  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.
