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Optimal regularity for quasiminimal sets of codimension one in $\R^2$ and $\R^3$

Camille Labourie, Yana Teplitskaya

Abstract

Quasiminimal sets are sets for which a pertubation can decrease the area but only in a controlled manner. We prove that in dimensions $2$ and $3$, such sets separate a locally finite family of local John domains. Reciprocally, we show that this property is a sufficient for quasiminimality. In addition, we show that quasiminimal sets locally separate the space in two components, except at isolated points in $\R^2$ or out a of subset of dimension strictly less than $N-1$ in $\R^N$.

Optimal regularity for quasiminimal sets of codimension one in $\R^2$ and $\R^3$

Abstract

Quasiminimal sets are sets for which a pertubation can decrease the area but only in a controlled manner. We prove that in dimensions and , such sets separate a locally finite family of local John domains. Reciprocally, we show that this property is a sufficient for quasiminimality. In addition, we show that quasiminimal sets locally separate the space in two components, except at isolated points in or out a of subset of dimension strictly less than in .

Paper Structure

This paper contains 15 sections, 30 theorems, 193 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Definitions
  3. Standard results
  4. Ahlfors-regularity
  5. Uniform rectifiability
  6. No isolated components
  7. Interface points
  8. Local finiteness
  9. In the plane
  10. In the three-dimensional space
  11. Applications
  12. Quasiminimal sets and John domains
  13. Components of Omega minus K are local John domains
  14. A sufficient condition for quasiminimality
  15. The dimension of junction points

Key Result

Lemma 3.3

Let $K$ be a $M$-quasiminimal set in $\Omega$. For all $x_0 \in K$ and $r > 0$ such that $B(x_0,r) \subset \Omega$,

Figures (6)

  • Figure 1: A topological competitor of $K$ in a ball $B$.
  • Figure 2: An example of quasiminimal set with different blow-up limits at the origin.
  • Figure 3: An illustration for Corollary \ref{['lem_plane_separation']}.
  • Figure 4: The competitor in Proposition \ref{['prop_isolated']}.
  • Figure 5: Merging $V_2, V_3, V_5$ into $V_1$ within $B$.
  • ...and 1 more figures

Theorems & Definitions (68)

  • Definition 2.1
  • Remark 2.2
  • Remark 2.3: Motivation
  • Remark 2.4: Quasiminimal sets in fracture mechanics
  • Remark 2.5: Example
  • Remark 3.1
  • Remark 3.2
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • ...and 58 more