Periodic homogenization of elliptic operators with drifts

This thesis studies the homogenization of second-order elliptic operators in the divergence form. In particular, we study the boundary value problem formulated as follows:\[\label{problem}\begin{cases}\mathfrak {L}_{\delta} u_{\delta} + \lambda u_{\delta} = f_{\delta} \quad \text {in} \quad \Omega, \\\qquad \quad \; u_{\delta} = 0 \quad \; \text{on} \quad \partial \Omega,\end{cases}\]where $\lambda \in \mathbb{R}$, $\delta>0$ and $\Omega$ is a nonempty open set in $\mathbb{R}^{d}$ with $d\geq 3$. Here $\mathfrak{L}_{\delta}$ is a second-order elliptic operator and $f_{\delta}$ is a source term which represents a real-valued function on $H^{1}_{0}(\Omega)$. In the initial case, $\mathfrak{L}_{\delta}$ is defined by \[\mathfrak{L}_{\delta}= - \text{div}(A^{\delta} \nabla u + C^{\delta}u ) + B^{\delta} \nabla u + k^{\delta} u\]Here $A^{\delta}$ denotes a diffusion matrix with bounded measurable coefficients. $B^{\delta}$ and $C^{\delta}$ are the drift terms and $k^{\delta}$ is a potential, with all components in variable exponent Lebesgue spaces.A second case considers the operator $\mathfrak{L}_{\delta}$, given by\[\mathfrak{L}_{\delta}= - \text{div}(A^{\delta} \nabla u ) + B^{\delta} \nabla u \]where the components of the drift terms $B^{\delta}$ are assumed to be in Lorentz spaces. To consider these homogenization problems, we define the periodic unfolding operator within the framework of both variable exponent Lebesgue spaces and Lorentz spaces. Furthermore, we extend the known equivalence of two-scale convergence and the weak convergence of unfolded sequence to these settings.