Homogenization estimates of operator type for an elliptic equation with quasiperiodic coefficients (Q2516042)

In the paper under review, the authors provide homogenization estimates of operator type for elliptic equations of divergent and nondivergent types in \(\mathbb{R}^d\), with \(d \geq 2\). As a first case, they consider the (divergent) equation \[ u_\epsilon \in H^1(\mathbb{R}^d)\, ,\quad (\mathcal{A}_\epsilon + 1)u_\epsilon=f\, , \quad f \in L^2(\mathbb{R}^d)\, , \] with \[ \mathcal{A}_\epsilon =-\mathrm{div}(a(\cdot/\epsilon)\nabla)\, , \] where \(a(\cdot)\) is a convenient quasiperiodic matrix. Under suitable assumptions, they provide estimates of the type \[ \|u_\epsilon-u\|_{L^2(\mathbb{R}^d)}\leq \epsilon C\|f\|_{L^2(\mathbb{R}^d)}\, , \] where \(u\) is the solution of the homogenized equation, and of the type \[ \|u_\epsilon-v_\epsilon\|_{H^1(\mathbb{R}^d)}\leq \epsilon C\|f\|_{L^2(\mathbb{R}^d)}\, , \] where \(v_\epsilon\) is otained by adding a corrector to the function \(u\). Then they turn to consider the nondivergent equation \[ u_\epsilon \in H^1(\mathbb{R}^d)\, , \quad (\mathcal{A}_\epsilon+1)u_\epsilon=f\, , \quad f \in L^2(\mathbb{R}^d)\, , \] with \[ \mathcal{A}_\epsilon =-a_{ij}(\cdot/\epsilon)\frac{\partial^2}{\partial x_i \partial x_j}\, , \] where, as above, \(a(\cdot)\) is a convenient quasiperiodic matrix. Then again, under suitable assumptions, they provide an estimate of the type \[ \|u_\epsilon-v_\epsilon\|_{H^1(\mathbb{R}^d)}\leq \epsilon C\|f\|_{L^2(\mathbb{R}^d)}\, , \] where \(v_\epsilon\) is the solution of an (\(\epsilon\)-dependent) averaged equation.