Compactly supported BV cutoffs for finite-variation L² functions

By [F2], choose $\chi \in C_c^{\mathrm{\infty }}({\mathbb{R}}^2)$ with $0\le \chi \le 1$, $\chi =1$ on $B(0,1)$, $\mathrm{supp}\chi \subset B(0,2)$, and let ${\chi }_R(x)=\chi (x\mathrm{/}R)$ for integers $R\ge 1$. This proves existence of a family with ${\sup }_{R}R\mathrm{\parallel }\mathrm{\nabla }{\chi }_R{\mathrm{\parallel }}_{\mathrm{\infty }}<\mathrm{\infty }$. Now fix any family satisfying all four conditions in the statement, write $C={\sup }_{R}R\mathrm{\parallel }\mathrm{\nabla }{\chi }_R{\mathrm{\parallel }}_{\mathrm{\infty }}$, and put $v_R={\chi }_{R}u$. Then $v_R\in L^2$ has compact support, hence is in $L^1$, and $v_R\to u$ in $L^2$ because ${\chi }_R=1$ on $B(0,R)$ and $u\in L^2$.

For every scalar $\psi \in C_c^{\mathrm{\infty }}({\mathbb{R}}^2)$, local integrability of $u$ and the definition of distributional derivative give $-\int {\chi }_{R}u{\mathrm{\partial }}_i\psi =-\int u{\mathrm{\partial }}_i({\chi }_R\psi )+\int u\psi {\mathrm{\partial }}_i{\chi }_R=\int {\chi }_R\psi d{D}_{i}u+\int u\psi {\mathrm{\partial }}_i{\chi }_R\mathrm{.}$ Thus $D{v}_R={\chi }_{R}Du+u\mathrm{\nabla }{\chi }_{R}dx$. The variation of the first term is at most $\mathrm{\mid }Du\mathrm{\mid }({\mathbb{R}}^2)$, and the second has variation at most $\frac{C}{R}{\int }_{B(0,2R)\setminus B(0,R)}\mathrm{\mid }u\mathrm{\mid }dx\le C^{\mathrm{\prime }}\mathrm{\parallel }u{\mathrm{\parallel }}_{L^2({\mathbb{R}}^2\setminus B(0,R))},$ by Cauchy–Schwarz and the annulus area bound $\mathrm{\mid }B(0,2R)\setminus B(0,R){\mathrm{\mid }}^{1\mathrm{/}2}\le C^{\mathrm{\prime }\mathrm{\prime }}R$. This error tends to zero. Since $v_R\in L^1$ and its derivative is a finite vector Radon measure, $v_R\in BV({\mathbb{R}}^2)$, and [F1] yields ${\mathrm{lim\hspace{0.17em}sup}}_{R}TV(v_R;{\mathbb{R}}^2)\le \mathrm{\mid }Du\mathrm{\mid }({\mathbb{R}}^2)=T{V}_2(u;{\mathbb{R}}^2)$. The estimate applies to the arbitrary controlled family fixed in step 1.1 and proves the statement.