example of a $BV$ function which is not $W^{1,1}$

The following example presents a function $u\in BV(\mathrm{\Omega })\setminus W^{1,1}(\mathrm{\Omega })$.

Example 1.

Let $\mathrm{\Omega }:=(-1,1)\times (-1,1)\subset {ℝ}^2$. We will show that the function

belongs to $BV(\mathrm{\Omega })$.Given $\varphi \in C_c^1(\mathrm{\Omega },{ℝ}^2)$, $\varphi =({\varphi }^1,{\varphi }^2)$, one has

${\displaystyle {\iint }_{\mathrm{\Omega }}}u(x,y)\mathrm{div}\varphi (x,y)𝑑x𝑑y$$={\displaystyle {\int }_{-1}^1}\left[{\displaystyle {\int }_0^1}{\varphi }_x^1(x,y)𝑑x\right]𝑑y+{\displaystyle {\int }_0^1}\left[{\displaystyle {\int }_{-1}^1}{\varphi }_y^2(x,y)𝑑y\right]𝑑x$

$={\displaystyle {\int }_{-1}^1}{\varphi }^1(1,y)-{\varphi }^1(0,y)dy+{\displaystyle {\int }_0^1}{\varphi }^2(x,1)-{\varphi }^2(x,-1)dx$

$=-{\displaystyle {\int }_{-1}^1}{\varphi }^1(0,y)+0=-{\displaystyle \int \varphi (x,y)𝑑\mu (x,y)}$

if we choose $\mu :=({\mu }^1,{\mu }^2):=({\mathscr{H}}^1\mathrm{⌞}(\{0\}\times (-1,1)),0)$. So we notice that $u\in BV(\mathrm{\Omega })$ and $Du=\mu$ is singular with respect to the Lebesgue measure $ℒ$.