Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the Lavrentiev gap for manifold-valued maps

Carlo Alberto Antonini, Filomena De Filippis, Cintia Pacchiano Camacho

TL;DR

This work investigates smooth approximation and the Lavrentiev gap for manifold-valued maps in Musielak–Orlicz spaces $W^{1,\varphi}(\mathsf{M},\mathsf{N})$ with nonuniform, double-phase growth. It extends classical density results by permitting $x$-dependent growth and establishes density of smooth maps under integral-growth and topological assumptions; additionally, it proves density under a $\text{k}$-connected target when the growth is sufficiently controlled, ensuring the absence of Lavrentiev phenomena. The authors identify sharp conditions under which density fails, providing a vectorial Lavrentiev counterexample in the double-phase setting, thereby clarifying the necessity of the local growth constraint. Overall, the paper delivers a comprehensive framework for the density of smooth manifold-valued maps in nonstandard growth spaces and clarifies the interplay between growth, topology, and Lavrentiev gaps. This advances the understanding of regularity and approximation for variational problems with nonhomogeneous, anisotropic energies on manifolds.

Abstract

We investigate the validity and the failure of modular density of smooth maps on compact manifolds.

On the Lavrentiev gap for manifold-valued maps

TL;DR

This work investigates smooth approximation and the Lavrentiev gap for manifold-valued maps in Musielak–Orlicz spaces with nonuniform, double-phase growth. It extends classical density results by permitting -dependent growth and establishes density of smooth maps under integral-growth and topological assumptions; additionally, it proves density under a -connected target when the growth is sufficiently controlled, ensuring the absence of Lavrentiev phenomena. The authors identify sharp conditions under which density fails, providing a vectorial Lavrentiev counterexample in the double-phase setting, thereby clarifying the necessity of the local growth constraint. Overall, the paper delivers a comprehensive framework for the density of smooth manifold-valued maps in nonstandard growth spaces and clarifies the interplay between growth, topology, and Lavrentiev gaps. This advances the understanding of regularity and approximation for variational problems with nonhomogeneous, anisotropic energies on manifolds.

Abstract

We investigate the validity and the failure of modular density of smooth maps on compact manifolds.

Paper Structure

This paper contains 6 sections, 7 theorems, 102 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Properties of Young functions
  5. Musielak-Orlicz spaces
  6. A few useful results

Key Result

Theorem 1.1

Let $\mathsf{M}$ and $\mathsf{N}$ be Riemannian manifolds as in M1, N1. Let $\varphi: \mathsf{M} \times [0,\infty) \to [0,\infty)$ be a function satisfying v1-v4 and denote by Assume either or Then, $C^{\infty}(\mathsf{M},\mathsf{N})$ is dense in $W^{1,\varphi}(\mathsf{M},\mathsf{N})$.

Theorems & Definitions (15)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.1
  • Corollary 1.3
  • Theorem 1.4
  • Definition 2.1: $j$-connected manifolds
  • Definition 2.2: Vanishing web oscillations HIMO
  • Remark 2.1
  • Theorem 2.3
  • proof
  • ...and 5 more