, Malkeil Shoshan, Changyou Wang

Abstract

The purpose of this note is to present a positive answer to an open problem proposed in the recent book \cite{Brezis-Mironescu} by H. Brezis and P. Mironescu. It has been stated in this book {\it Sobolev Maps to the Circle} as Proposition 4.3. We demonstrate, in particular, the value of the least mass of the area minimizing integral rectifiable currents with a given boundary equals to the infimum of areas among smoothly immersed submanifolds with the same boundary, under the assumption that the boundary is that of a smooth submanifold.

On a Problem Posed by Brezis and Mironescu

Abstract

The purpose of this note is to present a positive answer to an open problem proposed in the recent book \cite{Brezis-Mironescu} by H. Brezis and P. Mironescu. It has been stated in this book {\it Sobolev Maps to the Circle} as Proposition 4.3. We demonstrate, in particular, the value of the least mass of the area minimizing integral rectifiable currents with a given boundary equals to the infimum of areas among smoothly immersed submanifolds with the same boundary, under the assumption that the boundary is that of a smooth submanifold.
Paper Structure (7 sections, 14 theorems, 72 equations, 5 figures)

This paper contains 7 sections, 14 theorems, 72 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Notation and Preliminaries
  3. Cutting out a tubular neighborhood
  4. Applying Spherical Inversion
  5. Attaching with cones
  6. Proof of Theorem 1.1
  7. An example

Key Result

Theorem 1.1

For any $T \in \mathcal{F}_0^l$, $0\le l\le N-2$, the two definitions of least mass spanned by $T$ are equivalent, i.e. minimizing the areas of smoothly immersed and oriented $(l+1)$-dimensional submanifolds in $\Omega\times \mathbb{R}$ whose boundary is $T$ is equivalent to minimizing the masses of

Figures (5)

  • Figure 1: Sketch for the case $N \geq 4$, $l = N-2$, $\Gamma$ is a $(l+1)$ integer rectifiable current, $\mathcal{S}$ has Hausdorff dimension $(l-2)$.
  • Figure 2: An $\varepsilon$-tubular neighborhood of $\mathcal{S}$.
  • Figure 3: Removing the tubular neighborhood of the singular set results in $A = M_0 \cup T \cup \Gamma \setminus (E \cup S)$
  • Figure 4: A sketch of the image of $A$ after spherical inversion - not to scale
  • Figure 5: Attaching with cones

Theorems & Definitions (25)

  • Theorem 1.1
  • Remark 1.2
  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3: Coarea Formula
  • Definition 2.4
  • Lemma 2.5: Slicing
  • Theorem 2.6: Federer-Fleming Compactness
  • Theorem 2.7
  • Theorem 2.8: Monotonicity
  • ...and 15 more