and vanishing on $\Gamma_D$ in the sense of trace, and one can easily check that $V$ is Hilbertian, and the norm can be legally set as ${\left\lVert \cdot \right\rVert}_V={\left\lvert \cdot \right\rvert}_{1,\Omega}$ when $\Gamma_D \neq \varnothing$. Following the notations in [@Han2017], we denote $V_j=L^2(\Gamma_C)$ as the main space for hemivariational inequality and $\gamma_j \in \mathcal{L}(V,V_j)$ as trace operator from $V$ to $V_j$. We point here that the split $\Gamma= {    \sbox{\myboxA}{$\m@th\Gamma_D$}    \setbox\myboxB\null    \ht\myboxB=\ht\myboxA    \dp\myboxB=\dp\myboxA    \wd\myboxB=0.75\wd\myboxA    \sbox\myboxB{$\m@th\overline{\copy\myboxB}$}    \setlength\mylenA{\the\wd\myboxA}    \addtolength\mylenA{-\the\wd\myboxB}    \ifdim\wd\myboxB<\wd\myboxA    \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA}    \else
    \hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB}    \fi} \cup {    \sbox{\myboxA}{$\m@th\Gamma_N$}    \setbox\myboxB\null    \ht\myboxB=\ht\myboxA    \dp\myboxB=\dp\myboxA    \wd\myboxB=0.75\wd\myboxA    \sbox\myboxB{$\m@th\overline{\copy\myboxB}$}    \setlength\mylenA{\the\wd\myboxA}    \addtolength\mylenA{-\the\wd\myboxB}    \ifdim\wd\myboxB<\wd\myboxA    \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA}    \else
    \hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB}    \fi} \cup {    \sbox{\myboxA}{$\m@th\Gamma_C$}    \setbox\myboxB\null    \ht\myboxB=\ht\myboxA    \dp\myboxB=\dp\myboxA    \wd\myboxB=0.75\wd\myboxA    \sbox\myboxB{$\m@th\overline{\copy\myboxB}$}    \setlength\mylenA{\the\wd\myboxA}    \addtolength\mylenA{-\the\wd\myboxB}    \ifdim\wd\myboxB<\wd\myboxA    \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA}    \else
    \hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB}    \fi}$ must be regular enough to guarantee that $\gamma_j$ is compact, and normally it is true because that fractional Sobolev space $H^{1/2}(\Gamma_C)$ is compactly embedded into $L^2(\Gamma_C)$ (refer [@Nezza2012] for more details).

To describe the periodic structure, we denote $Q=(-1/2, 1/2)^d$ as a representative cell, and call a function $f$ 1-periodic, it means: $$f(\bm{x}+\bm{z}) = f(\bm{x}) ~~~~ \forall \bm{x} \in {\mathbb{R}}^d \text{ and } \forall \bm{z} \in {\mathbb{Z}}^d ,$$ and we also use a superscript $\epsilon$ for $f(\bm{x})$ to represent 