The approximation of BV functions

Question

From Evans & Gariepy 's book, I learned that generally for any $u\in BV(R^n)$, we can find $u_n\in BV(R^n)\cap C^\infty (R^n)$ such that $$ \lim_{n\to\infty} \|u_n-u\|_{L^1(R^n)} = 0$$ and $$\lim_{n\to\infty}|Du_n|(R^n)= |Du|(R^n)$$ However, later in this book, they start to use $u_n\in BV(R^n)\cap C_c^\infty(R^n)$ to approximate $u$ without any future reference. I don't understand why they could replace $C^\infty$ function with $C_c^\infty$ function.

I know in Sobolev space we could use $C_c^\infty$ function because $W^{1,p}(R^n)=W_0^{1,p}(R^n)$. Is it something similar happens in $BV$ space? I am not sure, or at least I can not prove it myself. Any help is really welcome!

Answer 1

If you have an approximating sequence of $C^\infty$ functions $\phi_i$, then you can turn them into an approximating sequence of $C^\infty_0$ functions by multiplying by appropriate cut-off functions $\psi_i\in C^\infty_0$ such that

• $\| \nabla \psi_i\|< C$ pointwise uniformly bounded,
• $0\leq\psi_i\leq1$,
• $\psi_i \equiv 1$ on $B_i(0)$, and
• $\|\psi_i\phi_i - \phi_i\|_{L^1} < \frac{1}{i}$. This gives $L^1$ convergence.

Here, note that if $f\in BV$ and $\psi\in C^\infty_0$ then $\psi f\in BV$ and $D(\psi f) = \psi Df + f D\psi$ as vector valued Radon measures.

To show convergence, it is important to note that we can get estimates off of bounded measurable sets $K$; for any Borel set $U$, we have $\lim\limits_{r\to \infty} |Du|(U\setminus B_r) = 0$ and \begin{align} |D\psi_i \phi_i|( U\setminus B_r) & \leq |D\phi_i|(U\setminus B_r) + \|\phi_i\nabla\psi_i\|_{L^1(U\setminus B_r)},\\ & \leq |D\phi_i|(U\setminus B_r) + C\|\phi_i\|_{L^1(U\setminus B_r)},\\ &\leq |Df|(U\setminus B_r)+ |Df - D\phi_i|(U\setminus B_r) + C\|f\|_{L^1(U\setminus B_r)} + C\|f - \phi_i\|_{L^1}. \end{align} Hence we have that $\limsup\limits_{i\to\infty}|D\psi_i \phi_i|(U\setminus B_r)\leq |Df|(U\setminus B_r)+ C \|f\|_{L^1(U\setminus B_r)}$. Therefore, given $\epsilon > 0$, we may find $R$ large enough such that $\limsup\limits_{i\to\infty}|Df - D\psi_i\phi_i|(U\setminus B_R) <\epsilon$. Note that here we have used the globalness of the space (we aren't looking at $BV_{\text{loc}}$).

Now note that $|Df - D\psi_i\phi_i|(U\cap B_R) = |Df - D\phi_i|(U\cap B_R)$ for $i \geq R$, because then $\psi_i \equiv 1$ on $B_R$. Therefore, we get $\limsup\limits_{i\to \infty}|Df - D\psi_i\phi_i|(U) < \epsilon, $ and the functions $\psi_i\phi_i\in C^\infty_0$ give a new approximating sequence.