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Suppose we have, for an open bounded set $\Omega \subset \mathbb{R}^n$:

  1. A function $u \in L^p(\mathbb{R}^n) \cap C(\mathbb{R}^n)$.
  2. A sequence of mollifiers $(\rho_n) \subset C_c^{\infty}(\mathbb{R}^n)$.
  3. A sequence of functions $(\xi_n) \subset C_c^{\infty}(\mathbb{R}^n)$ with $0 \leq \xi_n(x) \leq 1$ defined as: $$ \xi_n=\xi(\frac{x}{n})= \left\{\begin{matrix} 1 & if \ |\frac{x}{n}| \leq 1\\ 0 & if \ |\frac{x}{n}| \geq 2 \end{matrix}\right.$$

For $\xi \in C_c^{\infty}(\mathbb{R}^n)$ and $0 \leq \xi(x) \leq 1$.

I want to show that the sequence $(\rho_n * u)(\xi_n)_{|_\overline{\Omega}}$ converges uniformly to $u_{|_{\overline{\Omega}}}$.

I have been able to prove these facts:

  • Using Dominated Convergence Theorem we clearly see that for any $g \in L^p(\Omega)$ we have $ \xi_n g_{|_{\overline{\Omega}}}\to g_{|_{\overline{\Omega}}} $ in $L^p(\Omega)$.

    • Using basic results of convolution and regularization we have that, as $\overline{\Omega}$ is compact and $u \in L^p(\mathbb{R}^n) \cap C(\mathbb{R}^n)$ $\rho_n * u_{|_{\overline{\Omega}}} \to u_{|_{\overline{\Omega}}}$ uniformly.

    • As $\overline{\Omega}$ is bounded, suppose by a ball of radius $M$, we also have that $\xi_n g_{|_{\Omega}} \to g_{|_\Omega}$ uniformly on $\overline{\Omega}$, as for $n \geq M $ $\xi_n \equiv 1$ on $\overline{\Omega}$.

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The 2nd and 3rd of the facts you cite give the result. Indeed, you have proved that for any $\epsilon>0$ there is $N_1$ such that $\max_{\overline{\Omega}}|\rho_n *u-u|<\epsilon$ whenever $n>N_1$. Also, you know that there exists $N_2$ such that $\xi_n\equiv 1$ on $\overline{\Omega}$ whenever $n>N_2$.

Conclusion: $\max_{\overline{\Omega}}|(\rho_n *u)\xi_n-u|<\epsilon$ whenever $n>\max(N_1,N_2)$.

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