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Applied Analysis Seminar
Homogenization of Linear Elliptic Equations in Nondivergence-Form: Characterizations of Good Diffusion Matrices
Homogenization of Linear Elliptic Equations in Nondivergence-Form: Characterizations of Good Diffusion Matrices
Organizer
Speaker
Timo Sprekeler
Time
Friday, January 7, 2022 2:00 PM - 3:00 PM
Venue
1110
Online
Zoom 388 528 9728
(BIMSA)
Abstract
In this talk, we discuss the periodic homogenization of linear elliptic equations of the form
$-A(x/\varepsilon):D^2 u^{\varepsilon} = f$ subject to a Dirichlet boundary condition. We
characterize good diffusion matrices $A$, i.e., those for which the sequence of solutions
converges at a rate of $\mathcal{O}(\varepsilon^2)$ in the $L^{\infty}$-norm to the solution
of the homogenized problem. Such diffusion matrices are considered “good” as the optimal
rate of convergence in the generic case is only $\mathcal{O}(\varepsilon)$. First, we provide
a class of good diffusion matrices, confirming a conjecture posed by Guo and Tran in 2020.
Then, we give a complete characterization of diagonal diffusion matrices in two dimensions
and a systematic study in higher dimensions.
This talk is based on joint work with Xiaoqin Guo (University of Cincinnati) and Hung V. Tran
(University of Wisconsin Madison).