Zero-mean bounded-variation space ($B{V}_0(\Omega )$)

The zero-mean bounded-variation space isOn a bounded connected Lipschitz domain, removing the constant nullspace makes total variation coercive on this subspace by the Poincaré inequality for total variation.

On a bounded connected Lipschitz domain,It reduces to $\Vert u{\Vert }_{L^1}\le C_{\Omega }\mathrm{TV}(u)$ on $B{V}_0(\Omega )$.