The projection onto the null space of total variation operator

Question

Let $TV$ denote the total variation semi-norm over domain $\Omega\subset \mathbb R^2$ which is open bounded with smooth boundary.

Let $\mathcal N$ denote the null space of $TV$. That is, a function belongs to $\mathcal N$ should be a constant. Let $P$ denote the projection operator onto $\mathcal N$.

My question: let a function $u\in L^\infty$ be given. Then do we have $$ P(u) = \frac1{|\Omega|}\int_\Omega u\,dx $$ hold? i.e., the projection gives the average of $u$?

thank you!

Answer 1

The projection operator you wrote down is the projection onto $\mathcal N$ with respect to the $L^2$-scalar product: Let $u\in L^2(\Omega)$ and $v\in \mathcal N$ be given, that is, $v$ is constant. Then $$ \int_\Omega( u-P(u))v dx = v \cdot |\Omega| (P(u)-P(u))=0. $$