Homogenization Theory of Elliptic System with Lower Order Terms for Dimension Two
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Publication:6413733
DOI10.3934/CPAA.2023010zbMath1520.35005arXiv2210.06329MaRDI QIDQ6413733
Publication date: 12 October 2022
Abstract: In this paper, we consider the homogenization problem for generalized elliptic systems $$ mathcal{L}_{varepsilon}=-operatorname{div}(A(x/varepsilon)
abla+V(x/varepsilon))+B(x/varepsilon)
abla+c(x/varepsilon)+lambda I $$ with dimension two. Precisely, we will establish the $ W^{1,p} $ estimates, H"{o}lder estimates, Lipschitz estimates and $ L^p $ convergence results for $ mathcal{L}_{varepsilon} $ with dimension two. The operator $ mathcal{L}_{varepsilon} $ has been studied by Qiang Xu with dimension $ dgeq 3 $ in cite{Xu1,Xu2} and the case $ d=2 $ is remained unsolved. As a byproduct, we will construct the Green functions for $ mathcal{L}_{varepsilon} $ with $ d=2 $ and their convergence rates.
Smoothness and regularity of solutions to PDEs (35B65) Singular perturbations in context of PDEs (35B25) Homogenization in context of PDEs; PDEs in media with periodic structure (35B27) Boundary value problems for second-order elliptic systems (35J57)
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