Noncoercivity of total variation on constants
= Noncoercivity of total variation on constants
On a bounded domain of positive measure, the constant functions $u_k\equiv k$ satisfy $\operatorname{TV}(u_k)=0$ while $\|u_k\|_{BV}=k|\Omega|\to\infty$. 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.