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

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