Example of BV function that is not $p$-quasicontinuous

Question

It is true that a $W^{1,p}(\Omega)$ function is $p$-quasicontinuous, but it shouldn't be true for $BV(\Omega)$ functions.

• What is an example of BV function that is not $p$-quasicontinuous?

Answer 1

The relationship between capacity and Hausdorff dimension when $p < n$ is that $\newcommand{\Cap}{\mathrm{Cap}}\Cap_p(E) = 0$ if and only if $\dim_H(E) \le n-p$.

For a simple example, let $n=2$ and $1 < p < 2$. Any circle in the plane will have positive $p$-capacity.

Let $D$ be a disk and define $f(x) = 1$ if $x \in D$ and $f(x) = 0$ otherwise. Let $\Omega$ be an open set containing $\overline D$. Any open set $U$ with the property that $f_{\Omega \setminus U}$ is continuous must contain $\partial D$ and thus cannot have arbitrarily small $p$-capacity.

It is routine to show that this $f$ has bounded variation.