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I'm really annoyed with mollifier, I really never now when I can use it, and in what are there important. Here several example :

1) Let $f\in L^1_{loc}(\mathbb R)$. We want to show that if for all $\varphi\in \mathcal C_c^1(\mathbb R)$, we have $\int_\Omega f\partial _i\varphi=0$, then $f$ is constant a.e.

Proof : Let $B\subset \subset \Omega $ a ball and $\ell=dist(B,\Omega ^c)>0$. Let $\rho_n$ a standard mollifier. Then $\partial _i(\rho_n*f)=0$ when $n>2/\ell$, and thus $\rho_n*f$ is constant in $B$. Since $\rho_n*f\to f$, the claim follow.

Question : Where is $\rho_n$ is coming, why can we consider it, and in what is it interesting ?

2) Let $u,v\in W^{1,p}\cap L^\infty (\Omega )$ with $p\in [1,\infty ]$. Then $uv\in W^{1,p}(\Omega )$ and $$\partial _i(uv)=u\partial _i v+v\partial _i u.$$

Proof : Let $p<\infty $ and let $D\subset \subset \Omega $ an open set. Let $\rho_n$ a standard mollifier. Define for $n$ large enough $$u_n=\rho_n*u\quad \text{and}\quad v_n=\rho_n*v.$$

We have that $u_n\to u$ and $v_n\to v$ in $W^{1,p}(D)$ and $\|u_n\|_{L^\infty(D) }\leq \|u\|_{L^\infty (\Omega )}$ and $\|v_n\|_{L^\infty(D) }\leq \|v\|_{L^\infty (\Omega )}$. WLOG, assume $u_n\to u$ a.e. in $D$ and $\partial _iu_n\to \partial _i u$ a.e. in $D$. Then, in $D$, $$\partial _i(uv)=\partial _i u_nv_n+u_n\partial _i v_n\to \partial _i uv+u\partial _i v$$ in $L^p(\Omega )$. We have for $\varphi\in \mathcal C^1_c(\Omega )$, $$-\int_\Omega uv\partial _i\varphi=\lim_{n\to \infty }-\int_{\Omega }u_nv_n\partial _i \varphi$$ $$=\lim_{n\to \infty }\int \partial (u_nv_n)\varphi=\lim_{n\to \infty }\int (u_n\partial _iv_n+v_n\partial _iu_n)\varphi=\int_\Omega u\partial _i v+v\partial _iu)\varphi.$$ Therefore $uv\in W^{1,p}(\Omega )$ and $\partial _i(uv)+u\partial _i v+v\partial _i u$.

Question : I'm not sure to understand all the proof, but as previous, what is doing $\rho_n$, why can we consider it, and how it works ?

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