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

TL;DR

This paper develops a homogenization framework for elliptic equations with possibly unbounded singular drifts and potentials, using the unfolding method, and applies it to diffusion processes.

Contribution

It introduces a homogenization approach for elliptic equations with singular, unbounded coefficients using the unfolding method, extending previous results to more general operators.

Findings

Established homogenization results for operators with unbounded drifts and potentials.

Derived effective equations describing the limit behavior of diffusion processes.

Demonstrated the applicability of the unfolding method to complex elliptic operators.

Abstract

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