Homogenization of diffusion processes with singular drifts and potentials via unfolding method

Abstract

This work is concerned with homogenization problems for elliptic equations of the type \[ \begin{cases} \mathfrak{L}_δ u_δ + λu_δ = f_δ \qquad \text{in} \;\; D, \\ \qquad \quad \;\, u = 0 \qquad \, \text{on} \;\; \partial D, \end{cases} \] where $\hat{I}´>0$, $\hat{I}»\in \mathbb{R}$, $D$ is a bounded open set in ${\mathbb{R}}^d$, and $f_{\hat{I}}´\in H^{-1}(D)$. The operator ${\mathfrak{L}}_{\hat{I}}´u=-\mathrm{div}\left(A^{\hat{I}}´\nabla u+C^{\hat{I}}´u\right)+B^{\hat{I}}´\nabla u+k^{\hat{I}}´u$ involved uniformly bounded diffusion coefficients $A^{\hat{I}}´$, where drifts $B^{\hat{I}}´$, $C^{\hat{I}}´$, and potential $k^{\hat{I}}´$ are possibly unbounded. An application to homogenization of the corresponding diffusion processes is also discussed.

24 pages