# Singular perturbation of an elastic energy with a singular weight

## TL;DR

This paper investigates the effects of singular perturbations on elastic energies with singular weights, revealing multi-scale pattern formation, energy distribution, and convergence of minimizers to structured Young measures.

## Contribution

It introduces a new analysis of elastic energy minimization under singular weights, deriving energy scaling laws and convergence results for minimizers.

## Key findings

- Energy of minimizers is uniformly distributed across the sample.
- Minimizers converge to Young measures supported on functions with slopes ±1.
- Pattern formation occurs at multiple scales depending on the perturbation.

## Abstract

We study the singular perturbation of an elastic energy with a singular weight. The minimization of this energy results in a multi-scale pattern formation. We derive an energy scaling law in terms of the perturbation parameter and prove that, although one cannot expect periodicity of minimizers, the energy of a minimizer is uniformly distributed across the sample. Finally, following the approach developed by Alberti and M\"{u}ller in 2001 we prove that a sequence of minimizers of the perturbed energies converges to a Young measure supported on functions of slope $\pm 1$ and of period depending on the location in the domain and the weights in the energy.