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Convex minimizing curves of the scaling-invariant nonlocal Willmore energy

Giovanni Giacomin, Armin Schikorra

Abstract

We consider the scaling-invariant nonlocal Willmore energy, defined via the nonlocal mean curvature by Caffarelli, Roquejoffre and Savin. Our main result is the existence of minimizers in the class of convex $C^1$-curves.

Convex minimizing curves of the scaling-invariant nonlocal Willmore energy

Abstract

We consider the scaling-invariant nonlocal Willmore energy, defined via the nonlocal mean curvature by Caffarelli, Roquejoffre and Savin. Our main result is the existence of minimizers in the class of convex -curves.
Paper Structure (14 sections, 37 theorems, 378 equations, 2 figures)

This paper contains 14 sections, 37 theorems, 378 equations, 2 figures.

Table of Contents

  1. Introduction and main results
  2. Preliminaries and Notation
  3. Sobolev spaces
  4. The fractional mean curvature
  5. The maximum principle for the nonlocal mean curvature
  6. Small energy controls BMO-parametrization
  7. Consequence of VMO-control: BiLipschitz-control and graph
  8. Sobolev control
  9. Self-repulsiveness of Willmore energy for convex curves
  10. Convex curves with finite energy are continuously differentiable
  11. Proof of Theorem \ref{['MainTheorem-23']}
  12. Facts about convex curves in R2
  13. Integration formula for the mean curvature -- Proof of Proposition \ref{['salame']}
  14. The covering argument: Locally uniform smallness

Key Result

Theorem 1.1

For any $s \in (0,1)$ there exists a minimizer of $\mathscr{W}_s$ in $X$.

Figures (2)

  • Figure 1: The barrier $G$ from \ref{['lm:barrierestimates']}
  • Figure 2: The geometry in the proof of \ref{['le:cvdfqw']}

Theorems & Definitions (69)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Theorem 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4: Sobolev inequality
  • proof
  • Lemma 2.5
  • proof
  • ...and 59 more