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

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

###### Example 1.

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

 $u(x,y)=\begin{cases}1,\quad\text{if x\geq 0}\\ 0,\quad\text{if x<0}\end{cases}$

belongs to $BV(\Omega)$. Given $\phi\in C_{c}^{1}(\Omega,\mathbb{R}^{2})$, $\phi=(\phi^{1},\phi^{2})$, one has

 $\displaystyle\iint_{\Omega}u(x,y)\mathrm{div}\phi(x,y)\,dxdy$ $\displaystyle=\int_{-1}^{1}\left[\int_{0}^{1}\phi^{1}_{x}(x,y)\,dx\right]dy+% \int_{0}^{1}\left[\int_{-1}^{1}\phi^{2}_{y}(x,y)\,dy\right]dx$ $\displaystyle=\int_{-1}^{1}\phi^{1}(1,y)-\phi^{1}(0,y)\,dy+\int_{0}^{1}\phi^{2% }(x,1)-\phi^{2}(x,-1)\,dx$ $\displaystyle=-\int_{-1}^{1}\phi^{1}(0,y)+0=-\int\phi(x,y)\,d\mu(x,y)$

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

Title example of a $BV$ function which is not $W^{1,1}$ ExampleOfABVFunctionWhichIsNotW11 2013-03-22 15:12:59 2013-03-22 15:12:59 paolini (1187) paolini (1187) 5 paolini (1187) Example msc 26B30