Correctors and field fluctuations for the -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 micro-structure.