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

TL;DR

This paper develops a corrector theory to approximate gradient fields in periodic composites with materials exhibiting different power law behaviors, providing bounds on local singularities and field amplification.

Contribution

It introduces a novel corrector framework for the $p_{ abla}(x)$-Laplacian with rough exponents in the sublinear growth case, enabling multi-scale analysis of microstructure effects.

Findings

Bounds on local singularity strength for gradient fields

Multi-scale measures of field amplification

Applicable to materials with exponents between -1 and 0

Abstract

A corrector theory for the strong approximation of gradient fields inside periodic composites made from two materials with different power law behavior is provided. Each material component has a distinctly different exponent appearing in the constitutive law relating gradient to flux. 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. The results in this paper are developed for materials having power law exponents strictly between -1 and zero.