 Noncoercivity of total variation on constants

On a bounded domain of positive measure, the constant functions $u_k\equiv k$ satisfy $\mathrm{TV}(u_k)=0$ while $\Vert u_k{\Vert }_{BV}=k|\Omega |\to \infty$. Thus the total variation seminorm on a domain is neither coercive in the $BV$ norm nor strictly convex. Additional control of the mean can remove this obstruction; the Poincaré inequality for total variation supplies the needed bound on suitable connected domains.