Hypertemperature effects in heterogeneous media and thermal flux at small-length scales

We employ an enriched microscopic heat conduction model that can account for size effects in heterogeneous media. Through, physically, relevant scaling arguments we improve the regularity of the corrector in the classical problem of periodic homogenization in the three-dimensional setting and, in doing so, we clarify the intimate role correctors play in measuring the difference between the heterogeneous solution (microscopic) and the homogenized solution (macroscopic). Moreover, if the data are of the form $f = div F$ with $F \in L^3 (Ω, R^3)$, then we can prove the classical corrector convergence theorem as well