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Yet another proof of the density in energy of Lipschitz functions

Danka Lučić, Enrico Pasqualetto

Abstract

We provide a new, short proof of the density in energy of Lipschitz functions into the metric Sobolev space defined by using plans with barycenter (and thus, a fortiori, into the Newtonian-Sobolev space). Our result covers first-order Sobolev spaces of exponent $p\in(1,\infty)$, defined over a complete and separable metric space endowed with a boundedly-finite Borel measure. Our proof is based on a completely smooth analysis: first we reduce the problem to the Banach space setting, where we consider smooth functions instead of Lipschitz ones, then we rely on classical tools in convex analysis and on the superposition principle for normal $1$-currents. Along the way, we obtain a new proof of the density in energy of smooth cylindrical functions in Sobolev spaces defined over a separable Banach space endowed with a finite Borel measure.

Yet another proof of the density in energy of Lipschitz functions

Abstract

We provide a new, short proof of the density in energy of Lipschitz functions into the metric Sobolev space defined by using plans with barycenter (and thus, a fortiori, into the Newtonian-Sobolev space). Our result covers first-order Sobolev spaces of exponent , defined over a complete and separable metric space endowed with a boundedly-finite Borel measure. Our proof is based on a completely smooth analysis: first we reduce the problem to the Banach space setting, where we consider smooth functions instead of Lipschitz ones, then we rely on classical tools in convex analysis and on the superposition principle for normal -currents. Along the way, we obtain a new proof of the density in energy of smooth cylindrical functions in Sobolev spaces defined over a separable Banach space endowed with a finite Borel measure.
Paper Structure (11 sections, 6 theorems, 37 equations)

This paper contains 11 sections, 6 theorems, 37 equations.

Table of Contents

  1. Introduction
  2. General overview.
  3. The proof strategy
  4. Some additional comments
  5. Preliminaries
  6. Metric and measure spaces
  7. Banach spaces and 1-currents
  8. Metric Sobolev spaces
  9. Sobolev spaces via relaxation
  10. Sobolev spaces via plans
  11. Main results

Key Result

Theorem 2.1

Let $\mathbb{V}$ be a finite-dimensional Banach space. Then for every acyclic current $T\in{\bf N}_1(\mathbb{V})$ there exists $\boldsymbol{\pi}\in\mathfrak M_+(C([0,1];\mathbb{V}))$ concentrated on non-constant Lipschitz curves of constant speed such that $({\rm e}_+)_\#\boldsymbol{\pi}=(\partial T

Theorems & Definitions (23)

  • Theorem 2.1: Superposition principle
  • proof
  • Definition 2.2: Cylindrical function
  • Remark 2.3
  • Definition 2.4: Sobolev space via relaxation of Lipschitz functions
  • Definition 2.5: Sobolev space via relaxation of cylindrical functions
  • Definition 2.6: Plan with barycenter
  • Lemma 2.7
  • proof
  • Definition 2.8: Sobolev space via plans
  • ...and 13 more