BV extension and translation bounds

The zero extension is in $BV({\mathbb{R}}^n)$ by [F1]. Its $L^1$ norm equals $\mathrm{\parallel }u{\mathrm{\parallel }}_{L^1(\mathrm{\Omega })}$, and [F1] bounds its full-space variation by $C_{\mathrm{\Omega }}(\mathrm{\parallel }u{\mathrm{\parallel }}_1+TV(u;\mathrm{\Omega }))$; its support is contained in $\bar{\mathrm{\Omega }}$ by definition.

Let first $v\in C_c^{\mathrm{\infty }}({\mathbb{R}}^n)$. The fundamental theorem [F4] along the segment from $x$ to $x+h$ gives $v(x+h)-v(x)={\int }_0^{1}h\cdot \mathrm{\nabla }v(x+th)dt$. Integrate the absolute value in $x$, apply Tonelli [F5], and use translation invariance [F3] to obtain $\mathrm{\parallel }v(\cdot +h)-v{\mathrm{\parallel }}_1\le {\int }_0^1\mathrm{\mid }h\mathrm{\mid }{\int }_{{\mathbb{R}}^n}\mathrm{\mid }\mathrm{\nabla }v(x+th)\mathrm{\mid }dxdt=\mathrm{\mid }h\mathrm{\mid }{\int }_{{\mathbb{R}}^n}\mathrm{\mid }\mathrm{\nabla }v\mathrm{\mid }dx\mathrm{.}$

For general $v\in BV({\mathbb{R}}^n)$, choose $v_k\in C_c^{\mathrm{\infty }}$ with $v_k\to v$ in $L^1$ and $\int \mathrm{\mid }\mathrm{\nabla }{v}_k\mathrm{\mid }\to TV(v)$ by [F2] under [A1]. Translation is an $L^1$ isometry [F3], so $v_k(\cdot +h)-v_k\to v(\cdot +h)-v$ in $L^1$. Passing to the limit in step 1.2 proves the translation inequality.