Past exam of the mathematics course of the University of Cambridge / 2014 / iii / Paper 64 / 2 / i / Solution

For a scalar $u\in BV(\Omega )$, put $E_t=\{x\in \Omega :u(x)>t\}$. The coarea formula for BV functions is the measure identityfor every Borel $A\subseteq \Omega$. For almost every real $t$, $E_t$ is a finite-perimeter set, with $\mathrm{Per}(E_t;A)=|D{\chi }_{E_t}|(A)$. In particular $\mathrm{TV}(u)={\int }_{\mathbb{R}}\mathrm{Per}(E_t;\Omega )dt$. This is relative perimeter: it counts interfaces inside $\Omega$, and does not add the boundary jump of a zero extension.