TY - JOUR
T1 - Boundary Homogenization of a Class of Obstacle Problems
AU - Li , Jingzhi
AU - Liu , Hongyu
AU - Tang , Lan
AU - Wang , Jiangwen
JO - Annals of Applied Mathematics
VL - 2
SP - 240
EP - 260
PY - 2022
DA - 2022/04
SN - 38
DO - http://doi.org/10.4208/aam.OA-2022-0001
UR - https://global-sci.org/intro/article_detail/aam/20456.html
KW - Homogenization, boundary obstacle, correctors, asymptotic analysis.
AB - <p style="text-align: justify;">We study the homogenization of a boundary obstacle problem on
a $C^{1,α}$-domain $D$ for some elliptic equations with uniformly elliptic coefficient
matrices $\gamma.$ For any $\epsilon \in \mathbb{R}_+,$ $∂D=\Gamma ∪Σ,$ $\Gamma ∩Σ=∅$ and $S_{\epsilon}\subset Σ$ with suitable assumptions, we prove that as $\epsilon$ tends to zero, the energy minimizer $u^{\epsilon}$ of $\int_D
|\gamma ∇u|^2dx,$ subject to $u≥\varphi$ on $S_{\epsilon},$ up to a subsequence, converges weakly in $H^1
(D)$ to $\tilde{u},$ which minimizes the energy functional $$\int_D
|\gamma∇u|^2+ \int_Σ
(u−\varphi)^2\_\mu (x)dS_x,$$ where $\mu (x)$ depends on the structure of $S_{\epsilon}$ and $\varphi$ is any given function in $C^∞(\overline{D}).$</p>