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A Global Method for Relaxation for Multi-levelled Structured Deformations

Ana Cristina Barroso, José Matias, Elvira Zappale

Abstract

We prove an integral representation result for a class of variational functionals appearing in the framework of hierarchical systems of structured deformations via a global method for relaxation. Some applications to specific relaxation problems are also provided.

A Global Method for Relaxation for Multi-levelled Structured Deformations

Abstract

We prove an integral representation result for a class of variational functionals appearing in the framework of hierarchical systems of structured deformations via a global method for relaxation. Some applications to specific relaxation problems are also provided.
Paper Structure (10 sections, 10 theorems, 124 equations)

This paper contains 10 sections, 10 theorems, 124 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Approximation theorem for hierarchical (first-order) structured deformations
  5. The global method
  6. Applications
  7. $2$-level (first-order) structured deformations
  8. Homogenization problems
  9. Functionals arising in the analysis of second-order structured deformations
  10. Multi-levelled structured deformations

Key Result

Lemma 2.1

Let $\lambda$ be a non-negative Radon measure in $\mathbb R^N$. For $\lambda$ a.e. $x_0 \in \mathbb R^N$, for every $0 < \delta < 1$ and for every $\nu \in \mathbb S^{N-1}$, the following holds so that

Theorems & Definitions (18)

  • Lemma 2.1
  • Definition 2.2
  • Theorem 2.3: Approximation Theorem in $SD(\Omega)$
  • Definition 2.4
  • Theorem 2.5: Approximation Theorem for $(L+1)$-level structured deformations
  • Remark 3.1
  • Theorem 3.2
  • Remark 3.3
  • Lemma 3.4
  • proof
  • ...and 8 more