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A notion of $s$-fractional mass for $1$-currents in higher codimension

Marco Cicalese, Tim Heilmann, Andrea Kubin, Fumihiko Onoue, Marcello Ponsiglione

Abstract

In this paper we propose a notion of $s$-fractional mass for $1$-currents in $\R^d$. Such a notion generalizes the notion of $s$-fractional perimeters for sets in the plane to higher codimension one-dimensional singularities. Remarkably, the limit as $s\to 1$ of the $s$-fractional mass gives back the classical notion of length for regular enough curves in $\R^d$. We prove a lower semi-continuity and compactness result for sequences of $1$-currents with uniformly bounded fractional mass and support. Moreover, we prove the density of weighted polygonal, closed and compact oriented curves in the class of divergence-free 1-currents with compact support and finite fractional mass. Finally, we discuss some possible applications of our notion of fractional mass to build up purely geometrical approaches to the variational modeling of dislocation lines in crystals and to vortex filaments in superconductivity.

A notion of $s$-fractional mass for $1$-currents in higher codimension

Abstract

In this paper we propose a notion of -fractional mass for -currents in . Such a notion generalizes the notion of -fractional perimeters for sets in the plane to higher codimension one-dimensional singularities. Remarkably, the limit as of the -fractional mass gives back the classical notion of length for regular enough curves in . We prove a lower semi-continuity and compactness result for sequences of -currents with uniformly bounded fractional mass and support. Moreover, we prove the density of weighted polygonal, closed and compact oriented curves in the class of divergence-free 1-currents with compact support and finite fractional mass. Finally, we discuss some possible applications of our notion of fractional mass to build up purely geometrical approaches to the variational modeling of dislocation lines in crystals and to vortex filaments in superconductivity.
Paper Structure (5 sections, 16 theorems, 100 equations)

This paper contains 5 sections, 16 theorems, 100 equations.

Table of Contents

  1. A notion of $s$-fractional mass for $1$-current and its properties
  2. Asymptotic behavior of the $s$-fractional mass and $s$-fractional curvature for smooth curves as $s \uparrow 1$
  3. Compactness
  4. Characterization of the boundary of the sets of finite $s$-fractional perimeter in $\mathbb{R}^2$
  5. Approximation theorem

Key Result

Proposition 1.1

Let $E \subset \mathbb{R}^2$ be an open bounded set with $C^1$ boundary. For all $s \in (0,1)$ where we identify $\partial E$ as an element of $\Gamma$ in the obvious way.

Theorems & Definitions (36)

  • Proposition 1.1
  • proof
  • Proposition 1.2: First variation of $s$-fractional mass
  • proof
  • Definition 1.3: $s$-fractional curvature
  • Lemma 1.4
  • proof
  • Definition 1.5
  • Remark 1.6
  • Remark 1.7
  • ...and 26 more