Correctors and Field Fluctuations for the $p_{ϵ}(x)$-Laplacian with Rough Exponents

We provide a corrector theory for the strong approximation of fields inside composites made from two materials with different power law behavior. The correctors are used to develop bounds on the local singularity strength for gradient fields inside micro-structured media. The bounds are multi-scale in nature and can be used to measure the amplification of applied macroscopic fields by the microstructure.