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In the same article by Benoit Perthame: (related to this question A limit involving a regularizing kernel) I encountered an inequality, which I didn't manage to prove.

It is like this:

$$ \int_{\Bbb{R}^d}\left[ |u^0(x)|-\int_{\Bbb{R}}\left(\chi(\xi,u^0(x))\star \varphi_\varepsilon \right)^2d \xi\right]dx \leq C \|u^0\|_{BV}\cdot\varepsilon $$

where $$ \chi(\xi,u)=\begin{cases} 1 & {0\leq \xi\leq u} \newline -1 & u \leq \xi \leq 0 \newline 0 & \text{otherwise} \end{cases}$$

and $\varphi_\varepsilon$ is a regularization kernel in $x$ and $u_0$ is regular enough for all objects to be well defined.

In the article, the inequality is stated as obvious, and no indication, reference or attempt to prove it is made. It is possible to prove that the LHS tends to zero as $\varepsilon \to 0$. Still, I cannot get the majorization by $\varepsilon$ in the RHS, which clearly depends only on $\varphi_{\varepsilon}$.

Do you have some ideas in proving this inequality? Thank you.

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