scaling $f^\epsilon(\bm{x})\coloneqq f(\bm{x}/\epsilon)$ if $f$ is 1-periodic. $H^k_\sharp(Q)$ or $W^{k,p}_\sharp(Q)$ with “$\sharp$” means this functional space is the completion of smooth 1-periodic functions with respect to the $H^k(Q)$ or $W^{k,p}(Q)$ norm. We have a fundamental theorem for $f^\epsilon$:

\[thm:oscillating converge\] Let $1\leq p < \infty$, $\forall f \in L^p_\sharp(Q)$, then $f^\epsilon \rightharpoonup \mathcal{M}_Q f= 1/{\left\lvert Q \right\rvert}\int_Q f$ in $L^p(\omega)$. Here $\omega$ is an arbitrary bounded open subset in ${\mathbb{R}}^d$.

It is customary to write $C$ as a positive constant, and $C(p_1,\cdots,p_n)$ indicates that $C$ depends on $p_1,\cdots,p_n$.

Then we introduce (Clarke) generalized directional derivative and subdifferential (see [@Denkowski2003]).

Let $\varphi: X \rightarrow {\mathbb{R}}$ be a locally Lipschitz function. For $x, h \in X$, the *generalized directional derivative* of $\varphi$ at $x$ along the direction $h$, denoted by $\varphi^0(x; h)$ is defined by $$\varphi^0(x; h) \coloneqq \limsup_{y\rightarrow x, \lambda \downarrow 0} \frac{\varphi(y+\lambda h)-\varphi(y)}{\lambda} = \inf_{\epsilon, \delta>0} \sup_{\substack{{\left\lVert x-y \right\rVert}_X<\epsilon \\ 0<\lambda< \delta}} \frac{\varphi(y+\lambda h)-\varphi(y)}{\lambda} .$$ The *generalized subdifferential* of $\varphi$ at $x \in X$, is the nonempty set $\partial \varphi (x) \subset X^*$ defined by $$\partial \varphi(x) \coloneqq \{ x^* \in X^*: {\left\langle x^*,h \right\rangle} \leq \varphi^0(x; h), \forall h \in X \}.$$

From now on, the matrix function $$A^\epsilon(\bm{x})={\left[ 