Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Energy-minimizing torus-valued maps with prescribed singularities, Plateau's problem, and BV-lifting

Giacomo Canevari, Van Phu Cuong Le

Abstract

In this paper, we investigate the relation between energy-minimizing torus-valued maps with prescribed singularities, the lifting problem for torus-valued maps in the space BV, and Plateau's problem for vectorial currents, in codimension one. First, we show that the infimum of the $W^{1,1}$-seminorm among all maps with values in the $k$-dimensional flat torus and prescribed topological singularities $S$ is equal to the minimum of the mass among all $\textit{normal}$ $\mathbb{R}^k$-currents, of codimension one, bounded by $S$. Then, we show that the minimum of the $BV$-energy among all liftings of a given torus-valued $W^{1,1}$-map $\textbf{u}$ can be expressed in terms of the minimum mass among all $\textit{integral}$ $\mathbb{Z}^k$-currents, of codimension one, bounded by the singularities of $\textbf{u}$. As a byproduct of our analysis, we provide a bound for the solution of the integral Plateau problem, in codimension one, in terms of Plateau's problem for normal currents.

Energy-minimizing torus-valued maps with prescribed singularities, Plateau's problem, and BV-lifting

Abstract

In this paper, we investigate the relation between energy-minimizing torus-valued maps with prescribed singularities, the lifting problem for torus-valued maps in the space BV, and Plateau's problem for vectorial currents, in codimension one. First, we show that the infimum of the -seminorm among all maps with values in the -dimensional flat torus and prescribed topological singularities is equal to the minimum of the mass among all -currents, of codimension one, bounded by . Then, we show that the minimum of the -energy among all liftings of a given torus-valued -map can be expressed in terms of the minimum mass among all -currents, of codimension one, bounded by the singularities of . As a byproduct of our analysis, we provide a bound for the solution of the integral Plateau problem, in codimension one, in terms of Plateau's problem for normal currents.
Paper Structure (13 sections, 16 theorems, 261 equations)

This paper contains 13 sections, 16 theorems, 261 equations.

Table of Contents

  1. Introduction
  2. Notations and preliminaries
  3. Preliminary notions
  4. Currents with coefficients in normed groups and Plateau's problems
  5. Push-forward of currents
  6. The Hodge-star operator and the codifferential
  7. Convolution of currents
  8. Convolution of currents with mollifiers
  9. Convolution of currents with vector-valued maps and differential forms
  10. The distributional Jacobian
  11. Energy-minimizing maps with prescribed singularities and mass minimization for normal currents
  12. BV-liftings of torus-valued maps and mass minimization for integral currents
  13. Appendix

Key Result

Theorem 1

Let $S$ be a $\mathbb{Z}^k$-integral flat boundary, of dimension $(d-2)$. Assume that $S$ has finite mass and compact support and the condition hp:H is satisfied. Then, for any $p\in [1, \, \infty]$, one has

Theorems & Definitions (58)

  • Theorem 1
  • Remark 1
  • Theorem 2
  • Theorem 3
  • Remark 2
  • Remark 3
  • Definition 2.4
  • Remark 2.5
  • Definition 2.6
  • Definition 2.7
  • ...and 48 more