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

TL;DR

This paper develops a corrector theory to accurately approximate and bound the local singularity strength of gradient fields in micro-structured media with rough exponents, aiding understanding of field amplification in composites.

Contribution

It introduces a novel corrector framework for the $p_{ ext{epsilon}}(x)$-Laplacian with rough exponents, providing multi-scale bounds on field singularities in composite materials.

Findings

Bounds on local singularity strength established

Multi-scale analysis of field amplification achieved

Corrector theory improves approximation accuracy

Abstract

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.