Title: Homogenization of diffusion processes with singular drifts and potentials via unfolding method Show Chinese title

Title: 通过展开方法对具有奇异漂移和势的扩散过程进行均质化

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. Show Chinese abstract

Abstract: 本工作涉及椭圆方程的均质化问题，其类型为\[ \begin{cases} \mathfrak{L}_{\delta} u_{\delta} + \lambda u_{\delta} = f_{\delta} \qquad \text{in} \;\; D, \\ \qquad \quad \;\, u = 0 \qquad \, \text{on} \;\; \partial D, \end{cases} \]，其中$\delta > 0$,$\lambda \in \mathbb{R}$,$D$是$\mathbb{R}^{d}$中的一个有界开集，且$f_{\delta} \in H^{-1}(D)$。 算子$ \mathfrak{L}_{\delta} u = -{\rm div} \left( A^\delta \nabla u + C^\delta u \right) + B^\delta \nabla u +k^\delta u $涉及一致有界的扩散系数$A^\delta$，其中漂移项$B^\delta$，$C^\delta$和势能$k^\delta$可能是无界的。也讨论了其对应扩散过程的均质化应用。