Multi-level structured deformations: relaxation via a global method approach

TL;DR

The paper develops a comprehensive global method for relaxation of energies in multi-level structured deformations, establishing integral representations in the spaces $HSD_L^p(\Omega)$ and $SD_{L,p}(\Omega)$ via blow-up analyses of a Dirichlet-type functional $m$. It provides explicit relaxed densities $f$ and $\Phi$ (and their multi-level variants $f_{2,p}$, $\Phi_{2,p}$) in terms of local minimization problems, and demonstrates that the approach recovers classical Choksi–Fonseca results under weaker hypotheses, while also enabling iterated relaxation across hierarchical levels. The framework unifies macro-, sub-macroscopic, and disarrangement effects and applies to both homogeneous and inhomogeneous bulk energies, with careful treatment of different integrability regimes ($p>1$ vs $p=1$). The results offer a versatile tool for analyzing energies in materials with hierarchical microstructures and disarrangements, with potential applications to crystals, composites, and multi-scale elasticity.

Abstract

We present some relaxation and integral representation results for energy functionals in the setting of structured deformations, with special emphasis given to the case of multi-level structured deformations. In particular, we present an integral representation result for an abstract class of variational functionals in this framework via a global method for relaxation and identify, under quite general assumptions, the corresponding relaxed energy densities through the study of a related local Dirichlet-type problem. Some applications to specific relaxation problems are also mentioned, showing that our global method approach recovers some previously established results.

Theorems & Definitions (15)

• Theorem 2.1: Approximation Theorem - Del Piero & Owen
• Theorem 2.2: Alberti
• Theorem 2.3: Approximation Theorem - Choksi & Fonseca
• proof
• Definition 4.1
• Theorem 4.2
• proof
• Theorem 5.1
• proof
• Remark 5.2