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

TL;DR

This work addresses homogenization of elliptic Dirichlet problems with potential singular drifts by combining unfolding with a regular lower bounded semi-Dirichlet form framework. It develops δ-dependent forms with unbounded drifts, establishes well-posedness and a priori estimates, and derives two-scale homogenization results, including explicit cell formulas for $A^{\mathrm{eff}}, B^{\mathrm{eff}}, C^{\mathrm{eff}}, k^{\mathrm{eff}}$ and convergence of $u_\lambda^\delta$ to $u_0$ with a corrector $u_1$. The main contributions extend homogenization to unbounded drift and singular potential terms and connect PDE limits with diffusion-process convergence via Mosco convergence. This enhances the theoretical toolkit for multiscale diffusion in heterogeneous media and informs stochastic-process descriptions in periodic environments.

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 $δ> 0$, $λ\in \mathbb{R}$, $D$ is a bounded open set in $\mathbb{R}^{d}$, and $f_δ \in H^{-1}(D)$. The operator $ \mathfrak{L}_δ u = -{\rm div} \left( A^δ\nabla u + C^δu \right) + B^δ\nabla u +k^δu $ involved uniformly bounded diffusion coefficients $A^δ$, where drifts $B^δ$, $C^δ$, and potential $k^δ$ are possibly unbounded. An application to homogenization of the corresponding diffusion processes is also discussed.

Theorems & Definitions (29)

• Lemma 2.1
• Remark 2.1
• Proposition 2.1
• Theorem 2.1
• Corollary 2.1
• Theorem 2.2
• Corollary 2.2
• Remark 2.2
• Theorem 2.3
• proof