Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 326 / 2 / 1 / Solution

For $u\in L^1(\Omega )$, its total variation seminorm on a domain isThe corresponding bounded-variation space and its zero-mean subspace areIf $u_{\Omega }=|\Omega {|}^{-1}{\int }_{\Omega }udx$, the Poincaré inequality for total variation isIn particular, $\Vert u{\Vert }_{L^1}\le C_{\Omega }\mathrm{TV}(u)$ for $u\in B{V}_0(\Omega )$.