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

We propose an enriched microscopic heat conduction model that can account for size effects in heterogeneous media. Benefiting from physically relevant scaling arguments, we improve the regularity of the corrector in the classical problem of periodic homogenization of linear elliptic equations in the three-dimensional setting and, while doing so, we clarify the intimate role that correctors play in measuring the difference between the heterogeneous solution (microscopic) and the homogenized solution (macroscopic). Moreover, if the data are of form $f=\mathrm{d}\mathrm{i}\mathrm{v}\mathit{F}$ with $\mathit{F}\in {\mathrm{L}}^3(\mathrm{\Omega },{\mathbb{R}}^3)$, then we recover the classical corrector convergence theorem.